Advances in Heat Transfer, Volume 4

Advances in

HEAT TRANSFER Edited by James P. Hartnett

Thomas F. Irvine, Jr.

Department of Energy Engineering University of Illinois at Chicago Chicago, Illinois

State University of N e w York at Stony Brook Stony Brook, Long Island N e w York

Volume 4

@ 1967 ACADEMIC PRESS

New York

-

London

COPYRIGHT 0 1967, BY ACADEMIC PRESS INC.

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LIST OF CONTRIBUTORS E. R. G. ECKERT, Department of Mechanical Engineering, University of Minnesota, Minneapolis, Minnesota A. J. EDE, National Engineering Laboratory, East Kilbride, Scotland

C. FORBES DEWEY, JR., University of Colorado and Joint Institute for Laboratory Astrophysics, Boulder, Colorado

JOSEPH F. GROSS, Department of Geophysics and Astronomy, R A N D Corporation, Santa Monica, California E. PFENDER, Department of Mechanical Engineering, University of Minnesota, Minneapolis, Minnesota ROBERT SIEGEL, Lewis Research Center, National Aeronautics and Space Administration, Cleveland, Ohio ALICE M. STOLL, Aerospace Medical Research Department, U. S. Naval Air Development Center, Johnsville, Warminster, Pennsylvania

1 Present address: Department of Mechanical Engineering, University of Aston in Birmingham, England.

V

PREFACE In the preface to Volume 1, we noted that heat transfer research has grown at an amazing rate during the past decade, primarily due to problems associated with the growth of the atomic energy industry and the aerodynamics and astronautics efforts throughout the world. We also noted that while the results of these research efforts are normally published as individual articles in national and international journals, it is often difficult for the nonspecialist, or even the specialist, to make engineering use of these individual papers. It was our hope that review articles which start from widely understood principles and develop the topics in a logical fashion would be of value to the engineering and scientific communities. The interest aroused by the first three volumes of “Advances in Heat Transfer” seems to us to be an indication that this function is being fulfilled.

J. P. HARTNETT T . F. IRVINE, JR.

October, 1967

vii

Advances in Free Convection A. J. EDE* National Engineering Laboratory East Kilbride. Scotland

I. Introduction . . . . . . . . . 11. Laminar Flow . . . . . . . . A. The Classical Problem of the Flat Plate B. The Vertical Circular Cylinder . . . C. Nonuniform Surface Temperature . . D. Nonuniform Physical Properties . . E. Nonsteady Conditions . . . . . F. Very Low Grashof Numbers . . . G. Effect of Vibration . . . . . . 111. Turbulent Flow . . . . . . . . A. Instability of Laminar Flow . . . . B. Turbulence . . . . . . . . Symbols . . . . . . . . . . References . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . .

. . .

1 4 4 . 1 7 . 22

. .

. . . . .

. . . . . . . . . . . .

28 33 4 3 4 7 5 0 5 0 5 4 62 62

I. Introduction

T h e average treatise on heat transfer dismisses free convection in a single chapter. It is, nevertheless, potentially as large a subject as forced convection, and has developed rapidly within the last decade. The present review acknowledges this situation by confining its attention to one aspect only, that of free convection on a vertical surface. T h e fluid is assumed to be of infinite extent and devoid of any motion or temperature variations other than those associated with the free convection; the surface is assumed to be free of any obstruction which might disturb the flow. Although papers falling within this category are still appearing, it would seem that the vein has now been almost worked out, and that attention is being increasingly diverted

* Present address: Department of Mechanical Engineering, University of Aston in Birmingham, England. An excellent survey of a much wider field was prepared by Ostrach (111) in 1959. 1

2

A. J. EDE

elsewhere. T h e subject has therefore reached a convenient stage for a general review. T h e early development of the subject was characterized by the appearance of many papers dealing with experimental work ; theoretical papers were rare. I n recent years the position has been reversed, and most recent publications have been largely concerned with analytical work. T h e considerable ingenuity displayed by early mathematicians has largely given way to solution by computer. Progress is faster but the papers are not so attractive to read! Analytical work has been preponderantly concerned with laminar flow, no

' Y

FIG.1. Coordinate system.

doubt because turbulence is less tractable. Fortunately laminar flow is very important in free convection. Velocities are comparatively low, and the leading edge is not a serious source of disturbance. I n this review laminar flow in all its aspects will be taken first, and instability and turbulence will be examined later. T h e system of coordinates is displayed in Fig. 1. T represents the absolute temperature of the fluid, and takes the values T, at the surface and T , in the bulk fluid. 0 denotes the excess of the fluid temperature at a point over the bulk temperature, i.e., T - T , ;it can of course be negative, in which case the direction of the convective flow will be reversed. No further reference will be made to this possibility.

ADVANCES IN FREE CONVECTION

3

In much analytical work it is convenient to dedimensionalize the variables, and this can be done in many ways. In some papers a succession of transformations is carried out, and the variety of nomenclature then employed adds considerably to the difficulty of following the argument and of comparing one paper with another. A common device is to allot special symbols to the basic variables, calling them 3, a, 19or A’,U, 0 in order that x , y, 8, etc. can be reserved for the dimensionless variables in which the greater part of the analysis is to be conducted. Sometimes an author will leave his final conclusions in terms of these transformed variables, and it may be suspected that he was more interested in the elegance of his mathematics than in the usefulness of his results. It is impossible in a review paper to maintain a consistent set of symbols and yet cover all possibilities, and symbols will sometimes be defined where they appear without regard for the fact that they may already have been used with a different significance. The basic symbols x, y, u, zi, 8, etc. will however always refer to the straightforward dimensional variables. Application of dimensional analysis to free convection on a flat vertical surface reveals that the Nusselt number Nu, = h x / k is a function of the Grashof number Gr, =glg8,x3/v2 and the Prandtl number Pr = q / k . Here h is the local heat-transfer coefficient at the position x ; the average coefficient h for the whole surface is also of interest, and can be used to form a Nusselt number NuL = h L / k and a Grashof number GrL = glg8, L3/v2.The Rayleigh number Ra = G r - P r is also widely used. For the vertical circular cylinder, the only other shape which will arise in this review, generality can be preserved by introducing a further group D / L , and the diameter can be used to form alternative Nusselt and Grashof numbers NuD = h D / k and Gr, = glgdWD3/v2.These usages arose from the early work in which the surface temperature 8, was nearly always assumed to be uniform. When the surface heat flux qzu is assumed uniform, a modified Grashof number Gr,+ = glgq,,x4/v2can be employed. In later work with more complicated boundary conditions these groups are not quite so useful. All analytical work is based on the following set of equations, representing the conservation of mass, momentum, and energy in a fluid moving under the influence of a body force : ap+ at

a ax] (pu,) = 0

4

A. J. EDE

F a y -+hi axj axi

ax,

auj) - 32(aU,)~j

+r)-

axj

(5)

In all the cases discussed here the flow either is two-dimensional or has cylindrical symmetry, so that only two momentum equations have to be considered. If there are no sources of heat within the fluid, Q is omitted. If viscous dissipation can be neglected, the term prefaced by 7 in Eq. (5) is omitted. If the flow is steady terms in a/& are omitted. The boundary conditions specify that the temperature or heat flux is some function of x at y = 0, and is zero at infinity. The velocity components are zero aty = 0 and infinity. The spatial derivatives of temperature and velocity are zero at infinity. The boundary conditions for velocity differ from their forced convection counterpart, and have the effect that the velocity and temperature profiles are quite different in shape. Velocity changes from zero through a maximum to zero again, whereas temperature varies monotonically from maximum to zero. T h e peculiarity which distinguishes free convection from forced, as far as analytical work is concerned, is that the equations of momentum and energy are coupled ; they cannot be treated separately. The motion is, of course, directly caused by the transfer of heat, and has no independent existence. As a consequence the principle of superposition cannot be used to develop solutions for complicated situations from solutions for simple, idealized cases. 11. Laminar Flow

A. THECLASSICAL PROBLEM OF THE FLAT PLATE The flow is assumed to be steady ;the surface temperature is uniform and the physical properties of the fluid (apart from density) are unaffected by temperature. Dissipation is neglected and there are no heat sources within the fluid. Two principal methods of analysis have been developed: direct solution of the differential equations, with the aid of similarity transformations, and solution of the equations in an integral form, by assuming plausible expressions for the velocity and temperature profiles. Both procedures assume that the profiles are of essentially the same form for all values of x. T h e set of equations previously quoted, with the simplifications mentioned, were assembled by Oberbeck (1) in 1879. He aligned his axes so that one was vertical, giving F , = -g, F2 = F3 = 0, and put p = po(l -PO). I t

5

ADVANCES IN FREECONVECTION follows that ap/ax = --pop aO/ax, etc., and if p become

= -pogx

+p', the equations

au + av + aw = p u -ae + v -ae + w ax ay aZ ( ax ay axas>

~- -

(

av

av

ax

ay

ae

ae

po u - + v - + w -

(

poc u - + a - - + w -

ax

ay

-

avj--+-+;: $) az - q (;;: ae - k az )-

;

--

(-+-+7 ay ax aa 2x 20

a20

Oberbeck's attempts to obtain a solution as a power series in/3 do not include any cases falling within the scope of this survey. Two years later Lorenz (2) greatly simplified these equations by means of certain sweeping assumptions, and reduced them to the following two, which he solved in series : O=jSgPoO+T pocue = kL

(d2uldy2)

(d26/dy*)

(11) (12)

His solution leads to the following expression for the Nusselt number : Nu,

= 0.548(GrL.Pr)''4

(13)

a result which is astonishingly close to experimental data and more accurate solutions. The simplifications introduced, however, have the effect that the local heat transfer coefficient is not a function of x, which is contrary to experience.

1. The Dzfeerential Equation Method It is necessary to advance 50 years, passing over some not very successful theoretical work and the gradual emergence of dimensional analysis, to reach the next stride forward. This was achieved by Pohlhausen, who in collaboration with Schmidt and Beckmann (3)solved the Oberbeck equations in a less primitive form : azl

av

.&+ay=o

A. J. EDE

6

au (. + pcu-++( iz P

au

D a y ) = PgPe

i:)

=K-

azu

+ 77 ay2

a2e

ay

T h e zero suffix to p has now been dropped as its variation with temperature is accounted for by the introduction of /3. T o obtain these equations they applied Prandtl's boundary-layer approximations, assuming that the effects of the free convection were confined to a thin layer of fluid adjacent to the heated surface (as their experiments clearly showed). A more rigorous derivation of these simplified equations has since been given by Ostrach (4). I n view of the continuity equation (14) Pohlhausen introduced a stream function such that u = a+/ay, v = -a+/ax, and then sought a similarity transformation in terms of a new variable = Axmy". It is found, on subthe equations become stitution, that if m = -*, n = 1, A = (g/3p2/4q2)1/4,

+

c

f"'+ 3fs" - 2f '2 + g = 0 g" + 3Pr .fg'

+

=0

(18)

where f andg are the functions of 5 in which and 0 are expressed. It follows from this that the temperature and velocity distributions can be expressed in terms of y/xli4, and this accorded with the experimental measurements made by Schmidt and Beckmann. Pohlhausen attempted a solution of these equations in series, but encountered convergence difficulties and was therefore compelled to take boundary values of f n and g' from the velocity and temperature gradients obtained experimentally, and solve numerically for a single value of the Prandtl number, namely 0.733, appropriate to air. The heat-transfer coefficient resulting from this analysis is a function of x, and the results can be expressed as follows : or

Nu,

= 0.39(Gr,.Pr)114

NuL = 0.52(GrL.Pr)li4

After Saunders (5) and Schuh (6) had obtained further solutions by approximate methods which did not have to fall back on experimental data, Ostrach (4) virtually concluded the study of the Schmidt and Beckmann equations by obtaining exact solutions for eight values of Pr, from 0.01 to 1000, by means of a computer. The solution may be put in the form or

Nu,

=

(3A/4)(Gr,.Pr)li4

NuL = A(GrL-Pr)Ii4

ADVANCES IN FREECONVECTION

7

where A is a function of Pr. Further calculations of the same type have been made by Sugawara and Michiyoshi (7), Sparrow et al. (8),and Gebhart (9). Le Fevre (10) considered the extreme cases where Pr tends to zero or infinity. Starting with Eqs. (17) and (18), a change of variables leads to Pr(fl”’ +gl) +flfl’’ - +flz +gl g1”

=0

+ gl‘fl = 0

(23) (24)

which facilitates the task since for Pr -+ 0 Eq. (23) becomes 4f12

f1f1”-

+g1 = 0

(25)

and for Pr -+03 it becomes

+g1 = 0 (26) Solution by computer gives Eq. (22) with A = 0.670 327 for Pr -+ w and A = 0.800 544Pr’/4 for Pr + 0. fl”’

TABLE I COMPUTER SOLUTIONS OF THE CLASSICAL PROBLEM Pr

A

0 0.01 0.03 0.09 0.5 0.72 0.733 1.o 1.5 2.0 3.5 5.0 7.0 10 100 1000 10,000

0.800 564Pr1I4 0.240 279 0.308 0.377 0.496 0.516 492 0.517508 0.534705 0.555 059 0.568033 0.589916 0.601 463 0.611 035 0.619 0.653 349 0.665 0.668 574 0.670 327

co

0.849 126Pr11Z 0.080 592 0 0.136 0.219 0.442 0.504630 0.507 890 0.567 140 0.651 534 0.716483 0.855 821 0.953 956 1.05418 1.168 2.191 40 3.97 7.091 30 0.710 989Pr114

Table I lists the best values at present available for A in Eq. (22). The value of the derivativeg’ at 5 = 0 is also tabulated. This is the generalized form of or (32/2Prli4/4)A. the heat-transfer coefficient and is equal to N~,/(Gr,/4)l/~ It will be seen that, for Pr > 100, A is almost constant, in conformity with

8

A. J. EDE

the long-recognized fact that, for so-called “creeping motion,” where the inertial terms can be omitted from the momentum equations, dimensional analysis indicates that the solution must have the general form Nu =f(Gr-Pr), so that A must be constant in Eq. (22). It has also been argued that, for very low Prandtl numbers, viscosity should not appear in the general solution, which must therefore have the form Nu = j(Gr.Pr2). The solution for Pr + 0 is evidently in conformity with this conclusion. Le Fevre also proposed an empirical expression which fits the computer solutions very closely and facilitates interpolation to other values of the Prandtl number :

A 4 = Pr/{2.43478 + 4.884Pr1/’ + 4.952 83Pr) A simpler form of slightly reduced accuracy is A 4 = 2Pr/5( 1 + 2P1-l~’+ 2Pr) T h e various solutions are displayed in Fig. 2.

log,,Pr

FIG.2. Flat plate: comparison of theoretical solutions.

2. The Integral Equation Method With the work of Ostrach, the classical problem may be regarded as completely solved. I t is, however, of more than merely historical interest to examine alternative methods that have been proposed, and in particular the integral equation method, because they may be used for more complicated problems. Following the ideas of von Karman (11) and others, the usual boundary layer equations (15) and (16) are formally integrated with respect

ADVANCES IN FREECONVECTION

9

t o y across a boundary layer of thickness 6, assumed to be the same for both velocity and temperature. They become

These equations can also be obtained quite simply from first principles (12). In order to proceed further it is necessary to adopt approximate expressions for u and 0 as functions of y. The following simple polynomials give profiles which correspond quite well with experimental results : = u(y/6) (1 -y/ti)2, e = e,(i - y / q 2 (31) where U and 6 are as yet undetermined functions of x. On substituting these expressions into the integrals, Eqs. (29) and (30) reduce to

T h e variables can be separated by means of the substitutions U = p x m , 6 = qx", and the following solution is obtained :

U = 5.17v(Pr + 20/21)-'~2(g/3B,/v2)'~2 x1j2 6 = 3.93Pr-'I2(Pr + 20/21)"4(g~8,/v2)-'~4x ' / ~

whence, in the usual form, Nu, = 0.51P1-'/~(Pr + 20/21)-'/4 GrL'4 or NuL = 0.68Pr'i4(Pr + 20/21)-'/4(GrL.Pr)1/4

(34) (35) (36)

(37) For air, this gives NuL = 0.55Rat/4,which is in adequate agreement with the more exact solution. This method was first used in free convection by Squire (13). I t has the advantage of simplicity, and gives a solution containing the Prandtl number explicitly. It is of course only approximate, and various improvements have been proposed. Merk and Prins (14) used more complicated expressions for the velocity and temperature profiles. Their solution has the form of Eq. (22) with A = 1.887y, where (Pr + 1 . 3 4 1 ) ~ ~1.292 x 1OP2(Pr+ 0.1602)y4+ 2.199 x 10W = 0 (38) Sugawara and Michiyoshi (15) introduced three modifications. In their

A. J. EDE

10

first method they assumed third-degree polynomials for u and 0, and in order to establish the coefficients they went back to the differential equations, from which they obtained ~

By this means they were able to arrive at expressions for the profiles which involved only one unknown parameter, namely 6, and this was found by solving the energy equation in the integral form. In their second method they assumed that the vertical component of velocity, u, was a function of temperature only. This implies that isothermals are also lines of constant u, which, in view of the Pohlhausen analysis and experimental results, is a reasonable assumption. By means of this transformation the differential equations were put into a form which could be solved by assuming polynomial expressions for 0. T h e integral energy equation is then solved as usual for 6. In their third attempt, Sugawara and Michiyoshi considered the possibility that the two boundary layers for velocity (6) and temperature (6') might be of significantly different thickness. The expressions for u and 0 obtained by their second method were put in terms of the two different deltas, and substituted into the integral equations, which were then solved for 6 and x = S'lS on the assumption that the latter was independent of x. Fujii (16) took a somewhat similar line, but used profiles expressed in terms of exponentials :

where s is determined from the boundary conditions. The solution then proceeds as usual. The first form of temperature profile gives acceptable results only for Pr < 0.1, but the second is reasonably satisfactory for all Prandtl numbers. Brindley (17) has applied Meksyn's asymptotic expansion procedure to the similarity solution. Starting with Eqs. (17) and (18) he expressed f and g as infinite power series in f ; consideration of the boundary conditions leads to

f=

130

C r=2

02

g=l+

a,.?,

C a,.? r=l

These are put into the equations and, by equating to zero the sums of coefficients of corresponding powers of (, the values of the a's and a's are determined in terms of a 2 and a 4 .Writing F and G for the series expressions for f and g and substituting for f,f and g in Eqs. (17) and (18) gives I,

fiit+3Ff"=2F'2-G g" + 3Pr * Fg' = 0

(41) (42)

ADVANCES IN FREECONVECTION

11

which can be integrated to provide expressions for f and g in terms of exponential functions of the integral of F. These functions and their integrals are evaluated by a lengthy procedure, at the conclusion of which the boundary conditions at infinity are used to determine a2 and a 4 .Numerical results, based on the first three terms of the expansion, are obtained by computer for Prandtl numbers of 0.01,0.733, 7.0, 10, 100, and 1000. They agree quite well with the more exact solutions except at Pr = 0.01. As with the other methods which have been described in this section, this is not claimed to have any superiority over the Pohlhausen-Ostrach procedure, but it can be used in more difficult cases (combined forced and free convection is discussed in the paper). All of these solutions can be put into the form of Eq. (22) with A given as a function of Pr. Some of them are compared in Fig. 2. In considering the application of such analyses to fluids having Prandtl numbers differing considerably from unity, the assumptions made concerning boundary-layer thickness should be borne in mind. The boundary-layer approximations require that both boundary layers shall be thin ; some solutions assume that both are of the same thickness. When the Prandtl number is large the velocity boundary layer extends beyond the thermal boundary layer.

3. Comparison with Experimental Data A large number of experimental observations are available for testing these solutions of the classical problem. In making the comparison it should be realized that there are many difficulties in reproducing the idealized situation assumed in the theoretical treatment. Perfect precision requires that the only motion in the fluid shall be that resulting from the presence of the heated surface; in practice, convection currents caused by outside influences or other parts of the apparatus, and disturbances such as draughts, are extremely difficult to suppress entirely. While the heat put into the experimental body may easily be determined with precision, some is usually lost by processes other than by free convection from the surface under consideration. T h e rate of heat transfer by free convection in gases, particularly, is quite low, and comparatively large amounts of heat may be lost by conduction along supports, leads, and so on, and by radiation, and accurate estimation of these losses can be difficult. The “infinite volume of fluid at a uniform temperature” may in fact be a comparatively small bulk of fluid in a container, and it may be difficult to attain a really steady state or to be sure that the fluid temperatures measured are truly representative and entirely unaffected by the presence of the heated surface. A finite surface must have edges, and their shape and size is evaded in the theoretical treatment so far considered. It is difficult to insure that the flow is substantially two-dimensional.

A. J. EDE

12

Apart from such practical difficulties, the temperature distribution on the surface demands attention. The classical theory assumes that it is uniform. Some experimenters have attempted with varying success to achieve this condition, but most have simply produced a situation corresponding approximately to that of a uniform heat flux at the surface. Before considering the experimental data, therefore, it is desirable to examine the extent to which this factor is likely to affect the results. A useful guide can be obtained from a theoretical paper by Sparrow and Gregg (It?), who showed that a similarity solution can readily be obtained for the uniform heat flux case. T h e treatment is essentially the same as that for the uniform temperature ’~ problem, except that the similarity variable now has the form y / ~ linstead o f y / ~ ” Results ~. were obtained by computer for Prandtl numbers of 0.1, 1, 10, and 100, and mean Nusselt numbers were calculated on the basis of (a) an average temperature difference for the whole surface, and (b) the temperature difference halfway up the surface. In all cases they were found to be slightly higher than those for a uniform temperature ; the ratios are listed in Table 11. Since most experimenters have used method (b) in analyzing their data, it can be seen that this lack of correspondence between idealization and experiment is not likely to be a serious source of difficulty. TABLE I1

TEMPERATURE UNIFORMHEAT-FLUX~

COMPARISON BETWEEN U N I F O R M

AND

Ratio of Nusselt numbers

Pr

(a) Mean

(b) At L / 2

0.1 1 10 100

1.08 1.07 1.06 1.05

1.02 1.015 1.01 1 .oo

a The table gives the ratio of the Nusselt number for a uniform heat-flux to the Nusselt number for a uniform temperature.

The largest single group of data is concerned with the vertical flat plate in air. The available data are plotted in Fig. 3, and can be seen to form a smooth curve, concave upwards. The straight line represents Ostrach’s solution ; it is tangential to the curve, representing the data reasonably well for Rayleigh numbers between lo6 and lo8,but deviating from the curve at

ADVANCES I N FREE CONVECTION

13

either end. Even in the region of best agreement, almost all the experimental points lie above the line, and since the commonest source of experimental error-uncontrolled air currents-usually leads to increased heat transfer this is understandable. The divergence at high Rayleigh numbers may be due to the development of turbulence, and that at low Ra to a thickening of the boundary layer to such an extent that the boundary-layer approximation becomes invalid.

FIG.3. Flat plate: experimental data for air.

Data are also available for comparison with the theoretical velocity and temperature profiles. Figures 4a and 4b compare the Ostrach solution with the data of Schmidt and Beckmann. T h e agreement may be regarded as satisfactory, but a closer look will be taken later. Another important difference between theory and experiment is concerned with the physical property data required for calculating the various groups. In the theoretical work all properties other than density are assumed to have constant values, but no real experiment can be conducted under isothermal conditions and all properties vary to some extent with temperature. It is therefore necessary to decide at what “reference” temperature the physical properties should be evaluated. Most experimental work in gases is not sufficiently precise to warrant close attention to this point, but Eichhorn (29) has made careful measurements of velocity profiles near a heated plate

14

A. J. EDE

FIG.4. Flat plate: velocity and temperature profiles in air. [From Ostrach

(4.1

ADVANCES IN FREECONVECTION

15

38 cm high, using illuminated dust particles, and has compared them with the Ostrach solution, paying particular attention to the question of the reference temperature. He found that using the surface temperature or the bulk fluid temperature altered the position of the data points by about So$, and the precision of his data was quite sufficient for this to be appreciable. After a detailed discussion of the observations of Schmidt and Beckmann (3) and of Ostrach ( 4 ) on this question he concluded that the best reference temperature (expressed as the excess temperature) was 0.83 times the surface temperature. In a theoretical paper which will be discussed later Sparrow and Gregg (20) advocated a corresponding figure of 0.62, but they added that this was not greatly superior to 0.5. The latter corresponds to the simple arithmetic mean of the surface and bulk temperatures, and is often called the “film” temperature. Eichhorn’s results (19)indicate that the velocity in the outer regions of the boundary layer is a little higher than expected, particularly towards the lower part of the surface, and he suggests that possible explanations include stray aircurrents, a starting-length effect, and failure of the boundary-layer approximations at small distances from the leading edge. A useful description of the techniques available for measuring temperature and velocity distributions in free convection is given by Kraus (21). A smaller amount of data are available for fluids other than air, and can be used for examining the effect of the Prandtl number. The data for each fluid can be represented reasonably well by the same type of equations as Eq. (22). T h e data for all fluids can therefore be correlated by plotting A = NuL/Ra2l4against Pr, provided the Rayleigh numbers are limited to a range within which Eq. (22) is valid. Figure 5 illustrates the difficulty of obtaining data of sufficient precision to test the effect of the Prandtl number. Only the small quantity of data for liquid metals reveals a significant effect, and this is in line with the theoretical prediction. In view of the great preponderance of data for air, individual points have not been plotted for this fluid ; instead, the vertical line indicates the region within which almost all the points would lie, the circle indicates the point representing a leastsquares fit of the data, and the cross indicates a mean value obtained from the relevant data of Saunders (22), probably the most precise and comprehensive individual set of results. In calculating the dimensionless numbers plotted in Fig. 5 the physical properties have been taken at the arithmetic mean of the surface and bulk fluid temperatures. Other reference temperatures have been tried but in view of the scatter of the data this is not a very profitable exercise. The matter becomes very important however for liquids of large Prandtl number because of the marked effect of temperature on viscosity. In Fig. 5 a group of data obtained with oil is plotted with physical constants taken also at the

A. J. EDE

16

bulk and surface temperatures. Correlation is evidently poor when the bulk temperature is used, but there is little to choose between the mean and surface temperatures. Agreement with the Ostrach solution is slightly better when the mean temperature is used. This is not conclusive however since the velocity boundary layer for aviscous liquid, as already mentioned, may be thicker than is consonant with the boundary-layer approximation, so that close agreement may not be expected. Also shown in Fig. 5 is the curve corresponding to the interpolation equation (28). A few measurements have been made of velocity and temperature profiles in fluids other than air. Goldstein and Eckert (23) used a Zehnder-Mach

L Ede Elhacd v Sounders Mercury

2

I

0

I

2

3

%o R

FIG.5. Flat plate: experimental data, all fluids.

interferometer to make detailed observations of temperature profiles in water (Pr = 6.4). T h e boundary condition was that of a uniform heat flux, so the results were compared with the results of the Sparrow and Gregg analysis for the appropriate Prandtl number. A very satisfactory agreement was obtained (Fig. 6). Szewczyk (24)measured temperature distributions in water by means of a thermocouple probe, and also obtained excellent agreement with theory. Wilke et ul. (25) measured the rate of muss transfer from vertical plates, using electrolysis and rates of solution of organic solids. The Grashof numbers varied from lo4 to lo9, and the Schmidt numbers from 500 to 80,000. A general correlation of NuL = 0.66(Gr12.Sc)'i4 was obtained. This provides, in effect, data at a much higher Prandtl number than any obtained so far in heat transfer measurements, and the coefficient agrees satisfactorily with the theoretical value of 0.67 (Table I).

ADVANCES IN FREE CONVECTION

17

FIG.6 . Flat plate: temperature profiles in water. [From Goldstein and Eckert (23).]

B. THEVERTICAL CIRCULAR CYLINDER For sufficiently large values of D / L , the cylinder can be regarded as effectively the same as the flat plate. When D / L is small enough for the curvature to be significant analysis is more difficult and has not been brought to the same degree of completeness. An early attempt by Elenbaas (26) was based on the Langmuir (27) stationary-film hypothesis. By an ingenious train of argument, drawing on the consideration that any relation for the cylinder should reduce to that for a flat surface as D --f m, he deduced that NuD exp(-2/NuD)

= 0.6(D/L)i'4Rag4

(43) the numerical coefficient being determined empirically. Apart from unconventional procedures such as this the majority of attempts have been based on similarity methods, applied either to the differential equations or to their integrated form. Sparrow and Gregg (28) used the former method. The boundary-layer equations in cylindrical coordinates are

a(Yzl)- 0 -+-ax ay

(44)

A. J. EDE

18

The following substitutions are made :

f(5, C) = c3 x-1‘4 $4

g(5, C) = e/e,

where 2 mm, the indicated plasmas emit blackbody radiation in the visible range of the spectrum.)

Very strong absorption (K,’ > 1) occurs for resonance lines for which the absorption coefficient K,‘ is so high that a layer thickness L of a fraction of a millimeter is already sufficient for complete absorption. In the immediate neighborhood of such a resonance line, the absorption coefficient may be a factor of 10* smaller so that layer thicknesses of lo6 cm or more are required for complete absorption. Finkelnburg and Peters ( 4 ) calculated the conditions for which the continuous radiation of a laboratory plasma would approach a blackbody radiator ( I , 2 0.9BV).Figure 4 shows the result of their calculation for five different gases. By considering only singly ionized species of these gases, all curves merge into a common curve which corresponds to an ionization degree of 100n;,. Above this common curve a laboratory plasma with a layer thickness of 2 mm or larger would be a cavity radiator at the plasma temperature. Argon, for example, with an ionization potential of 15.8 V would fall

PLASMA HEATTRANSFER

24 1

between the curves for helium and hydrogen and become a blackbody radiator for temperatures T > 2 x 104”K and pressures p > 100 atm. Cesium with the lowest ionization potential would require a minimum pressure of about 5 atm to become a cavity radiator at 5000°K. Conceivably, there are other physical processes in plasmas which also lead to the emission of continuous radiation. Neutral atoms or molecules of certain elements, for example, may have an affinity for electrons. This causes free-free and free-bound radiation by similar mechanisms as described for the interaction of positive ions with electrons. The recombination process corresponds in this case to the formation of negative ions. Another process which may be responsible for the generation of continuous spectra is the chemical reaction between neutral particles in the plasma. Such a reaction may be considered as a “recombination” process with the corresponding “recombination” continuum. This type of chemical reaction plays an important role in reentry plasmas as well as in plasmas emanating from rocket exhausts. In the latter case the situation may become rather complex because of the numerous combustion products involved. Finally, in the presence of a magnetic field in the plasma, electrons which are forced into an orbital motion around the magnetic flux lines give rise to a continuous radiation which is called cyclotron radiation. However, the number of collisions which the electrons suffer in the plasma has to be small compared with the number of electron orbits. This requirement is usually expressed by the relation w e r e= Ac/rL

1

(24)

w e is the electron cyclotron frequency, T, the average time interval between two electron collisions, A, the mean free path length of the electrons, and rL the average Larmor radius. Since this review deals essentially with rather dense plasmas and moderate magnetic field intensities, cyclotron radiation will not be of importance. The contribution of radiation to the total heat transfer between a plasma and a bounding solid wall increases with increasing pressure, increasing scale, and especially with increasing temperature. Figure 5 presents, as an example, the total intensity of the radiant flux leaving an air plasma volume element per unit solid angle. The intensity is normalized with the dimensionless density of the plasma on the basis that the intensity increases with the power 1.7 of the density. The enthalpy and the equivalent flight velocity &fz are listed on the abscissa of the diagram. The figure has been taken from Morris P t al. (7). The points present the results of shock tube and ballistic experiments. The line indicates the total continuum radiation generated by free-free and free-bound transitions of electrons. It has been obtained by extrapolation of measured values.

E. R. G. ECKERT AND E. PFENDER

242

In Sections I11 and IV of this review, radiation is not included in the discussions on boundary layer flow. It has to be added to the convective heat transfer as long as the photon mean free path is large compared to the boundary layer thickness. For the situation that the photon mean free path is of the same order as the boundary layer thickness, the radiative and the convective heat transfer interact mutually and have to be considered together. For ENTHALPY ( kJ/kg 1

.5

.I

5

I

0x10~

( M I S PROORAM A > Z o 0 0 A )

- 2 -

L

SHOCK TUBE WTA

t Id-

0

x

A NEREM.R.H. 0 HOSHIZAKI, H.

n

c3 5 -

e-

2

J-

fld

5

-

-

5-

0

FLAO5,R.F.

PAQE,W.A. (BALLISTIC OATA)

i

5 2-

pt Idc)

5-

W

y 2 -

9 lo5B 5 2 lo41

0

I 5

I 10

I 15xd

FLIGHT VELOCITY ( m/sec 1

FIG.5 . Comparison of normalized radiation intensity data of air (7).

information on methods to analyze this situation, the reader is referred to the literature, for instance, to the contribution by Cess (8). Regardless of whether or not radiation from a plasma constitutes an important heat transfer mechanism, it is frequently used for diagnostic purposes in a number of different spectroscopic measurements ( 5 ) . B. SURVEY OF STEADY, DENSE PLASMAS Before we proceed to a discussion of plasma properties and their influence on heat transfer, a classification of steady, dense (collision-dominated)

PLASMA HEATTRANSFER

243

plasmas will be undertaken. Transient plasmas will not be considered in this survey. There are essentially three different ways by which steady plasmas may be generated : either by electrical means, by combustion processes, or by shock waves. Plasmas generated by shock waves may sometimes be considered as quasi-steady plasmas, as, for example, in reentry simulation studies or in the reentry process itself. The generation of steady high-temperature plasmas is restricted to electrical methods which may be subdivided into electrical discharges with electrodes and electrodeless discharges. The commonly used discharge type with electrodes, which represents a rather simple means for producing high-temperature, high-density plasmas, is the electric arc, which attracted increasing interest during the past 20 years. Electric arcs are used for basic research as well as for many applications. Figure 6 shows a number of various arcs in a temperature sequence. The temperatures in this diagram should be understood as approximate maximum values. The arc types which fall below a maximum temperature of 104"K in Fig. 6 are designated as low-intensity arcs and they are, in general, operated at current levels I < 50 A. High-intensity arcs with axis temperatures T > 104"K ( I > 100 A) fill a rather wide gap in the temperature scale. Among these arc types, the cascaded arc, first described by Maecker (9),found widespread interest as a research tool for the experimental determination of plasma transport properties, and for flow and heat transfer studies. The highest steady-state temperature level of about 100 x lo3O K was recently achieved by Mahn et al. (10)in a magnetically confined hydrogen arc at a pressure of about & atm. Even though high-frequency (or rf-) discharges have the advantage of being free from possible electrode contaminations, this discharge type has not yet found as widespread an application as the electric arc, although it seems feasible to generate with rf-discharges atmospheric pressure plasmas which are in the same temperature range as high-intensity arc plasmas. However, several disadvantages of the rf-discharge, such as poor coupling to the power source, restriction of the heating process by the skin effect, stability of the plasma, etc., may be responsible for its limited adoption in the laboratory and for applications. Combustion-generated plasmas became particularly interesting for magnetohydrodynamic (MHD) applications. Because of their relatively low temperature, these plasmas are frequently seeded with alkaline metals or their compounds which have a rather low ionization potential. In this way reasonable electrical conductivities can be achieved in spite of the relatively low temperatures. Shock tubes of different designs are used to generate high-temperature, high-density plasmas which are especially interesting for basic studies. The

E. R. G. ECKERT AND E. PFENDER

244

temperatures observed behind the shock front are in the same range as in high-intensity arcs. A comparison of shock heated with arc-generated plasmas reveals a considerable advantage of the former in that they are almost homogeneous. In reentry simulation studies and during the actual reentry phase of a space vehicle, plasmas are generated in a similar manner, covering a rather wide pressure range (0.01 < p < 100 atm). T CKK)

loo xid

50 40

30

20

f-

MAGNETicALLY CONFINED ARCS (HYDROGEN)

-WATER

-

-GAS

WHIRL STABILIZED ARCS

WHIRL STABILIZED SHOCK WAVE GENERATED

ARCS PLASMAS

CASCADED ARCS

-HIGH

FREQUENCY DISCHARGES

-HIGH

CURRENT ARCS IN AIR

10

7 5

4 3

-LOW -HG

CURRENT ARCS IN AIR OR NOBLE ARCS GASES

-

-ALKALINE METAL ARCS CHEMICAL ROCKET EXHAUSTS SEEDED FLAME GENERATED PLASMAS

FIG.6. Placement of various plasmas on the temperature scale.

Plasmas which are of interest in this review are frequently referred to as thermal plasmas, indicating that thermalprocesses govern the plasma state in contrast to cold plasmas (for example, glow discharges) in which thermal effects of the heavy plasma constituents are not significant. The question whether such a thermal plasma is in a thermodynamic equilibrium state or not is of importance for the correct interpretation of the observed plasma phenomena. Therefore, the next section will deal with the approach to thermodynamic equilibrium in a plasma.

C. THEAPPROACH TO THERMODYNAMIC EQUILIBRIUM I N A PLASMA Many considerations in plasma physics and technology are based on the assumption of thermodynamic equilibrium in the plasma. Because of the

PLASMAHEATTRANSFER

245

fundamental importance of this concept and its bearings on plasma heat transfer, the main facts will be discussed in this review. A more comprehensive treatment of this subject may be found in the work of Finkelnburg and Maecker (3),Griem (5), Unsold (ZI), and Drawin and Felenbok (12).

1. The Plasma in Perfect Thermodynamic Equilibrium Thermodynamic equilibrium prevails in a uniform, homogeneous plasma volume if kinetic and chemical equilibria as well as every conceivable plasma property are unambiguous functions of the temperature which in turn is the same for all plasma constituents and their possible reactions. More specifically, the following important conditions must be fulfilled. (a) The velocity distribution functions for particles of every species r which exists in the plasma, including the electrons, follows a MaxwellBoltzmann distribution :

vr is the velocity of particles of species r, m, is their mass, and T is their temperature, which is the same for every species r , and which is, in particular, identical with the plasma temperature. (b) The population density of the excited states of every species r follows a Boltzmann distribution [see Eq. (3)] :

nr(gr,r/zr)exp ( - X r . s l k B (26) T h e excitation temperature T which appears explicitly in the exponential term and implicitly in the terms n, and 2, of this relation is identical with the plasma temperature. (c) T h e particle densities (neutrals, electrons, ions) are described by the Saha-Eggert equation which may also be considered as a mass action law : %,r =

x r f l represents

the energy which is required to produce a ( r + 1)-times ionized particle from a r-times ionized particle. Equation (11) is identical with Eq. (27) for r = 0. The ionization temperature T in this equation is identical with the plasma temperature. A similar relation holds for the dissociation process, which is of importance for plasmas which contain, in addition, molecular species. (d) The electromagnetic radiation field is that of blackbody radiation of the intensity B , as described by the Planck function [see Eq. (16)]

246

E. R. G. ECKERT AND E. PFENDER

B

2hV3 1 -~ ' c 2 exp(hv/ksT) - 1 =-..

~

The temperature of this blackbody radiation is again identical with the plasma temperature. I n order to generate a plasma which follows this ideal model as described by Eqs. (25) to (28), the plasma would have to dwell in a hypothetical cavity whose walls are kept at the plasma temperature or the plasma volume would have to be so large that the central part of this volume, in which thermodynamic equilibrium prevails, would not sense the plasma boundaries. In this way the plasma would be penetrated by blackbody radiation of the same temperature. An actual plasma will, of course, deviate from these ideal conditions. The observed plasma radiation, for example, will be much less than the blackbody radiation because most plasmas are optically thin over a wide wavelength range as pointed out in the first section of this review. Therefore, the radiation temperature of a plasma deviates appreciably from the kinetic temperature of the plasma constituents or the already mentioned excitation and ionization temperatures. In addition to radiation losses, plasmas suffer irreversible energy losses by conduction, convection, and diffusion which also disturb the thermodynamic equilibrium. Thus, laboratory plasmas as well as some of the natural plasmas cannot be in a perfect thermodynamic equilibrium state. In the following sections, deviations from thermal equilibrium and their significance will be discussed.

2. The Concept of Local Thermal Equilibrium Since an actual plasma does not exhibit a homogeneous distribution of its properties (for example, in temperature and density), equilibrium considerations can only be applied locally. For this reason the concept of local thermodynamic (or thermal) equilibrium (LTE) was introduced which is less restrictive than the definition of perfect thermodynamic equilibrium. In this sense L T E may be considered as a special case of the more general concept of thermal equilibrium. L T E requires that collision processes and not radiative processes govern transitions and reactions in the plasma and that there is a microreversibility among the collision processes ;in other words, a detailed equilibrium of each collision process with its reverse process is required. Steady state solutions of the respective collision rate equations will then yield the same energy distribution pertaining to a system in complete thermal equilibrium with exception of the rarefied radiation field. L T E requires further that local gradients of the plasma properties (temperature, density, heat conductivity, etc.) are sufficiently small so that a given particle which diffuses from one location to another in the plasma finds sufficient time to equilibrate, i.e., the

PLASMA HEATTRANSFER

247

diffusion time should be of the same order of magnitude as the equilibration time. From the equilibration time and the particle velocities an equilibration length may be derived which is smaller in regions of small plasma property

FIG.7. Temperature profiles at various cross sections of a high mass flux argon plasma jet (z:distance from nozzle exit). 10

8

6

2

0

0

I

2 3 RADIUS ( mm 1

4

FIG.8. Isotherms of a low mass flux argon plasma jet ( z : distance from nozzle exit).

gradients (for example, in the center of an electric arc). Therefore, with regard to spatial variations LTE is more probable in such regions. Heavyparticle diffusion and resonance radiation from the center of an inhomogeneous plasma source help to reduce the effective equilibration distance

248

E. R. G. ECKERT AND E. PFENDER

in the outskirts of the source. Figures 7 and 8 show a temperature profile and the isotherms of argon plasma jets (13)as examples of spatial variations of the temperature within a plasma. In the following a systematic discussion of the important assumptions for L T E will be undertaken based on actual plasmas among which the electric arc appears as the most appropriate source. a . Kinetic Equilibrium. It may be safely assumed that each species (electron gas, ion gas, neutral gas) in a dense high-temperature plasma will assume a Maxwellian distribution. However, the temperatures defined by these Maxwellian distributions may be different from species to species. Such a situation which leads to a two-temperature concept will be discussed for an arc plasma.

PRESSURE (mrnHg)

-

FIG.9. Electron and gas kinetic temperatures in an arc plasma (20).

T h e electric energy which is fed into an arc is dissipated in the following way. T h e electrons according to their high mobility pick up energy from the electric field which they partially transfer by collisions to the heavy plasma constituents. Because of this continuous energy flux from the electrons to the heavy particles, there must be a “temperature gradient” between these two species, so that Te T , T, is the electron temperature, and T , the temperature of the heavy species, assuming that ion and neutral gas temperatures are the same. In the two-fluid model of a plasma, defined in this manner, two distinct temperatures T , and T, may exist. The degree to which T , and T, deviate from each other will depend on the thermal coupling between the two species. The difference between these two temperatures can be expressed by the following relation (3):

’

PLASMA HEATTRANSFER

249

T,- Ta=4 m -_____ (A,eE)2 Te g m e (3/2kBTe)' ma is the mass of the heavy plasma constituents, A, the mean free path length of the electrons, and E the field intensity. Since the mass ratio ma/8m, is for hydrogen already about 230, the amount of (directed) energy (A,eE) which

the electrons pick up along one mean free path length has to be very small compared to the average thermal (random) enengy 3/2kBT, of the electrons. Low field intensities, high pressures (A, lip), and high temperature levels are favorable for a kinetic equilibrium a'mong the plasma constituents. At low pressures, for example, appreciable deviations from kinetic equilibrium may occur. Figure 9 shows in a semischematic diagram how electron and gas temperature separate in an electric arc with decreasing pressure. For an atmospheric argon high-intensity arc with E = 13 V/cm, A, = 3 x cm, mA/me= 7 x lo4, and T, = 30 x 103"K,the deviation between T, and Ta is only 2% (3). 6. Excitation Equilibrium. In order to determine the excitation equilibrium every conceivable process which mqy lead to excitation or deexcitation has to be considered. We will restrict ourselves in this discussion to the most prominent mechanisms which are collisional and radiative excitation and deexcitation.

-

Excitation (1) electron collision (2) photoabsorption

Deexcitation (1) collision of the second kind (2) photoemission

We saw that for the case of a perfect thermodynamic equilibrium microreversibilities have to exist for all prbcesses; i.e., in the above scheme excitation by electron collision will be balanced by the reverse process, namely, collisions of the second kind and excitation by photoabsorption processes will be balanced by photoemission processes which include spontaneous and induced emission. T h e population of excited states is given by a Boltzmann distribution [see Eq. (26)]. The microreversibility for the radiative processes holds only if the radiation field in the plasma reaches the intensity B, of blackbody radiation. However, actual plasmas are over most of the spectral range optically thin, so that the situation for excitation equilibrium seems to be hopeless, Fortunately, if collisional processes dominate, photoabsorption and photoemission processes do not have to balance; only the sum on the left-hand side and the right-hand side of the scheme above have to be equal. Since the contribution of the photoprocesses to the number of excited atoms is almost negligible under this condition, the excitation process is still close to LTE.

25 0

E. R. G. ECKERT AND E. PFENDER

c. Ionization Equilibrium. For the ionization equilibrium we will again only consider the most prominent mechanisms which lead to ionization and recombination.

Ionization (1) electron collision (2) photoabsorption

Recombination (1) three-body recombination (2) photorecombination

In a perfect thermodynamic equilibrium state with cavity radiation, a microreversibility among the collisional and radiative processes would exist and the particle densities would be described by the Saha-Eggert equation. Without cavity radiation the number of photoionizations is almost negligible, requiring instead of the microreversibility a total balance of all processes involved. Photorecombinations, especially at lower electron densities, are not negligible. Elwert (14) showed that the frequency of the three remaining elementary processes is only a function of the electron density leading, for n, = 7 x 1015 cmP3, to the same frequency of these elementary processes. The result is an appreciable deviation between actual and predicted values [from Eq. (27)]of the electron densities. Only for values n, > 7 x 1015 does the Saha-Eggert equation predict correct values. For smaller electron densities the Corona formula (14) has to be used which considers only ionization by electron impact and photorecombination :

In this equation a is Sommerfeld’s fine-structure constant, 5, the number of valence electrons, n the principal quantum number of the valence shell, the ionization energy of hydrogen, andg a constant with a value between 1.4 and 4 (14). T h e particle concentrations in low-intensity arcs at atmospheric pressure, for example, have to be calculated with this formula. Significant deviations from Saha-equilibrium may also occur in the fringes of high-intensity arcs and plasma jets generated by arcs. Another important chemical reaction in a plasma which is generated from a molecular gas is the dissociation process. The considerations for this reaction are very similar to ionization and will, therefore, not be reiterated here. In summary, we found that L T E exists in a steady optically thin plasma when the following conditions are simultaneously fulfilled : (u) The different species which form the plasma have a Maxwellian

distribution. (/3) Electric field effects are small enough, and the pressure and the temperature are sufficiently high so that T , = T,.

PLASMA HEATTRANSFER

25 1

( y ) Collisions are the dominating mechanism for excitation (Boltzmann

distribution) and ionization equilibrium (Saha-Eggert equation). (6) Spatial variations of the plasma properties are sufficiently small. Besides the conditions for the two extreme cases, namely L T E (based on Saha ionization equilibrium) and Corona equilibrium, we are also interested in the region between these two limiting cases. In this range three-body recombination as well as radiative recombination and deexcitation is significant. Several authors (14-18) present theories for ionization equilibrium over the entire range of radiative-collisional elementary processes. In particular, Bates et al. (15) report detailed calculations of optically thin and optically thick hydrogen plasmas. We will discuss their results for the optically thin case. If a is the combined collisional-radiative recombination coefficient and S the corresponding ionization coefficient, rate equations may be established which describe the effective rate of population and depopulation. The rate of population of the ground state is described by

(%c)

=

meni

POP

In this relation no,orepresents the number of neutral hydrogen atoms in the ground state ; n, and n iare electron and ion densities, respectively. The rate of depopulation of the ground state is given by

Under steady-state conditions

or Figure 10 shows a state diagram for values of S / a as a function of the electron density with pressure and electron temperatures as parameters mV3) . high electron densities [plotted from data of Bates et al. ( 1 5 ) ] At pairs of L T E and non-LTE curves plotted for the same electron temperature merge. At low electron densities mP3) the non-LTE curves merge into curves valid for Corona equilibrium. The divergence of the non-LTE curves from the L T E curves at lower pressures and/or lower electron temperatures and densities shows how large the deviation from L T E may become in such parameter ranges. Taking values for 1atm it can be seen that L T E is closely approached for electron temperatures in the interval 14,000 < T, < 28,000"K.

252

E. R. G. ECKERTAND E. PFENDER

Deviations of this kind from L T E may be found, for example, in arc plasma regions adjacent to walls where the electron density drops appreciably. T h e situation becomes even more complicated in the immediate vicinity of a wall where a thin layer, called a plasma sheath, separates the actual plasma from the wall. I n this sheath strong deviations from quasi neutrality may be found. Because of the general importance of plasma

Id8

lozo 102' loz2 ELECTRON DENSITY (I?) NON-LTE ----- -. LTE HYDROGEN 10"

loz3

loz4

~

FIG.10. State diagram for hydrogen in L T E and non-LTE (15).

sheaths for the interaction between a plasma and neighboring walls and in particular for plasma heat transfer, the next paragraph will be devoted to a brief discussion of plasma sheaths and their formation.

D. PLASMA-WALL BOUNDARIES AND PLASMA SHEATHS Every solid component (electrodes and walls) of a plasma device which is exposed to a hot plasma has to be protected by appropriate cooling. In

PLASMA HEATTRANSFER

253

engineering and laboratory devices the plasma is frequently constricted or enclosed by walls and/or electrodes which are water-, transpiration-, or radiation-cooled. In steady-state operation the walls and electrodes carry a continuous heat flux driven by the temperature difference between the plasma and these cooled components which extend their cooling effect into the plasma in a thin layer which may be described as a temperature boundary layer. T h e thickness of this boundary layer comprises many free path lengths of the electrons or heavy particles. At the bottom of this boundary layer, overlying the solid wall, a plasma sheath is formed with a thickness in the order of a Debye length.

Sheath Formation For the following consideration, we assume an electrically conducting but insulated wall “in contact” with a dense plasma, a situation found, for example, in a wall-stabilized high intensity arc at atmospheric pressure. Since there is no net current flow to an electrically insulating wall, electron and ion currents reaching the wall have to compensate each other. Recombination of the electrons and ions occurs at the wall surface and, for a cold wall, also to a certain degree in the plasma close to the wall surface. T h e corresponding concentration gradient of the electrons and ions causes these particles to diffuse toward the wall. The diffusion coefficient of the electrons is much higher than that of the ions, which would mean an electron current larger than the ion current. Such a situation which cannot exist on an insulated wall is prevented by the fact that the wall assumes a negative potential which retards the electrons and accelerates the ions until both fluxes are of equal magnitude. This requires that quasi-neutrality can no longer exist close to the wall. Deviations from quasi-neutrality, however, are restricted to a very thin layer which is called a sheath (Fig. 11).The fact that there is no net charge carrier flow to the surface can be expressed mathematically in the following way, assuming a one-dimensional situation with many collisions occurring in the sheath :

i = j , + j i = (eneve+ enipi)Ex + eD,dn , dx

-

dni dx

EDi - = 0

(36)

In this relation, n, indicates the particle density of the electrons, ni the particle density of the ions, pe and p i are the electron and ion mobility, respectively, Ex is the electric field intensity in the direction normal to the surface, and D, and Di are the electron and ion diffusion coefficients into the predominantly neutral gas. The electron mobility and diffusion coefficients are much larger than the corresponding values for the ions. The gradient of the electric field intensity is connected with the space charge c(ni - n,) by the Poisson equation

E. R. G. ECKERT AND E. PFENDER

254

Since charge imbalances can only occur over a distance in the order of a Debye shielding length, AD, the thickness of the sheath will also assume a value in this order of magnitude.

WALL

0

v I

7

L

7

L

1 I

I

II I

I I

-c

FIG.11. Temperature boundary layer and sheath formation.

*

Assuming that T , T, and considering shielding by electrons only in a singly ionized gas, the Debye shielding length may be expressed by

This shielding length depends on the heavy-particle temperature and the electron density. Any relation between the electron density and the temperature T, will depend on the thermodynamic state of the plasma. For an estimate of the order of magnitude of the Debye length in an atmospheric argon plasma at a temperature of 104"K,LTE will be assumed in spite of the fact that Eq. (38) does not hold for LTE. The electron density in such a plasma is approximately 3 x 1 O I 6 cmp3 and the corresponding Debye length is about cm. T h e Debye length is, in this case, small compared to the mean 4x free path length of the electrons (about 3 x lo-' cm) and the atoms (about 5 x lop5cm). The buildup of a space charge can, therefore, not occur in the

PLASMA HEATTRANSFER

255

plasma itself, since Coulomb forces effectively prevent the separation of ions and electrons and keep the plasma neutral (Fig. 11). Near an electrically insulated surface the electron density will be much smaller and the Debye length as well as the sheath thickness, according to Eq. (38), much larger. T h e Saha equation certainly does not apply in this region and too little is known about the electron density to make a quantitative prediction of the sheath thickness possible. In most weakly ionized high-pressure plasmas, the sheath is, therefore, probably collision dominated and Eq. (36) can be applied. Sheaths in highly ionized plasmas at moderate or low pressures may, on the other hand, be considered as collisionless and the following relations exist for this condition. An estimate of the thickness of a collisionless sheath yields (29)

T h e potential drop across the sheath, assuming that there are no collisions in the sheath, can be found from the requirement that electron and ion currents balance (29):

T , is the electron temperature in the plasma and ma is the mass of the ions (or atoms). For a net current flow to the originally insulated surface, the thickness d, of the collisionless sheath will depend on the potential difference across the sheath and on the net current drawn to the surface. A net current flow to the surface may be achieved by biasing the surface with respect to the plasma. With a positively biased surface the situation may be compared with that of a space-charge-limited thermionic vacuum diode by identifying the edge of the plasma sheath with the thermionically emitting cathode. The random electron current reaching the sheath from the plasma corresponds in this analogy to the electron current emitted from the cathode. As in a diode, the current to the biased surface depends on the applied potential across the collisionless sheath and on the thickness of the sheath (20) :

T,is the electron temperature at the edge of the sheath. The last term in the bracket of Eq. (41) represents a correction term for the common spacecharge equation which accounts for the initial velocities of the electrons

256

E. R. G. ECKERT AND E. PFENDER

entering the sheath. A very similar relation holds for a negatively biased surface which draws an ion current of the density

where Tirepresents the ion temperature at the edge of the sheath. T h e potential imposed on the surface is essentially confined to the sheath in which a space charge of the opposite sign absorbs the electric field. This fact has a very important consequence for the applicability of Langmuir probes in plasmas (21). T o what degree the existence of the sheath, whether collisionless or collision dominated, influences the heat transfer to a wall depends on the value of the ratio of the sheath thickness d, to the boundary layer thickness 6. T h e influence of the sheath will be small as long as the ratio hD/S is small in order of magnitude. In the analyses discussed in Sections I11 and IV, the influence of the sheath on heat transfer is assumed to be negligible.

E. THERMODYNAMIC AND TRANSPORT PROPERTIES OF PLASMAS For plasma heat transfer a number of thermophysical properties of the plasma are of importance which may be subdivided into thermodynamic and transport properties. In this review these properties will be briefly discussed for argon plasmas.

1. Plasma Composition and Thermodynamic Properties For the calculation of the chemical composition of a plasma in thermodynamic equilibrium (see Fig. 1) as a function of the temperature, a system of simultaneous nonlinear equations has to be solved. This system of equations remains essentially the same for any given gas or gas mixture from which a plasma may be generated. The appropriate equations are obtained from the conservation-of-mass law na = C nua r

(43)

where nu is the total number density of particles of a given species ( a ) , n/ represents the number density of atoms or ions of the same species ( a ) , and r indicates the ionization stage ;the condition for quasi neutrality n, = C rn/ r

(44)

PLASMA HEATTRANSFER

257

and the mass-action law which is, in this case, expressed by the Saha-Eggert relations [Eq. (27)] which connect the particle densities of the different ionization stages with the electron densities

(45)

'

TEMPERATURE

fi)

i x 10'

FIG.12. Composition of an argon plasma in thermodynamic equilibrium (22).

If the plasma contains r different ionization stages of a certain species ( a ) and if there are N different species in the plasma, then the number of unknown particle densities is rN + 1 where the electron density is the additional unknown. Since there are ( Y - 1) Saha-Eggert equations for each species and one conservation-of-mass relation, the total number of equations is r N . The last required relation is provided by Eq. (44).From this complete set of equations the equilibrium composition of any given plasma may be calculated. As an example, Fig. 12 shows the equilibrium composition of an

258

E. R. G. ECKERT AND E. PFENDER

argon plasma at atmospheric pressure. As pointed out in the preceding section, actual plasmas deviate more or less from a perfect thermodynamic equilibrium which, of course, will also cause deviations from the equilibrium plasma composition.

TEMPERATURE (K)

FIG.1 3 . Mass density of an argon plasma (22).

Having the composition of a plasma calculated, the thermodynamic properties of the plasma (mass density, internal energy, enthalpy, entropy, and specific heat) may be found from the contribution of the different plasma constituents to the overall properties. Figures 13-17, taken from Cambel (22), demonstrate examples of such equilibrium properties for argon plasmas as a function of the temperature. Another frequently used method for illustration of the most important thermodynamic properties in a single diagram is by means of enthalpy-entropy charts (Mollier diagrams). Figure 18 shows

PLASMA HEATTRANSFER

259

such a Mollier diagram for an argon plasma in perfect thermodynamic equilibrium whereas Fig. 19 refers to an argon plasma based on an ionization equilibrium according to Elwert (14).Both diagrams are valid for a temperature range lo4< T < lo5OK and are plotted with temperature, mass density, and total pressure as parameters (23).At high pressures there is a reasonable

r

5x18

2-

'01

-

I

X 0

\ X 7

>

5-

W

a W

2 W

d 2z

E f

5E 0

10

15

20

TEMPERATURE

25

30

-.

35xld

fK 1

FIG.14. Internal energy of an argon plasma (22).

agreement of the different thermodynamic properties of Fig. 19 with the corresponding values in Fig. 18 up to temperatures of about 5 x 104"K. As discussed in the preceding section, there is a predomination of collisional processes in this parameter range so that even without blackbody radiation (photo processes neglected) thermodynamic equilibrium may be closely approached. At pressuresp < 1 bar, Corona equilibrium prevails causing the isotherms to be parallel to the abscissa. For the temperature range T < 2 x 104"K,which is of particular interest in plasma applications, Fig. 20

E. R. G. ECKERT AND E. PFENDER

260

shows thermodynamic properties for an argon plasma in perfect thermodynamic equilibrium (24). 2. Plasma Transport Properties

Transport phenomena in plasmas encompass the flow situation of every plasma constituent, namely electrons, ions, and neutrals including radiation fluxes under the influence of driving “forces” as, for example, electric fields, 5

2

loB

a

X

5

\

X 7

t J

9!-

2

z

w

10‘

5

0

10

15 20 25 TEMPERATURE fK)

30

35x10~

FIG.15. Enthalpy of an argon plasma (22).

temperature-, pressure-, density-, and velocity-gradients. In order to describe the transfer of electrical charge, mass, momentum, and energy within the plasma and from the plasma to its surroundings, characteristic transport properties have been defined as, for example, electrical conductivity, heat conductivity, viscosity, and diffusivity. For the calculation of these transport properties different methods have been proposed. For further information, the reader is referred to the work of Finkelnburg and Maecker (3) and Cambel(22).

26 1

PLASMA HEATTRANSFER

---l--/

3 / / -

0

10

15

20

25

30 x 10’

TEMPERATURE CK)

FIG. 16. Entropy of an argon plasma (22).

TEMPERATURE lo5OK) is the excessive specific heat flux to the walls of the container including the electrodes. From the survey in Fig. 6 it follows, for example, that a water-cooled constricted arc provides higher maximum temperatures than a free-burning arc because this maximum depends on the power input ZE per unit length of the arc column (25). For a given current I , the field intensity E and therefore the power input I E per unit length is a function of the heat flux to the surroundings. This flux is appreciably higher for a constricted arc column.

PLASMA HEATTRANSFER

263

T h e difficulty in increasing the temperature can be seen by a simple consideration of the conditions in a fully developed rotationally symmetric arc column within a water-cooled constrictor. Radiation from the arc column will be neglected so that energy losses occur by radial conduction only. Balancing the power input to the arc per unit volume with the radial heat flux, we obtain the Elenbaas-Heller differential equation

j E + div (kgrad 2')

5

10

15

=

0

20

(46)

25

30

ENTROPY (kJ/k Y,, the electrical conductivity vanishes. T h e temperature at r = yo may be To. Double integration of Eq. (47) with the condition that the temperature gradient is equal to zero at 1' = 0 results in the relation

264

E. R. G. ECKERT AND E. PFENDER

T o solve the right-hand integral, a relation k = f ( T ) is required. Since we are interested in extremely high temperatures, the conducting core may be considered as fully ionized and Spitzer’s formula k T5/*may be used as

-

DENSITY (hg/m’l

FIG.20. Enthalpy-density diagram of an argon plasma in thermodynamic equilibrium (24.

an approximation (26). Assuming, additionally, the temperature To to be small compared to T,,, results in the relation with

IE

- Td!:

I = n r o2 ’

(49)

(50) To raise the maximum temperature by a factor of 3 requires about a 50 times higher power input according to Eq. (49). Radiation, which has been neglected, makes this factor even larger. Calculations by Maecker (25)for

PLASMA HEATTRANSFER

265

hydrogen and gases similar to hydrogen indicate that temperatures T > 3 x 104"Kare not feasible in this way even if one provides this power because there is no material which is able to withstand the high specific wall heat fluxes. The highest permissible specific heat flux to a water-cooled

1

/

I

ARGON I ATM

3 10

5 TEMPERATURE fK )

FIG.21. Electrical conductivity of an argon plasma (22).

copper wall is about 19 kW/cm2.Steady plasmas at higher temperature levels will only be feasible if appropriate means can be found to reduce the heat flux to the walls. A reduction of the wall heat flux can be achieved either by a reduction of the heat conductivity or by reducing the temperature gradient at the wall which determines the heat flux into the wall. Two possibilities have been proposed by which the heat conductivity can be reduced (10). One of these possibilities considers the fact that k V$ for a Knudsen number Kn < 1 with the tube radius as reference length. A plasma maintained in a low-

-

E. R. G. ECKERT AND E. PFENDER

266

-

ARGON IATM

5

L

//I

10 TEMPERATURE CK)

15x10’

FIG.22. Thermal conductivity of an argon plasma (22).

TEMPERATURE

(K)

FIG.23. Viscosity of an argon plasma (22).

PLASMA HEATTRANSFER

267

pressure environment would meet this condition. T h e second method uses a superimposed magnetic field parallel to the axis of the arc which reduces the heat conductivity in the radial direction. A reduction of the wall heat fluxes by reducing the temperature gradient at the wall can also be implemented with a transpiration-cooled wall. T h e last two methods will be briefly discussed based on some recent publications. 0.5 0.4

W

a

2 0.2 W m h

0.I TEMPERATURE

(.K)

FIG.24. Pressure increase in a magnetic field (28).

1. Reduction of the Heat Conductivity by M e a m of a Magnetic Field For the following discussion, a rotationally symmetric steady highpressure plasma column will be considered which is enclosed in a watercooled tube. Energy losses are compensated by ohmic heating which is provided by an axial electric current. Such a column exchanges steady diffusion currents in the radial direction, consisting of electrons and ions flowing toward the wall, and the neutral gas particles flowing toward the center of the plasma column as described in Section I1,A. For steady-state conditions there is no net electric current in the radial direction and the total mass flux in the radial direction has also to vanish. By superimposing an axial magnetic field, the diffusion of the charged particles will initially be reduced whereas the diffusion of the neutral particles is not influenced until again a steady state is reached with a somewhat higher pressure in the plasma column. Wienecke (27) and Witkowski (28) calculated the pressure increase in the column of a high-pressure arc under the influence of an axial magnetic field. Wienecke assumed LTE in the plasma whereas Witkowski based his

268

E. R. G. ECKERT AND E. PFENDER

calculations on Corona equilibrium. Figure 24 shows a typical result of their calculations. For sufficiently high temperatures (fully ionized plasma) the heat conductivity perpendicular to the magnetic field lines becomes rather small (k T-''') as shown by Braginskii (29) and Feneberg (30). Not only the

-

L

15x10~

HYDROGEN

5x10' BAR

0' 0

I

1

2

4

6

'-f

8x10'

TEMPERATURE (OK)

FIG.25. Influence of a magnetic field on the heat conductivity (30).

electron heat conductivity k, and the ion heat conductivity kiare reduced but also kI (see Fig. 2), which represents the diffusion transport of ionization energy, is appreciably influenced by the magnetic field (31). T h e strong effect of a magnetic field on the heat conductivity is demonstrated in Fig. 25 which was taken from Feneberg (30). Estimates for such favourable conditions show that plasma temperatures of lo5"K are feasible with a power input of only 3 kW/cm using hydrogen at 0.1 atm and applying a magnetic field of 2 W/m2 (10).

PLASMA HEATTRANSFER

269

2. Reduction of Wall Heat Fluxes by Transpiration Cooling In a cylindrical plasma column enclosed in a nonconducting porous transpiration-cooled tube, the temperature profile assumes an entirely different shape compared with that obtained in a water-cooled constrictor (Fig. 26). The temperature gradient at the wall becomes much smaller

HYDROGEN

-TRANSPIRATION-COOLED

-

.---WATER-COOLED

rr

0.25 cm

= 60.000%

,-Tmoi

-

-

-

0

.2

.4

.6

NORMALIZED

.8

10 .

RADIUS

FIG.26. Temperature profiles of a water-cooled and transpiration-cooled, constricted arc (32).

because the cold gas transpiring through the wall into the tube causes a convective energy transport in a direction opposite to that of the conduction heat flux. The energy which would, without transpiration cooling, flow to the wall is now intercepted and used to heat the transpiring gas. Anderson and Eckert (32) show that maximum temperatures of about 60 x lo3"K can be reached in the axis of a transpiration-cooled arc using a power input of about 50 kW/cm in atmospheric hydrogen. The power input is, of course, very high because the gas transpiring through the wall into the arc has to be heated, as mentioned before. However, the wall heat fluxes

270

E. R. G. ECKERT AND E. PFENDER

which constitute the primary restriction for the maximum temperatures obtainable in constricted arcs with water-cooled walls are in this case almost negligible. A more detailed discussion of the transpiration-cooled arc follows in Section IV,A.

111. Plasma Heat Transfer in the Absence of an Externally Applied Electric or Magnetic Field

A. QUALITATIVE CONSIDERATIONS This area of heat transfer has found attention mainly in connection with the reentry problem as was mentioned in the Introduction. We will first attempt to evaluate qualitatively how much such a heat transfer in an ionized gas is expected to differ from heat transfer in an ordinary gas at low temperature and we will do this for the example of rotationally symmetric stagnation flow. Heat transfer of a fluid with constant properties flowing with a rotationally symmetric laminar boundary layer over a surface near a stagnation point can be calculated from the following equation (33): Nu = 0.76 Re'/* Prn.4

(51) (for an explanation of the symbols, see the Nomenclature). The Nusselt number, Nu, is a dimensionless expression for the heat transfer coefficient. The symbol Re denotes the Reynolds number and Pr the Prandtl number. Relation (51) also holds for a gas as long as the product pp of density p and viscosity p as well as the Prandtl number are constants. The specific heat cp of the gas is allowed to vary arbitrarily with temperature when the heat transfer coefficient hi,combined in the Nusselt number Nu = hicpL/k with the specific heat cp, the thermal conductivity k, and a characteristic length L, is defined with an enthalpy difference as driving potential according to the equation (33): q r u = hi(G - j z u ) (52) qw, in this equation, denotes the heat flux to the wall surface per unit area and time, ie the stagnation enthalpy of the gas outside the boundary layer, and i, the enthalpy which the gas has in immediate proximity of the wall surface. It is easily checked that for a gas as defined above Eq. (51) leads to the same result regardless of at which temperature (or enthalpy) the properties are introduced into the dimensionless parameters Nu, Re, or Pr. In a two-component gas mixture, energy is transported not only as heat by thermal conduction, but also as enthalpy carried along by the individual components in their interdiffusion process. The relative intensity of the two transport mechanisms is described by the Lewis number, Le, defined as

PLASMA HEATTRANSFER

27 1

Le = pep Dlk (53) with p indicating the density, cp the specific heat at constant pressure, k the thermal conductivity of the mixture, and D the mass diffusion coefficient for the two components. For a gas mixture with a Lewis number equal to one, Eqs. (51) and (52) describe the total heat transfer process including diffusional and conductive energy transport. This applies even when chemical reactions between the components occur within the boundary layer, provided the enthalpies in Eq. (52) include the reaction enthalpy, and it holds regardless of whether or not local composition equilibrium exists within the boundary layer, and whether or not some relaxation process is involved which delays the reactions, as long as the proper enthalpies outside the boundary layer and at the wall surface are introduced in Eq. (52). Equation (51) can still be used to obtain approximate heat transfer coefficients for a gas with property variations different from the ones mentioned above, provided the thermodynamic and transport properties are introduced at a properly selected enthalpy with a value between the extremes occurring within the boundary layer. The arithmetic mean between i, and iw has been found to be a fairly good approximation. For a gas with a Lewis number different from one, corrections have been established by various authors. Fay and Riddell (34),for instance, found that the heat flux increases when the gas under consideration has a Lewis number larger than one and that this increase can, with good approximation, be accounted for in dissociated air by adding the term qc =

k (Len- 1)iD c* L

(54)

to the heat flux calculated with Eqs. (51) and (52). The exponent n was found to be 0.52 for equilibrium composition and 5 for frozen flow within the boundary layer. iD indicates the dissociation enthalpy. When the temperature is sufficiently high for thermal ionization, electrons and ions appear in addition to neutral particles in the gas. The Prandtl number Pr = p c p / k is now strongly reduced because of electron conduction and may drop at sufficiently high temperatures to values as low as 0.01 which are comparable to those of metals. The thermal conductivity is correspondingly increased. One might assume that the heat transfer coefficient, which according to Eq. (51) is proportional to k".6,increases drastically for such a situation. This, however, is not the case for a cold catalytic wall because the surface will always be separated by a nonionized or poorly ionized layer from the hotter part of the boundary layer. This layer will extend through the major portion of the boundary layer when equilibrium composition between charged and neutral particles is established within the boundary layer, because the temperature at which ionization starts will be, in

272

E. R. G. ECKERT AND E. PFENDER

general, closer to the temperature in the hot plasma outside the boundary layer than to the wall surface temperature. This nonionized layer will have a relatively low conductivity and will act as a thermally insulating layer between the ionized gas and the wall surface. In a chemically frozen boundary layer, conditions will be somewhat different. The concentration of ionized particles within the boundary layer is then determined by the diffusion process and will drop in a more or less linear way from the value outside the boundary layer to the value zero at the catalytic wall surface. One has therefore to expect that heat transfer will be somewhat larger for a frozen boundary layer than for an equilibrium boundary layer. Only a noncatalytic surface together with a nearly frozen condition within the boundary layer can establish an electron density with finite values through the whole width of the boundary layer and electronic conduction is then expected to increase the heat transfer to the wall surface by an order of magnitude. There is also another factor involved in heat transfer from an ionized gas which tends to reduce the heat flux. In the interdiffusion of electrically charged and uncharged particles in an ionized gas, the electrons and ions move as if they were joined together. This diffusion process is called ambipolar diffusion. The diffusion coefficient D for this process is assumed to be approximately by a factor of two larger than the diffusion coefficient for the corresponding neutral particles. It therefore maintains its order of magnitude in the Lewis number [Eq. (53)], whereas the thermal conductivity increases, possibly by an order of magnitude or more. This causes the Lewis number to assume values smaller than one. Correspondingly the correction term qc given by Eq. (54) is negative and reduces the Nusselt number. In summary, it may therefore be expected that the Nusselt number and the heat flux to the surface are not strongly influenced by the ionization process and that the heat flux in a boundary layer at composition equilibrium will be somewhat smaller than in a frozen boundary layer. More detailed analyses and experiments which verify our conclusions will be discussed in a later section. Before doing so, the basic equations describing combined diffusion and conduction processes will be developed.

B. BASICTRANSPORT EQUATIONS In this section an ionized gas will be considered which is macroscopically at rest. Such a gas consists, in general, of molecules, atoms, ions, and electrons, each again possibly comprising several species. Diffusion processes will set in when the concentration of the components varies locally. T h e analysis of this diffusion process is extremely involved and simplifications are therefore usually introduced. Such simplifications have been discussed,

PLASMA HEATTRANSFER

273

for instance, in the review paper by Chung (I). In the papers which will be discussed in the next section, the “binary diffusion model” is utilized. This model replaces, with regard to diffusion, the actual gas by a two-component mixture, one component comprised of the molecules and the other of the atoms, ions, and electrons. T h e justification for the use of such a model is the fact that at sufficiently high densities Coulomb forces prevent the ions and electrons from being separated by a diffusion process. The atoms and ions, on the other hand, differ less in their molecular weight than both differ from the molecular weight of the molecules. The binary diffusion model for this reason was found to give useful results and will be used in the basic equations describing mass and heat fluxes. A diffusional mass flux can be generated by a concentration gradient, by a pressure gradient, and by temperature gradients. In a boundary layer, diffusion occurs in a direction in which pressure gradients are negligibly small. For this reason pressure diffusion will be disregarded. Diffusion caused by an electric field tends to separate the electrons and the ions. Coulomb forces acting between the particles themselves, however, counteract this tendency and diffusion by body forces will therefore also be neglected. T h e remaining diffusion by concentration gradients and by temperature gradients is described in the two-component gas mixture by the following equation (35):

In this equation it is assumed that a concentration and temperature gradient exists in y direction. j D l denotes the mass flux per unit time and area of component one in this direction. w 1 is the mass fraction of component one, w 2the mass fraction of component two. T denotes the temperature, p the density of the mixture, D the binary mass diffusion coefficient, and CL the thermal diffusion ratio. The mass flux j D 2of component 2 is by definition equal to -jDl. An energy flux E per unit time and area will also be present. Its magnitude is described by the following equation : E =

-k

aT -

aY

+ (i, - i2)jD1- CLRTMM2 lM p 1

The first term in this equation describes energy transport by heat conduction. The second term gives the energy transport by enthalpy interdiffusion and the third term describes energy transport by the diffusion thermo effect. h denotes the thermal conductivity, il and i2 are the enthalpies of component 1 and component 2, respectively. R is the gas constant and M the molecular weight of the mixture. M I and M 2 denote the molecular weights of the two

E. R. G. ECKERT AND E. PFENDER

274

components, respectively. Equation (56) can also be written in a different way by introducing the mass flux given by Eq. (55),

T h e two equations (56) and (57) indicate that care is to be taken in specifying a thermal conductivity for a two-component mixture. The transport property k used in Eq. (56) describes the energy transport under the condition that the mass fluxjD1is zero. One can, however, also interpret the term in the square bracket of Eq. (57) which is multiplied by the temperature gradient as a thermal conductivity describing the energy transport under the condition that no concentration gradient exists. The analyses which will be discussed in the following section neglect the thermal diffusion and diffusion thermoprocesses assuming that they contribute little to the overall mass and energy transport.' Equations (55) and (56) or (57), respectively, simplify thus to

The difference between the two conductivities discussed above has now disappeared. However, an index f is added to the symbol k , indicating that the first term on the right-hand side of Eq. (59) considers energy transport due to heat conduction only, and does not include the energy transport by enthalpy interdiffusion described by the right-hand term of the equation. In various analyses the energy flux is written differently, replacing the temperature T in Eq. (59) by the mixture enthalpy i in the following way. The enthalpy of component one is described by the equation

il =

I"

rpl dT

+ilo

(60)

'The zero enthalpy i l o has to be included when chemical reactions occur between the two components. The enthalpy of the mixture is correspondingly

i = w i, + w2i2 = w

jcDldT + w2

and its differential

+

d i = w 1cpldT + w 2 r p Z d T dwl 1

cp2dT

+w

il

+ w 2iZ0

(61)

jrpl dT + dw, 1 rp2dT+ i l o d w l+ izodw2

Eckert (36) and Sparrow et al. (37) show that this is not always the case.

PLASMA

HEATTRANSFER

275

With the specific heat of the mixture and utilizing Eq. (60), the last equation becomes di = c p f d T (il- i2)dw I (62)

+

Introducing this enthalpy differential changes Eq. (59) to

T h e Lewis number in this equation is defined as

Indices f are again added to the thermal conductivity and to the specific heat, indicating that they describe the enthalpy and the energy transport, respectively, measured while the temperature changes, with the concentration, however, held constant. They are referred to as frozen properties. The last equation can also be written as

with the frozen Prandtl number Prf= pcpjp/k,. This equation is of advantage because in many cases the Lewis number of a gas mixture differs but little from the value one. An analysis based on the assumption of a Lewis number equal to one serves then as a good approximation to actual conditions. For such a gas, the second term in Eq. (63) is equal to zero, simplifying the equation for the energy flux considerably. Many analyses of reentry heat transfer assume local chemical equilibrium between the various species within the boundary layer. This means that the local mass fraction is, at a prescribed pressure, a function of temperature and consequently of enthalpy. Equation (63) can then be written in the following form :

T h e whole term within the brackets is now a parameter, depending on the local state in the gas, and can be used to define a Prandtl number, Pr,, according to the equation

This equation now describes, together with the equilibrium Prandtl number, the total energy flux in a gas under the condition of local chemical

E. R. G. ECKERT AND E. PFENDER

276

equilibrium and is useful in heat transfer calculations which are based on that condition. A comparison of Eqs. (63) and (64) shows that the difference between the frozen Prandtl number and the equilibrium Prandtl number vanishes for a gas with a Lewis number equal to one. T h e equations for the mass flux and energy flux in this section will now be utilized to obtain the boundary layer equations for a two-component gas mixture.

FIG.27. Rotationally symmetric boundary layer coordinates.

C. LAMINAR BOUNDARY LAYER EQUATIONS T h e equations describing conservation of mass, momentum, and energy for plane and rotationally symmetric, laminar boundary layers of a twocomponent gas mixture can be written in the following form :

a (py”u) +

ax

a (py”v) = 0

aY

PLASMA HEATTRANSFER

277

T h e coordinates x and y as well as the length r are explained in Fig. 27. T h e exponent n has to be set equal to zero for plane flow and equal to one for rotationally symmetric flow. The velocity components u and + are also indicated in Fig. 27. Parameters without an index refer to the mixture. The index one refers to component one. The symbol K ~ in, Eqs. (66) and (68), denotes the rate of mass generation of component one per unit volume and time by a chemical reaction. The differential quotient dpldx is written in total differentials because the pressure variation is negligibly small within the boundary layer in a directionnormal to the surface, according to boundary layer theory. The mass flux ratej,, is now introduced into Eq. (66) leading to

Two energy equations can be written depending on which of the equations (59) or (63) is introduced into Eq. (68) :

The symbol io in Eq. (70a) indicates the total enthalpy which includes kinetic energy. One of these equations is used in analyzing heat transfer processes in a gas which is not in composition equilibrium. For equilibrium, the energy equations can again be simplified to

These are the equations used in the analyses discussed in the next section. They will have to be modified in the presence of a magnetic field and when electrons enter or leave the surface on which the boundary layer borders.

278

E. R. G. ECKERTAND E. PFENDER

Boundary conditions have to be described at the wall surface and at the outer border of the boundary layer. At the outer border they are prescribed by the state of the main body of the gas. At the wall surface, both velocity components are prescribed to have a value zero in the following section. This excludes, for instance, ablation of wall material. The temperature T is in this section set equal to the wall surface temperature and a catalytic wall is assumed, so that the gas composition is the equilibrium composition at the wall surface temperature.

D. RESULTS OF REENTRY STUDIES Analyses and experiments on heat transfer in the reentry problem have been primarily concerned with rotationally symmetric flow near the stagnation point on a blunt body. This is the location where, in the range of laminar boundary layers, heat transfer coefficients reach the highest values. On the other hand, it is also a situation for which the boundary layer equations can be solved in a comparatively simple way by transforming them to total differential equations. T h e terms udp/dx and ~ ( d u / d y )or~ @jay) [(l - 1/Pr)p (ajay)(u2/2)] can be neglected near the stagnation point in the energy equations (68) or (69) and no difference exists between total and static enthalpies or temperatures. Results of such calculations are presented in Fig. 28. T h e ordinate of the figure is the parameter Nu/(Re)’” defined in the following way :

This parameter is plotted over the approach velocity of the air stream in meters per second defined by the equation V m= (2ieo)1’2. Hoshizaki (38), whose results are shown in the figure as the dashed line, performed his analysis for an equilibrium boundary layer of air and CO,, respectively. Correspondingly, he used the energy equation (71b) with properties reported by C. F. Hansen for air and by J. L. Raymond and M. Thomas for CO-,. His analysis for air covered a pressure range from 0.001 to 100 atm and wall temperatures between 300 and 3000°K. T h e results could be represented within +6”,, by the single line shown in Fig. 28. The results for CO-, correlated on the same line for wall temperatures higher than 500%. Fay and Kemp (39) performed their calculations for nitrogen. They considered a frozen condition as well as an equilibrium condition within the boundary layer. In the first case they used the energy equation (70b) and in

PLASMA HEATTRANSFER

279

the second case (71b). With regard to diffusion, they simplified the actual situation by utilizing the binary diffusion model. One component in this model is comprised of the molecules and the other component of the atoms, ions, and electrons. The thermodynamic and transport properties were calculated on the basis of kinetic theory. The Lewis number was in this way determined to be 0.6 in the temperature range between lo3 and 104"K.A temperature of 300°K was assumed for the wall surface. T h e results are

O.* 0.6

f

OS4

2

0.2

0.I

1

4

6

8

10

12

FLIGHT VELOCITY, V,

14

16

18x10'

(m/sec)

FIG.28. Calculated heat transfer parameter for rotationally symmetric stagnation point

flow: a, air and C O , , after Hoshizaki (38);b, N2, after Pallone and van Tassel (40);c , air, after Pallone and van Tassel (40);d, air [after Cohen (4Oa)l.The point symbols are for N2, [after Fay and Kemp (39)l.

plotted in Fig. 28 with the open symbols indicating an equilibrium condition and the full symbols denoting a frozen condition. In agreement with the qualitative discussions presented in the preceding section, it is observed that the heat transfer in the frozen boundary layer is larger than in the equilibrium boundary layer, with the difference increasing with increasing flight velocity. Pallone and van Tassel1 (40)also made calculations on heat transfer in the region of a rotationally symmetric stagnation point with boundary layer flow

280

AND E. PFENDER E. R. G. ECKERT

of nitrogen and air and with the assumption of equilibrium dissociation and ionization. They found that they could represent their results for nitrogen with an accuracy of &,5% by the following equation :

where Nu indicates the Nusselt number, Re the Reynolds number, Pr the Prandtl number, p the density, p the viscosity, V , the upstream velocity (flight velocity), p the upstream pressure, andp, = 1 atm. The index w refers to conditions at the wall surface and e to conditions outside the boundary layer. T h e dimensionless parameters Nu, Re, and Pr are based on properties at the wall surface. For a velocity V , below 11,300 m/sec, the last term in the above equation has to be replaced by one. Thermodynamic and transport properties published by Yos were used in these calculations. Corresponding calculations based on properties by Hansen resulted in heat transfer parameters for air which are again represented with good accuracy by the equation

The dependence on flight velocity V , was found to exist only for values V , > 9900 m/sec. Correspondingly, the last term in the above equation has again to be replaced by one for velocities below this value. The heat transfer parameters represented by these equations are also presented in Fig. 28. Experiments to measure heat transfer in the stagnation region of a blunt object under conditions simulating reentry were performed, mainly in shock tubes, by a number of investigators. Figure 29 summarizes earlier results and compares them with the results of analyses. The experimental difficulties made a certain amount of scatter unavoidable. In general, it can be stated, however, that agreement exists between the results of analyses and of experiments. Some disagreement among the analytical results obtained by various investigators is mainly due to the uncertainty in the transport properties. An analysis by Scala (41)was not included in Fig. 28 or 29 because it appears today that some of the transport properties used were wrong. This analysis results in heat transfer parameters up to twice as large as the ones shown in Fig. 28. They were originally supported by experimental results published by Warren and associates. Later refined measurements by Gruszcynski and Warren (421, however, resulted in values which agree with those in Fig. 29. Other recent measurements are also in agreement with the values in this figure. Park (43)calculated heat transfer of an ionized argon boundary layer for axisymmetric stagnation flow and laminar flat plate flow. He considered

PLASMA HEATTRANSFER

28 1

equilibrium as well as frozen conditions, the latter one for a catalytic wall. The energy equations (70a) and (71a), respectively, were used. The heat transfer parameter Nu/Re'i2 was again defined by Eq. (72). This parameter is presented as a function of the equivalent flight velocity V , [defined by V , = (2ie0)'/2]in Fig. 30 for the rotationally symmetric stagnation flow. Inserted in Fig. 30 are also the results of experiments which Park performed in a plasma jet wind tunnel. T h e experimental results indicate that the boundary layer was in a frozen condition. Inserted, in addition, as dashed lines, are the results of the analysis by Fay and Kemp (39)and as shaded area experimental results by Rose and Stankevics (44).Both of these results apply 1.0 0.9 0.8-

0.7 0s 0.5 -

-

I

-

-

-

-

0

ammo

to air. T h e figure shows that Nusselt numbers are not too different for argon and air. Ionization does not change the Nusselt number very much and the Nusselt number for a frozen condition is in all gases which were investigated larger than for an equilibrium condition, with the difference increasing with increasing flight velocity. T h e flight velocity as it has been used in the preceding figures is defined of the approaching gas stream, according to the through the total enthalpy ien equation i,O =

v:/2

and, is therefore, actually a statement of the gas enthalpy in the free stream. T h e enthalpy of the gas corresponding to the wall surface temperature is in all cases negligibly small compared to the total gas enthalpy so that the

E. R. G. ECKERT AND E. PFENDER

282

abscissa in these figures presents with good approximation also the difference

i, - i, which is used in Eq. (52)as the driving potential. T h e abscissa should be interpreted in this way if the figures are used to determine Nusselt numbers for stationary applications, for instance, in arc technology. Figure 31 presents as an example the frozen as well as the equilibrium Prandtl number of argon at a pressure of 0.1 atm. Both of these parameters have been used in the integration of the energy equations for frozen and equilibrium flow, respectively.

I

A

0.8

A

A

0

&

I 0

0.6

5P

---

/FROZEN

0'4

-

0.2

0.I

8

10

12

14

F L I G H T VELOCITY,V,

16

(m/sec)

18x lo3

FIG.30. Calculated and measured heat transfer parameter for rotationally symmetric stagnation point flow of argon [after Park (43)].

IV. Heat Transfer in the Presence of an Electric Current

A. ELECTRICALLY INSULATING SURFACE An ionized gas is electrically conducting and permits, therefore, a current to flow. Heat transfer to the wall adjacent to such a gas depends strongly on whether or not the wall is electrically insulating or carrying an electric current. On an insulating surface, heat transfer is mainly influenced by the electric current in the plasma owing to the fact that ohmic heat is generated and that a corresponding heat source termj2/a or o E has to be added to the energy equations (70) and (71). The existence of local kinetic equilibrium between the energy of the electrons and the heavy particles is generally assumed even though there is some question as to how well this condition

PLASMA HEATTRANSFER

283

is approximated in the regions of lower temperature near the wall. Heat fluxes can become quite large and in this way pose serious design problems. Such a situation will be discussed for the example of a constricted arc.

The Constricted Arc T h e constricted arc has become a very useful tool in research especially for the determination of properties of plasmas and it is also used as a means of generating high-temperature ionized gases for hypersonic wind tunnels

0.I

4poo

woo

~000

lop00

TEMPERATURE (k)

12pOo

FIG.31. Frozen and equilibrium Prandtl number for argon at 0.1 atm pressure [after Penski (44a)l.

and for space propulsion devices. The identifying characteristic of this arc is the achievement of high temperatures by constricting the arc in a tube, thereby increasing the electric energy dissipation per unit volume. T h e temperature achieved in such an arc is limited by the heat transfer to the constrictor walls. Usually water cooling is used for this purpose. Recently, however, attention has also been directed towards the application of transpiration cooling. Figure 32 sketches such arcs with a water-cooled and transpiration-cooled constrictor. The constrictor consists in the first case of a stack of water-cooled segments insulated electrically from each other. In the second case the constrictor is a porous tube through which the working gas is injected. A number of analyses have been reported (45-50) which investigate the performance characteristics of the water-cooled constricted

284

E. R. G. ECKERTAND E. PFENDER

arc. A few analyses (32,51,52)are also available for the transpiration-cooled arc. With sufficient constrictor length, the temperature field becomes developed in the downstream section of the constrictor in the sense that the temperature varies only radially maintaining its values in the axial direction. This is the case in the water-cooled constrictor when all of the electric energy dissipated within this section is transferred radially to the constrictor wall. In the transpiration-cooled constrictor, the developed stage is achieved ANODE (+)

WATER

WATER

TRANSPIRING GAS

FIG.32. Constricted arc with water cooling and transpiration cooling.

when all of the electric energy is used to heat the gas injected through the porous wall to the temperature of the axially moving gas stream. The velocity field also becomes developed in the sense that the velocity profiles at various cross sections are similar to each other. Most of the analyses are restricted to the developed condition. This will also be the case in the following discussion which is essentially based on the work of Anderson and Eckert (32). T h e existence of complete local thermodynamic equilibrium, of rotationally symmetric laminar flow, and of negligible energy dissipation by internal friction is also postulated. The gas properties entering the analysis are assumed to be functions of temperature only, which implies that the pressure differences in the flow field are moderate. ‘The following equations describe, under these assumptions, conservation of mass, of momentum in axial direction, and of energy for both constricted arc types :

PLASMA HEATTRANSFER

a@,

l a

-r-ar (YPc,)+pz

ac, aZ

ac ap aT az

pvz-+/Jur-2+-=

=o

(75)

1a ac, -- Yp-

+aE2-P,.=pz',-

285

al. ai

ar

1 (77)

T h e coordinate system is indicated in Fig. 32 and the symbols are explained in the List of Nomenclature. The conservation equation for momentum in the radial direction was also included in the analysis of Anderson and Eckert (32). It was, however, found that the radial pressure variation determined by this equation is small in the cases which will be discussed later on. 'The term P, on the left-hand side of Eq. (77) describes the energy loss of the gas by radiation, assuming the gas to be optically thin. This term also was comparatively small for the cases which were investigated. For the water-cooled constrictor, the system of equations simplifies because in the developed regime the radial velocities z-,. are equal to zero. This has the important consequence that the convective term on the righthand side of Eq. (77) is zero, which means that the energy equation is completely decoupled from Eqs. (75) and (76). T h e resulting equation becomes the well-known Elenbaas-Heller equation when the radiative term is neglected. T h e temperature field can therefore be calculated without knowledge of the velocity field. This calculation is usually performed by introduction of the heat conduction potential '2 = J kdT as the independent variable instead of the temperature. I t has been found that the inner wall surface temperature T,, and the magnitude of the heat flux 9,' per unit area which can be absorbed by the water-cooled constrictor wall are the parameters which limit the performance of this arc type. T h e proper boundary conditions are therefore

Figure 33 presents some results of the described analysis considering hydrogen at 1 atm pressure as the working gas. T h e properties of hydrogen have been obtained from the literature. Two heat transfer processes determine the maximum allowable heat flux, the heat conduction through the constrictor wall and the heat transfer from the wall to the cooling water. T h e temperature drop in the constrictor wall is appreciable. For a heat flux of 10 kW/cm' ( 3 x 10' Btu/ft? hr), for example, the temperature drop in a l-mm thick wall of pure copper or silver is 250°C. For this reason, only metals of very high conductivity can be used for the constrictor. Local boiling occurs on the water side of the constrictor wall and the permissible heat flux in this process

E. R. G. ECKERT AND E. PFENDER

286

is determined by the burn-out heat flux. A heat flux density of order 10 kW/cm2 is the maximum so far achieved. A limit for the wall temperature is, on the other hand, prescribed by the melting point of the material of which the constrictor wall is formed. T h e melting point of copper is, for instance, 1083°C. From Fig. 33 it can be seen that for these values a maximum gas temperature at the axis of the constrictor tube of approximately 40,000"K can be reached.

:----r--

36

I

35

1

!

[r,.0.25cm/

i

1

v)

z 33 w

t-

z

n J

w LL

32

;;

31

45poO'K

---

30

~

,

I

I

FIG.3 3 . Performance data of water-cooled constricted arc in hydrogen, 1 at m [after Anderson and Eckert (.??)I.

‘The transpiration-cooled constrictor operates with a finite radial flow velocity -cr leaving the constrictor wall. Therefore the right-hand term in Eq. (77) has to be maintained and the three conservation equations (75)-(77) are interrelated. Figure 34 presents numerical solutions of these equations for the boundary condition of a locally uniform velocity -cr at the constrictor surface and for the situation that all of the heat flowing toward the constrictor wall is returned to the gas in the interior of the constrictor tube by the gas injected through the porous wall. Again, hydrogen at 1 atm pressure is the working gas. A limitation to the performance of the transpiration-cooled

PLASMA HEATTRANSFER

287

constricted arc is set by the permissible inner wall surface temperature. Use of a ceramic material will allow fairly high values for this temperature. Another restriction is given by the rate with which gas is injected through the porous constrictor wall. Figure 35 shows how the inner constrictor surface 56

I

1

I

I

54

-

I

I

52

1

E

V

F

k

40 z W

V,

c

i 44

~

1

42

I

I

40 0

500

I

000

I CURRENT (AMP)

I

woo

2300

FIG.34. Performance data of transpiration-cooled constricted arc in hydrogen, 1 atni [after Anderson and Eckert (32)l.

temperature and the electric power input per unit length determine the rate of fluid injection per unit length. The average axial mass velocity Fz increases linearly with the coordinate z.This determines the tube length at which sonic velocities are reached. On the other hand, a certain tube length is required to achieve a developed temperature field and the balance of both factors determines the maximum allowable injection rate. Other factors like transition to turbulence and magnetic pressure do not appear to impose more restricting limitations.

E. R. G. ECKERT AND E. PFENDER

288

Figure 36 compares normalized enthalpy and mass velocity profiles for the two types of arcs. T h e enthalpy profile of the transpiration-cooled arc is characterized by a small value of the gradient at the constrictor wall. This means that the energy flux away from the tube axis is, for the most part, arrested by the radial mass flow towards the tube axis. T h e heat flux into the constrictor wall is therefore much smaller than for the water-cooled arc. T h e mass velocity profiles, on the other hand, indicate that the majority of the gas moves in the transpiration-cooled constrictor through a ring-shaped cross section adjacent to the tube wall. T h e mass averaged enthalpy flux, 0.12.ld'

I

1 7

0.10

8 0.08

.-sI

'€ 0.06

!i

FIG.35. Performance data of transpiration-cooled constricted arc in hydrogen, 1 atm [after Anderson and Eckert ( 3 2 ) ] .

therefore, is, at the exit of the transpiration-cooled arc, not larger than for the water-cooled arc, even when the axial temperature is much higher. T h e advantage of the first system lies in the high temperature in the region close to the tube axis. The results in Figs. 33-36 apply to a constrictor diameter of 0.5 cm. They hold for other diameters as well, as long as the radiative heat transfer is negligible, if the electric field strength E is scaled proportionally to the

PLASMA HEATTRANSFER

289

3.8X I 0 0 3.6 3.4 3.2

3.0 2.6 2.6 2.4

.

2 2.2

3

n

t

-1

2.0

4 3 x a

a

2I 1.6 t5 1.6 14 12 I .o

08 0.6 0.4 0.2 0

0

0.2

0.4

06

NORMALIZED RADIUS

Od

I.0

FIG.36. Comparison of enthalpy profiles and axial mass velocity protiles for watercooled and transpiration-cooled arcs in hydrogen, 1 atm [after Anderson and Eckert (.??)].

constrictor radius r7L,and the mass injection rate m per unit length inversely proportional to r7c.This scaling law follows readily from the conservation equations.

B. ELECTRICALLY CONDUCTING SURFACE (ELECTRODE)

1. Basic Considerations T h e heat transfer process becomes very involved to an electrically conducting surface, especially when an appreciable gradient of the electric

290

E. R. G. ECKERTAND E. PFENDER

potential and a flow of electrons exists normal to the surface as in the case of electrodes. A fall space with a thickness of the order of a mean molecular path length exists close to the surface and the ionized gas which is neutral in the bulk flow loses its neutrality within the sheath. T h e corresponding space charges create very strong gradients of the electric potential in this region, and the energy distribution among the various particles is far from equilibrium. However, even outside of the sheath within the temperature boundary layer, thermal equilibrium does not exist. The gradient of the electric potential is large enough to add, between collisions, considerable energy to the electrons and as a consequence the average energy of the electrons is higher than the energy of the heavy particles. T h e energy distribution among the electrons and among the heavy particles is still fairly close to a Maxwell-Boltzmann distribution ; as a consequence, one can define an electron kinetic temperature T, and a kinetic heavy particle temperature T . A model which describes conditions in the neighborhood of an electrically conducting surface with current flow into or out of the surface consists then of a binary gas mixture, namely, an electron gas and a heavy particle gas. One has then two sets of conservation equations with coefficients which describe the mutual interactions of the two gases. A simpler model was developed by Kerrebrock (53).H e starts out with an energy balance determining in an approximate way the electron temperature :

T h e definition of the symbols in this equation is contained in the List of Nomenclature. a indicates a dimensionless constant, the value of which is estimated by Kerrebrock to be of order 2. T h e drift velocity u,of the electrons is connected with the current density j by the equation j = en,u, neglecting the contribution of the ions to the current density. T h e collision frequency v, between electrons and heavy particles is given by the relation v, = e2n,/m,a. Introducing these expressions into Eq. (78) and solving for the electron temperature results in the equation

in which m, denotes the mass of the heavy particles, kB Boltzmann’s constant, e the electron charge, and n, the electron number density. Equation (79) which describes the difference between electron and gas temperature as a function of the current density, is identical with Eq. (30) of Section 11. Kerrebrock also assumes that the electron number density n, is the same as for a plasma in thermodynamic equilibrium at the temperature T, and is therefore given by Saha’s equation [see Eq. (27)]. T o describe with these

PLASMA HEATTRANSFER

29 1

equations the energy exchange within a boundary layer, the continuity equation (65) and the momentum equation (67) are maintained. T h e energy equation is

T h e two new terms in this equation are the last ones on the right-hand side.

I

2

3

RADIUS ( m m )

4

5

FIG.37. Profiles of heavy-particle temperature (7’)and electron temperature ( T , )for constricted arc in helium [after Pytte and \Villiams (541.

j 2 / u denotes Ohm’s energy dissipation and the last term the convective transport of the enthalpy of the electron gas. With Eqs. (79) and (27), the electron temperature can be determined as a function of the current density j and the heat transfer through the boundary layer can be calculated as a function of j . Kerrebrock performed such an analysis for a boundary layer on a flat plate by the integral technique. A similar calculation with a somewhat extended Eq. (78) was performed by Pytte and Williams (54) for a water-cooled constricted helium arc with a I-cm constrictor diameter, assuming that the constrictor wall is slightly biased and that therefore a flux

292

E. R. G. ECKERT AND E. PFENDER

of electrons enters the constrictor surface. From the results of this onedimensional calculation, Fig. 37 has been selected in which the variation of the gas temperature T and the electron temperature T, is plotted over the radial distance r from the arc axis. T h e parameter on the curve is the current density j , at the constrictor wall. I t is interesting to observe how, with increasing current density, the electron temperature exceeds the gas temperature more and more over an appreciable part of the tube cross section. T h e discussion up to now was concerned with the energy flux within the ionized gas close to the electrically conducting surface. An analysis of heat transfer to the surface itself has, in addition, to consider the energy conditions within the sheath and the energy which is freed when the electrons enter the solid surface, an energy which is given by the “work function” and which is analogous to a condensation energy. This will be discussed in connection with the experimental investigations.

2. Experimental Studies ‘Two problems arise in the development of engineering devices utilizing electrically generated plasmas (electric arcs). Firstly, the total heat flux to a water-cooled current-carrying surface represents in many applications the predominant energy loss which diminishes the efficiency of the particular plasma device. Therefore, efforts have been made to reduce such losses. Transpiration cooling has been considered because part of the heat flux to a surface is thus recovered by the gas transpiring through this surface. The second problem is connected with the specific heat flux to a currentcarrying surface which may become so high that local melting occurs, in spite of applying the most efficient water-cooling system which is able to remove about 19 kW/cm2.This situation has been experienced in high-pressure arcs even at moderate currents using working fluids with high heat conductivity (for example, hydrogen). The high pressure as well as the high heat conductivity of the working fluid lead to a constricted anode attachment resulting frequently in anode failure. The properties of the anode fall space and of the boundary layer apparently have a decisive influence on the current transition and therefore also on the local heat transfer to the anode. A cold surface favors a constricted attachment in order to keep the electron temperature and with it the electric conductivity sufficiently high in spite of the intense cooling of the boundary layer. High-temperature surfaces may permit a more diffuse current transition. In order to prevent too high specific heat loads at the surface, the anode arc terminus may be moved with high velocities over this surface distributing the heat flux to a larger area. Heated anodes offer another possibility to keep specific heat fluxes within reasonable limits. In the following sections anode heat transfer studies will be reviewed

PLASMA HEATTRANSFER

293

which have been carried out on various arc geometries and under various test conditions. For many of these investigations, energy transfer models have been proposed in order to determine the contribution of the different energy transfer modes to the total anode heat flux. Detailed anode heat transfer studies are available for arcs operated in an argon atmosphere and for water-cooled copper anodes which permit rather accurate overall and local heat transfer measurements. Therefore, this review will only deal with results obtained from such anodes. a. Free-Burning Arcs with Plane Anodes. The characteristic arc geometry of the free-burning, high-intensity arc shows a well-known bell-shaped stationary arc column (3, 55). T h e reason for the steadiness of this arc column is the strong cathode jet generated by magnetohydrodynamic forces in the cathode region (56). This region acts as an electromagnetic pump

q,e-o

qro

h,(Ie--lw)

Ma

q,

q,-o

FIG.38. Energy balance of anode surface element.

drawing gas from the surrounding and ejecting it towards the anode in the form of a jet. T h e cathode jet has an important influence on the convective part of the anode heat flux, especially at small arc lengths and high currents. T h e energy transfer to the anode involves a variety of transfer mechanisms which can be described by an energy transfer model (57-59) indicated in the scheme of Fig. 38. T h e energy transferred to the anode consists of: ( a ) Thermal and kinetic energy of the electrons comprising the arc current and penetrating the anode surface,

+

q, = ( j/e) ( i k B T, e U0) (81) q, is the energy flux per unit area connected with the current flow into the surface, j the current density, e the electronic charge, kB the Boltzmann constant, 7,the electron temperature at the border of the arc column, and U , the anode fall voltage. ( p ) Heat flux by condensation of the electrons which is proportional to the anode work function @*, w, .Pa (82) ( y ) Convective heat transfer from the hot plasma through the boundary layer, = U i , 4) (82a) L -

(IL"",

~

E. R. G. ECKERT AND E. PFENDER

294

(6) Radiative heat transfer from the arc plasma qra. Electrons suffer elastic collisions with heavy particles in the boundary layer and in the anode fall space, a process which will reduce that part of the electron current stagnation enthalpy which is transferred to the anode. However, the number of elastic collisions in the thin fall region is not large enough to transfer an appreciable amount of energy to the heavy particles since the energy transfer per collision is very small. The energy loss of the electrons to the heavy particles will therefore be essentially only the amount required for ion generation in this region. But the ion current density, in turn, is only on the order of one per cent of the total current density, so that the energy loss of the electrons traveling through the anode fall space may be neglected.

-

I

I

1

I

1

3

4

5

6

7

5 d

0

I

2

RADIUS ( m m )

FIG.39. Radial distribution of energy transfer at anode surface (59).

T h e energy carried away from the anode surface consists of: ( E ) T h e heat conducted away from the anode surface by the cooling water denoted by q. (Z;) T h e heat radiated away from the anode surface to the environment denoted by qye. Under steady state conditions the energy balance for a surface element of the anode can be written as

+ qre + q a b l =

qj

+ qo,,+ + Qva

qcon,.

(83)

In normal operation, the anode surface is kept well below the melting point ; therefore, qreand qablare negligible. All the other quantities besides U , and @, can be experimentally determined. T h e results of such measurements on a water-cooled anode in a current range 50-150 A (57-59) indicate that the heat flux due to the electron flow into the anode is the predominant heat transfer mechanism. A typical diagram of such measurements is shown in

PLASMA HEATTRANSFER

295

Fig. 39. The convective heat transfer was calculated with Eq. (52) assuming that this equation holds for a current-carrying surface. Measurements by Nestor (60) for currents up to 300 A show essentially the same results. Overall energy balances demonstrate that up to 85 yoof the arc power input is transferred to the anode (with electrode gaps up to 20 mm) and that the percentage of energy transfer to the anode is almost independent of the current. Another interesting finding of these studies is the behavior of the current density which maintains its maximum value at the anode center regardless of the total current (100 A < I < 400 A). T h e arc column and especially the anode attachment region spread correspondingly out with increasing current, and it appears that the arc adjusts its attachment area to that current density which provides a sufficiently high electron temperature and electric conductivity. T h e corresponding anode heat fluxes by Schoeck (57, 59) and Schoeck and Eckert (58) are in the range of 4 to 6 kW/cm2. T h e convective heat flux density increases with increasing current corresponding to a nearly linear increase of the velocity with which the cathode jet approaches the anode reaching a value of 180 mjsec at 150 A and 6 mm cathode to anode distance. I n a more recent thesis by Eberhart (64, the current range has been extended to 1100 A. T h e percentage of energy transferred to the anode is, up to 1100 A, again independent of the current. However, the electron heating of the anode which predominates at lower currents is surpassed by gaseous convection at very high currents depending upon the electrode gap. This fact is at least in part a consequence of another finding which indicates that the local current density at the stagnation point even decreases with increasing current. b. Cathode Axis Parallel to a Plane Anode. When the cathode is oriented parallel to the anode surface, one obtains quite a different shape of the plasma column as shown in Fig. 40. The cathode jet now impinges against an anode jet which is generated by a current density gradient toward, and in the immediate vicinity of, the anode. T h e reason why the arc column tends to attach to the anode in a rather contracted manner is probably that a lowtemperature gas layer is present on the anode surface. This constriction is usually not as strong as in the cathode region (62). The heat flux per unit area is expected to have a distribution curve similar to the one shown in Fig. 39, with, however, a smaller attachment area and a correspondingly larger maximum. From the photos it appears that it may be 100 times as high. 'I'he local heat flux at the anode may, under such conditions, surpass the highest permissible value and it is, therefore, fortunate that a superimposed axial gas flow frequently causes the arc to fluctuate, leading to a moving anode arc terminus. This spreads the anode heat flux over a wider surface area a n d reduces the maximum temperature of the anode appreciably.

E. R. G. ECKERTAND E. PFENDER

296

An estimate of the reduction in the maximum temperature which is obtained in this way can be found from the following model. Consider a semi-infinite solid with a point heat source moving with constant velocity ZI along a straight path on its surface. The temperature field created in this way

PLASMA COLUMN CATHODE JET

FIG.40. Scheme and photograph of impinging jets in argon (64).

in the solid material is steady when viewed by an observer who moves along with the heat source. Surfaces of constant temperature have a shape similar to half an egg shell and are described by the following equation (63):

e=-8=

[ p>(pP2 -+ 1)]

1 exp -27rPe”

( T - T,)k2 PCPVQ

VY

’

Per=-,

ff

Pe,=-

vx ff

in which 6 denotes a temperature parameter and Pe a dimensionless parameter which is frequently used in heat transfer analysis and is called the Peclet number. T is the local temperature in the solid at the radial distance r from the heat source, T , the temperature at r = 00, k the thermal conductivity, p the density, and cp the specific heat of the solid. q denotes the heat flux released by the point source per unit time, x the projection of r onto a straight line through the heat source in the direction of its movement, and CL

PLASMA HEATTRANSFER

297

the thermal diffusivity of the material. The curves in Fig. 41 present the intersections of the isothermal surfaces with the plane surface of the solid. Equation (84) can also be interpreted as describing the temperature of the material when, instead of the point source, a situation is considered in which ,

aw

e =

(T-Te)kz Pcpvq

\

,

pe

r

vx ;x, pe, = a

FIG.41. Temperature field of a moving point heat source on the surface of a semi-infinite

body.

the heat flux q is distributed over any of the isothermal surfaces. The equation describing the temperature field 8, of a stationary heat source can be brought into the following form :

8, = 1/2~rPe, (85) which contains the same parameters as Eq. (84). T o compare the temperature increase by a heat source of radius r moving with the velocity v with the temperature increase created by a stationary heat source of the same size, one can now calculate the parameter Pe, and determine with it 8, from Eq. (85) and 8 from Fig. 41, respectively. In this way the values 8/8, in Table I TABLE I Per

sje,

1.1 0.69

2.25

0.56

3.9 0.49

4.82 0.45

6.7 0.42

10

0.38

were obtained. It can be seen that the maximum temperature increase in the solid material is reduced to about one-third of the value created by a stationary heat source of equal strength when the Peclet number Pe, has a value of 10. The fluctuating arc repeats its movement down the anode almost periodically and the maximum temperature in the solid material will be somewhat

298

E. R. G. ECKERT AND E. PFENDER

larger than the value obtained by the estimate described above. The model suggested here can certainly be refined, taking into account the fact that the anode usually is a thin wall cooled on the back side by water and considering a heat flux distribution as presented in Fig. 41. I t is, however, felt that it 0.6

0.4

?* 0“

0.2

0

0

2.5

!j.o

7.5

10

0

2.5

X (cm)

(01 VELOCITY DEPENDENCE

5A X (cm)

7.5

10

( b ) PRESSURE DEPENDENCE

I = K)OAmp,S -7mm,V= 5Om/sec

I =IOOAmp,S =7mm,P*IOOmmHq

o

0.4

S=6mm

?*

0”

0.2

0

0

2.5

5.0

7.5

10

X (cm)

(c) CURRENT DEPENDENCE S= 7 mm, P*38OmmHg, V= 100m/rrc

0

2.5

!LO 7.5 X tern)

10

(d) ELECTRODE GAP DEPENDENCE

I =100omp,P=380mmHg,V=100m/rrc

FIG.42. “Local” heat flux distribution at plane segmented anode (argon).

already correctly describes, in its present form, the order of magnitude. No measurements of the distribution curve of the heat flux or of the temperature distribution in the anode have been performed to date with which the above analysis can be compared. Some indication of the relief provided by the arc movement is obtained by the results of some tests in which the anode was composed of segments with a width of 1.25 cm measured in the flow direction. The results are presented in Fig. 42 in which Qs presents the flux into an individual segment and Q A the total anode heat flux. Figure 43 shows the

PLASMA HEATTRANSFER

,

n e-(1 (hg’)’ +fg’ = 0

- t s ) g 2 + t,]}

(54) (55)

Equations (54)-(57) are valid for axisymmetric flow if (1is set equal to zero and j is unity.

2. A = O When the free-stream-flow direction coincides with the x direction, then A = 0 and t, = 1. For this case, the boundary-layer equations reduce to

(y)’ +fy

(x;;)’+fZ’’

=

pv’2 - (1 - t w )e - t,]

(58)

=0

3 . No Mass Transfer I n the special case where no mass transfer takes place in the boundary layer, Z1= 0 and clw= 0, and the boundary conditions (including the Eckert-Schneider condition) are ignored’ :

(xf”)’ +fj”

8’ ) +fe’

(A Pr ’

=

=

p { j -~ ( 1 - t,) e - t w }

( (jr

- 1)

( 12.:,)f”f’

(61)

(

$:2)m

‘

(62)

1 Although the solution for the concentration ratio z1 is z~ = 0, the solution of Eq. (43) for the normalized concentration z has a nonvanishing solution and a nonzero gradient z’ at the wall.

328

C. FORBES DEWEY,JR., AND JOSEPHF. GROSS

Equations (61) and (62) are the ordinary two-dimensional Prandtl boundarylayer equation. 4. p - T If the viscosity is directly proportional to the temperature, h = 1, and Eqs. (61) and (62) reduce to

p +jf”= p p

-

(1 - tw)e + tw>

(63)

5. P r = l Equations (63) and (64) reduce to a form in which the dissipation term of the energy equation is zero :

f ” +fy= p { y 2 - (1 - t w )e + tw> el’

+jet

=

o

(65)

(66)

6. p = O Finally, if the pressure-gradient parameter is zero, the momentum equation is uncoupled from the energy equation; Eq. (65) becomes f‘“+ff”= 0

(67)

and by inspection, the solution of the energy equation is

e =ff

(68)

E. COMPUTATION OF BOUNDARYLAYER PROPERTIES Heat-transfer rates and skin friction are related by the similarity transformations to the derivatives O’(O), f ”(0),and g’(0). The heat transfer from the stream to the wall is given by 4w = + k w ( a T / a Y ) w (694 After the proper substitutions have been made, the expression for the heat transfer in similarity coordinates is

and the local heat transfer coefficient is ch = %~/[prn

-

Hw)]

(70)

SOLUTIONS OF BOUNDARY-LAYER EQUATIONS

329

The skin friction for the x direction can be described in similarity coordinates as follows :

The spanwise coefficient is

The total component of skin friction is the vector sum of the two perpendicular components T~ and 7,: 7=

PW

rkJ

(2W2 [{f”(0)12U,2 + {g’(0)l2w,z1112

~w ue ~~

(73)

and the local skin friction coefficient is defined by

c,= 2r/pmurn2

(74)

Several integral relations have also been tabulated. They are

Il = (1 I2=

-W l ( 1 ) -W ) 1

jOrnf’(l- f 0 4

-

(1 - W d 3 ) + W )

(75) (76)

where

These integrals can be related to the boundary-layer-thickness parameters expressed in similarity coordinates. Displacement thickness :

Momentum thickness :

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C. FORBES DEWEY, JR.,

AND JOSEPH

F. GROSS

F. RANGEOF SOLUTIONS AND PARAMETERS The parameters for the systems of equations under consideration here are O < p < 5 andp=m pressure-gradient parameter Pr = 0.5, 0.7, 1.0 Prandtl number Sc = Pr (i.e., Le = 1) Schmidt number o = 0.5, 0.7, 1.0, or temperature-viscosity-law parameter 0.01 < s < 0.3 O,z’(0)I (90) X[f ”(O), g’(O), B’(O), z’(0)+ A~’(0)l (91) These integrations give the columns of M . T h e new guesses for the initial condition are corrected by calculating: - d S = M-’eO (92) It was established that convergence was very difficult for cases involving high/? (/?2 4).I n these cases, the limit accuracy was relaxed, and this is indicated in each table as needed. V. Applications of the Concept of Local Similarity A. GENERAL DISCUSSION The numerical results tabulated in this chapter are rigorously applicable only when the numerous similarity requirements listed in Section I1 are satisfied. In considering the diverse applications of compressible laminar boundary-layer theory, it is a rare occurrence indeed when all of these conditions are met. A most important question then arises: What approach

334

C. FORBES DEWEY, JR.,

AND JOSEPH

F. GROSS

should be used in predicting the behavior of the nonsimilar laminar boundary layer ? T h e copious literature relating to this question offers four basic types of approach. The first is to abandon the similar solutions entirely and adopt approximate techniques such as integral and series solutions containing free parameters. The most successful of the integral approaches appears to be that developed by Tani (34).The transcendental approximation proposed by Hanson and Richardson (35) and the “improved approximation ” technique of Yang (36)also appear very promising. Related to the integral methods is the powerful “strip method” proposed by Pallone (37),which follows closely the inviscid flow integral method of Belotserkovski and Chuskin (38). A second type of nonsimilar calculation employs a strictly numerical approach. The complete nonsimilar boundary-layer equations are used and a new set of calculations is performed for each particular problem. Examples of this approach may be found in the works of Smith and Clutter (39)and Flugge-Lotz and Baxter (40, 41). Although numerically satisfying, these calculations are extremely expensive and intractable to generalization. The third type of approach, that of Lees (42),has been most successful in capturing the spirit of the use of similar solutions in situations where exact similarity does not exist. He observed that under certain circumstances, notably when there is a highly cooled body in hypersonic flow, the local pressure-gradient parameter /3 had a negligible effect on the heat transfer to the surface. Many elaborations of this approach have been proposed to improve Lees’ simple result to provide more accurate numerical estimates of heat transfer, skin friction, and boundary-layer thickness. Moore (21) gives a lucid summary of one group of these results. Additional ideas for modifying local similarity not discussed by Moore may be found in the papers of Beckwith and Cohen (9),Smith (43),and Kemp et al. (44). This third approach is saddled with one difficulty: it is necessary to make one or more implicit ad hoc approximations regarding the contribution of the nonsimilar terms in the complete boundary-layer equations. In each of the papers cited in the previous paragraph, the question is not “Are similar solutions applicable ?” but rather “Which similar solution should be used ?” A number of methods have been proposed for choosing the similar solutions most appropriate to the local inviscid flow conditions, to wall temperature, and to boundary-layer history. In the context of the present discussion, this question necessitates the judicious choice of values for the eight similarity parameters. These values are usually determined [e.g. Beckwith and Cohen (9)] by satisfying one or more integral conservation equations exactly using assumed profiles obtained from similar solutions. Such procedures are closely related to the integral method of Thwaites (45) and to the more

SOLUTIONS OF BOUNDARY-LAYER EQUATIONS

335

recent use by Lees and Reeves (46, 47) of a family of similarity profiles generated by Stewartson (32). The fourth, and in many ways most satisfying, type of approach to nonsimilar boundary-layer calculations may be traced to the work of Meksyn (28). His underlying premise is very powerful: if the boundary layer is at all times very nearly described by a similar solution, then the direct effects of the nonsimilar terms may be calculated by asymptotically expanding the full boundary-layer equations in terms of small parameters which measure the departure of the solutions from similarity. I n this way, the accuracy of local similarity methods is explicitly determined by using the full nonsimilar equations. We shall demonstrate shortly that the linearized equations governing the departure from similarity depend only on the local similarity parameters and, consequently, need be computed only once.

OF THE BOUNDARY-LAYER EQUATIONS B. ASYMPTOTIC EXPANSION

For purposes of illustration, we shall limit our consideration to the “incompressible” momentum equation

f,,,+ff,, + P(5)

-f,’I

251f,fE, -fEf,,l

(93) where 5, 7, and /3(5) are as defined in Section I1 and the subscripts 71 and 4 denote derivatives. Equation (93) may be obtained from the general equations by assuming Pr = w = t, = t, = l. We also assume thatf, = 0, making the three boundary conditions for Eq. (93) =

Merk (48)was the first to expand the complete nonsimilar momentum equation in terms of a small parameter. Subsequently Bush (49) pointed out that Merk’s derivation neglected important terms in the correction equation. The remainder of this section will be concerned with Bush’s equations and their approximate solution. In the spirit of Meksyn’s approximation, we look for solutions to Eq. (93) when the right-hand side is small. The key to an appropriate expansion is the inversion introduced by Merk. We change variables from [ t , ~ /3(5)] ; to [p, 7 ; 5(P)] so that the streamwise momentum equation becomes

f,,,+ff,, + P[1 -#I

where

f(P, C) =f,(P,

0)

4P)[f,fp, -fpf,,I = 0; f,(P, = 1 =

4 P ) = 25PY5) = 2t(P)/t’(P)

(96) (97) (98)

336

C. FORBES DEWEY,JR., AND JOSEPH F. GROSS

If E is zero, the momentum equation reduces to the Falkner-Skan similarity equation with /3 as a single parameter. For small E we may perform an asymptotic expansion off(/3,7) of the form

f(P9 7) =fo(B, d + E(B>fl(B,rl) +

(99)

* * *

Then the derivativesf9 andfp are

+

+ ‘’’

(100) (101) As Bush pointed out, the term ~’(jl)is, in general, of order unity and may be expressed as f 9 = (f0)T

(fl),

fp = (fo)p+ 48)(fdp + - + E ’ ( P ) f t + * *

1

d

4 3 = B’(5)@”)I where

* *

= 2[1 +

E(P) = 5/3”(5)//3’(0= - 5(/3)5”(/3)/[4’(/3>12

(102) (103)

Substitution of the asymptotic sequence [Eq. (99)] into the momentum equation and boundary conditions produces a hierarchy of equations, the first two of which are (primes onfo andfl denote differentiation with respect to 7). Order unity =0 fo”’ +fofo” /3[1 (104) fo(8, 0) =fo’(P, 0) = 0 ; f O ’ ( B , a)= 1 Order E fl”’ + f O f l ” - AlfO’fl’ + A2fO”fl = @(Arl) (105) =0 fl(8, 0) = f t ’ ( B , 0)

+

=f*’(B,

In Eqs. (105) the terms (Al, A2,@) are A1=2+2/3+2E A2=3+2E (106) @(B, 7) =fo’(fo’>p - (f0)pfo” Note that two independent parameters (8, E ) appear in the first-order equation. Equations (104) represent the similar solutions of Falkner and Skan (50). Equations (105) represent the first correctionf, to the velocity profile which arises from nonsimilar terms. For example, the skin-friction derivative f”(/3, 0) is expressed as

0) + .fl”(B, 0) + * ’ * (107) where fo”(/3, 0) is the local similarity solution corresponding to the local value of /3, andfl”(/3, 0) is the correction obtained as a solution of Eqs. (10.5).

f”(B,0) =fo”(B,

SOLUTIONS OF

BOUNDARY-LAYER EQUATIONS

337

It is apparent that the correction will be of order E as long asfl”(/?, 0) is of the same order asfo”(p, 0) and E is small.

C. DETERMINATION 0 ~ f ~ ( /), 3 ,BY SUCCESSIVE APPROXIMATIONS We proceed to a consideration of the first-order equations for fl(p,v), Eqs. (105). The differential equation is linear with homogeneous boundary conditions and may be solved numerically. The primary difficulty arises in computing the inhomogeneous term @(p,7) which contains derivatives of fowith respect to both77 andp. This is a difficultterm to calculate numerically because a number of similarity solutions in the neighborhood of p must be known with high precision. The exact numerical calculation off, appears possible but has not been attempted. T h e technique adopted here is to substitute a transcendental approximation forfo(/3, 7) which allows the coefficients of the differential equation and to be expressed in terms of known functions. the forcing function @(p,~) Following an earlier paper by Bush (52),we representfo by the relation

This relation is found to be in excellent agreement with exact solutions forfo and appears quite adequate for our present purposes. I t is convenient to ) ( u , t ) and define a new dependent transform coordinates from ( 8 , ~to variable g(u, t ) according to the relations

a a,

= a -a

at,

[ 2 t L]

-a= a ’ @ ) - - + -

ap

2

0.23084

Substitution of these transformations into Eq. (105) results in a linear differential equation for g(u, t ) which contains derivatives of g with respect to t up to third order. The boundary conditions for this equation are (primes denote differentiation with respect to t ) g(., 0) = g’(u, 0) = g’(u, t.) = 0

(114)

C. FORBES DEWEY, JR.,

338

AND JOSEPH

F. GROSS

The differential equation for g(a,t ) is now integrated formally three times with respect to t , using the boundary conditions given by Eq. (114). The resulting integral equation is

g(., t ) = +t‘g’’(., 0) - J - ; J l M a ,t ) dt + A3 -A4

J-;j J- J-dt)g(a, t )dt + Y l ( t )

J-;J- Jo(t)g(a,4dt

(115)

T h e coefficients A3(a)and A4(a)are given by

A ~ ( u=)4 + 2E

+ 2/3,

A4(a)= 6

+ 4E + 2/3

(116)

and the terms J n ( t )are the nth integrals of the error function :

n=-1,0

J-,(X)

=

2

~

d;exp(-

x2),

Jo(x)= erf(x)

(1 17a)

n = 1,2,.., (117b) The term Y u , ( tis) defined by

where

A method of successive approximations is now applied to Eq. (115), substituting trial functions f ( a ,t ) for g(a,t ) in the three integrals and continuing until a trail function f is found that agrees satisfactorily with the function g computed from the integral equation. This scheme differs from Picard’s method in that the sequence of trial functionsf used in the integrals are suitably chosen integrable functions of t rather than the functions g(a,t ) obtained in the previous iteration step. Picard’s method converges absolutely, but in practice it usually cannot be continued analytically beyond one or two iterations. In the present scheme, convergence depends on the choice of trial functionsf but the integrals may be evaluated in closed form.

1. Evaluation of the Velocity Profile In this analysis based on the simplified momentum equation [Eq. (93)] we are most interested in the correction fl”(/3, 0) to the velocity gradient a t

SOLUTIONS OF BOUNDARY-LAYER EQUATIONS

339

the wall. It is therefore more convenient to work with the first derivative of Eq. (115), which is (after some rearrangement)

g’(.Pt)=

tg”(‘,o) +A4

1:1

- A 5 J l ( t ) g ( a , t ) + A6

Jo(t)g’(a,t)dt+ Y3(4 A5 = 3 +2E,

where

J:Jl(t)g’(a,t)dt

A6 =

(120)

2+2E

(121)

In following the technique of successive approximations, the terms g and g’ on the right-hand side of Eq. (120) must be replaced by the trial function

‘i’ I

g (a, m) = constant

FIG.3. Behavior of g andg’.

2 and its derivative 2’. The behavior of g and g‘ may be inferred from the boundary conditions g(u,0) = g’(u,0) = g’(u, m ) and is sketched in Fig. 3 . The behavior of Y3(t)and the two integrals appearing in Eq. (120) may be inferred from the behavior of g’ and the quantities Jn(t). Suppose we decompose the trial function f ( u , t ) into two parts, so that

f ( a , t ) = C(a).(t) (123) We then define ~ ( tto) be unity as t + 00, making f ( a , a)= C(a) # 0. The boundary conditions g(u, 0) = 0 and g’(u, 0) = 0 are automatically satisfied for all suitable trial functions f ( u , t ) ; the boundary condition g’(u, m) = 0 serves to evaluate both C(u)and g”(u, 0). The terms appearing on the righthand side of Eq. (120) have the following asymptotic forms :

340

C. FORBES DEWEY, JR.,

AND JOSEPH

F. GROSS

The terms (yl, y z ,73) are numerical constants which are determined by the choice of the trial function ~ ( t ) . Substituting Eqs. (124)-(127) into Eq. (120) and applying the boundary condition g’(a, m) = 0, we obtain the following formulas for C(a)and g”(0):

g”(% 0) = c(a)[(I

+ As)- A4Y21 - 6( 0, /3 is increasing and the local similarity value, fo”(/3, 0), would neglect “relaxation” effects and overestimate the shear at the wall. The correction reduces the skin-friction value by an amount proportional to E . The small magnitude (0.053) of the correction term is somewhat surprising, but this value is probably accurate to within about i20%.’ VI. Solutions for Large Values of the Pressure-Gradient Parameter p In using the concept of local similarity in highly accelerated flows, it is useful to have solutions of the laminar boundary-layer equations for large values of /3. The mathematical difficulties encountered in the limit /3+ 1may be illustrated by rewriting the streamwise momentum equation [Eq. (40)] in the form

B[(hf”)’+ff”] 1

-

(f’2 -

tw)9 -

Here X is redefined as the density-viscosity ratio at the wall (pp/pwpw). In the limit ,8 + a,Eq. (131) reduces from third to first order: 8+

m

The transverse-momentum equation and the energy equation remain of second order. As originally pointed out by Coles (52) and as later elaborated upon by Beckwith and Cohen ( 9 ) this leads to a singular perturbation problem in which the thickness of the velocity layer is of order /3-’/’ with respect to the total enthalpy layer and transverse velocity layers of order unity.2a The method of solution is similar to that employed in deriving a uniformly valid approximation to the Navier-Stokes equations in the limit of large Reynolds number [see Kaplun and Lagerstrom (54, Lagerstrom and Cole (55),and the recent book by Van Dyke (56)]. A. THEOUTERLIMITEQUATIONS Since the mathematical justification of this singular perturbation solution (more popularly called an inner- and outer-expansion procedure) has been 2 Bush estimated the correction term to be 0.204, but his result was obtained by asymptotically expanding an approximate solution rather than obtaining an approximate solution to the exact first-order equation. Zu Discussion of this problem also appears in an abbreviated version in Lagerstrom’s article (53).

342

C. FORBES DEWEY, JR.,

AND JOSEPH

F. GROSS

discussed in detail by Coles (52),our purposes will be served by a merely cursory development of the governing equations. The present analysis extends the work of Beckwith and Cohen (9) to include a power-law temperature-viscosity relation, a constant but nonunit Prandtl number, and the mass transfer at the surface. Assume that appropriate outer representations of the dependent variables f , 8, and g are of the following forms:

f

=fo

+(wp)fl+

* * *

e = eo + (i/djj)el + -

(133) g =go + ( 1 / d p ) g l+ * * * Substituting these representations into the momentum equations and the and ~ m a l l e r the ,~ energy equations and dropping all terms of order F1l2 following “outer” equations are obtained : +.

1

1‘2

(134)

+fogol = 0

(136) The appropriate boundary conditions are found by (a) requiring that the outer equations satisfy the exact outer boundary conditions, and (b) by exact matching of the inner representations with the outer. The results may be written fo(0)=fw, eo(0) =go@) = 0, e o ( a ) =go(a) = 1 (137) Only one boundary condition onf o may be satisfied by the outer equations, because the outer limit equation forforeduces to first order. The condition3“ f’+ 1 as q + a is automatically satisfied by Eq. (134). The no-slip condition, f‘ + 0 as q + 0, must be satisfied by an inner solution which is valid in a region of extent j g - l i 2 with respect to the scale of the outer solution. in the In this approximation, the density-viscosity ratio X = (pp/prUpL,,) outer layer becomes (Ago’)’

3 T h e outer limit equations are properly obtained by applying the limit /3 + m to the full equations expressed in outer variables, with 7 held fixed. This gives identical results to those cited here. 3 O T h e exact boundary condition.

SOLUTIONS OF BOUNDARY-LAYER EQUATIONS

343

where

As v + 0, h + [(l - u2)/ts]w-1since fo’(0) = (tW/tJ1/* and fo‘(0)is nonzero in general.4 The energy and transverse momentum equations are coupled through the implicit appearance of both Bo andgo in Eqs. (135) and (136). If Pr = 1, 0, = g o , and the number of coupled ordinary differential equations is reduced from three to two. Setting the viscosity-temperature exponent w and the Prandtl number Pr equal to unity reproduces the equations of Beckwith and Cohen (9). If u2 = 0 (i.e., the local Mach number is zero), t, = 1, and the transverse momentum and energy equations are again uncoupled. One interesting case was pointed out by Coles and we extend his rzsult to generalized compressible flow. Let t, = t, = 1, A = 0 so that u2 = ( Um2/2He) ( U J U , ) ~ = ul.Then h = (1 - u ~ ) ~ andfa‘ -’ = 1 for all 7. In taking this limit with Pr # 1, the product of (&’) and (1 - tW)-’ approaches Oo’ so that the energy equation becomes

O,,” + Oo’(v+fw)Pr(l

-

0 ~ ) ’ - ~ [1 ul(l - Pr)]-l

with the boundary conditions

=0

(139)

eo+o

v+o,

eo+i

rl+co,

If we define the new variable x and the constants xo and W by

x = (v/v‘W)

-

xo

xo =- f w l r n W = [ l - u1 (1 - Pr)][Pr(l - ( T ~ ) ~ - ~ ] - - I then Eqs. (139) and (140) are satisfied by the solution

e = [erf ( x / d )+ erf (xo/z/Z)]/[l + erf (xo/zr2)1

(142)

For xo < 0, we note the identity erf

(-x) =

-

erf

(x)

(143)

Equations (134)-( 136) with the boundary conditions of Eq. (137) represent a two-point boundary-value problem for the complete solution of the outer limit equations. The derivatives Oo’(0) and go’(0) along with the 4 It should be noted that the inner and outer expansion procedure breaks down in the limit t , + 0, because the outer solution forfo satisfies the exact boundary conditions for the complete equations and the inner solution forf’ is simply zero.

344

C. FORBES DEWEY, JR.,

AND JOSEPH

F. GROSS

integrals I t , 12,11(1),11(2),and 11(3)evaluated usingf,, do, andgo in place of

f,6, andg are given in Table VI. The integral IIis identically equal to zero. Beckwith and Cohen (9) calculated several of these quantities for the case of Pr = o = l,fw= 0, and nonunit values of t,.

B. THEINNER LIMITEQUATIONS Inasmuch as the order of the energy and transverse momentum equations is not reduced in taking the limit fl + co, the outer equations [Eqs. (135)(137)] represent complete solutions for the total enthalpy and transverse shear profiles for large /I. The termsfo and (fi2)’,which appear in the outer ’ in a region which equations, differ from the exact solutionsf and ( f 2 ) only smaller 2 in extent than the region of applicability of the outer equais F1j tions. Therefore, the outer equations asymptotically represent the complete solutions for 6 and g as /I + co, and in this limit the inner solutions for 6 and g are identically zero. The no-slip condition f‘(0)= 0 is satisfied by the inner limit equation forf; the inner equations for 6 andg, as noted previously, reduce to 6 = g = 0. T o examine the inner streamwise-momentum equation, it is necessary to introduce a new independent variable 75 and an inner representation for f: so that

q =4 3 %

+ (1/4F)f0(75) + .

f=fw

Jo’(?l)=fo‘(~)

*.

= U/%

(144) (145)

After rearrangement, the inner equation forfo’ becomes

and, dropping terms of order /Pand smaller, the result is where

(Aj?)’

- (&)2

+ tw/ts

=0

(147)

The boundary conditions onf;are found from the exact boundary conditions at the wall and the matching condition5 that the inner and outer representaa, with 75 large but fixed. The following tions off agree in the limit fl+ boundary conditions for the inner equation result:

Jo’(0) = 0,

o’(m) = ( t w / 4 ) 1 ’ 2

(149)

5 T h e matching condition applied here is elaborated by Van Dyke: inner representation of “outer representation” = outer representation of “inner representation.”

SOLUTIONS OF BOUNDARY-LAYER EQUATIONS

345

Since Eqs. (147) and (148) involve onlyfor and notfO, they represent a well-posed, second-order, two-point boundary-value problem. A more symmetrical form may be obtained by one final transformation : t = (tw/ts)”47j

Then the problem may be written (Aso’)’ - so2

+ 1 =0

with the boundary conditions s0(O) = 0,

and the definitions

h = [I

so( m)

=

1

(152)

- aso2]o-1

(153)

a = (I1 cos2(I/ts

(154) Values of sor(0) satisfying Eqs. (151)-(152) are given in Table VII for several values of w , (I, and For the special case of w = 1, Coles (52) pointed out that an analytic solution may be found in the form so = 1

3 sech2[t/dZ+ tanh-I

di]

(155) Using either the analytic solution for w = 1 or the numerical solutions for w # 1, the surface skin-friction derivativef,” is then found by reversing the previous transformations ; -

In general sor(0)depends on the three parameters 02,w, and t,; for w = 1, the value of so’(0) is $. VII. Discussion The large number of solutions listed in Tables I-VII suggests many possibilities for numerical correlations and comparisons with approximate results. Although an exhaustive discussion of these possibilities goes beyond our present purposes, we present selected examples in Figs. 4-14 of the use of the present solutions in understanding the influence of the similarity parameters on heat, mass, and momentum transfer. The local skin friction and heat transfer coefficients C,, and C, are defined by Eqs. (70) and (73). Following Dewey (ZO), they may be placed in a more symmetric form by using the Reynolds number

Re,, x

= Pe ue X I P e

(157)

Our thanks go to J. Avoesty of the RAND Corporation for suggesting a simple numerical quadrature technique for solving Eq. (1 51).

C. FORBES DEWEY, JR.,

346

AND JOSEPH

F. GROSS

and the dimensionless streamwise coordinate

With these definitions, the quantities CpLand C, become

+ (gw‘)2sin2A]

1/2

0.5

3.5

0.7

1 .o

1

tw = 0.6

3.0

2.5

w =

0.5 0.7

I .o

I

tw=0.15

2.0

1.5

I .o

0

0.5

1 .o

1.5

8

2.0

2.5

3.0

FIG.4. Variation of the skin friction coefficient with pressure gradient.

In Eqs. (159) and (160), only the terms appearing in the square brackets depend upon the similarity parameters ; the terms preceding the square brackets represent the external flow conditions and wall temperature. T h e Reynolds analogy function j = 2CdCf (161) is simply the ratio of the square brackets appearing in C,l and C, and consequently depends only upon the similarity parameters. In Fig. 4, the expected increase in the skin friction coefficient with increasing pressure gradient parameter /3 is shown. The relative increase in Cj/(Cj)ppois much greater for higher wall temperatures. This occurs

SOLUTIONS OF BOUNDARY-LAYER EQUATIONS

347

because the boundary layer is thicker and the response to increasing /3 is proportionately higher. Figure 5 shows similar behavior for the heat transfer coefficient. We have used the wall gradient parameter O,’[(l - tw)/(taw- t,)] in forming the heat transfer coefficient. As shown previously (4), 0,‘ varies greatly with /3 for

FIG.5. Variation of the heat transfer coefficient with pressure gradient.

t, > 0.6 because tawvaries with /3 if u # 0.6 The percentage increase here is less than for the skin friction but the trends with /3 and t, are the same. Lees (42) argued that the high density in the boundary layer near the wall for low wall temperatures “insulates” the wall from pressure gradient effects and the influence of /3 on C, is greatly reduced. This behavior is shown with decreasing values of t,. T h e next two figures deal with the Reynolds analogy function j = 2C,/Cf. Li and Gross (57)had earlier shown that large deviations from unity could occur in a hypersonic boundary layer even for t, e 1 and /3 = 0. Figure 6 demonstrates that j decreases with increasing /3, the decrease being faster with increasing wall temperature. This result can be easily explained by examining the limiting equations for /3 --f co. This heattransfer coefficient C, approaches an asymptote as /3 + co, whereas for large 6 It is particularly important to determine the proper adiabatic wall temperature in cases with mass injection because fa, is greatly decreased by injection (see Table 111). T h e assumption Pr = 1 predicts t., = 1 under all conditions, which is seriously in error for large injection and high local Mach numbers.

C. FORBES DEWEY,JR.,

348

AND JOSEPH

F. GROSS Pr = 0.7

f,=

0

t, = 1.0 QI

=o

t,=0.15

t W = 0.6

0.5

0

0.2

0.4

B

0.6

0.8

1 .o

FIG.6. Variation of the Reynolds analogy function with pressure gradient.

/l the skin-friction coefficient increases as /ll/*.The approach to this limiting

behavior is more rapid with larger wall temperatures. Similar behavior is shown in Fig. 7 for values of the hypersonic parameter u= 0, 0.5, 1.0.

0.15

0.60

SOLUTIONS OF BOUNDARY-LAYER EQUATIONS

349

Figure 8 shows the well known boundary-layer property that heat transfer is decreased by mass injection at the surface. The heat transfer reduction caused by a given value of fw decreases with increasing p. The reason will be explained below in the discussion on limiting solutions for large /3. Finally, the sweep angle parameter t, is varied in Fig. 9. When the sweep angle A is zero, there is skin friction only in the x direction [see Eq. (160)l. As the sweep angle increases, the span-wise term [g'(O) sin A] 1 .o

0.8

0.2

0

-0.2

-0.4

-0.6

-0.8

fw

FIG.8. Variation of the heat transfer coefficient with mass transfer.

contributes increasingly to the skin-friction coefficient. If the free-stream cos A], direction is sufficiently oblique, thex component of C,, [f"(0)(uc/um) no longer exerts an appreciable influence. The quantity [l +{g"(O) sin Alf"(0)cos

plotted in Fig. 9 is a relative measure of the contribution of the two skinfriction components, and approaches a limiting value as t, -to. Whalen(58) reported a similar result for displacement thickness for Pr = w = 1. I t is very difficult to obtain exact numerical solutions of the laminar boundary-layer equations for values of /3 greater than 2. The reason is simply that the singular behavior of f'(7) near the wall which exists for -+ m becomes dominant even for moderate values of 8. Conversely, the results obtained for p = m should be good representations of the behavior of the

C. FORBES DEWEY, JR.,

350

AND JOSEPH

F. GROSS

1 .o

-

0.9

E r4

6

-z

0.8

I

0

r, +

-"

-

0.7

C

0 m

+

c

0.6

u

1 .o

0.5

I

0

0.8

1

I

I

0.6

0.4

1

I

0.2

0

'I

FIG.9. Variation of skin friction coefficient with sweep angle. 1.25

Limit solutions

so

lo) = -3/4

1.20

1

1.151 1 .I5 0

I

I

I

I

II

I

I

0.5 a = -

u , cosl

.t

t S

FIG.10. Inner solutions for p

+ a.

I

'

I

'

1 .o

35 1

SOLUTIONS OF BOUNDARY-LAYER EQUATIONS

boundary layer for large but finite values of p. One of the important results that we wish to demonstrate is that by combining the exact numerical results for /3 < 5 given in Tables I-V and the limiting solutions for /3 -+ m given in Tables VI and VII, it is possible to estimate accurately the skin friction and heat transfer derivativesf,”, gWf,and 0,‘ for all positive values of /I. T h e skin friction results for i3 = m are displayed in Fig. 10. The influences of the two parameters w and a on the inner solutions for so’(0)are seen to be 2 .o

1.5

..-* C

0

L L

.C I

Y bl

1 .o

I

0

!

l

l

l

l

l

t

l

0.5 Limit parameter,

l

1 .o

l

l

l

l

1.5

-l/2

p

FIG.1 1 . Variation of skin friction with wall temperature for large 8.

relatively small. For 0.5 G w G 1.0 and 0 < a G 1.0, the value of so’(0) may be found by interpolation to better than 0.25%. The skin friction parameter fWff/3-’/* (t,/tJ-’/’’ approaches the limit of ~ ~ ’ as ( 06 )-+ m . T h e difficulties that were observed previously in calculating exact solutions for /?2 2 imply that this limit is approached very rapidly with increasing /I.This supposition is borne out by Figs. 11-13, where the skin friction parameter is shown as a function of p’/2. The limit parameter /3-1’z is suggested by the ordering procedure used to obtain the inner and outer equations. Solid lines indicate exact numerical solutions and dashed lines are extrapolations. Figure 11 illustrates the approach of the skin friction parameter to its asymptotic limit so’(0)for different wall temperatures, The limiting value is approached most rapidly for high wall temperatures. This result is to be expected from the behavior of the outer equations. Large values of t, increase the magnitude of the velocity difference across the inner layer,

352

C . FORBES DEWEY, JR.,

AND JOSEPH

F. GROSS

whereas for t, + 0, the distinction between the inner and outer layers breaks down and no proper limit is obtained. I t has also been found empirically that exact solutions are more difficult to obtain for large t, . The approach of the skin-friction parameter to its limiting value with increasing /?is illustrated in Fig. 12 for several values of the sweep parameter t, . From a numerical point of view, the accuracy of the present extrapolation procedure increases with increasing sweep. 2‘o

-

/

Limit parameter,

,3333

p-”’

FIG.12. Variation of skin friction with t , for large p.

I t is of interest to examine the behavior of the inner and outer equations with mass injection at the wall. Applications of these results include mass transfer cooling of rocket nozzles [Back and Witte, (59)]and blunt hypervelocity vehicles. The outer equations determine the heat transfer derivative 8,’ and they contain the boundary conditionfo(v -+ 0) =fw . Thus, surface mass transfer (f, < 0) acts to reduce 8,’ and consequently surface heat transfer even in the limit13 + a.On the contrary, the skin friction derivative fw” is found from the inner equation forfo’(0), Eq. (147), and the solution of this equation is independent of the value off,. In highly accelerated flows, therefore, the effects of blowing on skin friction become negligible in the limit + 00 with f, fixed. Figure 13 illustrates the behavior of the skin friction parameter as a function of the injection parameter 5”.The limit is approached smoothly for all J c , and accurate estimations of ffU” with decreasing values of /Pg-li2

353

SOLUTIONS OF BOUNDARY-LAYER EQUATIONS 1.6-

l

I

I

-

4

1.4

-

/-. --------

I

s

IQ

:a L

1.2

-

1

1.0

.C

..d

-\

-

C

p--o:

1.1547 /

0.6’ 0

-

-

R = 0=0.7 t, = 0.6

-

a2=0

-

0.8 -

-

-

-0.4

-

c

t

Limit

/ -

t, = 1

’

I

’

I

’

0.5

I

I

I

’

1 .o

’

I

I

FIG.14. Variation of heat transfer with wall temperature for large B.

’

1.5

354

C. FORBES DEWEY, JR., AND JOSEPHF. GROSS

may be obtained for all ,6 by comparing the exact solutions for ,6 < 5 and the inner limit solutions of Fig. 10. The wall heat transfer is related by the modified Stewartson and HowarthDorodnitsyn transformations (Eqs. (10) and (11)) to 9,'. Inasmuch as go and go represent the complete solutions for 9 and g as ,6 + co, the values of %,'(O) and go'(0) represent the asymptotic limits of 6,' and g,' as ,6+ 00. I t should be emphasized that 9,' and g,' become independent of /3 as ,6 + co, demonstrating that the heat transfer predicted by a local similarity analysis in highly accelerated flows approaches a limiting value. Figure 14 shows the typical behavior of the heat-transfer derivative 6,' with increasing values of ,6. The approach of 9,' to its limiting value for ,6 + co, is seen to be smooth and (at least for the case of u1 = 0) monotonic. In an earlier paper (4,we demonstrated that the proper parameter to use in comparing different heat transfer calculations is %,'[(l- t,)/(t,, - t,)]. For u1 = 0 and t, = 1, the adiabatic wall temperature tawis unity for all Pr so that the heat transfer parameter reduces to 6,'. Although we have not been able to prove it analytically, it appears that tau = 1.0 for all values of Pr, t,, w , and u in the limit ,6 = co. This is a very surprising result and should be examined further. The boundary-layer displacement thickness 6" is defined by the relation

where the integrals I,and I 2 are given in the list of symbols. For ,6 = co, the integral I l is identically zero and the integral I z [Eq. (75)]

is recorded in Table VI. In the absence of sweep, the velocity profile is monotonic and 0 1 for some range of 7 and I 2becomes negative. With sweep, therefore, the displacement thickness may be either positive or negative, depending upon the particular parameters being considered. Numerical comparison between the calculated results for moderate /3 and the present results for ,B = a, suggests that 6" monotonically decreases with increasing 8. In the special case when ,6 = m and t, = t, = 1,f' = 1 and 6" is of the order of /3-'" times the scale of the boundary displacement layer thickness for ,6 = 0. The numerical results given in Table V display the variation of boundarylayer properties for various Prandtl numbers near unity. These results may be used to estimate recovery factors for Pr > 1 by combining the present

SOLUTIONS OF BOUNDARY-LAYER EQUATIONS

355

results with the asymptotic solution of Narasimha and Vasantha (60) for Pr s 1. Although these authors explicitly solved the problem of a flat plate (/I= 0) with o = 1 and low Mach numbers, there is no reason why a similar analsyis could not be conducted for /I > 0 and general compressible flow. As Narasimha and Vasantha demonstrate, interpolations accurate to better than 3% in the recovery factor can be made between first-order asymptotic solutions for Pr s 1 and exact numerical results for Pr of order unity. I n concluding this discussion, it is advisable to point out that the numerical values listed for the wall derivativesf,”, Ow’, and g,’ allow reconstruction of the complete velocity and total enthalpy profiles by standard numerical integration techniques. Whereas the boundary layer equations themselves, including the boundary conditions at 7 = 0 and 7 --f co, represent a twopoint boundary-value problem, one may solve the equations as an initialvalue problem if the wall derivativesf,”, Ow’, andg,‘ are known. Furthermore, by using the wall derivatives given here, multi-term expansions of the boundary layer equations may be constructed for small q.This property is useful in cases where additional “nonclassical” behavior occurs near the surface, for example, in MHD boundary layers where an electrostatic potential sheath exists for small 7.

SOLUTIONS OF BOUNDARY-LAYER EQUATIONS KEY I SIMILAR SOLUTIONS FOR

w = Pr =

357

1 t

8

t

o

all

X

X

I'

X

X

I'

X

X

"

X

X

I'

X

X

"

X

X

'I

X

X

-0.8485

'I

X

X

-0.8755

I'

X

X

f"

0.0 (a)

0.05 0.1 0.2

0.0 -0.1414 -0.2828 -0.4243 -0.5657 -0.7071 -0.7782

0.0 0.0 0.0

0.25 0.2857

0.3 0.4

0.0 0.0 -0.2 -0.4 -0.6 0.0 0.0 -0.2 -0.4 -0.6

0.5

0.0

-0.5

0.75

0.0

("All

1.0 1 .o 0.1539 0.3333 0.6250 1.0 0.1000 1 .o 1 .o 1.0 1 .o 1 .o 1.0 1 .o

1 .o 1 .o

0.1000 0.1539 0.3333 0.6250 1 .o 0.1539 0.3333 0.6250 1 .o 0.1000 0.3333 1.0

0.15

0.2

0.b

0.5 0.6

0.8

1.0

X

X

X

X

X X

X

x x x

X

X

X

X

X

X

X

X

X

x

X

X X X

X X

X X

x x x x

x x x x

x x x x

x x x x

x x x

x x x

x x x x x

x x x x x

x x x x x

X

x

X

x x x x x x x x

x

X

x x x x

X

x

x

x

X

X

X

X

X

X

X

X

x

x

x

x

X

X

X

X

X

X

X

X

X

x x x

X

X

X

X

solutions for Pr =

x ~JJ E

x

1 and

B

x

= 0 are

2.0

x

linear in tW.

X

X

X

C. FORBES DEWEY, JR.,

358

AND JOSEPH

F. GROSS

KEY I (Continued) -

8

fw

1 .o

0.0

-0.5

-1.0

I *4

0 .o

1.5

0.0

1.8 2.0

2.4

2.8

3.4 4 .O 5.0

0.0 0.0

0.0

0.0

0.0

0.0 0.0

t

0.1000 0.1539 0.2500 0.3333 0.5000 0.6250 1 .o 0.1539 0.3333 0.6250 0.8333 1 .o 0.1539 0.3333 0.6250 1.0

o

t"

0.15

9.2

0.4

0.5

0.6

0.8

1.0 X

X

X

X

X

X

X

X

X X

X

X

X

X

x

X

X

X

x

x

x

x

x

2.0

x

x

x

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

x X

0.1000

0.1539 0.3333 0.6250 1.0 0.1000 0.1000 0.1539 0.3333 0.6250 1 .o 0.1539 0.3333 0.6250 1 .o 0.1000 0.1539 0.3333 0.6250 1 .o 1 .o 1.0 1.0

X

X

X

X

X

X

X

x

X

X

x

x

x

x

x

x

x

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

x X

x

x

x

x X

x

x

x X

X

X

X

X

X

X

X

X

X

X

X

X

X X

X X

X

X

X

X

X

X

X X

X

X

X

X

X

X

X

X

SOLUTIONS OF BOUNDARY-LAYER EQUATIONS

359

KEY I1 SIMILAR SOLUTIONS FOR A POWER-LAW VISCOSITY RELATION, Pr = O , f , , = 0

0 .o

0.5

0.1

1 .o

0.1

0.2

all

all

all

0.0 0.5 1.0 0.0 0.5 1.0

...

... ... ... ... ... ... ... ... ...

x x x

x x x

x x x

x

x x x

x x x

x x x

x

x x x

x x x

x x x

x

x

x

x

0.5

1 .o

1.0

0.1

0.3333

0.5000 0.6250 1.0

1.0 1.0 1.0 1.0

1 .o

1.0

1.0

0.5

0.3333

1.0 1.0 1.0

1.0 1.0

1 .o

1.0 1.0 1.0 1 .o

1.0 1.0 1.0

...

x

0.3333

1.0

1.0 1.0

...

X

X

0.1

0.6250 1 .o 0.3333 0.5000

0.6250 1 .o

0.2857

0.0

0.5 1 .o

0.5

0.1

1 .o

1.0 1.0 1.0

...

... e . .

X X

X

X

X

X

X X X

X

X

X

X

X

X

X

x

x

x

x

x

x

X

X

X

X

X

x

X

X

x

X

x

X

X

X

X

X

X

X

X

x

X

x

X

X X

X

0.6250 1.0

1.0 1 .o

0.3333

1.0 1.0 0.0 0.5 1.0

1.0 1.0

... ... ...

X

1.0 1.0 1.0 1.0

1.0 1.0 1.0

X

X

0.5000 0.6250 1 .o

X

X

1 .o

1.0

0.6250 1 .o

0.3333

... ...

x

X

x

X X X

X

X

X

x

X

x

x

X

X

X

X

x

x

x

x

x

x

X

X X

C. FORBES DEWEY, JR.,

3 60

AND JOSEPH

F. GROSS

KEY I1 (Continued)

I .05 0 04

0.5

0.3333

0.6250 1.o

0.1

0.3333

1.0

1.0

0.0 0.5 1.0

.1

I

0.0

1 .o

0.3333 0.6250

1.0

0.5

0.5

0.3333 1 .o

0.15

1.o

-

1.1

taw

--

X

X

X

X

X

X

X

1.0

X

0.0 0.5 1.0

... ...

X

...

X

1.0

0.0

1.0

0.0 0.5

X

x

X

0.0 0.5 1.0

... ... ...

1.0 1.0

1.0 0.0 0.25 0.5

0.5

0.0 0.5 1.o

0.0 1.0

0.5

1.0

0.5

0.1

1 .o

0.0 0.5

... ... ... ... ... ... ...

X

x

x

X

X X

X

X

X

X

X

X X

X

X

x x

X

x

x

x

x

X X

X

X X

X X

X

x

x

x

x

x

x

x

X X

X X

x

X

X

X

x

X

X

x

X

x

x x x

x x x

x x x

x

x

x

x

x

x

x

x

X

x

x

x

X

x x

x x

x x

X X

X

X X

x

X

x X

x x

X

x

X X

X

x

X

X

X

... ... ... ... *.. ... ... ... ... ...

X

X

0.5

1.0 1.0 0.0 0.5 1 .o

1.o

1 .o

1.0

X

0.8

1.o

.6

X

0.6

0.1

.4

X

1.0 1.0

.15

0.5

1 .o

0.6250

-

tW

X X

x

X

x X

X

X

X

X

X

X

X

x X

X

SOLUTIONS OF BOUNDARY-LAYER EQUATIONS

361

KEY I1 (Continued)

I

t

1 .o

0.5

0.7

1.0

0.3333 0.6250 1 .o

0.3333 0.6250 1 .o

1.0 1.0 0.0 0.25 0.5 0.6 0.: 1 .o

0.0 0.0

1.0 1.0 0.0 0.5 0.9 1 .o

0.0 0.0

1.0

0.0

0.1539 0.3333 0.6250 1 .o

1.0 1.0 0.0

0.6

1.4

0.7

1.0

1.5

0.5

1.0

0.0 0.25

0.7

1.0

0.5 0.0

1.8

0.5

1.0

0.6

2.0

0.5

1.0

0.7

1.0

0.0 0.25 0.0

3.0

0.5

0.7

1.0

1.0

0.0 0.0

...

... ... ... ... ...

... ... ... ...

X

X

X

X

X

X

X

x

X

X

X

X

x x x x

X

X

X

X

X

X

X

x x x x

x x

x x

X X

x

x

x

x

X X

X

x

x x x x

x

x x x x

X

x

X

x X

X X

X

X

X

X

X

X

X

x

x

x

x

x

x

x

x

x x

X

x

X

X

X

x

X X

X

x X

X

x

...

x

x

x

.*.

X

x

x

x

x

X

X

X

x

x

x

x

x X

x

x x

x x

x x

x

X X

X

... ... ... ...

x

X

0.0 0.0

... ... ... ... ... ...

x

X

X

7

C. FORBES DEWEY, JR.,

362

AND

JOSEPH F. GROSS

KEY I11 S I M I L A R SOLUTIONS FOR

Pr= 0.7, f, # 0, ts = 1

tw

05 0. 15 0.0

0.5

0.0 -0.2 -0.4

-0.6 0.7

0.0

1 .o 1 .o

1 .o 1.o

-0.4

0.5

0.0

0.7

0.4

0.5.

0.0

X

0.5 1.o

X

X

X

X

0 .o 0.5 0.0 0.5

0.7

1 .o

0.0

1.o 1.o

-0.6 0.0

-0.2

-0.4

X

X

X

1.o 1.o 1.o 1.o

-0.4

X

X

0.0 -0.2 -0.4 -0.6 -0.2

X

X

-0.6

-0.4

X

X

X

1.o 1.o 1.o

-0.2

X

X

1.o

0.2

aw -

0.0

1.o

-0.6

I. 0

0.5

1.o

-0.2

0.4 0. 6

1.o

1.o

0.0 0.5 1.o

X X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X X

X

X

X

0.0 0.5 1.0

X

0.0

X

0.5 1 .o

X

X

X

X

X

X

X

X

SOLUTIONS OF BOUNDARY-LAYER EQUATIONS

363

KEY I11 (Continued)

P

UI

0.4

0.7

fw

01

-0.6

0.0 0.5

.G5

X

1 .o

0.5

0.7

0.0 -0.2 -0.4 -0.6 -0.8 -1.0 -1.2

-1.4 0.75

0.7

0.0 -0.2

-0.4 -0.6 1.0

0.5

0.0

-0.2

-0.4 -0.6 0.7

1.0

0.0 -0.2

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.5 0.5 0.5 0.5

0.0 0.0 0.0 0.0 0.0

-0.4 -0.6 -0.8 -1.2 -1.4 -1.6

0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.0 -0.2 -0.4 -0.6 -0.8 -1.0 -1.2

0.0 0.0 0.0 0.0 0.0 0.0 0.0

tw 0.15 9.4 @,h 1.0

X

X

x

x

x

x

X X

X

X

X

X

X X

'

X

x

x

x

x X

X X

X

x

x

x X

X

X

X

X

X

X

X

x

x

X

X

X

X

X

X

X

x

x

x

X

X

X

X

X X

X

x

x

x X X X X

X

X

X

X

X

X

X

X

X

X

X X

X

X

X

-1.4 -1.5

0.0 0.0

X

X

-1.6

0.0

X

X

364

C. FORBES DEWEY, JR.,

AND JOSEPH

F. GROSS

KEY I11 (Continued)

P

W

2.0

0.5

fw

91

0.0

0.0 0.0

-0.4 0.7

-0.2

-0.4

5.0

0.5

0.0 0.0

-0.6

1 .o 0.0

-0.2

1.0

-0.6

1.0

X

X X

x x

x x

x

X X I

I

365

SOLUTIONS OF BOUNDARY-LAYER EQUATIONS

P 0.0

01 0.5

0.0 0.5 1.0

0.7

0.0 0.5 1.0

1.0

0.0 0.5 1.0

Pr 0.7 0.5 0.7 0.5 0.7 1.o

0.7 0.5 0.7 1 .o

-5

.6

- 8 1.0

1.1

2.0

taw

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

x

X

X

X

X

X

X

X

X

x

X

X

X

X

X

X

X

X

0.5

X

X

X

X

0.7

X

X

X

X

X

X

X

X

X

X

X

1 .o

0.1

0.15

0.1

0.0

0.7

0.2

0.5

1.0 1.0 0.0 1.0

0.7

X

0.0 0.5 1.0 1.0 0.0 1.0

0.7

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X X

X X

X

X

0.7 0.7 0.7

X X

X

1.0 0.7

X

1.0

X

X

X

X

X

x X

x

*Notes: a) Solution 6 for Pr = 1 apply for all 01 b) For PI linear ii

X

X

0.7

1.0 0.7

X X

1.0

0.7

X

X

0.7

1.0

X

X

X

0.5 0.7 1 .o

1.0 1.0 0.0

X

X

0.7

0.5 0.1 1.0

0.0

.4

X

0.10

1.0

.2

X

1.0

0.3

.15

X

0.0

0.7 1.0

.1

X

1.0

0.2057 0.5

X

0.7 1.0

0.05

0.7 1.0

0 .05

X

X

X

X

x

x

X

X

x

x

x

X

X

X

X

x X

x

x

X

C. FORBES DEWEY, JR.,

366

AND JOSEPH

F. GROSS

KEY IV (Continued)

0 .05

0.4

0.5

0.1

0.0 0.5 1.0

0.0 0.5 1.0

1.0

1.0

0.5

0.0

0.5

0.1

0.25 0.5 0.6 0.8 1.0

0.1 1 .o 0.1 0.1 0.7 0.1 0.1

0.0

0.5

0.5 1.0

1.0

0.0

1.0

0.15

1.0

0.5 0.1

0.5

X

0.0

0.5

X

2.0

X

X X

x

x

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

x

x

X

aw

X

X

X X

X

x

x X

X

X

t

X

x

X

-

X X

X

x

x X

X

X

X

x

X

X

X

X

X

X

X X

X

X

x

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X X

X

x

X

x

X

x

x

x

X

X

X

X

X

X

X

X

X

X

X

X

X

X X

X

X

X

x

x

X

X

X

x

X

X

X

X

X

X

X

X

X

X

x

x

x

x

x

x X

X X

X

X

X

x X

X

0.7

X

X

X

X

0.7

X

X

X

0.25

X

x

X

0.5 0.6

X

x

X

X

x

X

x

x

X

X

x

X

X

0.8 1.0

1.1

X X

X

0.1 1 .o

0.1 0.1 0.7

.8 1 . 0

X

0.1

0.5

.6

X

X

0.5 0.1

1.0

.5

X

0.1

0.7 0.7 1.0

.4

X

X

0.5

0.0 0.5 0.0

.2

X X

0.1 1 .o 0.1

0.5 0.1 1 .o

.15 X

0.1 0.7 0.1 0.1 1 .o 0.1 0.1

0.0

0.5 0.5

.0.7 0.1 0.1

.1

-

x X

X

X X X

X

X

X

X

SOLUTIONS OF BOUNDARY-LAYER EQUATIONS

367

KEY IV (Continued)

B 1.0

(u

0.7

1.0

1.4

1.5

1.8

2.0

b,

Pr 10 .o5

0.0

0.5 0.7 I 1.0 ~x

0.5 0.9 1.0

0.7 I 0.7 0.7

0.0

0.5 0.7 1.0

0.5 0.6 0.8 0.7 0.6

0.7 0.7

0.5

0.7

0.0

0.25

0.7 0.0 1.0 0.0

0.7 1.0

0.5

0.6 0.8

0.7

0.0

0.5 0.7

0.7

0.7

0.0

0.5

.4

.5

X X

-6

1.0

1.1

X

x

x

X

X

X

X

X

X

X

X

X

X

X

X

X

X X

X

X

x

x

x x

.8

2.0

X

X

X

X

x x

X

X

X

x x

X

X

X X

x

x

X

X X X

X

X

X

X

X

X

X

X

x

X

X

X

X

X

x

X

x

0.7

0.25

.2

I

0.7

0.7

.15

X

0.7

0.5

0.5

.1

X

X

X X X

X

x

x

x

X

x

x

X

X

x

x

X

X

X X

X

X

X

X

X

X

X

X

X X

X

X

X

X

X

X

X

X

X

X

X

0.5

0.7 0.5

1.0

0.0

1.0

X

2.4

1.0

0.0

1.0

X

X

X

2.8

1.0 0.0

1.0

X

X

X

3.0

0.5 0.7

0.0 0.0

0.7 0.7

3.4

1.0

0.0

1.0

X

X

4.0

1.0

0.0

1.0

X

X

X

5.0

1.0

0.0

1.0

X

X

X

X

x

X

x

x

x

x

x

x

X

X

X

X

X

X

X

X X

X

C. FORBES DEWEY, JR.,

368

AND JOSEPH

F. GROSS

tW

B

0.0

S

0.01

fw

u1 0 . 0 5 0 . 1 5 0 . 2 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8

1.0

0.0

0.0

(h)

-0.2 0.03 0.05 0.1

0.3

-0.6 0.0

0.01 'Two

X

X

X

X

X

X

1.0

X

X

X

0.0

X

X

X

X

0.0

x

0.5 1.o

X

x

0) (1))

x

(b)

X X

X

X

X

-0.2 -0.6

0.0 0.0

X

X

X

X

X

X

0)

1.0

X

X

0.0 0.0 0.0 0.0 0.0 - 0 . 2 0.0 -0.6 0.0

X

X

X

(I>)

X

X

X

X

X

X

0.0 0.9 -0.2 1.0 0.0 0.0 0.0 0.0 0.0 0.0 - 0 . 5 0.0 0.0 0.0 0.9

0.0 0.0

taw

X

0.5

0.0 -0.2 -0.6 0.0 0.0

2.0

x

X

0.0

0.3

X

x

0.01

0.05 0.1 0.2

X

X

0.05

0.01

X

X

0.5

1.0

0.0 0.0

X

X

0.0 1.0 0.0 0.0

0.4

0.02 0.3

X

0.5 1 .o

1.1

(b )

X

(h)

0))

X X

X

X

X

X

X

X

X

X

X

X

(h) (1))

(b) X

X X

X

X

X

X X

x X

X

X X X

s e t s of answers are given f o r each c a s e .

The f i r s t s e t

SOLUTIONS OF BOUNDARY-LAYER EQUATIONS

369

KEY VI SOLUTION OF

THE

ic

fw

t

3

0.5

0.0

0.1000

0.9 1.0

0.3333

0.9 1.0

OUTERLIMITEQUATIONS FOR /3 +a,

(?I

2

1.0 0.0

0.5 1.0 0.0

0.5 1.o

0.9 1.0

1.0

0.9 1.0

1.0 0.0

0.5 -0.2

1.0 0.0

0.5 -0.4

1.0

0.9

1.0

0.7

0.0

-0.2

-0.4

1.0

0.0

1.0 0.0

Pr 0.7 0.7 0.7 0.7

0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7

0.5

0.7 0.7 0.7

1.0

0.7

1.0

0.c 0.5

0.7 0.7

0.3333

0.9 1.0

1.0 0.0 0.5

0.7 0.7 0.7

1.0

0.9

1.0 0.0

0.7

1.0

0.5

0.7

0.9 1.0

1.0 0.0

0.7

0.1000

1.0

1.0

0.1000 0.1536 0.3333 0.6250 1 .o

0.9

0.9

1.0

1.0 1.0

1.0 1.0

1.0

0.7

0.5

0.7 0.7

1.0

0.7

0.0

0.5

0.7 0.7

1.0

1.0

X

X

1.0

1.0

X

X

1.0

1.0 1.0 1.0

X

X

X

X

X

X

1.0 1.0

6)An analytic solution is available for the special case =; t

ts= 1

w

TABLE I

0

SIMILAR SOLUTIONS FOR w = Pr = 1

t

B 0

I

t

ts

W

fll(o)

e’(o)

= gl(o)

I~

I2

I

1

0 1 .o

-0.1414

1

- 0.2 82 8

0 1.0

1

0

-0.4243

1

-0.5657

1

0

, ~

0.4696 0.4696

0.4696 0.46%

0.4696 1.686

0.4696 0.4696

1.686 1.686

1.686 1.686

1.217 1.217

0.3700 0.3700

0.3700 0.3700

0.5114 1.896

0.5114 0.5114

1.896 1.896

1.896 1.896

1.385 1.385

0.2766 0.2766

0.2766 0.2766

0.5594 2.163

0 -5594

0.5594

2.163 2.163

2.163 2.163

1.603 1.603

0 1 .o

0.1907 0.1907

0.1907 0.1907

0-6150 2.516

0.6150 0.6150

2.516 2.516

2.516

2.516

1.901 1.901

0 1 .o

0.1143 0.1143

0.1143 0.1143

0.6800 3 -021

0.6800 0.6800

3.021 3.021

3.021 3.021

2.341 2.341

1 .o

‘ 1

-0.7071

1

0 1.0

0.0502 0.0502

0.0502 0.0502

0.7573 3.862

0.7573 0.7573

2.862 3.862

3.862 3.862

3.105 3.105

-0.7782

1

0 1 .o

0.0243 0.0243

0.0243 0.0243

0.8025 4.628

0.8025 0.8025

4.628 4.628

4.628 4.628

3.825 3.825

-0.8485

1

0 1 .o

0.0048 0 -0048

0.0048 0.0048

0.8533 6.404

0.8533 0.8533

6-403 6.403

6.403 6.403

5.550 5.550

-0.8755

1

0 1 .o

0.0003 0 0003

0 -0003 0.0003

0.8621 9.591

0.8621 0.8621

9.591 9.591

9.591 9.591

8.729 8.729

-

0

?i

n

:Tf 0

arn P

TABLE I (Continued)

B

fw

tw

ts

0.05

0

1

0 0.5 1 .o

0.4848 0.5082 0.5311

0.4733 0.4773 0.4812

0.4453 1.023 1.593

0 -4626 0.4570 0.4514

1.675 1.663 1.652

1.654 1.623 1.593

1.208 1.199 1.191

0.1

0

1

0.15 0.4 0.6 1 .o

0.5123 0.5347 0.5523 0.5870

0.4788 0.4825 0.4854 0.4909

0.5910 0.8668 1.085 1.516

0.4532 0.4479 0.4439 0.4354

1.658 1.647 1.639 1.624

1.607 1.579 1.558 1.516

1.196 1.188 1.182 1.176

0.2

0

0.1539

0

0.5 1 .o

0.7839 1.222 1.611

0.5349 0.5845 0.6221

0.3600 0.7895 1.180

0.3158 0.1958 0.0777

1.503 1.393 1.320

1.101 0.7174 0.4077

1.081 0.9999 0.9456

0.3333

0 0.5 1 .o

0.6238 0.8506 1.058

0.5039 0.5344 0.5593

0.3761 0.8479 1.287

0.3981 0.3418 0.2855

1 585 1.508 1.450

1.383 1.154 0.9607

1.142 1.084 1.042

0.625

0 0.5 1.0

0.5543 0.6836 0.8047

0.4891 0.5082 0.5248

0.3842 0.8817 1.355

0.4314 0.4014 0.3712

1.627 1.576 1.533

1.515 1.373 1.247

1.173 1.134 1.103

1

0 0.5 1 .o

0.5233 0.6070 0.6867

0.4821 0.4951 0.5069

0.3881 0.8995 1.392

0.4457 0.4271 0.4082

1.648 1.612 1.580

1.576 1.480 1.392

1.188 1.161 1.138

0.1

0

1.017 1.756 2.401

0.5737 0.6425 0.6921

0.3304 0.7123 1.054

- 0.0248

0.1905

1.412 1.280 1.199

0.7389 0.1840 -0.2584

1.014 0.9169 0.8573

0.25

0

0.5 1 .o

-0.2332

-

w -4 to

TABLE 1 (Continued)

B 0.2857

t f "

tw

0

1

0 0.2 0.4 0.6 (L. 8 1 .o

0.5419 0.5883 0.6334 0.6774 0.7205 0.7627

0.4862 0.4932 0.4998 0.5061 0.5121 0.5179

0.3629 0.5593 0.7517 0.9406 1.126 1.309

0.4382 0.4285 0.4186 0.4087 0.3987 0.3887

1.636 1.617 1 * 599 1.582 1.566 1.551

1.542 1.491 1.443 1.396 1.352 1.309

1.179 1.165 1.151 1.139 1.127 1.116

-0.2

1

0 0.2 0.4 0.6 0.8 1 .o

0.4032 0.4547 0 5042 0.5520 0.5983 0.6434

0.3511 0.3604 0.3690 0.3769 0.3843 0.3913

0.3929 0.6183 0.8367 1.049 1.257 1.460

0.4909 0.4781 0.4651 0.4522 0.4392 0.4262

1.910 1.875 1.843 1.815 1.789 1.766

1.793 1.716 1.646 1.580 1.519 1.460

1.400 1.373 1.348 1.327 1.307 1.289

-0 -4

1

0 0.2 0.4 0.6 0.8 1.0

0.2758 0.3335 0.3876 0.4392 0.4886 0.5362

0.2301 0.2425 0.2534 0.2631 0.2720 0.2802

0.4252 0.6877 0.9379 1.178 1.411 1.637

0.5543 0.5370 0.5197 0.5025 0.4854 0.4684

2.279 2.209 2.151 2.102 2.059 2.021

2.133 2.009 1.901 1.805 1.717 f .637

1.707 1.651 1.605 1.566 1.532 1.501

-0.6

1

0

0.1629 0.2279 0.2867 0.3414 0.3931 0.4426

0.1273 0.1436 0.1568 0.1682 0.1782 0.1873

0.4591 0.7712 1.060 1.332 1.593 1.843

0.6318 0.6076 0.5839 0.5608 0.5381 0.5159

2.810 2 -656 2.545 2.457 2.385 2.324

2.628 2.402 2.227 2.081 1.955 1.843

2.169 2.038 1.945 1.872 1.812 1.762

0.2 0.4 0.6 0.8 1 .o

-

TABLE I (Continued)

B 0.3

0.4

t

fW

0

W

1

0 0.2 0.4 0.6 0.8 1 .o 2.0

0.5448 0.5931 0.6402 0.6860 0.7309 0.7748 0.9829

0.4868 0.4941 0.5009 0.5074 0.5136 0.5195 0.5457

0.3591 0.5539 0.7446 0.9318 1.116 1.297 2.165

0.4371 0.4270 0.4168 0.4065 0.3961 0.3857 0.3334

1.634 1.614 1.596 1.578 1.562 1 547 1.484

1.537 1.484 1.434 1.386 1.341 1.297 1.099

1.178 1.163 1.149 1.136 1.124 1.113 1.066

-

0

1

0 0.2 0.4 0.6 0.8 1 .o

0.5639 0.6254 0.6850 0.7429 0.7993 0.8544

0.4908 0.4997 0.5079 0.5157 0.5231 0 5300

0.3350 0.5198 0.7001 0.8765 1.049 1.219

0.4299 0.4175 0.4050 0.3923 0.3795 0.3667

1.623 1.599 1.577 1.557 1.538 1.521

1.505 1.441 1.381 1.325 1.271 1.219

1.170 1.152 1.135 1.120 1.107 1.094

-0.2

1

0 0.2 0.4 0.6 0.8 1 .o

0.4246 0.4924 0.5571 0.6194 0.6795 0.7378

0.3565 0.3681 0.3785 0.3881 0.3970 0.4052

0.3591 0.5689 0.7716 0.9680 1.159 1.346

0.4809 0.4648 0 4485 0.4322 0.4157 0.3993

1.889 1.846 J.809 1.776 1.747 1.720

1.743 1.649 1.565 1.487 1.414 1.346

1.384 1.351 1.322 1.297 1.275 1.254

0 0.4 0.6

0.2959 0.4413 0.5077 0.5711 0.6321

0.2363 0 2643 0.2757 0.2861 0.2956

0.3842 0.8547 1.074 1.286 1.491

0.5423 0.4994 0.4780 0.4567 0.4357

2.242 2.096 2.042 1.995 1.954

2.062 1.791 1.681 1.582 1.491

1.678 1.561 1.518 1.480 1.448

-0.4

1

0.8

1 .o

-

-

w

TABLE I (Continued)

P

0.4

0.5

-0.6

0

1

0.2 0 .& 0.6 0.8 1 .o

0.1809 0.2648 0.3400 0.4097 0.4756 0.5384

0.1340 0.1532 0.1686 0.1816 0.1931 0.2034

0.4094 0.6923 0.9524 1.197 1.431 1.656

0.1

0 0.5 1 .o

1.365 2.600 3.661

0.6211 0.7113 0.7739

0.1539

0 0.5 1 .o

1.089 1 966 2.728

-

0.3333

0 0.5 1 .o

-

0.625

1

0.6171 0.5878 0.5590 0.5307 0.5031 0.4760

2.741 2.572 2.453 2.362 2.288 2.225

2.520 2.267 2.074 1.915 1.778 1.656

0.2704 0 5882 0.8688

0.0133 -0.3417 -0.6827

1.315 1.169 1.085

0.2987 -0.4635 -1.075

0.9427 0.8355 0.7738

0.5828 0.6577 0.7111

0.2827 0.6257 0.9314

0.1703 -0.0671 -0.2988

1.393 1.256 1.172

0.6761 0.0812 -0.3942

1.000 0.8985 0.8374

0.7812 1 248 1.662

0.5328 0.5833 0.6215

0.3000 0.6856 1.037

0.3311 0.2195 0.1073

1.509 1.398 1.324

1.138 0.7649 0.4623

1.086 1.003 0.9480

0 0.5 1.0

0.6438 0.9167 1.165

0.5070 0 * 5410 0.5684

0.3097 0.7244 1.110

0.3961 0.3371 0.2769

1.577 1.494 1.432

1.366 1.122 0.9165

1.136 1.074 1.028

0 0.15 0.2 0.4 005 0.6 0.8 1 .o 2.0

0.5811 0.6368 0.6550 0.7262 0.7609 0.7952 0.8623 0.9277 1.235

0.4942 0.5020 0.5045 0.5140 0.5185 0.5228 0.5311 0.5390 0.5729

0.3146 0.4474 0.4909 0.6625 0.7469 0.8301 0 9941 1.155 1.920

0.4238 0.4131 0.4095 0.3949 0.3876 0.3802 0.3653 0.3503 0.2746

1.613 1.592 1.586 1.561 1 * 550 1.539 1.519 1.500 1.424

1.477 1.422 1.404 1.337 1.304 1.273 1.212 1.155 0.8988

1.162 1.147 1.142 1.124 1.115 1.107 1.092 1.078 1.022

0

1

-

-

2.111 1.968 1.869 1.793 1.732 1.681

r

0

z m

8

r

TABLE I (Continued)

0

fW

0.5

-0.5

t

tW

0.1539

0

0.6921 1.651

0.2956 0.3896

0.3132 0.7224

0.1742 -0.1967

1.846 1.550

0.3333

0 0.5

0.4265 0.9556 1.395

0.2377 0.3081 0.3540

0.3394 0.8202 1.233

0.4173 0.2252 0.0458

2.117 1.814 1.660

1.548 0-8506 0.3782

1.588 1.346 1.225

0.625

0 0.5 1 .o

0.3067 0.6290 0.9003

0.2049 0.2581 0.2941

0.3560 0.8954 1.365

0.5219 0.4127 0.3081

2.312 2.028 1.874

1.974 1.428 1.060

1.744 1.515 1.392

1

0 0.5 1 .o

0.2512 0.7414 0.6594

0.1873 0.2290 0.2583

0.3654 0.9464 1 -460

0.5685 0.4981 0.4294

2 -433 2.179 2.031

2.209 1.764 1.460

1.843 1.635 1.517

0.1

0 0.5 1 .o

0.6515 0.7549 0.8254

1.260 1.109 1.024

0.0149 -0.8868 -1.617

0.9023 0.7914 0.7294

0 0.5 1.0

0.5485 0.6089 0.6534

0.2329 0.5136 0 * 7599 0.2605 0.6046 0.9152

-0.1147 -0.5742 -1.015

0.3333

1.632 3.278 4.685 0.8796 1.509 2.062

0.2935 0.1487 0.0030

1.472 1.347 1.268

1.011 0.5673 0.2101

1.058 0.9655 0.9067

1

0 0.15 0.4 0.5 0.6 1.0

0.6181 0 6948 0.8173 0 * 8648 0.9112 1.090

0.5012 0.5112 0.5262 0.5318 0.5371 0.5568

0.2748 0.3953 0.5896 0.6654 0.7402 1.031

0.4120 0.3984 0.3751 0.3656 0.3560 0.3174

1.594 1.568 1.531 1.517 1.504 1.460

1.423 1.355 1.250 1.211 1.173 1.031

1.148 1.129 1.101 1.091 1.081 1.048

0.5

1 .o

0.75

0

-

0.8092 -0.1245

1.373 1.140

w

cn

w

TABLE 1 (Continued)

. I

o\

B 1

t f "

0

t

s

W

0.1

0 0.5 1.0

1.850 3.859 5.567

0.6735 0.7865 0.8625

0.2063 0.4615 0.6842

-0.2130 -0.7556 -1.275

1.223 1.069 0.9846

-0.1898 -1.197 -2.019

0.8754 0.7624 0.7007

0.1539

0 0.5 1 .o 2.0

1.431 2.866 4 094 6.254

0.6253 0.7212 0.7870 0.8817

0 2166

0.4934 0.7370 1.173

0.0235 -0.3404 -0.6945 -1.370

1.309 1.158 1.072 0.9673

0.3071 -0.4747 - 1 * 105 -2.164

0.9382 0.8270 0.7640 0.6879

0.25

0

1.102 2.079 2.923

0.5815 0.6593 0.7143

0.2267 0.5277 0.7955

0.1957 -0.0320 -0.2587

1.398 1.256 1.172

0.7265 0.1385 -0.3327

1.003 0.8987 0.8363

0.5 1.0

0.9607 1 734 2.409

0 * 5603 0.6280 0.6770

0.2318 0.5466 0.8289

0.2652 0.0942 -0.0781

1.445 1.312 1.230

0.9194 0.4243 0.0271

1.038 0.9396 0.8787

0 0.5 1 .o 2 .o

0.8105 1.364 1.853 2.726

0.5358 0.5904 0.6313 0.6931

0.2379 0.5711 0.8735 1.425

0.3346 0.2218 0.1061 -0.1244

- 1.386

1.503

1.309 1.206

1.135 0.7498 0.4382 -0.0782

1.081 0.9938 0.9367 0.8613

0

0.5 1 .o

0.7475 1.206 1.615

0.5248 0.5728 0.6094

0 * 2408 0.5833 0.8965

0.3622 0.2730 0.1805

1.531 1.424 1.350

1.229 0.8962 0.6242

1.102 1.021 0.9671

0 0.5 1 .o

0.6823 1.041 1.364

0.5130 0.5530 0.5845

0.2439 0.5975 0.9240

0.3897 0.3239 0.2551

1.562 1.468 1.401

1.330 1.056 0.8287

1.124 1.054 1.004

0.5 1 .o 0.3333

0.5

0.625

0.8333

0

-

-

TABLE I (Continued) ~

B 1

t

fW

0

-0.5

f"(0)

O'(0) = g'(0)

I1

~~

~ ~ ( i )1 ~ ( 2 )

1~(3)

0.15 0.2 0.4 0.5 0.6 0.8 1.0 2.0

0.6489 0.7445 0.7755 0.8963 0.9548 1.012 1.124 1.233 1.737

0.5067 0.5183 0.5219 0.5357 0.5421 0.5482 0.5597 0.5705 0.6156

0.2456 0.3570 0.3934 0.5360

0.4033 0.3875 0.3821 0.3603

1.579 1.550 1.541

1.508

1.383 1.305 1.280 1.186

1.137 1.115 1.109 1.084

0.6056 0.6743 0.8089 0.9402 1.560

0.3491 0.3380 0.3153 0.2923 0.1760

1.493 1.479 1.454 1.430 1.340

1.142 1.099 1.018 0.9402 0.6010

1.073 1.063 1.043 1.026 0.9595

0.1539

0 0.5 1.0

0.9717 2.512 3.774

0.3391 0.4536 0.5249

0.2288 0.5475 0.8164

-0.0152 -0.5454 -1.030

1.690 1.399 1.262

0.7219 -0.8079 -1.635

1.250 1.023 0.9188

2

0.3333

0 0.5 1.0

0.5719 1.426 2.129

0.2678 0.3558 0.4115

0.2480 0.6236 0.9434

0.3277 0.0549 -0.2016

1.970 1.657 1.505

1.215 0.3895 -0.1807

1.471 1.222 1.104

0.625

0 0.5 1.0

0.3910 0.9146 1.349

0.2261 0 2945 0.3390

0.2603 0.6837 1.048

0.4744 0.3208 0.1719

2.186 1.875 1.717

1.732 1.083 0.6471

1.642 1.392 1.268

r2

0

0.5 1.0

0.3067 0.6669 0.9692

0.2031 0.2580 0.2950

0.2675 0.7255 1.125

0.5392 0.4414 0.34Ltl

2.327 2.035 1.878

2.024 1.485 1.125

1.756 1.519 1.394

0 0.5 1.0

0.5068 2.144 3.416

0.1298 0.2520 0.3212

0.2363 0.6063 0.9018

-0.0286 -0.7956 -1.444

2.277 1.685 1.478

0.4951 -1.199 -2.267

1.766 1.270 1.103

1

1

-1 .o

tW

0.1539

0

-

tn

P

2

0

5

%

P

7 P

M

0 C

*2 0

5

TABLE I (Continued)

B

-1 .o

1

1.4

0

0.3333

0 0.5 1 .o

0.2159 1.139 1.847

0.0699 0.1652 0.2184

0.2570 0.7202 1.071

0.4449 0.0174 -0.3663

2.918 2 -091 1.830

1.892 0.3794 -0.4481

2.319 1.601* 1.385

0.625

0 0.5 1 .o

0.0970 0.6740 1.112

0.0369 0.1126 0.1546

0.2701 0.8053 1.225

0.6648 0.3855 0.1513

3.604 2.484 2.170

2.963 1.343 0.6582

2.934 1.931 1.665

1

0 0.5 1 .o

0.0481 0.4519 0.7566

0.0202 0.0822 0.1168

0.2780 0.8805 1.351

0.7624 0.5713 0.4058

4.234 2.817 2.459

3.801 1.989 1.351

3.523 2.218 1.908

0

2.135 4.657 6.790

0.6990 0.8234 0.9059

0.1762 0.4025 0.5991

-0.3318 -0.9785 -1.596

1.183 1.027 0.9429

-0.4251 -1.561 -2.495

0.8463 0.7316 0.6703+

0.5 1 .o

1.677 3.566 5.172

0.6507 0.7591 0.8321

0.1791 0.4190 0.6286

-0.0687 -0.5174 -0.9538

1.264 1.108 1.021

0.0950 -0.8020 -1.531

0.9053 0.7901 0.7268

0.3333

0 0.5 1 .o

1.091 2.115 3 .OOO

0.5772 0.6555 0.7107

0.1924 0.4668 0.7107

0.2249 0.0144 -0.1984

1.409 1.266 1.180

0.7930 0.2264 -0.2279

1.011 0.9051 0.8419*

0.625

0

0.8233 1.435 1.974

0.5362 0.5928 0.6350

0.2004 0.5008 0.7728

0.3422 0.2330 0.1190

1.504 1.383 1.305

1.148 0.7644 0.4535

1.081 0.9915 0.9334

0.1

1 .o

0

0.1539

0

0.5 1 .o

*

Il

f"(0)

W

0.5

1.5

= g'(0)

t

ts

fW

-4

Convergence to 10

O'(0)

TABLE I (Continued)

B

t W

fii(o)

0 0.15 0.2 0.4 0.5 0.6 0.8 1 .o 2.0

0.6987 0.8278 0.8695 1.031 1.109 1.185 1.334 1.477 2.140

0 0.5

0.1

ei@)

I1

I2

Il(U

11(2)

5(3)

0.5147 0.5289 0.5333 0.5498 0.5574 0.5646 J . 5781 0.5906 0.6423

0.2049 0.3035 0.3356 0.4610 0.5220 0.5821 0.6996 0.8141 1.352

0.3914 0.3725 0.3661 0.3395 0.3259 0.3122 0.2844 0.2562 0.1128

1.558 1.524 1.514 1.476 1.460 1.444 1.416 1.390 1.294

1.326 1.235

1.122 1.096 1.088 1.060 1.048 1.036 1.015 0.9963 0.9251*

1 .o

2.368 5.345 7.854

0.7176 0.8504 0.9378

0.1549 0.3623 0.5387

-0.4203 -1.140 -1.842

1.156 0.9987 0.9148

-1.811 -2.847

0.8266 0.7107 0.649P

0 0.5 1 .o

2.471 5.660 8.342

0.7252 0.8616 0.9509

0.1464 0.3439 0.5144

-0-4568 -1.218 -1.945

1.146 0.9875 0.9039

-0.6615 -1.936 -2.991

0.8188 0.7026 0.6418*

0.1539

0 0.5 1.0

1.871 4.156 6.088

0.6681 0.7055 0.8635

0.1542 0.3694 0.5563

-0.1330 -0.6442 - 1.141

'1.236 1.076 0.9893

-0.0466 -1.025 - 1.824

0.8842 0.7666 0.7034

0.3333

0 0.5 1 .o

1.194 2.437 3.504

0.5891 0.6749 0.7345

0.1661 0.4132 0.6315

0.1072 -0.0423 -0.2848

1.385 1.235 1.148

0.7086 0.0925 -0.4016

0.9932 0.8825 0.8183*

0.625

0 0.5 1 .o

0.8837 1.629 2.281

0.5444 0.6073 0.6533

0.1734 0.4505 0.6892

0.3207 0.2095 0.0752

1.485 1.356 1.275

1.094 0.6839 0-3377

1.067 0.9713* 0.9111

fw

tS

= g'(0)

~~

1.5

1 .8

2

0

0

0

*Convergence

1

0.1

-4 to 10

.

1.206

1.097 1.046 0.9966 0.9026 0.8141 0.4268 -0.5935

TABLE I (Continued)

6 2

2 -4

fw 0

0

*Convergence

t"

tS

0 0.15 0.2 0.4 0.5 0.6 0.8 1 .o 2 .o

0.7386 0.8972 0 * 9483 1.146 1.241 1.333 1.513 1.687 2 -488

0.5206 0.5367 0.5417 0.5601 0.5686 0.5766 0.5915 0.6052 0.6615

0.1775 0.2673 0 * 2965 0.4101 0.4653 0.5194 0.6254 0.7284 1.193

0.3837 0.3626 0.3553 0.3254 0.3101 0.2945 0.2629 0.2309 0.0587

0.1539

0 0.5 1.o

2.001 4.574 6.741

0.6789 0.8018 0.8830

0.1394 0.3462 0.5131

-0.1722 -0.7022 -1.258

0.3333

0 0.5 1.o

1.264 2.666 3 864

0.5965 0.6871 0.7493

0.1504 0.3906 0.5838

0.625

0 0.5 1.o

0 9248 1.767 2.501

0.5495 0.6164 0 * 6648

1

0

0.7659 1.335 1* 838

0.5244 0.5757 0.6145

1

0.5 1.o

-4

t o 10 -3 ClCConvergence to 10

.

.

-

1.544 1.506 1.495 1.454 1.436 1.420 1.390 1.363 1.263

1.288 1.187 1.156 1.036 0.9804 0.9266 0.8245 0.7284 0.2908

1.111 1.082 1.074 1.044 1.030 1.018 0 9957* 0.9760* 0.9022*

1.219 1.058 0.9708

-0.1309 -1.119 -2.003

0.8718 0.7529* 0 6 898"

0.1805 -0.0631 -0.3389

1.370 1.217 1.129

0.6582 0 0406 -0.5072

0.9826 0.8691* 0.8044*

0.1573 0.4113 0.6387

0.3208 0.1881 0.0481

1.473 1.340 1.257

1.061 0.6213 0.2678

1.059 0.9593 0.8978*

0.1611 0.7309 0.6762

0.3791 0.3005 0.2151

1.535 1.422 1.347

1.265 0.9408 0.6762

1 .lo4 1.020* 0.9637

-

-

B 2.8

t

fW

0

t

W

1.0

2 * 816 6.774 0.083

0.7484 0.8961 0.9961

0.1209 0.2933 0.4412

-0.5687 -1.439 -2.268

1.115 0.9550 0.8721

0 1 .o

2.117 7.342

0.6877 0.8991

0.1443 0.4784

-0.1507 -1.355

0.3333

0 1.0

1.325 4.196

0 6026 0.7617

0.1379 0.5460

0.625

0

0.5 1 .o

0.9612 1.895 2 704

0.5538 0.6241 0.6744

1

0 0.5 1 .o

0.7901 1.421 1.978

1

0.5 1.0

0.1

0

0.1539

0.5

-0.8655 -2.268 -3.436

0.7961 0.6787 0.6186*

1.206 0 9562

- 0.0900 -2.148

0.8619* 0.6791*

0.1671 -0.3826

1.359 1.114

0.6179 -0 5904

0.9739 0.7933*

0.1444 0.3874 0.5979

0.3144 0.1741 0.0256

1.464 1.327 1.243

1.035 0.5778 0.2111

1.052 0.9494 0.8870*

0.5275 0.5817 0.6223

0.1480 0.4033 0.6341

0.3756 0.2928 0.2023

1.527 1.411 1.333

1.246 0.9088 0.6341

1.098 1.011* 0.9537

1.541 2.170

0.5893 0.6320

0.3697 0.5929

0.2835 0.1919

1.396 1.317

0.8699 0.5938

0.9416*

0.1205 0.3434 0.5447

0.3684 0.2762 0.1751

1.511 1.385 1.305

1.207 0.8393 0.5447

1.086 0.9918 0.9320**

0.1052 0.3096 0.4923

0.3647 0.2668 0.1586

1.502 1.368 1.288

1.184 0.7994 0.4923

1.079 0.9796 0.9196-

-

-

3-4

0

4

0

1

0 0.5 1 .o

0.8502 1.650 2.347

0.5346 0.5955 0.6401

5

0

1

0 0.5 1.0

0 * 8907 1.815 2.616

0 6042

*Convergence

. ,,Convergence to 10- 3 . to

-4 10

-

0.5389

0.6509

-

-

1 .ooo

SIMILAR SOLUTIONS FOR

A

w

TABLE I1 POWER-LAW VISCOSITY RELATION,Pr = O , f w = 0

fl'(0)

e'(o)

g'(0)

0.3490 0.4143 0.4399 0.4696

0.3034 0.3625 0.3861 0.4139

0.3734 0.4394 0.4642 0.4876

00

N

I1

I2

11(1)

11(2)

11(3)

0.3490 0.4143 0.4399 0.4696

0.4037 0.7886 1.089 1.686

0.3490 0.4144 0.4399 0.4696

1.411 1.571 1.626 1.686

1.411 1.571 1.626 1.686

1.185 1.303 1.343 1.385

0.2963 0.3341 0.3246 0.0

0.3734 0.4394 0.4642 0.4876

0.4608 0.8600 1.167 1.645

0.3735 0.4394 0.4642 0.4876

1.452 1.610 1.663 1.710

1.452 1.610 1.663 1.710

1.166 1.250 1.240 0.7665

0.3507 0.4328 0.4987 0.5208 0.5343

0.2527 0.3026 0.3045 0.2392 0.0

0.3507 0.4328 0.4987 0.5208 0.5343

0.3514 0.5519 0.9513

1.304 1.523 1.672 1.717 1.742

1.304 1.523 1.672 1.717 1.742

1.003 1.142 1.201

1.546

0.3494 0.4303 0.4924 0.5124 0.5239

0.9846

1.1

0.5491

1.521

0.5491

1.953

0.5301

1.753

1.753

L-000

0.15 0.4 0.6 1 .o 0.15 0.4 0.6 0.9162

0.3920 0.4353 0.4514 0.4696 0.4089 0.4512 0.4663 0 -4802

0.3428 0.3820 0.3969 0.4139 0.3250 0.3431 0.3263 0.0

0.3920 0.4353 0.4514 0 -4696

0.4414 0.8137 1.106 1.686

0.3921 0.4353 0.4514 0.4696

1.508 1.614 1.649 1.686

1.508 1.614 1.649 1.686

1.255 1.334 1.359 1.385

0.4089 0.4512 0.4663 0.4802

0.4970 0.8778 1.173 1.636

0.4089 0.4512 0.4663 0.4802

1.538 1.640 1.672 1.700

1.537 1.640 1.672 1.700

1.224 1.270 1.248 0.7691

? 0

0.5

0.7

all

all

0

0.5

-

1

-

0

-

0.5

-

0.15 0.4 0-6 1 .o 0.15 0.4 0.6 0.9151 0.05 0.15 0.4 0.6 0.8009

1.251

1.163

U M

"g

TABLE I1 (Continued) n

0

0.7

all

all

1

0.1

0.5

1

1

-

-

0.05 0.15 0.4 0.6 0.8187

0.3924 0.4451 0.4835 0 4960 0.SO34

0.2843 0.3136 0.3025 0.2420 0.0

0.3924 0.4451 0.4835 0.4960 0 5034

-

0.3963 0.5766 0.9637 1.253 1.580

0.3923 0.4448 0.4832 0.4947 0.5029

1.442 1.586 1.681 1.707 1,725

1.442 1.586 1.681 1.707 1.725

1.100 1.188 1.195 1.134 0.8037

1.1

0.5097

1.291

0.5097

1.982

0.5085

1.737

1.737

2.444

0.15 0.4 0.6 1 .o

0.4696 0.4696 0.4696 0.4696

0.4139 0.4139 0.4139 0.4139

0.4696 0.4696 0.4696 0.4696

0.5091 0.8554 1.132 1.686

0.4696 0.4696 0.4696 0.4696

1.686 1.686 1.686 1.686

1.686 1.686 1.686 1.686

1.385 1.385 1.385 1.385

0.15 0.4 0.6 0.9179

0.4696 0.4696 0.4696 0.4696

0.3739 0.3572 0.3281 0.0

0.4696 0.4696 0.4696 0.4696

0.5595 0.9057 1.183 1.623

0.4696 0.4696 0.4696 0.4696

1.686 1.686 1.686 1.686

1.686 1.686 1.686 1.686

1.326 1.301 1-259 0.7732

1

0.15 0.4 0.6 0.8357

0.4696 0.4696 0.4696 0.4696

0.3339 0.3006 0.2439 0.0

0.4696 0.4696 0.4696 0.4696

0.6097 0.9560 1.233 1.559

0.4696 0.4696 0.4696 0.4696

1.686 1.686 1.686 1.686

1.686 1.686 1.686 1.686

1.267 1.217 1.134 0.7729

1

0.05 0.15 0.4 0.6 0.7948

0.3780 0.4733 0.5655 0.6066 0.6376

0.2597 0.3112 0.3134 0.2435 0.0

0.3623 0.4476 0.5194 0.5450 0.5613

0.3187 0.5088 0.8753 1.149 1.409

0.3414 0.4197 0.4717 0.4840 0.4882

1.291 1.513 1.652 1.690 1.709

1.260 1.464 1.575 1.591 1.590

0.9912 1.124 1.166 1.106 0.8825

1.1

0.6789

1.644

0.5909

1.785

0.4852

1.709

1.566

2.185

0

-

0.5

-

TABLE I1 (Continued)

f"(0)

0.1

0.7

I2

11(3)

0.3333

1

1

0.15 0.8151

0.5523 0.7916

0.3304 0.0

0.4188 0.5580

0.5197 1.375

0.3964 0.3793

1.531 1.602

1.387 1.265

1.134 0.6223

0.5

1

1

0.15 0.814(

0.5191 0.7020

0.3257 0.0

0.4643 0.5435

0.5246 1.399

0.4142 0.4222

1.550 1.639

1.452 1.407

1.149 0.6643

0.15 0.813

0.5056 0.6651

0.0

0.4615 0.5373

0.5270 1.411

0.4212 0.4394

1.559 1.656

1.479 1.467

1.155 0.6794

0.2900 0.3206 0.3096 0-2463 0.0

0.4021 0.4573 0.5008 0.5164 0.5274

0.3614 0.5303 0.8823 1.150 1.428

0.3834 0.4320 0.4608 0.4655 0.4653

1.430 1.571 1.654 1.674 1.682

1.394 1.521 1.576 1.575 1.561

1.087 1.166 1.157 1.063 0.7109

0.625

0.2

I1

1

1

0.3237

1

1

-

0.05 0.15 0.4 0.6 0.8121

0.4202 0.4850 0.5492 0.5808 0.6083

1.1

0.6420

1.492

0.5447

1.659

0.4751

1.702

1.569

0.8969

1

1

1

-

0.15 0.4 0.6 0.8291

0.5085 0.5331 0.5525 0.5744

0.3381 0.3051 0.2459 0.0

0.4781 0.4824 0.4857 0.4894

0.5563 0.8715 1.121 1.403

0.4529 0.4460 0.4404 0.4341

1.660 1.648 1.638 1.627

1.610 1.576 1.549 1.520

1.240 1.174 1.071 0.6798

0.5

0.3333

1

1

0.15 0.790

0.6229 1.033

0.3336 0.0

0.4808 0.6313

0.4625 1.214

0.3458 0.3112

1.451 1.562

1.198 1.023

1.064 0.8013

0.625

1

1

0.5441 0.8240

0.3235 0.0

0.4668 0.5986

0.4713 1.286

0.3918 0.4105

1.489 1.653

1.349 1.319

1.095 0.7607

1

1

-

0.15 0.792 0.05 0.15 0.4

0.4015 0.5091 0.6243

0.2657 0.3188 0.3209

0.3718 0.4604 0.5370

0.2977 0.4752 0.8161

0.3366 0.4114 0.4550

1.285 1.507 1.640

1.230 1.419 1.499

0.9808 1.110 1.139

TABLE I1 (Continued)

0.2

0.5

0.7

1

0.3333

1

0.6818 0.7268

1.1

0.7913

1.748

0.6069

1.659

0.15 0.6 0.8123

0.6428 0.9193 1.024

0.3431 0.2732 0.0

0.4908 0.5747 0.5951

0.4769

0.2467 0.0

0.5655 0.5838

1.069 1.303

0.4608 0.4589

1.673 1.686

1.494 1.474

1.062 0.8130

0.4479

1.680

1.425

2.337

1.011 1.243

0.3568 0.3134 0.2787

,1.495 1.546 1.535

1.240 1.062 0.9563

1.098 0.9341 0.5252

0.5

1

0.15 0.8095

0.5828 0.8663

0.3352 0.0

0.4798 0.5727

0.4843 1.275

0.3892 0.3566

1.526 1.587

1.350 1.184

1.122 0.5789

0.625

1

0.15 0.6 0.8085

0.5581 0.7377 0.8004

0.3318 0.2567 0.0

0.4752 0.5471 0.5627

0.4875 1.047 1.290

0.4021 0.4028 0.3878

1.539 1.614 1.612

1.397 1.340 1.282

1.133 0.9888 0.6020

0.05 0.15 0.4 0.6 0.8064

0.4445 0.5202 0.6068 0.6548 0.6979

0.2950 0.3265 0.3147 0.2470 0.0

0.4103 0.4678 0.5155 0.5338 0.5465

0.3356 0.4925 0.8141 1.059 1.314

0.3766 0.4216 0.4426 0-4415 0.4350

1.421 1.560 1.633 1.646 1.653

1.358 1.470 1.495 1.473 1.439

1.076 1.150 1.136 1.036 0.6471

1.1

1

1

0.6 0.7897

0.7539

1.489

0.5607

1.657

0.4223

1.647

1.386

2.709

0.3333

1

0.15 0.8297

0.6768 1.001

0.3604 0.0

0.5112 0.5550

0.4959 1.207

0.3793 0.2769

1.565 1.458

1.316 0.9579

1.160 0.4980

0.625

1

0.15 0.8257

0.5840 0.7700

0.3475 0.0

0.4935 0.5215

O.SO82 1.260

0.4214 0.3670

1.615 1.540

1.475 1.251

1.199 0.5734

w

TABLE I1 (Continued)

0.2

1

1

1

0.2857

0.5

0.3333

1

1

0.625

1

1

1

0

-

0.5

-

1

0.7

-

0.3333

1

1

0.5

1

1

m

0.5424 0.5883 0.6633

0.3414 0.3085 0.0

0.4851 0.4926 0.5044

0.5143 0.8065 1.288

0.4395 0.4270 0.4057

1.639 1.618 1.586

1.550 1.491 1.397

1.219 1.141 0.6165

0.15 0 7933

0.6898 1.209

0.3436 0.0

0.4965 0.6565

0-4359 1.168

0.3175 0.2198

1.434 1.553

1.101 0.7817

1.044 0.6157

0.15 0.7887

0.5843 0.9251

0.3307 0.0

0.4787 0.6191

0.4461 1.210

0.3792 0.3727

1.481 1.632

1.296 1.196

1.081 0.7071

0.15 1 .o

0.4283 0-7627

0.3157 0.4534

0.3643 0.5179

0.3292 1.309

0.3342 0.3887

1.360 1.551

1.302 1.309

1.144 1.280

0.15 0.9071

0.4605 0.7683

0.3101 0.0

0.3934 0.5397

0.3743 1.282

0.3537 0.4019

1.406 1.595

1.328 1.335

0.5667

0.05 0.15 0.4 0.6 0.7849

0.4209 0.5371 0-6701 0.7401 0.7952

0.2710 0.3246 0.3266 0.2490 0.0

0.3808 0.4704 0.5505 0.5810 0.6012

0.2779 0.4503 0.7737 1.013 1.223

0.3299 0.4053 0.4428 0.4439 0.4374

1.268 1.502 1.631 1.661 1.667

1.199 1.385 1.446 1.426 1.393

0.9696 1.100 1.120 1.032 0.7890

1.1

0.8787

1.862

0.6294

1.548

0.4195

1.640

1.325

2.232

0.15 0.8093

0.7114 1.197

0.3521 0.0

0.5050 0.6200

0.4478 1.159

0.3278 0.2053

1.473 1.496

1.139 0.7527

1.075 0.4696

0.6315 10.9897

0.3421 0.0

0.4913 0.5928

0.4561 1.193

0.3710 0.3089

1.510 1.556

1.280 1.034

1.104 0.5290

0.15 0.4 0.8235

-

-

00

::::62

1.123

TABLE I1 (Continued)

0.2857

0.4

0.7

0.625

1

1

0.15 0.8049

0.5984 0.9027

0.3378 0.0

0.4853 0.5806

0.4598 1.209

0.3883 0.3504

1.526 1.585

1.339 1.157

1.117 0.5540

1

1

-

0.05 0.15 0.4 0.6 0.8022

0.4636 0.5475 0.6512 0.7118 0.7664

0.2988 0.3310 0.3194 0.2499 0.0

0.4171 0.4758 0.5259 0.5459 0.5603

0.3143 0.4657 0.7744 1.007 1.236

0.3707 0.4143 0.4298 0.4241 0.4133

1.411 1.553 1.624 1.635 1.635

1.329 1.433 1.439 1.402 1.356

1 A068 1.138 1.108 0 9885 0.6055

-

1.1

0.8401

1.565

0.5771

1.557

0.3950

1.621

1.286

2.709

1

1

1

-

0.15 0.4 0.6 0.8194

0.5684 0.6308 0.6793 0.7310

0.3438 0.3108 0.2473 0.0

0.4901 0.4999 0.5072 0.5148

0.4846 0.7611 0.9773 1.210

0.4299 0.4134 0.4001 0.3854

1.625 1.598 1.578 1.558

1 508 1.432 1.374 1.314

1 204 1.117 0.9918 0.5758

0.5

0.3333

1

0

0.15 0.8938

0.6910 1.434

0.3363 0.0

0.4393 0.6268

0.3396 1.081

0.2834 0.1360

1.311 1.410

1.024 0.5590

1.030 0.4240

0.5

0.15 0.8498

0.7224 1.431

0.3403 0.0

0.4643 0.6476

0.3656 1.087

0.2837 0.1264

1.346 1.452

1.012 0.5407

1.022 0.4055

1

0.15 0.7898

0.7703 1.412

0.3550 0.0

0.5146 0.6872

0.4069 1.084

0.2845 0.1311

1.418 1.523

0.9944 0.5552

1.023 0.5541

0

0.15 0.9407

0.5342 1 *044

0.3219 0.0

0.3972 0.5669

0.3279 1.156

0.3244 0.3039

1.342 1.484

1.204 1.009

1.092 0.5241

0.5

0.15 0.8738

0.5674 1.044

0.3221 0.0

0.4255 0.5893

0.3616 1.149

0.3359 0.3085

1.383 1 530

1.213 1.019

1.077 0.4872

0.625

1

-

-

TABLE I1 (Continued)

0.4

0.5

0.7

0.15 0.7829

0.6332 1.045

0.3393 0.0

0.4928 0.6439

0.4182 1.117

0.3648 0.3334

1.473 1.599

1.236 1.070

1.067 0.6992

0.15 1 .o

0.4540 0.8544

0.3192 0.4632

0.3686 0.5300

0.3094 1.219

0.3303 0.3667

1.347 1.521

1.273 1.219

1.133 1.257

0.15 0.9048

0.4887 0.8564

0.3141 0.0

0.3992 0.5533

0.3511 1.194

0.3483 0.3786

1.394 1.570

1.296 1.244

1.111 0.5232

0.05 0.15 0.4 0.6 0.7815

0.4416 0.5714 0.7260 0.8112 0.8782

0.2757 0.3318 0.3334 0.2515 0.0

0.3870 0.4824 0.5667 0.5995 0.6203

0.2694 0.4229 0.7270 0.9516 1.149

0.3321 0.3987 0.4291 0.4246 0.4135

1.289 1.499 1.624 1.651 1.659

1.188 1.349 1.386 1.351 1.306

0.9672 1.090 1.099 0.9983 0.7155

1.1

0.9837

1.924

0.6489

1.474

0.3884

1.644

1.218

2.562

0

0.15 0.8964

0.7382 1.428

0.3634 0.0

0.4742 0.6161

0.3620 1.072

0.3039 0.1408

1.385 1.390

1.071 0.5692

1.081 0.4242

0.5

0.15 0.8545

0.7613 1.418

0.3602 0.0

0.4911 0.6275

0.3841 1.070

0.3005 0.1346

1.406 1.414

1.055 0.5585

1.065 0.4054

1

0.15 0.8052

0.7938 1.403

0.3622 0.0

0.5212 0.6471

0.4162 1.068

0.2945 0.1234

1.450 1.458

1.029 0.5395

1.052 0.4289

0

0.15 0.9414

0.5807 1.040

0.3550 0.0

0.4369 0.5619

0.3527 1.152

0.3550 0.3027

1.429 1.475

1.274 1.007

1.152 0.5242

0.5

0.15 0.8763

0.6049 1.032

0.3457 0.0

0.4564 0.5743

0.3834 1.136

0.3594 0.3044

1.456 1.502

1.272 1.013

1.126 0.4875

0.625

1

1

0

-

0.5

-

1

-

0.3333

0.625

1

1

1

TABLE I1 (Continued)

0.4

0.7

0.625

1

1

0

0.5

0.4292 1.122

0.3726 0.3077

1.513 1.559

1.275 1.021

1 .I00 0.5053

0.3347 1.219

0.3687 0.3667

1.443 1.521

1.358 1.219

1.204 1.257

0.4344 0.4909 0.5159 0.5430

0.3749 0.6569 0.8701 1.185

0.3781 0.3955 0.3906 0.3728

1.475 1.544 1.554 1.551

1.366 1.362 1.318 1.235

1.166 1.176 1.119 0.5240

0.4 0.6 0.7974 1.1

0.4864 0.5007 0.7053 0.7811 0.8490 0.9451

0.3034 0.4246 0.3363 0.4853 0.3239 0.5389 0.2500 0.5609 0.5763 0.0 1.636 0.5953

0.2943 0.4358 0.7220 0.9380 1.151 1.460

0.3654 0.4063 0.4154 0.4050 0.3887 0.3616

1.404 1.546 1.608 1.614 1.617 1.603

1.300 1.393 1.377 1.326 1.265 1.172

1.058 1.126 1.092 0.9700 0.5624 2.887

0.15 0.8005

0.6473 1.025

0.3448 0.0

0.4971 0.6007

-

0.15 1.0

0.5027 0.8544

0.3597 0.4632

0.4129 0.5300

-

0.15

0.5281 0.6583 0.7386 0.8470

0.3422 0.3671 0.3500 0.0

1

0.4 0.6 0.905f 1

-

0.05 0.15

1

'

0.3333

1

1

0.15 0.8231

0.8319 1.383

0.3759 0.0

0.5366 0.5993

0.4289 1.033

0.3173 0.1439

1.500 1.363

1.098 0.5815

1.103 0.3952

0.625

1

1

0.15 0.817s

0.6727 0.9951

0.3563 0.0

0.5094 0.5520

0.4438 1.091

0.3887 0.2967

1.571 1.467

1.342 1.006

1.158 0.4799

1

0

0.15 1 .O

0.5901 0.8544

0.4327 0.4632

0.4926 0.5300

0.3806 1.219

0.4380 0.3667

1.619 1.521

1.514 1.219

1.333

0.15 0.9071

0.5948 0.8337

0.3901 0 .o

0.4942 0.5283

0.4156 1.173

0.4286 0.3646

1.614 1.525

1.487 1.221

1.261 0.5248

0.5

-

-

1.257

w

TABLE I1 (Continued)

Q 0

0.4

1

1

1

-

0.15 0.4 0.6 0.8147

0.5998 0.6824 0.7461 0.8125

0.3465 0.3134 0.2474 0.0

0.4959 0.5082 0.5173 0.5264

0.4515 0.7109 0.9129

0 -4192 0 3980 0.3809 0.3624

1.608 1.575 1.552 1.529

1-460 1.366 L.295 1.224

1.187 1.092 0.9564 0.5334

0.5

0.5

0.3333

1

0.5

0.3469 0.0

0.9453 0.3868

1.006 0.3726

-

0.2520 0.3218 0.3940 0.4266 0.4705 0.4791

1.155 1.250

0 2644 0.0638 0 2666 0.3274 0.3658 0.3685 0.3503 0.3431

1.328 1.424

0

0.4739 0.6662 0.2923 0.3720 0.4533 0.4899 0.5390 0.5486

0.3450 1.022

1

0.7824 0.15 0.8488 1.599 0.3518 0.05 0.4747 0.15 0.6472 0.4 0.7511 0.6 0.9277 1.0 1.1 0.9681

1.153 1.336 1.455 1.484 1.500 1.500

1.103 1.251 1.290 1.258 1.155 1.126

0.9820 1.125 1.212 1.232 I. 240 1.240

0.05 0.15 0.9523

0.3653 0.4909 0.9260

0.2517 0.3187 0.0

0.3038 0.3857 0.5494

0.1840 0.3127 1.142

0.2733 0.3245 0.3549

1.171 1.358 1.522

1.112 1.259 1.165

0.05 0.15 0.4 0.6 0.903 1.1

0.3827 0.5114 0.6867 0.7908 0.9266 1.007

0.2532 0.3171 0.3637 0.3539

1 -024

0.3189 0.4037 0.4878 0.5247 0.5635 0.5830

0.2001 0.3339 0.6147 0.8245 1.131 0.1325

0.2826 0.3444 0.3794 0.3786 0.3612 0 3446

1.194 1.385 1.507 1.537 1.552 1.553

1.124 1.271 1.303 1.264 1.179 1.117

0.05 0.15 0.8826

0.3913 0.5215 0.9280

0.2546 0.3173 0.0

0.3265 0.4128 0.5708

0.2077 0.3438 1.127

0.2874 0.3496 0 3645

1.206 1.399 1.566

1.130 1.277 1 * 185

0.9772 1.114 0.4897 0.9728 1.103 1.147 1.098 0.4935 2.080 0.9712 1.098 0.4953

0.25

-

0.5

-

0.6

-

0.0

1.125

0.1704 0.2946 0.5628 0.7652

-

-

-

-

-

TABLE I1 (Continued)

0.5

0.5

1

0.8

-

f"(0)

e'(o)

g'(0)

0.4138 0.4928 0.5475 0.9335

0.2603 0.2989 0.3204 0.0 0.2815 0.3387 0.3385 0.2521

1.1

0.4614 0 * 6009 0.7723 0.8688 0.9439 1.069 0.4252 0.5243 0.6713 0.7643 0.9277 0.9657

0.05 0.15 0.4 0.6 0.9040 1.1

0 4500 0.5513 0.6986 0.7913 0.9169 0 9920

0.05 0.15 0.4

0.5048 0.6073 0.7489

0.05 0.1 0.15

0.8393 1

0.7

1

0

-

-

0 -05 0.15 0.4 0.6 0.7762 1.1 0.05 0.15

0.4 0.6 1 .o

0.5

-

1

-

-

Il(3)

I1

I2

11(1)

0.3471 0.4028 0.4371 0 5906 0.3970 0.4957 0.5831 0.6156 0.6374 0.6666

0.2261 0.3017 0.3675 1.121

0.3007 0.3419 0.3638 0.3731

1.236 1.366 1.435 1.605

1.146 1.249 1.294 1.201

0 2496 1.386 0.6808 0.8997 1.080 1 404

0.3233 0.3871 0 -4149 0 * 4094 0.3958 0.3640

1.258 1.460 1.586 1.629 1 a634 1.632

1.155 1 -306 1.334 1.294 1.242 1.138

1.074 1.089 0.9863 0.7265 2.655

0 2008 0.3182 0.5779 0.7751 1.155 1.247

0.3248 0.3652 0.3823 0.3768 0.3503 0.3420

1.307 1.433 1.499 1.508

1.244 1.334 1.324 1.275 1.155 1.124

1.098 1.196 1.244 1 * 249 1.240 1.237

0.3013 0.3452 0.3710 0.3535 0.0 0.9870

0.3578 0.4164 0.4750 0.5018 0.5390 0 5464 0.3784 0.4388 0.4975 0.5238 0.5526 0.5674

1.253 1.340 1.321 1.266 1.170 1.105

1.075 1.157 1.162 I. 101 0.4941 2.030

0.430'5 0.4928 0.5488

0.3339 0.3733 0.3867 0.3784 0.3557 0.3379 0.3616 0 * 4004 0.4047

1.339

0.3071 0 3405 0.3277

0.2322 0.3560 0.6238 0.8258 1.122 1.308 0.2794 0.4134 0.6862

1.278 1.363

1.051 1.118

0.0

2.001

0.3109 0.3625 0 4142 0.4378 0.4705 0.4770

-

-

-

-

1.500

1.496 1.465 1.530 1.538 1.532 1.524

-

1.400 1 540 1 * 599

1.332

0 * 9680

1.052 1.090 0 -4989

-

0 9530

1.076

w

TABLE I1 (Continued)

0.5

0.7

1

0.75

1

1

1

-

0.8370 0.9146 1.029

0.0

1.700

0.5719 0.5886 0.6095

0.8938 1.087 1.386

0.3900 0.3711 0.3372

0.15 0.4 0.6 1.0

0.6136 0.7105 0.7850 0.9277

0.4358 0.4471 0.4555 0.5390

0.4964 0.5103 0.5205 0.5390

0 3609 0.6026 0 7906 1.155

0.15 0 -4 0.6 0.8111 1.1

0.6249 0.7237 0.7997 0.8773 0.9800

0.3485 0.3152 0.2474 0.0, 1.409

0 5003 0.5145 0.5248 0.5349 0.5478

0.4272 0.6743 0.8662 1.064 1.328

0.05 0.15 0.4 0.6 0.8994 1.1

0.4131 0.5618 0.7768 0.9092 1.083 1.189

0.3251 0.4129 0.5020 0.5419 0.5840 0.6061

0.05 0.15 0.4 0.6 1 .o

0.4557 0.5726 0.7568 0.8770 1.090 1.140

0.2576 0.3234 0.3721 0.3616 0.0 1.085 0.3150 0.3683 0.4232 0.4489 0.4849 0.4922

0.3629 0.4237 0.4862 0.5155 0.5568 0.5652

0.6 0.7934

0

-

1

-

0.5

1

0.5

-

0.7

1

0

-

9 N

1.1

1.1

0.2518

-

1.268 1.200 1.090

0.9366 0.5470 2.953

0.4331 0.4092 0.3897 0.3503

1.607 1.603 1.588 1.609 1.572 1.545 1.500

1.487 1.380 1.301 1.155

1.325 1.296 1.276 1.240

0.4113 0.3866 0.3667 0.3454 0.3161

1.596 1.559 1.533 1.508 1.478

1.425 1.318 1.238 1.159 1.058

1.174 1.073 0.9303 0.5038 2.702

0.1796 0 3000 0.5518 0.7391 1.008 1.183

0.2784 0.3367 0.3630 0.3549 0.3264 0.3017

1.092 1.223 1.224 1.165 1.052 0.9717

0.9608 1.086 1.121 1.064 0.4382 2.118

0.1800 0.2857 0.5183 0.6939 1.031 1.113

0.3207 0.3582 0.3682 0.3565 0.3174 0.3059

1.182 1.368 1.483 1.508 1.518 1.517 1.292 1.412 1.470 1.476 1.460 1.454

1.211

1.085 1.179 1.221 1.223 1.209 1.204

-

-

1.288 1.251 1.183 1.031 0.9922

I'ABLE I1 (Continued)

I2 0.75

1

0.7

0.5

1

0.5

-

0.05 0.15 0.4 0.6 0.9004 1.1

0.4833 0.6029 0.7885 0.9089 1.072 1.173

0.3059 0.3512 0-3787 0.3605 0.0

0.3850 0.4478 0.5108 0 5400 0.5719 0.5890

-

0.3333

1

0

0.05 0.15 0.8980

0.6684 0.9742 2.298

0.625

1

0

0.05 0.15 0.9429

0.4913 0.6981 1.580

1

0

-

0.05 0.15 0 -4 0.6 1 .o 1.1

0.4O25 0.5612 0.8060 0.9615 1.233 1.295

1.045 0.2862 0.3695 0.3626 0.4740 B.0 0.6980 0.2665 0.3280 0 * 3394 0.4198 0.0 0.6190 0.2584 0* 3004 0.3317 0.3844 0.4097 0.4730 0 4460 0 5140 0.4959 0.5705 0.5058 0.5817

0.05 0.15 0.9490

0.4187 0.5808 1.224

0.2588 0.3291 0.0

0.3132 0.3998 0.5829

0.05 0.15 0.4

0.4395 0.6057 0.8558

0.2612 0.3285 0.3787

0.3302 0 4204 0.5132

0.25

-

0.5

-

-

-

-

0.2072 0.3187 0.5591 0.7396 1 .oooo

1.169

0.3279 0.3690 0-3692 0.3542 0.3215 0.2963

1.325 1 -446 1 503 1.507 1 *496 1.485

-

1.215 1.287 1.240 1.166 1.044 0.9624

1.061 1.139 1.135 1.065 0.4385 2.062

-

0.1550 0.2546 0.8082

0.2033 0.2097 -0.1293

1.075 1.228 1.287

0.1496 0.2520 0.8788

0.2520 0.2948 0.1736

1.115 0.1280 1.378

0.9741 1.052 0.6162

0.9235 1.042 0.4018

0.1408 0.2437 0.4631 0.6272 0 9402 1.016

0.2617 0.31% 0 3430 0.3344 0.2923 0.2791

1.127 1.300 1.404 1 -426 1.430 1.429

1.053 1.175 1.167 1.102 0.9402 0.8978

0.9605 1 .O% 1.172 1.186 1.186 1.184

0.1514 0.2578 0 * 9289 0.1641 0.2746 0.5051

0* 2673 1.146 0.3230 1.323 0.2955 1.458 0.2755 1.172 0.3311 1.355 0.3507 1.465

1.059 1.180 0.9490

0.9556 1.084 0.3943

1.068 1.187 1.166

0.9516 1.074 1.102

-

0.7781 0 8644 0.7633 0.9629 - 0 -0618 0.2881

w

TABLE I1 (Continued)

0.5

-

0.6

-

0.8

-

~

1

0.5

1

1

~

-

0.6 0.8965 1.1

1.013 1.219 1.349

0.3675 0.0 1.135

0 * 5554 0.6000 0.6241

0.6760 0.9187 1 .080

0.3369 1.488 0 * 3000 1.495 0.2689 1.492

1.092 0 9601 0.8652

1.039 0.3992 2.149

0.05 0.1 0.15 0.8748

0.4499 0.5467 0.6181 1.218

0.2632 0.3047 0.3292 0.0

0.3390 0.3954 0.4310 0.6091

0.1700 0.2298 0.2825 0.9152

0.2799 0.3169 0.3356 0.3025

1.185 1.307 1.371 1.513

1.073 1.159 1.192 0.9655

0.9504 1.033 1.070 0.4013

0.05 0.1 0.15 0.8290

0.4774 0.5774 0.6501 1.218

0.2708 0.3115 0.3344 0.0

0.3632 0.4227 0.4599 0.6344

0.1848 0.2472 0 3016 0,9094

0.9490 1.029 1.063 0.4061

0.6 0.7640 1.1

0.5381 0.7166 0.9653 1.110 1.220 1.423

0 * 3014 0 3623 0.3634 0.2598

0.4277 0 * 5342 0.6465 0.6760 0.7006 0.7483

0.2068 0.3297 0.5532 0.7445

1.221 1.348 1.415 1 * 565 1.244 1.473 1.488 1.603 1.606 1.517

1.086 1.173 1.205 0.9788

0.05 0.15

0.2926 0.3302 0.3486 0 * 3090 0.3133 0.3745 0.3694 0.3577 0.3329 0.2860

0.4338 0.5111 0.6871

0.1784 0.2328 0.2701 0 2224 0 * 8031 -0.1164

1.208 1.299 1.265

0.3929 0.4609 0.6140

0.1735 0.2692 0.8765

0.2985 0.3208 0.1743

1.260 1 a366 1.368

1.084 1.109 0.6178

1.028 1.102 0.4019

0.3670 0.4295

0.1643 0.2614

0.3177 0.3532

1.280 1.396

1.186 1.253

1.076 1.167

0.4

0.7

9

P

-

0.0

2.402

0.3333

1

0

0.05 0.15 0.9006

0.7685 1.034 2.295

0.625

1

0

0.05 0.15

0.5761 0.7516 1.577

0.3367 0.3912 0.0 0.3206 0.3736 0.0

1

0

-

0.05

0.4821 0.6146

0.3183 0.3729

0.9435 0.15

-

0.8838

1.136

-

-

1 a087 1.221 1.163 1.104 1.027 0.8877

0.9266 1.049 1.016 0.8990 0.6067 2.484 3.8477 0.9573 '0.7915 1.012 -0.0368 0.2890

c)

*

ra

TABLE I1 (Continued) I

4

1

0.7

1

0

0.5

0.9

1

1

-

-

-

-

Wo)

ei(o)

g'(0)

0.4 0.6 1 .o 1.1

0.8317 0.9756 1.293

0.4302 0.4574 0.4959 0.5037

0.05 0.1s 0.4 0.6 0.8975 1.1

0.5120 0.6478 0.8671 1.012 1.207 1.331

0.15 0.4 0.6 0.8093

I1

I2

11(1)

I p )

11(3)

0.4950 0.5262 0.5705 0.5795

0.4741

0.6343 0.9402 1.014

0.3576 0.3413 0.2923 0.2783

1.669 1.450 1.430 1.424

1.197 1.116 0.9402 0.8964

1.205 1.203 1.186 1.181

0.3095 0.3560 0.3848 0.3658 0.0 1.091

0.3902 0.4550 0.5214 0.5526 0.5869 0.6057

0.1884 0.2908 0.5110 0.6759 0.9113 1.067

0.3235 0.3570 0.3560 0.3359 0.2955 0.2644

1.314 1.432 1.4W 1.470 1.457

1.187 1.248 1.180 1.092 0.9523 0.8578

1.051 1.126 1.115 1.039 0.3995 2.089

0.6940 0.9117 1.055 1.192

0.3506 0.3494 0.2822 0.0

0.4955 0.5617 0.5919 0.6160

0.3239 0.5505 0.7195 0.8902

0.3702 0.3612 0.3019

1.491 1.544 1.543 1.534

1.254 1.173 1.075 0.9683

1.095 1.037 0.8892 0.4096

0.05 0.5809 0.15 0.7188 0.4 0.9317 11.071 0.6 0.7779 1.185 1.1 1.380

0.3222 0.3576 0.3426 0.2551 0.0 1.970

0 ;548 0.~230 0.5891 0.6162 0.6372 0.6650

0.2275 0.3372 0.5616 0.7336 0.8753 1.130

0.3488 0.3810 0.3668 0.3379 0.3108 0.2547

1.387 1.528 1.562 1.579 1.549 1.527

1.202 1.262 1.175 1.075 0.988) 0.8266

1.023 0.8527 0.5073 3.035

1.272 1.077

0.1107 -1.207

0.9563 0.1603

1.233

0.3351

0.1539

1

0

0.15 0.8838

1.764 3.839

0.4779 0.0

0.6465 0.7797

0.2809 0.0612 0-7071 -0.7567

0.3333

1

0

0.1s 0.904s

1.132 2.290

0.4391

0.5730 0.6715

0.2962 0.7960

0.0

1.484

0.2436 1.420 -0.0978 1.236

1.026

1.08R

0.8397 1.094 -0.0008 0.2903

CA C

2

5

$

F U

>

F 2z

M

0

>

2 v,

w

s

TABLE I1 (Continued)

-~

1

1.4

1.5

1

0.7

0.5

0.625

1

1

0

-

1

0.6

-

1

0

0

-

0.25

-

0.5

-

0.15 0.9444

0.8444 1.572

0.4329 0.0

0.5321 0.6067

0.15 0 -4 1 .o

0.7104 0.8735 1.233

0.4473 0.4637 0.4959

0.5107 0.5308 0.5705

0.15 0.4 0.6 0.8721

0.3596 0.3830 0.3516

1.1

0.7200 0.9874 1.167 1.389 1.564

0.05 0.15 0.4 0.6 1 .o

0.4427 0.6304 0.9337 1.131 1.477

0.2628 0.3383 0.4200 0.4585 0.5119

0.05 0.15 0.9469

0.4608 0.6526 1.464

0.05

0.4842 0.6810 0.9917 1.192 1.453

0.2636 0.3361 0.0 0.2667 0.3362 0.3888 0.3763 0 .o

0.4719 0.5427 0.5765 0.6109 0.6343 0 3060 0.3928 0.4860 0.5298 0.5906 0.3197 0.4095 0.6045 0.3381 0.4319 0.5304 0.5760 0.6241

0.15 0.4 0.6 0.8922

0.0

1.295

-

0.2992 0.8733 0.2934 0.4921 0 9402

0.3657 0.1753

1.516 1.354

1.210 0.6204

1.207 0 4024

0.4171 0.3814 0.2923

1.572 1.521 1.430

1.395 1.246 0.9402

1.296 1.257 1.186

0.2640 0.4622 0.6100 0.8028 0.9587

0.3505 0.3403 0.3125 0 * 2653 0.2214

1.426 1.472 1.469 1.453 1.437

1.200 1.105 0.9983 0.8486 0.7246

1.101 1.071 0.9707 0.3577 2.341

0.1225 0.2123 0.4030 0 5448 0.8141

0.2590 0.3119 0.3291 0.3134 0.2562

1.110 1.277 1.374 1.391 1.390

1.022 1.128 1.092 1.008 0.8141

0.9465 1.078 1.148 1.159 1.155

0.1312 0.2242 0.8034

0.2640 0.3163 0.2585 0.2715 0.3234 0.3332 0.3111 0.2620

1.130 1.302 1.421

1.026 1.130 0.8216

0.9418 1 -066 0.3415

1.158 1.336 1.440 1.459 1.463

1.033 1.136

0.9384 1 SO56 1.075 1.004 0.3466

-

0.1419 0.2383 0.4389 0.5871 0.7937

1 .ON

0.9885 0.8311

-

TABLE I1 (Continued)

f"(0)

0.3138 0.3196 0.2561

1.262 1.415 1.390

1.150 1.021 0.8140

1.061 1.176 1.155

0 3083

0.3273 0.2993 0.1938

1.286 1.347 1.451 1.469 1.467

1.095 1.116 1.047 0.9388 0.6373

1.010 1 -044 1.048 0.9526 2.484

0.1419 0.5504 0.8140

0.1 0.15 0.4 0.6 1.1

0.6421 0.7354 1.080 1.303 1 788

0.3155 0.3414 0.3877 0.3616 1.425

0.4115 0.4496 0.5524 0.6000 0.6786

0.1856 0.2285 0.4182 0.5578 0.8857

0.05 0.15 0.4 0.6 1.0

0.4766 0.6893 1.043 1.276 1.687

0.2661 0.3432 0.4276 0.4677 0.5235

0.3103 0.3991 0.4956 0.5414 0.6052

0.1097 0.1906 0.3617 0.4884 0.7282

0.2573 0.3082 0.3196 0.2988 0.2308

1.097 1.261 1.352 1.367 1.363

0.9992 1.096 1.040 0.9445 0.7282

0.9364 1.065 1.131 1.140 1.134

0.25 -

0.05 0.15 0.9452

0.4962 0.7137 1.669

0.2672 0.3414 0 -0

0.3246 0.4167 0 6204

0.1172 0.2009 0.7180

0.2619 0.3119 0.2326

1.119 1.287 1.396

1.003 1.096 0.7348

0.9319 1.053 0.3063

0

0.05 0.15 0.4 0.6 1 .o

0.5640 0.7473 1.071 1.291 1.687

0.3270 0.3851 0.4484 0.4793 0.5235

0.3781 0.4449 0.5179 0.5537 0.6052

0.1264 0.2028 0.3690 0.4930 0.7282

0.3112 0.3417 0.3326 0.3046 0.2308

1.249 1.357 1.396 1.391 1.363

1.125 1.168 1.067 0.9561 0.7282

1.051 1 136 1.164 1.158 1.134

-

1.8

0.5

1

0.6

-

1

I1 (3)

0.3733 0.5420 0.5906

0

0.7

I1(2)

0.3233 0.4700 0.5119

1

1

I1(1)

0.5266 1.145 1.477

0.7

0.5

I2

0.05 0.6 1 .o

1.5

2

g'(0)

0

-

-

-

-

-

0.3241

-

w

TABLE I1 (Continued) I

n

3

0.5

0.7

Q

1

1

*Convergence

0

to

-

-

\o 00

0.15 0.4 0.6 1.0 0.05

0.15 0.4 0.6 1 .o

0* 7883 1.228 1.522 2.044 0.6255 0.8493 1.257 1.537 2.044

0.3503 0.4383 0.4805 0.5397

0.4082 0-5093 0.5578 0.6258

0.1616 0.3072 0.4142 0.6158

0.3036 0.3074 0.2796 0.1967

1.238 1.323 1.335 1.328

1.051 0.9723 0.8603 0.6158

1.047 1.109* 1.115*

0.3324 0.3926 0.4593 0.4923 0.5397

0.3849 0.4545 0.5319 0.5702 0.6258

0 1057 0.1711 0.3125 0.4176 0.6158

0.3083 0.3360 0.3194 0.2847 0.1967

0.1231 1.334 1.367 1.359 1.328

1.091 1.121 0.9973 0.8708 0.6158

1.037 1.118 1.141 1.133 1.107

-

1.107

P

hj

I2

L?

5e

TABLE 111

SIMILAR SOLUTIONS FOR Pr = O.7,fM,# 0, t, ~~

B 0

~

UI

0.5

0.7

-

-

t

I1

I2

11(1)

11(2)

11(3)

0.4328 0.5343

0.5519 1.546

0.4303 0.5239

1.523 1.742

1.523 1.742

1.142 0.9846

0.2093 0.0

0.2920 0.3897

0.6331 1.697

0.4888 0.5774

1.836 2.004

1.836 2.004

1.415 1.368

0.1262

0.1663 0.2563

0.7514 0.5620 1.898 0.6411

2.318

0.0

2.366

2.318 2.366

1.843 1.840

f"(0)

9'(0)

g'(0)

0.15 0.8009

0.4328 0.5343

0.3026 0.0

1

0.15 0.7755

0.2920 0.3897

1

0.15 0.7455

0.1663 0.2563

=1

0

1

-0.2 -0.4

1

-~

~

f"

=

W

-0.6

1

0.15 0.4 0.6

0.0629 0.1135 0.1315

0.0556 0.0720 0.0435

0.0629 0.1135 0.1315

0.9530 1.517 1.956

0.6548 0.6980 0.7123

3.224 2.981 2.934

3.224 2.981 2.934

2.672 2.440 2.444

0

0

0.15 1 .o

0.3920 0.4696

0.3428 0.4139

0.3920 0.4696

0.4414 1.686

0.3921 0.4696

1.508 1.686

1.508 1.686

1.255 1.385

0.5

0.15 0.9162

0.4089 0.4802

0.3250

0.4089 0.4802

0.4970 1.636

0.4089 0.4802

1.538 1.700

1.537 1.700

1.224 0.7691

0.15 0.8187

0-4451 0.5034

0.3136 0.0

0.4451 0.5034

0.5766 1.580

0.4448 0.5029

1.586 1.725

1.586 1.725

1.188 0.8037

0

0.15 1.0

0.2559 0.3305

0.2422 0.3108

0.2559 0.3305

0.5611 1.999

0.4559 0.3305

1.869 1.999

1.869 1.999

1.539 1.623

0.5

0.15

0.2718 0.3400

0.2281 0.0

0.2718 0.3400

0.6022 1.887

0.4718 0.5400

1.884 2.004

1.884 2.004

1.508 1.210

1

-0.2

0.9028

0.0

TABLE 111 (Continued)

P 0

=1

UJ

f"

0.7

-0.2

1

0.15 0.7925

0.3050 0.3612

0.2205 0.0

0.3050 0.3612

0.6625 1.757

0.5048 0.5605

1.901 2.011

1.901 2.011

1.458 1.228

-0.4

0

0.15 1 .o

0.1371 0.2049

0.1481 0.2126

0.1371 0.2049

0.7445 2.447

0.5371 0.6049

2.443 2.447

2.443 2.447

1.998 1.968

0.5

0.15 0.8868

0.1509 0.2128

0.1398 0.0

0.1509 0.2128

0.7580 2.239

0.5509 2 -422 0.6128' 2.438

2.422 2.438

1.957 1.763

1

0.15 0.7616

0.1795 0.2311

0.1369 0.0

0.1795 0.2311

0.7824 1.996

0.5792 0.6301

2.373 2.415

2.373 2.415

1.872 1.754

0

0.15 0.4 0.6

0.0433 0.0728 0.0843 0.0975

0.0620 0.0435 0.0941 0.0728 0.1063 0.0843 0.1201 0.0975

1.081 1.705 2.200 3.181

0.6433 0.6728 0.6843 0.6975

3.613 3.321 3.249 3.181

3.613 3.321 3.249 3.181

2.979 2.695 2.622 2.552

0.5

0.15

0.8682

0.0533 0.1033

0.0615 0.0

0.0533 0.1033

1.032 2.805

0.6533 0.7034

3.460 3.143

3.460 3.143

2.856 2.563

0.15 0.4 0.6 0.7248

0.0745 0.1041 0.1138 0.1177

0.0651 0.0693 0.0432 0.0

0.0745 0.1041 0.1138 0.1177

0.9795 1.591 2.068 2.364

0.6737 0.7026 0.7119 0.71 57

3.214 3.091 3.063 3.052

3.214 3.091 3.063 3.052

0.15 0.7897

0.5091 0.7268

0 .o

0.3188

0.4604 0.5m

0.4752 1.303

0.4114 0.4589

1.507 1.686

1.419 I .474

1.110 0.8130

0.15

0.3733 0.5874

0.2286 0.0

0.3250 0.4463

0.5289 1.390

0.4638 0.5021

1.779 1.901

1.670 1.653

1.343 1.107

-0.6

W

1.o

1

0.2

0.5

0 -0.2

1 1

0.7625

2.629 2.501

2.486 2.502

TABLE I11 (Continued)

B

UJ

f"

0.2

0.5

-0.4

0.5

W

3.2525 3.4579

0.1494 0.0

0.2056 0.3216

0.5983 1.476

0.5516

0.15 0.4 0.6 0.6882

0.1517 3.2611 0.3192 0.3406

0.0839 0.0975 0.0498 0.0

0.1080 0.1760 0.2039 0.2114

0.6917

0.6061

1b.111

0.15 0.8064

D.5202 0.6979

0.3265 0.0

0.4678 0.5465

-0.2

0.15 0.7765

0.3846 0.5614

0.2361 0.0

-0.4

0.15

0.2633 0.4352

0.1562

0.7400

0.15 0.4 0.6 0.6950

0

-0.6

0.4

t

0.15 0.7282

-0.6

0.7

01

0 -0.2 -0.4

0.5277

2.154 2.158

2.017 1.875

1.670 1.466

0.6180 0.6110 0.6115

2.701 2.541 2.485 2.498

2.526 2.302 2.193 2.165

2.158 1.984 1.916 1.855

0.4925 1.314

0.4216 0.4350

1.560 1.653

1.470 1.439

0.6471

0.3329 0.4122

0.5479 1.408

0.4749 0.4797

1.836 1.880

1.724 1.628

1.383 0.9855

0.2135 0.2910

0.6179

0.0

0.5395 0.5328

2.212 2.164

2.070 1.866

1.708 1.367

0.2487 0.3000 0.3219

0.1610

0.0896 0.0914 0.0484 0.0

0.1598 0.1793 0.1865

0-1150 0.7106 1.145 1.470 1.625

0.6184 0.6178 0.6041 0.5963

2.752 2.617 2.551 2.529

2.568 2.356 2.230 2.177

2-186 2.018 1.902 1.807

0.15 0.7815

0.5714 0.8782

0.3318 0.0

0.4824 0.6203

0.4229 1.149

0.3987 0.4135

1.499 1.659

1.349 1.306

1.090 0.7155

0.15

0.7522

0.4377 0.7383

0.2432

0.3499 0.4865

0.4634 1.202

0.4479 0.4509

1.749 1.847

1.568 1.448

1.300 0.9953

0.15 0.7172

0.3179 0.6060

0.1653

0.2330 0.3638

0.5129 1.257

0.5069 0.4946

2.078 2.072

1.857 1-623

1.582 1.296

0.0 0-0

1.427 1.586

1.511

1.150

B 0.4

t

f"

U

0.5

-0.G

1

0.15 0.6749

0.2160 0.4828

0 .o

0.1006

0 * 1363 0.2548

0.5746 1.313

0.5780 0.5464

2.521 2.347

2.250 1.842

1.971 1.628

0.7

0

0

0.15 1.0

0.5027 0.8544

0.3597 0.4632

0.4129 0.5300

0.3347 1.219

0.3687 0.3667

1.443 1.521

1.358 1.219

1.204 1.257

0.5

0.15 0.9058

0.5281 0 8470

0.3422 0.0

0.4344 0.5430

0.3749 1.185

0.3781 0.3728

1.475 1.551

1.366 1.235

1.166 0.5240

1

0.15 0.7974

0.5807 0.8490

0.3363 0.0

0.4853 0.5763

0.4358 1.151

0.4063 0.3887

1.546 1.617

1.393 1.265

1.126 0.5624

0

0.15 1 .o

0.3786 0.7377

0.2662 0.3727

0.2851 0.4052

0.3934 1.346

0.4211 0.3993

1.730 1.720

1.608 1.346

1.429 1.408

0-5

0.15 0.8880

0.4002 0.7202

0.2511 0-0

0.3054 0.4156

0.4259 1.280

0.4298 0.4083

1.753 1.754

1.608 1.373

1.391 0.8373

1

0-4468 0.15 0-7647 0.7109

0.2472 0.0

0.3531 0.4452

0.4760 1.207

0.4559 0.4281

1.800 1.816

1.615 1.416

1.340 0.8903

0

0.15 1 .o

0.2707 0.6321

0.1829 0.2903

0.1768 0.2956

0.4657 1.491

0.4845 0.4357

2.113 1.954

1.940 1.491

1.735 1.588

0.5

0.15 0.8656

0.1706 0.0

0-1944 0.4882 0.3022 1.383

0.4923 0.4495

2.122 1.996

1.930 1.540

1.696 1.171

1

0.15 0.7256

0.2875 0.6025 0.3263 0.5810

0.1687 0.0

0.2362 0.3264

0.5251 1.266

0.5155 0.4141

2.133 2.060

1.906 1.602

1.624 1.223

0

0.15 1 .o

0.1826 0.5384

0.1124 0.2175

0.0933 0.2034

0.5537 1.656

0.5611 0.4760

2.632 2.225

2.391 1.656

2.162

-0.2

-0.4

-0.6

1

W

-

1.800

TABLE 111 (Continued)

B 0.4

0.5

fw

0.7

0.1

f**(O)

e*(o)

g*(o)

0.1580 0.1830 0.2054

I~

I2

I10)

11(2)

11(3)

0.9262 1.193 1.493

0.5546 0.5308 0.4978

2.449 2.364 2.286

2.081 1.913 1.743

1.925 1.801 1-531

0.5872 0.5692 0.5415 0.5298

2.579 2.454 2.384 2.360

2.299 2.052 1.894 1.838

2.015 1.834 1.680 1.596

0.5

0.4 0.6 0.8367

0.3251 0.4078 0.4950

0.1308 0.1207 0.0

1

0.15 0.4 0.6 0.6772

0.2228 0.3517 0.4316 0.4601

0.1031 0.1391 0.5857 0.1018 0-1914 0-9521 0.0472 0.2151 1.222 0.0 0.2226 1.323

0

0.15 1 .o

0.5243 0.9277

0-3625 0.4164 0.4705 0.5390

0-3182 0.3652 0.3503 1.155

1.433 1.500

1.334 1.155

1.196 1.240

-0.2

0

0.15 1.0

0.4014 0.8126

0.2699 0.3810

0.2895 0.4153

0.3704 1.266

0.4163 0.3796

0.1689

1.711

1.572 1.013

1.414 1.014

-0.4

0

0.15 1.0

0.2942 0.7077

0.1874 0.2995

0.1822 0.3065

0.4331 1.391

0.4777 2.077 0.4121 -1.909

1.883 1.391

1.706 1.553

-0.6

0

0.15 0.4 0.6 1 .o

0.2057 0.3534 0.4482 0.6138

0.1178 0.1682 0.1927 0.2274

0.0990 0.1511 0.1771 0.2146

0.5079 0.8354 1.077 1.532

0.5518 0.5357 0.5096 0.4477

2.562 2.379 2.289 2.162

2.297 1.997 1.820 1.532

2.105 1.936 1.857 1.750

-0.8

0

0.5 1.0

0.3255 0.5314

0.1216 0.1656

0.0968 0.1410

1.081 1.689

0.5853 0.4866

2.718 2.447

2.186 1.689

1.979

-1.0

0

0.15 1 .o

0.0939 0.4604

0.0284 0.1148

0.0140 0.6933 0-0859 1.863

0.7472 0.5489

4-045 2.763

3.601 1-863

3.421 2.240

-1.2

0

0.15 0.4

0.0683 0.1774

0.0099 0.0341

0.0031 0.0169

0.7968 1.218

0.8700 0.7683

5.060 3.872

4.531 3.160

4.393 3.236

-0.6

0

2.211

+ 0

TABLE I11 (Continued)

P

B

fw

=1

-1.2

0

t

W ~~

0.5

0.7

0.75

0.7

0.5

1 .o

0.2563 0.4006

0.0499 0.0753

0.0281 0.0481

1. 513 2.052

0.6998 0.5745

3.506 3.105

2.672 2.052

2.896 2.536

0

1 .o

'3.3510

0.0464

0.0244

2.256

0.6229

3.467

2.256

2.862

0.5

0.15 0.9004

0.6029 1.072

0.3512 0.0

0.4478 0.5719

0.3187 1.0000

0.3690 0.3215

1.446 1.496

1.287 1.044

1.139 '3.4385

-0.2

0.5

0.15 0.8808

0.4771 0.9440

0.2618 0.3213 0.0 0.4471

0.3535 1 059

0.4119 0.3495

1.701 1.674

1.496 1.146

1.344 0.7242

-0.4

0.5

0.15 0.8560

0.3651 0.8222

0.1830 0.0

0.2127 0.3941 0.3354 1.120

0.4696 0.3825

2.027 1.882

1.764 1.267

1.612

-0.6

0.5

0.15 0.4 0.6 0.8240

0.2697 0.4588 0.5817 0.7064

0.1171 0.1455 0.1311 0.0

0.1256 0.1844 0.2134 0.2387

0.4411 0.7349 0.9500 1.179

0.5389 0 5076 0.4693 0.4221

2.446 2.284 2.200 2.126

2.111 1.798 1.604 1.415

1.965 I . 772 1.635 1.338

0

0

0.15 1.0

0.5612 1.233

0.3317 0.4959

0.3844 0.5705

0.2437 0.9402

0.3176 0.2923

1.300 1.430

1.175 0.9402

1.096 1.186

-0.2

0

0.15 1 .o

0.4461 1.121

0.2441 0.4090

0.2633 0.4497

0.2787 1.009

0.3674 0.3119

1.570 1.593

1.394 1.009

1.312 1.310

-0.4

0

0.15 1 .o

0.3469 1.018

0.1684 0.3298

0.1641 0.3428

0.3189 1.085

0.4279 0.3330

1.917 1.778

1.675 1.085

1.595 1.452

-0.6

0

0.4 0.6

0.5028 0.6559 0.9226

0.1763 0.2106 0.2591

0.1621 0.1987 0.2512

0.6238 0.8128 1.167

0.4791 0.4431 0.3556

2.174 2.094 1.984

1.693 1.496 1.167

1.783 1.709 1.612

-1.4

1

0.6

0

1 .o

1.022

. c -

TABLE 111 (Continued)

1

0.7

1

Q1

W

0

0

0.15 1 .o

0.6146 1.233

0.3729 0.4959

0.4295 0.5705

0.2614 0.9402

0.3532 0.2923

1.396 1.430

1.253 0.9402

1.167 1.186

-0.2

0

0.15 1.0

0.4947 1.121

0.2828 0.4090

0.3053 0.4497

0.2944 1.009

0.4004 0.3119

1.647 1.593

1.454 1.009

1.364 1.310

-0.4

0

0.15 1.0

0.3886 1.018

0.2028 0.3298

0.2002 0.3320 0.3428 1.085

0.4566 0.3330

1.965 1.778

1.707 1.085

1.617 1.452

-0.6

0

0.15 0.4 0.6 1.0

0.2983 0.5189 0.6647 0.9226

0.1350 0.1912 0.2191 0.2591

0.1175 0.1770 0:2072 0.2512

0.3744 0.6296 0.8166 1.167

0.5239 0.4894 0.4481 0.3556

2.366 2.197 2.109 1.984

2.028 1.704 1.503 1.167

1.945 1.791 1.716 1.612

-0.8

0

1 .o

0.8354 0.1976

0.1758

1.255

0.3799

2.212

1.255

1.793

-1.2

0

1.0

0.6858

0.1034

0.0731

1.452

0.4330

2.723

1.452

2.218*

-1.4

0

1.0

0.6229

0.0704

0.0429

1.561

0.4620

3.000

1.561

2.460*

-1.6

0

1 .o

0.5674 0.0457

0.0234

1.675

0.4924

3.287

1.675

2.721*

0

0

0.15 1 .o

0.7104 1.233

0.4473 0 -4959

0.5107 0.5705

0.2934 0.9402

0.4171 0.2923

1.572 1.430

1.395 0.9402

1.296 1.186

-0.2

0

0.15 1 .o

0.5842 1.121

0.3536 0.4090

0.3823 0.4497

0.3236 1.009

0.4607 0.3119

1.798 1.593

1.572 1.009

1.469 1.310

-0.4

0

0.15 1 .o

0.4688 1.018

0.2677 0.3298

0.2691 0.3428

0.3575 1.085

0.5112 0.3330

2.076 1.778

1.788 1.085

1.682 1.452

*convergencet o

10-4.

TABLE I I I (Continued)

1

2

1

0.5

0.7

-0.6

0

0.15 1 .o

0.3658 0.9226

0.1912 0.2591

0.1740 0.2512

0.3957 1.167

-0.8

0

0.15 1.0

0.2774 0.8354

0 e 1260 0.1976

0.0999 0.1758

0.4383 1.255

0.5702 0.3556 0.6392 0.3799

- 1 .o

0

0.15

0.2057

0.0742

0.0485

0.4856

-1.2

0

0.15 1.0

0.0374 0.1034

0.0187 0.0731

-1.4

0

0.15

0.1524 0.6858 0.1171 0.6229

0.0155 0.0704

-1.6

0

0.15

0.0960 0.5674

0

0

0.4 0.6 1 .o

1.043 1.276 1 a687 1.065 1.475

1 .o 1 .o

2.419 1.984

2.054 1.167

1.951 1.612

2.842 2.212

2.388 1.255

2.294 1 793

0.7201

3.366

2.813

2.738

0.5372 1.452

0.8151 0.4330

4.007 2.723

3.351 1.452

3.311 2.218*

0.0054 0.0429

0.5923 1.561

0.9248 0.4620

4.764 3.000

4.012 1.561

4.023 2.460*

0.0052 0.0457

0.0011 0.0234

0.6497 1.675

1.046 0.4924

5.602 3.287

4.764 1.675

4.840 2.721*

0.4276 0.4677 0.5235

0.4956 0.5414 0.6052

0.3617 0.4884 0.7282

0.3196 0.2988 0.2308

1.352 1.367 1.363

1.131 1 140

0.3048 0.3605

0.3174 0.3802

0.5552 0.8100

0.3544 0.2549

1.727 1.664

1.040 0.9445 0.7282 1.123 0.8100

-

-

1.134

-0.4

0

0.6 1 .o

-0.2

0

0.6

1 .o

1.180 1.578

0.3930 0.4384

0.4336 0.4863

0.5248 0.7678

0.3304 0.2425

1.557 1.506

1.039 0.7678

1.286* 1 243

-0.4

0

0.4 0.6

0.9597 1.181

0.2038

0.4094 0.4464

0.4792 0.6222

0.3915 0.3327

1.889 1.848

1.226 1.036

1.245 1.035

-0.6

0

0.15 0.4

0.4291 0.7594

0.1520 0.2140

0.1364 0.2032

0.2640 0.4556

0.5010 0.4481

2.206 2.047

1.807 1.459

1.815 1.673

'convergence

to

1.420* 1.365

-

PI

d

(v

0

. ?

PI

rl

4

v1

:

0.

SOLUTIONS OF BOUNDARY-LAYER EQUATIONS

0

m

00

4 4

d F - 4

n

m I

0

N h PI

h

N03

..

4 0

w m e mln

? ? '9

I4

U

h

4

U N

Y h

0

h

d

. . ? ?

A

N d d

y m

v1

m m dpl m w

d

Y 4 d

H

m N 0

w m

*

d

0

?

m

0

0

w

4

. . ;=. 0

w

06 d d 0

m

00

YN

m a m

h N N h Q v1

00

2.!:

*cow*

00

..

N +I

d

I4

n

0

-

U

v1

m o -3

0

N

N

m

0 03

h

N d

:?

WPI

47 04

F - 4 h

v)

00

-3m N N

. . f;. .

N-3

n

a l

-0, h

-

3

Y 0

ry

u

b"

2 3

m.

407

TABLE IV SIMILAR SOLUTIOXS FOR t,

0

0.5

0

0.5

l,f,, = 0

f"(0)

e"o)

g'(0)

0.15 0.4 0.6 1 .o

0.3490 0.4143 0.4399 0.4696

0.3034 0.3625 0.3861 0.4139

0 3490

0.5

0.15 0.4 0.6 0 8482

0.3816 0.4437 0.4662 0.4834

0.7

0.15 0.4 0.6 0.9151 0.15 0.4 0.6

0.7

I1

I2

I p )

I1(2)

0.4143 0.4399 0.4696

0.4037 0-7886 1.089 1.686

0.3490 0.4144 0.4399 0.4696

1.411 1.571 1.626 1.686

1.411 1.571 1.626 1.686

1.185 1.303 1.343 1.385

0.2443 0.2608 0.2287 0.0

0.3816 0.4437 0.4662 0.4834

0.3951 0.8234 1.163 1.584

0.3816 0.4437 0.4662 0.4834

1.478 1.623 1.670 1.705

1.478 1.623 1.670 1.705

1.274 1.332 1.268 0.7940

0.3734 0.4394 0.4642 3.4876

0.2963 0.3341 0.3246 0.0

0.3734 0.4394 0.4642 0.4876

0.4608 0.8600 1.167 1.645

0.3735 0.4394 0.4642 0.4876

1.452 1.610 1.663 1.710

1.452 1.610 1.663 1.710

1.166 1.250 1.240 0.7665

0.4532 0.5043 0.5232

0.2166 0.1716 0.0450

0.4532 0.5043 0.5232

0.4784 0.9359 1.251

0.4315 '1.532 0.4949 1.687 0.5106 1.721

1.532 1.687 1.721

1.240 1.252 1.174

0.05 0.15 0.4 0.6 0.8009

0.3507 0.4328 0.4987 0.5208 0.5343

0.2527 0.3026 0 3045 0.2392 0.0

0.3507 0.4328 0.4987 0 5208 0.5343

0.3514 0.5519 0.9513 1.251 1.546

0.3494 0.4303 0.4924 0.5124 0.5239

1.304 1.523 1.672 1.717 1.742

1.304 1.523 1.672 1.717 1.742

1.003 1.142 1.201 1.163 0.9846

1.1

0.5491

1.521

0.5491

1.953

0.5301

1.753

1.753

2 .ooo

-

1

=

0.5

0.7

-

-

-

TABLE IV (Continued) ~

0

0.5

all

1

f"(0)

e'(0)

g'(0)

1 .o 1.1

0.4223 0.4988 0.5264 0.5541 0.5550

0.4223 0.4988 0.5264 0.5541 0.5550

0.4223 0.5723 0.4988 0.9284 0.5264 1.192 0.5541 1.698 0.5550 1.871

0.15

0.4 0.6

0.7

I1

~~

~

I#)

I1(2)

I1(3)

0.4146 0.4734 0.4939 0.5179 0.5345

1.466 1.611 1.657 1.709 1.749

1.466 1.611 1.657 1.709 1.749

1.051 1.137 1.163 1.192 1.215

l2

0

0.7

0.15 0.4 0.6 1 .o

0.3920 0.4353 0.4514 0.4696

0.3428 0.3820 0.3969 0.4139

0.3920 0.4353 0.4514 0.4696

0.4414 0.8137 1.106 1.686

0.3921 0.4353 0.4514 0.4696

1.508 1.614 1.649 1.686

1.508 1.614 1.649 1.686

1.255 1.334 1.359 1.385

all

1

0.6 1. -0

0.4498 0.4696

0.4498 0.4696

0.4498 0.4696

1.167 1.686

0-4498 1.645 0.4696 1.686

1.645 1.686

1.196 1.217

0.5

0.7

0.15 0.4 0.6 0.9162

0.4089 0.4512 0.4663 0.4802

0.3250 0.3431 0.3263 0.0

0.4089 0.4512 0.4663 0.4802

0.4970 0.8778 1.173 1.636

0.4089 0.4512 0.4663 0.4802

1.538 1-640 1.672 1.700

1.538 1.640 1.672 1.700

1.224 1.270 1.248 0.7691

1

0.5

0.15 0.4 0.6 0.6729

0.4517 0.4854 0.4954 0.4978

0.2254 0.1793 0.0733 0.0

0.4517 0.4854 0.4954 0.4978

0.5519 0.9694 1.294 1.412

0.4512 0.4844 0.4941 0.4965

1.610 1.690 1.712 1.717

1.610

1.690 1.712 1.717

1.245 1.201 1.044 0.9324

0.05 0.15 0.4 0.6 0.8187

0.3924 0.4451 0.4835 0.4960 0.5034

0.2843 0.3136 0.3025 0.2420 0.c

0.3924 0.4451 0.4835 0.4960 0.5034

0.3963 0.3923 0.5766 0.4448 0.9637 . 0.4832 0.4947 1.253 0.5029 1.580

1.442 1.586 1.681 1.707 1.725

1.442 1 * 586 1.681 1.707 1.725

1.100 1.188 1.195 1.134 0.8037

1.1

0.5097

1.291

0.5097

1.982

1.737

1.737

2.444

0.7

0.5085

TABLE IV (Continued) ~

B 0

w

~

f"(0)

I1

I2

Il(1)

Il (2)

I1(3)

0.4386

0.6046

0.4369

1.555

1.555

1.118

0.4696 0.4696

0.4696 1.686

0.4696 0.4696

1 *686 1.686

1.686 1.686

1.217 1.217

0.4139 0.4139 0.4139 0.4139

0.4696 0.4696 0.46% 0.4696

0.5091 0.8554 1.132 1.686

0-4696 0.4696 0.46% 0.4696

1.686 1.686 1.686 1.686

1.686 1.686 1.686 1.686

1.385 1.385 1.385 1.385

0.4696 0.4696 0.4696 0.4696

0.3029 0.2763 0.2311 0.0

0.4696 0.4696 0.4696 0.4696

0.4671 0.8593 1.173 1.569

0.4696 0.4696 0.4696 0.4696

1.686 1.686 1.686 1.686

1.686 1.686 1.686 1.686

1.434 1.378 I . 283 0.7952

0.15 0.4 0.6 0.9179

0.4696 0.4696 0.4699 0.4696

0.3739 0.3572 0.3281 0.0

0.4696 0.4696 0.4699 0.4696

0.5595 0.9057 1.177 1.623

0.4696 0.4696 0.4684 0.4696

1.686 1.686 1.683 1.686

1.686 1.686 1.683 1.686

1.326 1.301 1.266 0.7732

0-5

0.15 0.4 0.6

0.4696 0.4696 0.4696

0.2301 0.1860 0.0955

0.4696 0.4696 0.4696

0.5812 0.9738 1.287

0.4696 0.4696 0.4696

1.686 1.686 1.686

1.686 1.686 1.686

1.300 1.188 0.9983

0.7

0.15 0.4 0.6 0.8357

0 * 4696 0.4696 0.4696 0.4696

0.3339 0.3006 0.2439 0 .o

0.4696 0.4696 0.4696 0.4696

0.6097 0.9560 1.233 1.559

0.4696 0.4696 0.4696 0.4696

1.686 1.686 1.686 1.686

I .686

1.686 1.686 1.686

1.267 1.217 1.134 0.7729

0.15

0.4696

0.4696

0.4696

0.6521

0.4696

1.686

1.686

1.217

g'(0)

Q1

Pr

0.7

all

1

0.15

0.4386

0.4386

1

all

1

0

1 .o

0.4696 0.4696

0.4696 0.4696

0

0.7

0.15 0.4 0.6 1 .o

0.4696 0.4696 0.4696 0.4696

0.5

0.5

0.15 0.4 0.6 0.8522

0.7

1

all

1

TABLE IV (Continued)

0.05

1

all

1

0 0.5 1 .o

0.4848 0.5082 0.5311

0.4733 0.4773 0.4812

0.4733 0.4773 0.4812

0.4453 1.023 1.593

0.4626 0.4570 0.4514

1.675 1.663 1.652

1.654 1.623 1.593

1.208 1.199 1.191

0.1

0.5

1

0.7

0.05 0.15 0.4 0.6 0.7948

0.3780 0.4733 0.5655 0.6066 0.6376

0.2597 0.3112 0.3134 0.2435 0.0

0.3623 0.4476 0.5194 0.5450 0.5613

0.3187 0.5088 0.8753 1.149 1.409

0.3414 0.4197 0.4717 0.4840 0.4882

1.291 1.513 1.652 1.690 1.709

1.260 1.464 1.575 1.591 1.590

0.9912 1.124 1.166 1.106 0.8825

0.7

1

0.7

0.05

0.4202 0.4850 0.5492 0.5808 0.6086

0.2900 0.3206 0.3096 0.2463 0.5096

0.4021 0.4573 0.5008 0.5164 0.5274

0.3614 0.5303 0.8823 1.150 1.428

0.3834 0.4320 0.4608 0.4655 0.4653

1.430 1.571 1.654 1.674 1.682

1.394 1.521 1.576 1.575 1.561

1.087 1.166 1.157 1.063 0.7109

0.15

0.4 0.6 0.8120

1

all

1

1

0.7

1.1

0.6420

1.492

0.5447

1.659

0.4751

1.702

1.569

0.8969

0.15

0.5123

0.4 0.6 1 .o

0.5347 0.5523 0.5870

0.4788 0.4825 0.4054 0.4909

0.4788 0.4825 0.4854 0.4909

0.5909 0.8668 1.085 1.516

0.4531 0.4480 0.4438 0.4355

1.658 1.647 1.639 1.624

1.607 1.595 1.558 1.516

1.196 1.188 1.182 1.170

0.15 0.4 0.6 0.8290

0.5085 0.5331 0.5525 0.5744

0.3381 0.3051 0.2459 0.0

0.4781 0.4824 0.4857 0.4894

0.5563 0.8715 1.121 1.403

0.4529 0.4460 0.4404 0.4341

1.660 1.648 1.638 1.627

1.610 1.576 1.549 1.520

1.240 1.174 1.071 0.6798

N

B

u

r

u1

Pr

tw

f"(0)

e'(0)

g'(0)

I1

12

Il(1)

Il(2)

I1(3)

0.15

0.7

0

0.7

0.6

1.145

0.4700

0.5420

0.5504

0.3196

1.415

1.021

1.176

0.2

0.5

1

0.7

0.05 0.15 0.4 0.6 0.7897

0.4015 0.5091 0.6243 0.6818 0.7268

0.2657 0.3188 0.3209 0.2467 0.0

0.3718 0.4604 0.5370 0.5655 0.5838

0.2977 0.4752 0.8161 1.069 1.303

0.3366 0.4114 0.4550 0.4608 0.4589

1.285 1.507 1.640 1.673 1.686

1.230 1.419 1.499 1.494 1.474

0.9808 1.110 1.139 1.062 0.8130

1.1

0.7913

1.748

0.6069

1.659

0.4479

1.680

1.425

2.337

0.05 0.15 0.4 0.6 0.8064

0.4445 0.5202 0.6068 0.6548 0.6979

0.2950 0.3265 0.3147 0.2470 0.0

0.4103 0.4678 0.5155 0.5338 0.5465

0.3357 0.4925 0.8141 1.059 1.314

0.3766 0.4216 0.4426 0.4415 0.4350

1.422 1.560 1.633 1.646 1.653

1.358 1.470 1.495 1.473 1.439

1.076 1.150 1.136 1.036 0.6471

1.1

0.7539

1.489

0.5607

1.657

0.4223

1.647

1.386

2.709

0 0.5

1.0

0.5233 0.6070 0.6867

0.4821 0.4951 0.5069

0.4821 0.4951 0.5069

0.3881 0.8995 1.392

0.4457 0.4271 0.4082

1.648 1.612 1.580

1.576 1.480 1.392

1.188 1.161 1.138

0.15 0.4 0-8235

0.5424 0.5883 0.6633

0.3414 0.3085 0.0

0.4851 0.4926 0.5044

0.5143 0.8065 1.288

0.4395 0.4270 0.4057

1.639 1.618 1.586

1.550 1.491 1.397

1.219 1.141 0.6165

0.15 1.0

0.4283 0.7627

0.3157 0.4534

0.3643 0.5179

0.3292 1.309

0.3342 0-3887

1.360 1-551

1.302 1.309

1.144 1.280

0.7

1

1

all

1

0.2857

0.5

0

0.7

1

0.7

0.7

9 r

0

z

m

G3 U

j3 L(

F!

* 3

L(

g m

zr

Fg (r,

TABLE IV (Continued)

0.2857

0.5

0.7

1

1

0.5

0.7

1

0.7

1

all

1

0.7

1

0.7

f"(0)

9"O)

g'(0)

0.15 0.9071

0.4605 0.7683

0.3101 0.0

0.05 0.15 0.4 0.6 0.7849

0.4209 0.5371 0.6701 0-7401 0.7952

1.1

I1

I2

0.3934 0.5397

0.3742 1-282

0.3537 0.4019

0.2710 0-3246 0.3266 0.2490 0.0

0.3808 0-4704 0.5505 0.5810 0.6012

0.2779 0.4503 0.7737 1.013 1.223

0.8787

1.862

0.6294

0.05 0.15 0.4 0.6 0.8022

0.4636 0.5475 0.6512 0.7118 0.7664

0.2988 0.3310 0.3194 0.2499 0.0

1.1

0.8401

0 0.2 0.4 0.6 0.8

I

p

11(2)

11(3)

1.406 1-595

1.328 1.335

1.123 0.5663

0.3299 0.4053 0.4428 0.4439 0.4374

1.268 1.502 1.631 1.661 1.667

1.199 1-385 1.446 1.426 1.393

0.9696 1-100 1.120 1.032 0.7890

1.548

0.4195

1.640

1.325

2.232

0.4171 0.4758 0.5259 0.5459 0.5603

0.3143 0.4657 0.7744 1.007 1.236

0.3707 0.4143 0.4298 0.4241 0.4133

1.411 1.553 1.624 1.635 1.635

1.329 1.433 1.439 1.402 1.356

1.068 1.138 1.108 0.9885 0.6055

1.565

0.5771

1.557

0.3950

1.621

1.286

2.709

1.o

0.5419 0.5883 0.6334 0.6774 0.7205 0.7627

0.4862 0.4932 0.4998 0.5061 0.5121 0.5179

0.4862 0.4932 0.4998 0.5061 0.5121 0.5179

0.3629 0.5593 0.7517 0.9406 1.126 1.309

0.4382 0.4285 0.4186 0.4087 0.3987 0.3887

1.636 1.617 1.599 1.582 1.566 1.551

1.542 1.491 1.443 1.396 1.352 1.309

1.179 1.165 1.151 1.139 1.127 1.116

0.15 0.4

0.5684 0.6308

0.3438 0.3108

0.4901 0.4999

0.4846 0.7611

0.4299 0.4134

1.625 1.598

1.508 1.432

1.204 1.117

TABLE IV (Continued)

fii(o)

ei(o)

gi(o)

I2

11(1)

11(3)

0.2857

1

1

0.7

0.6 0.8194

0-6793 0.7310

0.2473 0.0

0.5072 0.5148

0.9773 1.210

0.4001 0.3854

1.578 1.558

1.374 1.314

0.9918 0.5758

0.3

1

all

1

0 0.2 0.4 0.6 0.8 1 .o 2 .o

0.5448 0.5931 0.6402 0.6860 0.7309 0.7748 0.9829

0.4868 0.4941 0.5009 0.5074 0.5136 0.5195 0.5457

0.4868 0-4941 0.5009 0.5074 0.5136 0.5195 0.5457

0.3591 0.5539 0.7446 0.9318 1.116 1.297 2.165

0.4371 0.4270 0.4168 0.4065 0.3961 0.3857 0.3334

1.634 1.614 1.596 1.578 1.562 1.547 1.484

1.537 1.484 1.434 1.386 1.341 1.297 1.099

1.178 1.163 1.149 1.136 1.124 1.113 1.066

0.4

0.5

0

0.7

0.15 1.0

0.4540 0.8544

0.3192 0.4632

0.3686 0.5300

0.6094 1.219

0.3303 0.3667

1.347 1.521

1.273 1.219

1.133 1.257

0.5

0.7

0.15 0.9048

0.4887 0.8564

0.3141 0.0

0.3992 0.5533

0.3511 1.194

0.3483 0.3786

1.394 1.570

1.296 1.244

1.111 0.5232

1

0.7

0.05 0.15 0.4 0.6 0.7815

0.4416 0.5714 0.7260 0.8112 0.8782

0.2757 0.3318 0.3334 0.2515 0.0

0.3870 0.4824 0.5667 0.5995 0.6203

0.2694 0.4229 0.7270 0.9516 1.149

0.3321. 0.3987 0.4291 0.4246 0.4135

1.289 1.499 1.624 1.651 1.659

1.188 1.349 1.386 1.351 1.306

0.9672 1.090 1.099 0.9983 0.7155

1.1

0.9837

1.924

0.6189

1.474

0.3884

1.644

1.218

2.562

0

0.7

0.15 1 .o

0.5027 0.8544

0.3597 0.4632

0.4129 0.5300

0.3347 1.219

0.3687 0.3667

1.443 1.521

1.358 1.219

1.204 1.257

0.5

0.7

0.15 0.4

0.5281 0.6583

0.3422 0.3671

0.4344 0.4909

0.3749 0.6569

0.3781 0.3955

1.475 1.544

1.366 1.362

1.166 1.176

0.7

TABLE IV (Continued)

0.4

0.7

1

0.5

0.7

0.6 0.9058

0.7386 0.8470

0.3500 0.0

0.5159 0.5430

0.8701 1.185

0.3906 0.3728

1.554 1.551

1.318 1.235

1.119 0.5240

1

0.7

0.05 0.15 0.4 0.6 0.7974

0.4864 0.5807 0.7053 0.7811 0.8490

0.3034 0.3363 0.3239 0.2500 0.0

0.4246 0.4853 0.5389 0.5609 0.5763

0.2943 0.4358 0.7220 0.9380 1.151

0.3654 0.4063 0.4154 0.4050 0.3887

1.404 1.546 1.608 1.614 1.617

1.300 1.393 1.377 1.326 1.265

1.058 1.126 1.092 0.9700 0.5624

1.460

all

0

1.1

0.9451

1.636

0.5953

0.3616

1.603

1.172

2.887

1

0 0.2 0.4 0.6 0.8 1 .o

0.5639 0.6254 0.6850 0.7429 0.7993 0.8544

0.4908 0.4997 0.5079 0.5157 0.5231 0.5300

0 -4908 0.3350

0.4997 0.5079 0.5157 0.5231 0.5300

0.5198 0.7001 0.8765 1.049 1.219

0.4299 0.4175 0.4050 0.3923 0.3795 0.3667

1.623 1.599 1.577 1.557 1.538 1.521

1.505 1.441 1.381 1.325 1.271 1.219

1.170 1.152 1.135 1.120 1.107 1.094

0.7

0.15 1.0

0.5901 0.8544

0.4327 0.4632

0.4926 0.5300

0.3806 1.219

0.4380 0.3667

1.619 1.521

1.514 1.219

1.333 1.257

0.5

0.7

0.15 0.9074

0.5948 0.8337

0.3901 0.0

0.4942 0.5283

0.4156 1.173

0 * 4286 0.3646

1.614 1.525

1.487 1.221

1.261 0.5248

1

0.7

0.15 0.4 0.6 0.8147

0.5998 0.6824 0.7461 0.8125

0.3465 0.3134 0.2474 0.0

0.4959 0.5082 0.5173 0.5264

0.4515 0.7109 0.9129 1.125

0.4192 0.3980 0.3809 0.3624

1.608 1.575 1.552 1.529

1.460 1.366 1.295 1.224

1.187 1.092 0.9564 0.5334

"1

0.5

0.5

Pr

tw

0.5

0.15 0.4 0.6

0.4646 0.6386 0.7451

0.2823 0.3456 0.3742

0.3746 0.4546 0.4905

0.2303 0.5099 0.7282

0.3495 0.3837 0.3809

1.372 1.476 1.496

1.324 1.342 1.292

1.286 1.387 1.410

0.7

0.05 0.15 0.4 0.6 1 .o 1.1

0.3518 0.4747 0.6472 0.7511 0.9277 0.9681

0.2520 0.3218 0.3940 0.4266 0.4705 0.4791

0.2923 0.3720 0.4533 0.4900 0.5390 0.5486

0.1704 0.2946 0.5628 0.7652 1.155 1.250

0.2666 0.3274 0.3658 0.3685 0.3503 0.3431

1.153 1.336 1.455 1.484 1.500 1.500

1.103 1.251 1.290 1.258 1.155 1.126

0.9820 1.125 1.212 1.232 1.240 1.240

all

1

0.15 0.4 0.6

0.4846 0.6558 0.7572

0.3689 0.4517 0.4890

0.3689 0.4517 0.4890

0.3530 0.6109 0.7989

0.3081 0.3503 0.3578

1.301 1.433 1.471

1.185 1.243 1.227

0.9788 1.054 1.071

0.25

0.7

0.05 0.15 0.952:

0.3653 0.4909 0.9260

0.2517 0.3187 0.0

0.3038 0.3857 0.5494

0.1840 0.3127 1.142

0.2733 0.3345 0.3549

1.171 1.358 1.522

1.112 1.259 1.165

0.9772 1.114 0.4897

0.5

0.7

0.05 0.15 0.4 0.6 0.903C 1.1

0.3827 0.5114 0.6867 0.7908 0.9266 1.007

0.2532 0.3171 0.3637 0.3539 0.0 1.024

0.3189 0.4037 0.4878 0.5247 0.5635 0.5830

0.2001 0.3339 0.6147 0.8245 1.131 1.325

0.2826 0.3444 0.3794 0.3786 0.3612 0.3446

1.194 1.385 1.507 1.537 1.552 1.553

1.124 1.271 1.303 1.264 1.179 1.117

0.9728 1.103 1.147 1.098 0.4935 2.080

0.6

0.7

0.05 0.15 0.882t

0.3913 0.5215

0.2546 0.3173 0.0

0.3265 0.4128 0.5708

0.2077 0.3438 1.127

0.2874 1.206 0.3496 1.399 0.3645 1.566

L .277

1.130

0.9712 1.098 0.4953

0

0.9280

1.185

TABLE IV (Continued)

0.5

0.5

0.05 0.1 0.15 0.8393

0.4138 0.4928 0.5475 0.9335

0.2603 0.2989 0.3204

0.3471 0.4028 0.4371 0.5906

0.2261 0.3017 0.3675 1.121

0.3007 0.3419 0.3638 0.3731

1.236 1.366 1.435 1.605

1.146 1.249 1.294 1.201

0.9680 1.052 1.090 0.4989

0.05 0.15 0.4 0.6 0 - 7762

0.2815 0.3380 0.3385 0.2521

1.1

0.4614 0.5998 0.7723 0.8688 0.9439 1.069

2.001

0.3970 0.4938 0.5831 0.6156 0.6374 0.6666

0.2496 0.3980 0.6808 0.8997 1.080 1.404

0.3233 0.3908 0.4149 0.4094 0.3958 0.3640

1.258 1.479 1.586 1.629 1.634 1.632

1.155 1.315 1.334 1.294 1.242 1.138

0.9530 1.079 1.089 0.9863 0.7265 2.655

0.5

0.15

0.5099

0.3173

0.4161

0.2437

0.3881

1.462

1.408

1.370

0.7

0.05 0.15 0.4 0.6 1 .o 1.1

0.4252 0.5243 0.6713 0.7643 0.9277 0.9657

0.3109 0.3625 0.4142 0.4378 0.4705 0.4770

0.3578 0.4164 0.4750 0.5018 0.5390 0.5464

0.2008 0.3182 0.5779 0.7751 1.155 1.247

0.3248 0.3652 0.3823 0.3768 0.3503 0.3420

1.307 1.433 1.499 1.508 1.500 1.496

1.244 1.334 1.324 1.275 1.155 1.124

1.098 1.196 1.244 1.249 1.240 1.237

all

1

0.4 0.6 1.0

0 6825 0.7720 0.9277

0.4754 0.5022 0 5390

0.4754 0.5022 0.5390

0.6305 0.8110 1.155

0.3673 0.3665 0.3503

1.482 1.497 1.500

1.279 1.245 1.155

1.080 1.085 1.078

0.5

0.7

0.05 0.15 0.4 0.6 0.9040 1.1

0.4500 0.5513 0.6986 0.7913 0.9169 0.9920

0.3013 0.3452 0.3710 0.3535

0.3784 0.4388 0.4975 0.5238 0.5526 0.5674

0.2322 0.3560 0.6238 0.8258 1.122 1.308

0.3339 0.3733 0.3867 0.3784 0.3557 0.3379

1.339 1.465 1.530 1.538 1.532 1.524

1.253 1.340 1.321 1.266 1.170 1.105

1.075 1.157 1.162 1.101 0.4941 2.030

0.8

1

0.7

0

0.7

0.7

-

0.0

0.0

-

0.0

0.9870

P, 00

t

0.5

3

w

0.7

1

Ul

P=

1

0.5

0.7

all

f"(O)

e*(o)

g'(0)

0.15 0.4 0.6

0.6047 0.7484 0.8397

0.2377 0.1805 0.0407

0.05 0.15 0.4 0.6 0.7934 1.1

0.5048 0.6073 0.7489 0.8370 0.9146 1.029

0 0.1 0.15 0.2 0.4

I1

I2

11(1)

I I W

11(3)

0.4964 0.5481 0.5706

0.4006 0.7066 0.9314

0.4034 0.3951 0.3745

1.558 1.600 1.591

1.384 1.316 1.235

1.157 1.015 0.7591

0.3071 0.3405 0.3277 0.2518 0.0 1.700

0.4305 0.4928 0.5488 0.5719 0.5886 0.6095

0.2794 0.4134 0.6862 0.8938 1.087 1.386

0.3616 0.4004 0.4047 0.3900 0.3711 0.3372

1.400 1.540 1.599 1.607 1.603 1.588

1.278 1.363 1.332 1.268 1.200 1.090

1.051 1.118 1.076 0.9366 0.5470 2.953

2.0

0.5811 0.6184 0.6368 0.6550 0.7262 0.7609 0.7952 0.8623 0.9277 1.235

0.4942 0.4995 0.5020 0.5045 0.5140 0.5185 0.5228 0.5311 0.5390 0.5729

0.4942 0.4995 0.5020 0.5045 0.5140 0.5185 0.5228 0.5311 0.5390 0.5729

0.3146 0.4034 0.4474 0.4909 0.6625 0.7468 0.8301 0.9941 1.155 1.920

0.4238 0.4167 0.4131 0.4095 0.3949 0.3876 0.3802 0.3653 0.3503 0.2746

1.613 1.599 1.592 1.586 1.561 1.550 1.539 1.519 1.500 1.424

1.477 1.440 1.422 1.404 1.337 1.304 1.273 1.212 1.155 0.8988

1.162 1.152 1.147 1.142 1.124 1.115 1.107 1.092 1.078 1.022

0.5

0.15 0.4 0.6 1.0

0.5913 0.6955 0.7754 0.9277

0.3806 0.3913 0.3990 0.4130

0.4907 0.5067 0.5183 0.5390

0.2686 0.5385 0.7484 1.155

0.2570 0.4262 0.4011 0.3503

1.626 1.582 1.552 1.500

1.561 1.430 1.333 1.155

1.521 1.486 1.461 1.419

0.7

0.15 0.4 0.6 1.0

0.6136 0.7105 0.7850 0.9277

0.4358 0.4471 0.4555 0.4705

0.4964 0.5103 0.5205 0.5390

0.3609 0.6026 0.7906 1.155

0.4331 0.4092 0.3897 0.3503

1.609 1.572 1.545 1.500

1.487 1.380 1.301 1.155

1.325 1.296 1.276 1.240

1

tw

0.5 0.6 0.8 1.0 0

r U m

-2

TABLE IV (Continued)

B

0.5

0.75

W

1

u1

Pr

1

p Ip)

11(3)

0.4086 0.3760 0.3496 0.3409

1.599 1.556 1.525 1.516

1.425 1.293 1.195 1.165

1.189 0.9936 0.7003 0.5362

0.4272 0.6743 0.8662 1.064 1.328

0.4113 0.3866 0.3667 0.3454 0.3161

1.596 1.559 1.533 1.508 1.478

1.425 1.318 1.238 1.159 1.058

1.174 1.073 0.9303 0.5038 2.702

t"

fil(0)

e'(0)

g'(0)

0.5

0.15 0.4 0.6 0.6653

0.6160 0.7243 0.8071 0.8335

0.2440 0.1831 0.0693 0.0

0.4990 0.5154 0.5273 0.5309

0.4148 0.6966 0.9150 0.9852

0.7

0.15 0.4 0.6 0.8111 1.1

0.6249 0.7237 0.7997 0.8773 0.9800

0.3485 0.3152 0.2474 0.0 1.409

0.5003 0.5145 0.5248 0.5349 0.5478

I2

0.5

0.5

0.7

0.05 0.15 0.4 0.6 0.8994 1.1

0.4131 0.5618 0.7768 0.9092 1.083 1.189

0.2576 0.3234 0.3721 0.3616 0.0 1.085

0.3251 0.4129 0.5020 0.5419 0.5840 0.6061

0.1796 0.3000 0.5518 0.7391 1.008 1.183

0.2784 0.3367 0.3630 0.3549 0.3264 0.3017

1.182 1.368 1.483 1.508 1.518 1.517

1.092 1.223 1.224 1.165 1.052 0.9717

0.9608 1.086 1-121 1.064 0.4382 2.118

0.7

0

0.7

0.05 0.15 0.4 0.6 1 .o 1.1

0.4557 0.5726 0.7568 0.8770 1.090 1.140

0.3150 0.3683 0.4232 0.4489 0.4849 0.4922

0.3629 0.4237 0.4862 0.5155 0.5568 0.5652

0.1800 0.2857 0.5183 0.6939 1.031 1.113

0.3207 0.3582 0.3682 0.3565 0.3174 0.3059

1.292 1.412 1.470 1.474 1.460 1.454

1.211 1.288 1.251 1.183 1.031 0.9922

1.085 1.179 1.221 1.223 1.209 1.204

P N

TABLE IV (Continrred)

~

0.75

0.7

1

1

0.5

0

~~

0.5

all

0

a11

0.7

0.05 0.15 0.4 0.6 0.9044 1.1

0.4833 0.6029 0.7885 0.9089 1.072 1.173

0.3059 0.3512 0.3787 0.3605 0.0' 1.045

0.3850 0.4478 0.5108 0.5400 0.5719 0.5890

0.2072 0.3187 0.5591 0.7396

0

0.15 0.4 0.5 0 -6 1 .o

0.6181 0.6948 0.8173 0.8646 0.9112 1.090

0.5012 0.5112 0.5262 0.5318 0.5371 0.5568

0.5

0.15 0.4 0.6 1 .o

0.5440 0.7923 0.9521 1.233

0.7

0.05 0.15 0.4 0.6 1 .o 1.1

1

0.15 0.4 0.6 1 .o

1

1.169

0.3279 0.3690 0.3692 0.3542 0.3215 0.2963

1.325 1.446 1.503 1.507 1.496 1.485

1.215 1.28-7 1.240 1.166 1.044 0.9624

1.061 1.139 1.135 1.065 0.4385 2.062

0.5012 0.5112 0.5262 0.5318 0.5371 0.5568

0.2748 0.3953 0.5896 0.6654 0.7402 1.031

0.4120 0.3984 0.3751 0.3656 0.3560 0.3174

1.594 1.568 1.531 1.517 1.504 1.460

1.423 1.355 1.250 1.211 1.173 1.031

1.148 1.129 1.101 1.091 1.081 1.048

0.2891 0.3575 0.3894 0.4334

0.3857 0.4734 0.5142 0.5705

0.1968 0.4248 0.6005 0.9402

0.3473 0.3674 0.3516 0.2923

1.340 1.428 1.439 1.430

1.269 1.234 1.146 0.9402

1.261 1.349 1.364 1.364

0.4025 0.5612 0.8060 0.9615 1.233 1.295

0.2584 0.3317 0.4097 0.4460 0.4959 0.5058

0.3004 0.3844 0.4730 0.5140 0.5705 0.5817

0.1408 0.2437 0.4631 0.6272 0.9402 1.016

0.2617 0.3476 0.3430 0.3344 0.2923 0.2791

1.127 1.300 1.404 1.426 1.430 1.429

1.053 1.175 1.167 1.102 0.9402 0.8978

0.9605 1.096 1.172 1.186 1.186 1.184

0.5789 0.8204 0.9714 1.233

0.3826 0.4722 0.5137 0.5705

0.3826 0.4722 0.5137 0.5705

0.2868 0.4985 0.6518 0.9402

0.2921 0.3219 0.3196 0.2923

1.261 1.380 1.411 1.430

1.092 1.107 1.062 0.9402

0.9475 1.014 1.026 1.026

1.0000

TABLE IV (Continued)

1

0.5

0.25

0.7

0.05 0.15 0.9490

0.4187 0.5808 1.224

0.2588 0.3291 0.0

0.3132 0.3998 0.5829

0.1514 0.2578 0.9289

0.2673 0.3230 0.2955

1.146 1.323 1.458

1.059 1.180 0.9490

0.9556 1.084 0.3943

0.5

0.7

0-05 0.15 0.4 0.6 0.8965 1.1

0.4395 0.6057 0.8558 1.013 1.219 1.349

0.2612 0.3285 0.3787 0.3675 0.0 1.135

0.3302 0.4204 0.5132 0.5554 0.6000 0.6241

0.1641 0.2746 0.5051 0.6760 0.9187 1.080

0.2755 0.3311 0.3507 0.3369 0.3000 0.2689

1.172 1.355 1.465 1.488 1.495 1.492

1.068 1.187 1.166 1.092 0.9601 0.8652

0.9516 1.074 1. l o 2 1.039 0.3992 2.149

0.6

0.7

0.05

0.4499 0.5467 0.6181 1.218

0.2632 0.3047 0.3292

0.3390 0.3954 0.4310 0.6091

0.1700 0.2298 0.2825 0.9152

0.2799 0.3169 0.3356 0.3025

1.185 1.307 1.371 1.513

1.073 1.159 1.192 0.9655

0.9504 1.033 1.070 0.4013

0.2708 0.3115 0.3344 0.0

0.3632 0.4227 0.4599 0.6344

0.1848 0.2472 0.3016 0.9094

0.2926 0.3302 0.3486 0.3090

1.221 1.348 1.415 1.565

1.O% 1.173 1.205 0.9788

0.9490 1.029 1.063 0.4061

0.1

0.15 0.8748

0.7

0.0

0.8

0.7

0-05 0.1 0.15

0.8290

0.4774 0.5774 0.6501 1.218

1

0.7

0.05 0.15 0.4 0.6 0.7640 1.1

0.5381 0.7166 0.9653 1.110 1.220 1.423

0.3014 0.3623 0.3634 0.2598 0.0 2.402

0.4277 0.5342 0.6465 0.6760 0 7006 0.7483

0.2068 0.3297 0.5532 0.7445 0.8838 1.136

0.3133 0.3745 0.3694 0.3577 0.3329 0.2860

1.244 1.473 1.&88 1.603 1.606 1.517

1 .087 1.221 1.163 1.104 1.027 0.8877

0.9266 1.049 1.016 0.8990 0.6067 2.484

0.1 0.4 0.6

0.5341 8.151 0.9645

0.3060 0.3750 0.3992

0.4036 0.4934 0.5251

0. l G O l 0.4327 0.6061

0.3739 0.3824 0.3585

1.396 1.468 1.461

1.341 1.265 1.160

1.312 1.387 1.385

0

0.5

-

8 N

TABLE IV (Continued)

~

1

0.7

~~

-

~~~

_

_

0

0.7

0.5 0.15 0- 4 0.6 1 .o 1.1

0-4821 0.6146 0.8317 0.9756 1.233 1.293

0.3183 0.3729 0.4302 0.4574 0.4959 0.5037

0.3670 0.1643 0.4295 0.2614 0.4950 0.4741 0.5262 0.6343 0.5705 0.9402 0.5795 1.014

a1

1

0.15 0.6 1 .o

0.6381 0.9873 1.233

0.4310 0.5271 0.5705

0.4310 0.5271 0.5705

0.5

0.7

0.05 0.15 0.4 0.6 0.8975 1.1

0.5120 0.6478 1 505 1.012 1.207 1.331

0.3095 0.3560 1.024 0.3658 0.0 1.091

0.3902 0.1884 0.3235 0.4550 0.2908 0.3573 1.366 0 2288 0.1155 0.5526 0.6759 0.3359 0.5869 0.9113 0.2955 0.6057 1.067 0.2644

1.187 1.314 1.051 1.432 1.126 1.248 0 * 4640 0.4537 0.3749 1.039 1.484 1.092 0.9523 0.3995 1.470 0.8578 2.089 1.457

0.3118 0.6605 0.9402

0.3177 0.3532 0.3576 0.3413 0.2923 0.2783

1.280 1.396 1.449 1.450 1.430 1.424

1.186 1.253 1.197 1.116 0.9402 0.8964

1.076 1.168 1.205 1.203 1.186 1.181

0.3262 1.363 0.3268 1.438 0.2923 1.430

1.167 1.076 0.9402

1.007 1.040 1.026

-

0.9

0.7

0.15 0.4 0.6 0.809:

0.6940 0.9117 1.055 1.192

0.3506 0.3494 0.2822 0.0

0.4955 0.5617 0.5919 0.6160

0.3239 0.5505 0.7195 0.8902

0.3702 0.3612 0.3351 0.3019

1.491 1.544 1.543 1.534

1.254 1.173 1.075 0.9683

1.095 1.037 0.8892 0.4096

1

0.7

0.05 0.15 0.4 0.6 0.7775 1.1

0-5809 0.7188 0.9317 1.071 1.185 1.380

0.3222 0.3576 0.3426 0.2551 0.0 1.970

0.4548 0.5230 0.5891 0.6162 0.6372 0.6650

0.2275 0.3372 0.5616 0.7336 0.8753 1.130

0.3488 0.3810 0.3668 0.3379 0.3108 0.2547

1.387 1.528 1.562 1.579 1.549 1.527

1.202 1.262 1.175 1.075 0.9880 0.8266

1.026 1.088 1.023 0.8527 0.5073 3.035

TABLE I\' (Continued)

1

1

I

Pr

UI

0

all

f '1 (0)

I2 ~~

0.5

0.4 0.6 1.o

0.8524 0.9840 1.233

0.4037 0.4145 0.4334

0.5261 0.5422 0.5705

0.4458 0.6149 0.9402

0.4066 0.3691 0.2923

1* 535

1.495 1.430

1.315 1.182 0.9402

1.449 1.417 1.364

0.7

0 0.15 0.4 0.5 0 06 1.0

0.6071 0.7104 0.8735 0.9362 0.9976 1.233

0.4362 0.4473 0.4637 0.4696 0.4753 0.4959

0.4970 0.5107 0.5308 0.5382 0.5452 0.5705

0 1695 0.4376 0.2934 0.4171 0.4921 0.3814 0 5692 0.3669 0.6454 0.3522 0.9402 0.2923

1.607 1.572 1.521 1 SO4 1.487 1.430

1.494 1.396 1.246 1.191 1.138 0.9402

1.324 1.296 1.257 1.244 1.231 1.186

0

0.6489 0.7445 0.7755 0.8963 0.9548 1.012 1.124 1.233 1.737

0.5067 0.5183 0 * 5219 0.5357 0.5621 0.5482 0.5597 0.5705 0.6156

0.5067 0.5183 0.5219 0.5357 0.5421 0.5482 0.5597 0.5705 0.6156

0.2456 0.3570 0.3934 0.5360 0.6056 0.6743 0.8089 0.9402 1.561

1.579 1.550 1.541 1.508 1.493 1.479 1.454 1.430 1.340

1.383 1.305 1.280 1.186 1.142

1.137 1.115 1.109 1.084 1.073

1.099 1.018 0.9402 0.6010

1.063 1.043 1.026 0.9595

1

0.15 0.2 0.4 0.5 0.6 0.8 1.o 2 .o

-

-

0.4033 0.3875 0.3821 0.3603 0.3491 0.3380 0.3153 0.2923 0.1761

-

m

2 0

5

%

kc?: 1: U L-

?

r 8

M

0

5

2 0

5

TABLE IV (Continued)

1.4

1.5

0.6

0.7

0.1

0.5977

0.3107

0.4043

0.2042

0.3118

1.295

1.122

1.020

0.8

0.7

0.1

0.6320

0.3189

0.4343

0.2195

0.3248

1.341

1.135

1.018

0.7

0.6

0.7

0.15 0.4 0.6 0.8721 1.1

0.7200 0.9874 1.167 1.389 1.564

0.3596 0.3830 0.3516 0.01.295

0-4719 0.5427 0.5765 0.6109 0.6343

0.2640 0.4622 0.6100 0.8028 0.9587

0.3505 0.3404 0.3125 0.2653 0.2214

1.426 1.472 1.469 1.453 1.437

1.200 1.105 0.9983 0.8486 0.7246

1.101 1.071 0.9707 0.3577 2.341

0.5

0

0.7

0.05 0.15 0.4 0.6 1 .o

0.4427 0.6304 0.9337 1.132 1.477

0.2628 0.3383 0.4200 0.4585 0.5119

0.3060 0.3928 0.4860 0.5298 0.5906

0.1225 0.2123 0.4030 0.5448 0.8141

0.2590 0.3119 0.3291 0.3134 0.2562

1.110 1.277 1.374 1.391 1.390

1.022 1.128 1.092 1.008 0.8141

0.9465 1.078 1.148 1.159 1.155

0.25

0.7

0.05 0.15 0.9469

0.4608 0.6526 1.464

0.2636 0.3361 0.0

0.3197 0.4095 0.6045

0.1312 0.2242 0.8034

0 2640 0.3163 0.2585

1.130 1.302 1.421

1.026 1.130 0.8216

0.9418 1.066 0.3415

0.5

0.7

0.05 0.15 0.4 0.6 0.8922

0.4842 0.6810 0.9917 1.192 1.453

0.2667 0.3362 0.3888 0.3763 0.0

0.3381 0.4319 0.5304 0.5760 0.6241

0.1419 0.2383 0.4389 0.5871 0.7937

0.2715 0.3234 0.3332 0.2620

1.158 1.336 1.440 1.459 1.463

1.033 1.136 1.084 0.9885 0.8311

0.9384 1 .056 1.075 1.004 0.3466

0.05 0.15 0.4

0.5266 0.6863 0.9604 1.477

0.3233 0.3799 0.4407 0.5119

0.3733 0.4383 0.5082 0.5906

0.1419 0.2267 0.4117 0.8140

0.3138 0.3462 0.3427 0.2561

1.262 1.373 1.418 1.390

1.150 1.203 1.120 0.8140

1.061 1.149 1.181 1.155

0.5

0.7

0

0.7

1 .o

-

0.3211

T A B L E I V (Continued)

1.5

1.8

1

1

0.

0.6

0.7

0 0.15 0.2 0.4 0.5 0.6 0.8 1.0 2.0

0.6987 0.8278 0.8695 1.031 1.109 1.185 1.334 1.477 2.140

0.5147 0.5289 0.5333 0.5498 0.5574 0.5646 0.5781 0.5906 0.6423

0.1 0.15

0.6421 0.7354 1.080 1.303 1.788

0.3155 0.3414 0.3877 0.3616 1.425

0.4 0.6 1.1 2

0.5

0.5498 0.5574 0.5646 0.578i 0.5906 0.6423

0.2049 0.3035 0.3356 0.4610 0.5220 0.5821 0.6996 0.8141 1.352

0.3914 0.3725 0.3661 0.3395 0.3259 0.3122 0.2844 0.2562 0.1128

1.558 1.524 1 514 1.476 1.460 1.444 1.416 1.390 1.294

1.326 1.235 1.206 1.097 1.046 0.9966 0.9026 0.8141 0.4268

1.122 1.096 1.088 1.060 1.048 1.036 1.015 0.9963 0.9251

0.4115 0.4496 0.5524 0.6000 0.6786

0.1856 0.2285 0.4182 0.5578 0.8857

0.3083 0.3241 0.3273 0 2993 0.1938

1.286 1.347 1.451 1.469 1.467

1.095 1.116 1.047 0.9388 0.6373

1.010 1.044 1.048 0.9526 2.484

0.5147 0.5289 0.5333

-

-

0.8

0-7

0.1

0.6795

0.3249

0.4436

0.1991

0-3212 1.335

1.108

1.010

0

0.5

0.15 0-4 0.6

0.6650 1.024 1.263

0.2975 0.3712 0.4063

0.3994 0.4955 0.5412

0.1598 0.3368 0.4711

0.3454 0 3508 0.3209

1.304 1.377 1.381

1.207 1.122 0.9984

1.232 1.308 1.318

0.7

0.05 0.15 0.4 0.6 1 .o

0.4766 0.6893 1.043 1.276 1.687

0.2661 0.3432 0.4276 0.4677 0.5235

0.3103 0.3991 0.4956 0.5414 0.6052

0.1097 0.1906 0.3617 0 -4884 0.7282

0.2573 0.3082 0.3196 0.2988 0.2308

1.097 1.261 1.352 1.367 1.363

0.9992 1.096 1.040 0.9445 0.7282

0.9364 1.065 1.131 1.140 1.134

*Convergence

.

to 10- 4

-

*

-

~

0 2

-

~

~

o1

Pr

0.5

0.25

0.7

0.05 0.15 0.945

0.4962 0.7137 1.669

0.2672 0.3414

0.0

0.3246 0.4167 0.6204

0.1172 0.2009 0.7180

0.2619 0.3119 0.2326

1.119 1.287 1.396

1.003 1.096 0.7348

0.9319 1.053 0.3063

0.7

0

0.5

0.15 0 -4 0.6 1.o

0.7168 1.049 1.276 1.687

0.3333 0.3890 0.4163 0.4555

0.4419 0.5157 0.5523 0.6052

0.1673 0.3421 0.4748 0.7282

0.3821 0.3646 0.3268 0.2308

1.394 1.418 1.403 1.363

1.286 1.150 1.010 0.7282

1.316 1.347* 1.339* 1-310

0.7

0.05 0.15 0-4 0.6 1.o

0.5640 0.7473 1.071 1.291 1.687

0.3270 0.3851 0.4484 0.4793 0.5235

0.3781 0.4449 0.5179 0.5537 0.6052

0.1264 0.2028 0.3690 0.4930 0.7282

0.3112 0.3417 0.3326 0.3046 0.2308

1.249 1.357 1.396 1.391 1.363

1.125 1.168 1.067 0.9561 0.7282

1SO51 1.136 1.164 1.158 1.134

0.

0.5

0.15 0.4 0.6

0.7679 1. l o 7 1.345

0.2937 0.2994 0.2449

0.4734 0 5492 0.5865

0.2029 0.3899 0.5308

0.3621 0.3308 0.2838

1.421 1.442 1.427

1.217 1.053 0.8976

1.193 1.105 0.9170

all

1

0 0.15 0.2 0.4 0.5 0.6 0.8 1.0 2.0

0.7386 0.8972 0.9483 1.146 1.241 1.333 1.513 1.687 2.488

0 * 5206 0.5367 0.5417 0.5601 0.5686 0.5766 0.5915 0.6052 0.6615

0.5206 0.1775 0.5367 0.2673 0.5417 0 * 2965 0 * 5601 0.4101 0.5686 0.4653 0.5766 0.5194 0.5915 0.6254 0.6052 0.7284 0.6615 0.1211

0.3837 0.3626 0.3553 0.3254 0.3101 0.2945 0.2629 0.2309 0.0676

1.544 1.506 1.495 1.454 1.436 1.420 1.390 1.363 1.263

1.288 1.187 1.156 1SO36 0.9804 0.9267 0.8245 0.7284 0.3086

1.111 1.082 1.074 1.044 1.030 1.018 0.9957* 0.9760* 0.9023*

u)

1

*cmvergence t o 10-4.

tw

TABLE IV (Continued)

fii(o) 2.4

1

all

1

e@(o)

g@(o)

x1

I2

I I W

11(3)

0.5 1 .o

0.7659 1.335 1.838

0.5244 0.5757 0.6145

0.5244 0.5757 0.6145

0.1611 0.4309 0.6762

0.3791 0.3005 0.2151

1-535 1.422 1.347

1.265 0.9408 0.6762

1.104 1.020 0.9637

0

*

2.8

1

all

1

0 0.5 1.0

0.7901 1.421 1.978

0.5275 0.5817 0.6223

0.5275 0.5817 0.6223

0.1480 0.4033 0.6341

0.3756 0.2928 0.2023

1.527 1.411 1.333

1.246 0.9088 0.6341

1.098 1.011 * 0.9537

3

0.5

0

0.7

0.15 0.4 0.6 1 .o

0.7883 1.228 1.522 2.044

0.3503 0.4383 0.4805 0.5397

0.4082 0.5093 0.5578 0.6258

0.1616 0.3036 0.3072 0.3074 0.4142 0.2796 0.6158 0.1967

1.238 1.323 1.335 1.328

1.051 0.9723 0.8603 0.6158

1.047 1.109* 1.115* 1.107

0.7

0

0.7

0.05 0.15 0.4 0.6

0.6255 0.8493 1.257 1.537 2.044

0.3324 0.3926 0.4593 0.4923 0.5397

0.3849 0.4545 0.5319 0.5702 0.6258

0.1057 0.1711 0.3125 0.4176 0-6158

0.3083 0-3360 0.3194 0.2847 0.1967

0.1231 1.334 1.367 1.359 1.328

1.091 1-121 0.9973 0.8708 0-6158

1.037 1.118 1.141 1.133 1-10)

all

1

0 0.5

0.8221 .1.541 2.170

0.5314 0.5893 0.6320

0.5314 0.5893 0.6320

0.1325 0.3697 0.5929

0.3715 0.2835 0.1919

1.518 1.396 1.317

1.224 0.8699 0.5938

1.092 1.000 0.9416*

0

0.8502 1.650 2.347

0.5346 0.5955 0.6401

0.5346 0.1205 0.3684 0.5955 0.3434 0.2762 0.6401 0.5447 0.1751

1.511 1.385 1.305

1.207 0.8393 0.5447

1.086 0.9918 0.9320*

0.8907 1.815 2.616

0.5389 0.6042 0.6509

0.5389 0.6042 0.6509

1.502 1.368 1.288

1.184 0.7994 0.4992

1.079 0.9796 0.9196*

1 .o

3.4

1

4

1

1.o

all

1

0.5 1.0 5

1

*Convergence

all

1

0

0.5 1.0 to 10-4.

0.1052 0.3096 0.4923

0.3647 0.2668 0.1586

TABLE V SIMILAR SOLYTIONS FOR

0

0.01

0

0

0.15 0.4 0.6

Lo(=) 0.5

0.15 0.6 0.9152

1

0.15 0.6 0.7

A

SVTHERLAND VISCOSITY-TEMPERATURE RELATION, Pr = 0.7,t,

=

1

0.3564 0.3554 0.4162 0.4161 0.4407 0.4407 0.4696 0.4696

0.3100 0.3093 0.3643 0.3642 0.3868 0.3868 0.4139 0.4139

0.3564 0.3554 0.4162 0.4161 0.4407 0.4407 0.4696 0.4696

0.4105 0.4093 0.7909 0.7906 1.090 1.090 1.686 1.686

0.3563 0.3554 0.4162 0.4161 0.4406 0.4406 0.4696 0.4696

1.430 1.425 1.575 1.574 1.628 1.628 1.686 1.686

1.430 1.425 1.575 1.574 1.628 1.628 1.686 1.686

1.200 1.195 1.306 1.306 1.344 1.344 1.385 1-385

0.3804 0.3797 0.4644 0.4644 0.4871 0.4871

0.3019 0.3014 0.3247 0.3247 0.0 0.0

0.3804 0.3797 0.4644 0.4644 0.4871 0.4871

0.4681 0.4672 1.167 1.167 1.645 1.645

0.3804 0.3798 0.4643 0.4643 0.4871 0.4871

1.470 1.467 1.664 1.664 1.710 1.710

1.470 1.467 1.664 1.664 1.710 1.710

1.179 1.176 1 240 1.240 0.7667 0.7666

0.4327 0.4333 0.5125 0.5080 0.5197 0.5141

1.540 1.537 1.726 1.709 1.742 1 722

1.540 1.537 1.726 1.709 1.742 1.722

1 149 1.154 1.102 1.175 1.011 1.127

0.4326 0.4357 0.5126 0.5191 0.5198 0.5266

a S o l u t i o n independent of s .

0.3047 0.3050 0.2467 0.2378 0.1765 0.1609

0.4326 0.4357 0.5126 0.5191 0.5198 0.5266

0.5637 0.5558 1.285 1.239 1.438 1.384

-

-

-

0.5317 0.5174 0.5136

-0.5368 0.5159 0.5121 0.5486 0.5205 0.5184

B

S

fw

bl

tw

0

0.01

-0.2

0

0.15 0.4 0.6 l.O(a)

-0.6

0

0.15 0.4 0.6 l.O(a)

f"(0)

e'(0)

g'(0)

0.2221 0.2211 0.2794 0.2792 0.3028 0.3028 0.3305 0.3305

0.2109 0.2101 0.2632 0.2631 0.2848 0.2848 0.3108 0.3108

0.0229 0.0222 0.0598 0.0597 0.0767 0.0767 0.0975 0.0975

Il

I2

0.2221 0.2211 0.2794 0.2792 0.3028 0.3028 0.3305 0.3305

0.5369 0.5355 0.9754 0.9751 1.318 1.318 1.999 1.999

0.0376 0.0368 0.0802 0.0800 0.0982 0.0982 0.1201 0.1201

0.0229 0.0222 0.0598 0.0597 0.0767 0.0767 0.0975 0.0975

I

p

11(2)

5(3)

0.4221 0.4211 0.4794 0.4792 0.5028 0.5027 0.5305 0.5305

1.824 1.819 1.922 1.921 1.958 1.958 1.999 1.999

1.824 1.819 1.922 1.921 1.958 1.958 1.999 1.999

1.514 1.510 1.577 1.576 1.599 1.599 1.623 1.623

1.176 1.180 1.752 1.752 2.230 2.230 3.181 3.181

0.6228 0.6221 0.6598 0.6597 0.6768 0.6767 0.6975 0.6975

4.137 4.154 3.437 3.438 3,299 3.299 3.181 3.181

4.137 4.154 3.437 3.438 3.299 3.299 3.181 3.181

3.483 3.499 2.809 2.810 2.672 2.672 2.552 2.552

0.03

0

1

0.4

0.48GO 0.4939

0.3052 0.3025

0.4860 0.4935

0.9686 0.9398

0.4863 0.4854

1.677 1.667

1.677 1.667

1.183 1.211

0.05

0

1

0.15

0.4421 0.4452 0 -4823 0.4890 0.5007 0.5074

0.3134 0.3139 0.3045 0.3028 0.2492 0.2418

0.4421 0.4452 0.4823 0.4890 0.5007 0.5074

0.5759 0.5782 0.9640 0.9566 1.271 1.262

0.4422 0.4451 0.4823 0.4865 0.5007 0.5052

1.587 1.587 1.677 1.677 1.716 1.716

1.587 1.587 1.677 1.677 1.716 1.716

1.190 1.187 1.188 1.200 1.112 1.134

0.4074 0.4011 0.4309 0.4299

0.3559 0.3511 0.3778 0.3770

0.4074 0.4011 0.4309 0.4299

0.4577 0.4491 0.8095 0.8073

0.4074 0.4011 0.4309 0.4299

1.561 1.529 1.608 1.603

1.561 1.529 1.608 0.603

1.298 1.270 1.330 1.326

0.4 0.6 0.1

0

0

0.15 0.4

aSolution independent of s .

Lur 0.5317 0.5174 0.5136

-0.5317 0.5174 0.5136

-0.5761 0.7027 0.6208 0.5944 0.7384 0.6497

P N \o

L 2 0

'I'ABLE \' (Continued)

0

0.1

0

0.6

0

1 0.5

0.15 0.4

0.3

0

0.15

0

0.4 0.6 0.5

0.15 0.4 0.6 0.9151 0.15

1

0.6 ~~

-

0.4471 0.3928 0.4468 0.3926 0.4696 0.4139 0.4696 0.4139

0.4471 0.4468 0.4696 0.4696

1.100 1.099 1.686 1.686

0.4471 0.4468 0 -4696 0.4696

1.641 1.640 1.686 1.686

1.641 1.640 1.686 1.686

1.354 1.353 1.385 1.385

0.6210

0.4269 0.4224 0.4499 0.4495

0.3393 0.3358 0.3422 0.3419

0.4269 0.4224 0.4499 0.4495

0.5168 0.5108 0.8761 0.8753

1.264 1.246 1.268 1.267

0.7708

0.4140 0.4065 0.3983 0.3969 O.GO24 0.4020

0.4713 0.4615 0.4532 0.4513 0.4574 0.4569

0.5150 0.5021 0.8373 0.8332 1.115 1.114

1.591 1.570 1.637 1.635 1.715 1.668 1.656 1.648 1.663 1.660

1.591 1.570 1.637 1.635

0.4713 0.4615 0.4532 0.4513 0.4575 0.4569

0.4269 0.4223 0.4499 0.4495 0.4713 0.4615 0.4532 0.4513 0.4573 0.4569

1.715 1.668 1.656 1.648 1.663 1.660

1.412 1.371 1.365 1.358 1.367 1.367

0.4803 0.4738 0.4631 0.4624 0.4676 0.4677 0.4876 0.4876

0.3823 0.3773 0.3522 0.3517 0.3274 0.1274 0.0 0.0

0.4803 0.4738 0.4631 0.4624 0.4676 0.4677 0.4876 0.4876

0.5722 0.5639 0.8960 0.8947 1.177 1.177 1.645 1.645

0.4803 0.4738 0.4623 0.4630 0.4676 0.4677 0.4876 0.4876

1.726 1.697 1.671 1.668 1.678 1.678 1.710 1.710

1.725 1.697 1.670 1.668 1.678 1.678 1.710 1.710

1.356 1.333 1.291 1.289 1.252 1.252 0.7664 0.7664

0.4779 0.4792 0.4723 0.4788

0.3416 0.3416 0.2493 0.2430

0.4779 0.6203 0.4792 0.6216 0.4735 1.238 0.4788 1.240

0.4779 0.4792 0.4735 0.4785

1.725 1.723 1.692 1.693

1.725 1.723 1.692 1.693

1.300 1.296 1.135 1.134

aSoIution independent of s .

--

?

r

z

0

m

(I)

0.6722 0.9717 0.8437 0.7918 1.019 0.8837

r

0.8266

z

0.5000 1.101 0.8838

0

$

(I)

TABLE V (Continued)

B 0

S

fw

01

tw

0.3

-0.2

0

0.15 0.4

0.6 l.O(a)

-0.6

0

0.15 0.4 0.6 l.O(a)

0.4

0

0.05

1

0.15 0.4

1.1 _

_

~

~~~

f"(0)

e'(0)

g'(0)

0.3328 0.3227 0.3148 0.3129 0.3188 0.3183 0.3305 0.3305

0.3116 0.3036 0.2959 0.2944 0.2998 0.2994 0.3108 0.3108

0.1005 0.0914 0.0858 0.0841 0.0886 0.0882 0.0975 0.0975 0.5769 0.5808 0.7056 0.7130 0.9696 0.9706

~

a S o l u t i o n independent of s .

Il

I2

Il(U

11(2)

11(3)

0.3328 0.3227 0.3148 0.3129 0.3188 0.3183 0.3305 0.3305

0.6259 0.6131 1.012 1.007 1.337 1.336 1.999 1.999

0.5328 0.5227 0.5149 0.5129 0.5188 0.5183 0.5305 0.5305

2.033 1.984 1.979 1.971 1.983 1.980 1.999 1.999

2.033 1.984 1.979 1.971 1.983 1.980 1.999 1.999

1.656 1.613 1.613 1.605 1.614 1.612 1.623 1.623

0.1226 0.1138 0.1077 0.1061 0.1108 0.1104 0.1201 0.1201

0.1005 0.0914 0.0858 0.0841 0.0886 0.0882 0.0975 0.0975

1.018 1.015 1.675 1.675 2.187 2.187 3.181 3.181

0.7006 0.6914 0.6858 0.6841 0.6887 0.6882 0.6975 0.6975

3.225 3.205 3.247 3.247 3.227 3.227 3.181 3.181

3.225 3.205 3.247 3.247 3.227 3.227 3.181 3.181

2.596 2.577 2.620 2.620 2.599 2.599 2.552 2.552

0.3342 0.3363 0.3268 0.3270 1.683 1.819

0.4793 0.4854 0.5369 0.5496 0.6169 0.6298

0.4341 0.4352 0.7259 0.7215 1.495 1.475

0.4033 0.4063 0.4154 0.4203 0.3716 0.3793

1.542 1.545 1.609 1.610 1.644 1.639

1.395 1.394 1.377 1.380 1.187 1.202

1.131 1.128 1.085 1.098 3.076 2.732

% 0.9717 0.8437

2

0.7918

3

~

+I

-0.9717 0.8437 0.7918

--

5

$ m z

r2 %

m

0.7027

a

0.6208

2

0.5627

5 0

TABLE

0.5

0.01

0

0

0.15 0.4 0.6 1 .o (a)

-0.2

0

0.15 0.4 0.6 1 .o (a)

-0.6

0

0.15 0.4 0.6

0.02

0

0

0.2

z

V (Continued)

N

0.4907 0.4893 0.6585 0.6583 0.7623 0.7622 0.9278 0.9278

0.3422 0.3466 0.4208 0.4207 0.4691 0.4691 0.4803 0.4803

0.3803 0.3791 0.4602 0.4601 0.5032 0.5032 0.5420 0.5420

0.3909 0.3907 0.6170 0.6167 0.7846 0.7844 1.147 1.147

0.2625 0.2603 0.2864 0.2861 0.2677 0.2676 0.3463 0.3463

1.337 1.332 1.340 1.339 1.203 1.202 1.437 1.437

1.111 1.104 1.118 1.117 1.044 1.044 1.147 1.147

0.8468 0.8389 0.8351 0.8342 Oi6484 0.6482 1.114 1.114

0.3647 0.3631 0 * 5333 0.5330 0.6367 0.6367 0.8126 0.8126

0.2392 0.2383 0.3093 0.3092 0.3420 0.3420 0.3815 0.3815

0.2554 0.2543 0.3321 0.3319 0.3688 0.3688 0.4156 0.4156

0.3593 0.3582 0.6430 0.6427 0.8506 0.8505

1.648 1.643 1.661 1.661 1.644 1.644 1.684 1.684

1.512 1.508 1.447 1.446 1.383 1.383 1.265 1.265

1.356 1.352 1.340 1.339 1.332

1.265 1.265

0.3822 0.3811 0.3963 0.3961 0.3935 0.3935 0.3793 0.3793

0.1898 0.1884 0.3400 0.3398 0.4407 0.4406 0.6138 0.6138

0.1104 0.1094 0.1553 0.1552 0.1854 0.1854 0.2280 0.2280

0.0850 0.0842 0.1383 0.1381 0.1697 0.1696 0.2147 0.2147

0.5001 0.4973 0.8302 0.8300 1.074 1.074 1.532 1.532

0.2865 0.2867 0.5243 0.5241 0.5031 0.5031 0.4475 0.4475

1.896 1.890 2.364 2.364 2.275 2.275 2.158 2.158

1.763 1.763 1.989 1.989 1.813 1.813 1.532 1.532

1.486 1.489 1.931 1.931 1.847 1.847 1 740 1 * 740

0.5292 0.5273

0.3517 0.3507

0.4055 0.4042

0.3575 0.3560

0.3505 0.3493

1.402 1.397

1.297 1 *292

1.175 1.170

aSolution independent of s.

0.5317 0.5174 0.5136

-_ 0.5317 0.5174

1.332

0.5136

1.378 1.378

--

-

0'

P

m

t;!

U N

5a

14

P

0.5317 0.5174 0.5136

-0.5515

0 P

Rrn

TABLE V (Continued)

0

s

0.5

0.3

f"

0

tw

01

0

0.4 0.6 1 .o (a)

-0.2

0

0.15 0.4 0.6 1 .,(a)

-0.6

0

0.15 0.4 0.6

1

0.6968 0.6935 0.7740 0.7731 0.9278 0.9278

0.4406 0.4391 0.4529 0.4525 0.4803 0.4803

0 -4964 0.4944 0.5108 0.5102 0.5420 0.5420

0.6075 0.6037 0.7875 0.7863 1.147 1.147

0.3695 0.3670 0.3615 0.3608 0.3463 0.3463

1.491 1.481 1.469 1.466 1.437 1.437

1.297 1.290 1.242 1.240 1.147 1.147

1.150 1.143 1.136 1.134 1.114 1.114

0.4973 0.4793 0.5732 0.5698 0.6537 0.6527 0.8126 0.8126

0.3495 0.3367 0.3424 0.3405 0.3543 0.3538 0.3815 0.3815

0.3760 0.3612 0.3691 0.3668 0.3836 0.3830 0.4156 0.4156

0.3989 0.3976 0.6606 0.6575 0 8653 0.8643 1.265 1.265

0.4394 0.4509 0.4252 0.4235 0.4141 0.4136 0.3793 0.3793

1.745 1.771 1.721 1.715 1.712 1.710 1.684 1.684

1.624 1.630 1.502 1.496 1.424 1.423 1.265 1.265

1.441 1.450 1.403 1.398 1.397 1.396 1.378 1.378

0.2762 0.2617 0.3682 0.3652 0.4530 0.4522 0.6138 0.6138

0.1779 0.1685 0.1813 0.1795 0.1971 0.1966 0.2274 0.2274

0.1587 0.1482 0.1644 0.1623 0.1815 0.1810 0.2146 0.2146

0.5437 0.5362 0.8431 0.8410 1.080 1.080 1.532 1.532

0.5998 0.5888 0.5466 0.5447 0.5128 0.5124 0.4477 0.4477

2.587 2.552 2.396 2.391 2.296 2.294 2.162 2.162

2.306 2.271 2.008 2.003 1.824 1.823 1.532 1.532

2.073 2.041 1.941 1.937 1.860 1.859 1.750 1.750

aSolution independent of s.

0.8437 0.7918

_0.9717 0.8437 0.7918

-0.9717 0.8437 0.7918

--

z

TABLE V (Continued)

1

0.01

0

0.9

0.6889 0.6884 0.9225 0.9232 1.080 1.077 1.216 1.216

0.3577 0.3577 0.3573 0.3578 0.2915 0.2906 0.0135 0.0121

0.5201 0.5199 0.5768 0.5779 0.6227 0.6196 0.6493 0.6505

0.2919 0.2913 0.5547 0.5551 0.7333 0.7319 0.9023 0.9026

0.2376 0.2342 0.3678 0.3683 0.3466 0.3455 0.3135 0.3136

0.4913 0.4804 0.567 1.569 1.598 1.593 1.597 1.600

0.9650 0.9574 1.179 1.180 1.092 1.090 0.9885 0.9885

0.7918 0.7837 1.041 1.041 0.8959 0.8952 0.4309 0.4292

0.4

1.825 1.975

0.8582 0.9369

1.720 1.878

0.1881 0.1740

0.1057 0.0978

0.4019 0.3703

0.3958 0.3655

0.3462 0.3192

0.4

0.8176 0.8162 0.9664 0.9660

0.4184 0.4179 0.4498 0.4496

0.4825 0.4817 0.5181 0.5179

0.4682 0.4675 0.6296 0.6294

0.3494 0.3488 0.3368 0.3366

1.424 1.422 1.434 1.433

1.180 1.179 1.106 1.106

1.187 1.185 1.192 1.192

0.6795 0.6715 0.8784 0.8780 1.021 1.021

0.4550 0.4555 0.5543 0.5580 0.6526 0.6553

0.4566 0.4521 0.5113 0.5119 0.5580 0.5589

0.3003 0.2981 0.5064 0.5064 0.6602 0.6601

-0.1013 -0.1471 -0.2382 -0.2592 -0.2510 -0.2595

0.3296 0.2247 -0.0073 -0.0498 -0.0737 -0.0905

0.0345 -0.0863 -0.8562 -0 9269 -1.835 -1.877

0.15 0.4 0.6 0.8

-0.2 0.05

0

1

0

0.6 0.1

0

0

0.15 0.4 0.6

0.2

P

1.247 1.183 0.9784 0.9441 0.5849 0 5598

-

-

0.5450 0.5249 0.5192 0.5158 0.5267 0.5811 0.5644 0.7384 0.6497 0.6210

0

0

0.5

1.686 1.686

0.4311 0.4312

0.6205 0.6207

0.5346 0.5347

0.0818 0.0816

1.309 1.310

0.4055 0.4051

0.9468 0.9469

0.7805

-0.5

0

0.05

0.4925 0.4496

0.2520 0.2315

0.2712 0.2458

0.2870 0.2749

0.5333 0.5099

2.307 2.216

1 * 879 1.791

1.845 1.764

1.0426

TABLE V (Continued)

1

0.3

0

0

0.4 0.6

0.9

0.15 0.6

2

0.01

0

0

0.6

0.8554 0.8512 0.9926 0.9915

0.4476 0.4458 0.4965 0.4964

0.5140 0.5117 0.5375 0.5372

0.4848 0.4824 0.6520 0.6511

0.3705 0.3687 0.2582 0.2566

1.489 1.482 1.370 1.362

1.226 1.220 0.9425 0.9388

1.235 1.229 0.7262 0.7192

0.7397 0.7351 1.031 1.035

0.3716 0.3695 0.2743 0.2750

0.5242 0.5215 0.5597 0.5682

0.3452 0.3430 0.7070 0.7088

0.3945 0.3920 0.3244 0.3262

1.583 1.573 1.486 1.501

1.339 1.330 1.061 1.063

1.169 1.161 0.8859 0.8842

0.8694

1.334 1.334

1.031 1.030

0.5433 0.5433

0.5452 0.5451

0.2251 0.2253

1.272 1.272

0.7623 0.7629

0.5427 0.5446

0.5136

0.8437 0.7918 1.080

LI, 0

2 2 0

5

$ w 0

2

tl

TABLE VI SOLUTION OF THE OUTER LIMIT EQUATIONS FOR

fi

+m

-*

~

0.5

0

0-1

0.3333

1

~~~

~~~

0.9

1

0.7

0.4 0.6

1.128 1.249

0.8388 0.9283

-2.487 -3.383

0.8330 0.8262

-3.856 -5.055

0.6067 0.5952

1

0

0.7

0.4 0.6

1.128 1.249

0.8388 0.9283

-2.487 -3.383

0 8330 0.8262

-3.856 -5.055

0.6067 0.5952

0.5

0.7

0.4 0.6

0.9489 1.050

0.8077 0.8939

-3 -008 -4.056

0.9803 0.9771

-4.641 -6.043

0.6971 0.6876

0.9

1

0.7

0.15 0.4 0.6

0.4837 0.6181 0.6840

0.4421 0.5650 0.6253

-0.0764 -0.5016 -0.8623

1.269 1.344 1.348

-0.1279 -0.9260 -1.529

0.9456 0.9786 0.9723

1

0

0.7

0.15 0.4 0.6

0.6535 0.8351 0.9242

0.5016 0.6410 0.7094

-0.0039 -0.3297 -0.6099

0.9708 1.011 1.007

0.0087 -0.6133 -1.085

0.7648 0.7824 0.7734

0.5

0.7

0.5495 0.7022 0.7772

0.4758 0.6080 0.6729

-0.0485 -0.4268 -0.7489

1.129 1.189 1.190

-0.0761 -0.7898 - 1.329

0.8555 0.8821 0.8750

0

--

0.15 0.4 0.6

0.7

0.15 0.4 0.6 1 .o

0.4154 0.5309 0.5875 0.6676

0.3019 0.2538 0.1795 0.0

1.082 1.140 1.143 2.128

0.7833 0.5792 0.3866 0.0

0.9215 0.9653 0.9666 1.954

0.5

--

0.4965 0.6345 0.7022 0.7979

0.15 0.4 0.6 1.0

0.4175 0.5336 0.5905 0.6709

0.3789 0.4842 0.5359 0.6089

0.3310 0.2783 0.1968 0.0

1.262 1.342 1.351 2.342

0.8588 0.6350 0.4239 0.0

1.010 1.058 1.060 2.046

0.7

-

m

n

U

H

W

n U

H

W N

n

U

H

W d

N U

h

-

aJ

W 0

n

0

-

W

)r

3

h

LL

U

M

-

I

a"b'

10

b

U

'us 3

SOLUTIONS OF BOUNDARY-LAYER EQUATIONS

0 0 0 H 0 0 0 d

In

0

r-

4 0 0 0 d 0 0 0

m

0

I-

0000

. . . . . . . . . . . . . . , . . . . . d. o. 0. 0.

0004

0

PI

d 0 0 0 0 0 0 0

0

r-

I I

0

ln

I

0

0:

I I

I

0

P

0

I I

0

tn

I

I

0004

?P???P?? ???? ?'P???P?? ?P??

ln

0 0 0 4 0 0 0 H

tn

0

f-

ln

r-

I I

In

I

0

m 0

rl

I

0

Y

4

0

ln

0

TABLE VI (Continued)

0.5

-0.4

0.7

0

1

0.9

--

0.7

0.15 0.4 0.6 1.0

0.2040 0.2804 0.3181 0.3714

0.2012 0.2760 0.3130 0.3652

0.1

0.9

1

0.7

0.6877 1.131 1.235

1

0

0.7

0.15 0.4 0.6 0.15 0.4 0.6

0.5

0.7

0.9

1

0.7

1

0

0.7

0.5

--

0.3333

1

0

0.5268 0.8412 0.9182

0.5451 0.4399 0.3066 0.0 -1.516 -2.449 -3.274

0.9729 0.8233 0.7998

-2.505 -3.780 -4.865

0.7355 0.6049 0.5833

0.9269 1.131 1.235

0.6899 0.8412 0.9182

-1.347 -2.449 -3.274

0.8417 0.8233 0.7998

-3.780 -4.865

-2.231

0.6287 0.6049 0.5833

0.15 0.4 0.6

0.8395 1.021 1.114

0.7152 0.8698 0.9486

-1.572 -2.771 -3.668

0.9169 0.9058 0.8835

-2.576 -4.256 -5.435

0.6679 0.6494 0.6290

0.15 0.4 0.6 0.15 0.4 0.6

0.5855 0.7091 0.7719

0.5355 0.6483 0.7025

-0.0812 -0.4520 -0.7596

1.209 .1.211 1.189

-0.1396 -0.8341 -1.345

0.8938 1.050 0 8608

0.6966 0.8468 0.9231

0.5357 0.6500 0.7081

-0.0101 -0.3338 -0.6048

-0.0036 -0.6207 -1.075

0.8116 0.7929 0.7707

0.7

0.15 0.4 0.6

0.6308 0.7649 0.8329

0.5467 0.6623 0.7209

- 0.4041

-0-6938

1.036 1.024 0.9998 1.129 1.126 1 104

-0.0920 -0.7475 -1.230

0.8497 0.8358 0.8147

0.7

0.15 0.4 0.6 1 .o

0.5450 0.6582 0.7153 0.7979

0.4598 0.5530 0.5999 0.6676

0.3313 0.2637 0.1831 0.0

1.180 1.185 1.168 2.128

0.8471 0.6000 0.3941 0.0

0.9966 1.0000 0.9853 1 954

-0.0559

2.202 2.255 2.246 3.207

1.431 1.008 0.6619 0.0

1.684 1.680 1.655 2.604

-

TABLE VI (Continued)

2

U

fw

0.7

0

-0.2

tS

1

1

a

Pr

tw

0.5

0.7

0.15 0.4 0.6 1 .o

0.4931 0.5943 0.6454 0.7191

0.4495 0 5406 0.5865 0.6526

0.3389 0.2698 0.1873 0.0

1.289 1.304 1.289 2.252

0.8666 0.6138 0.4032 0.0

1.019 0.8082 1.008 1.976

0.9

0.7

0.15 0.4 0.6 1 .o

0.3887 0.4677 0.5075 0.5649

0.3810 0.4582 0.4971 0.5531

0.3998 0.3183 0.2210 0.0

1.618 1.647 1.634 2.594

1.022 0.7242 0.4757 0.0

1.203 1.207 1.189 2.151

0

0.7

0.15 0.4 0.6 1 .o

0.4227 0.5355 0.5926 0.6751

0.3729 0.4663 0.5133 0.5811

0.3714 0.2898 0.1996 0.0

1.345 1.317 1.285 2.228

0.9575 0.6613 0.4304 0.0

1.126 1.102 1.076 2.030

0.5

0.7

0.15 0.4 0.6 1.0

0.3929 0.4941 0.5452 0.6190

0.3663 0.4576 0.5036 0.5699

0.3789 0.2958 0.2038 0.0

1.452

0.9768 0.6750 0.4394 0.0

1.149 1.125 1.098 2.052

0.15 0.4 0.6 1.0

0.3261 0.4053 0.4451 0.5026

0.3208 0.3983 0.4373 0.4934

0.4394 0.3441 0.2374 0.0

1.778 1.776 1.749 2.692

1.131 0.7847 0.5115 0.0

1.331 1.308 1.279 2.227

0.15 0.4 0.6

0.3149 0.4247 0.4807 0.5619

0.2941 0.3863 0.4328 0.5001

0.4182 0.3193 0.2182 0.0

1.540 1.467 1.416 2.338

1.089 0.7311 0.4710 0.0

1.281 1.219 1.177 2.114

0.9

-0.4

1

0

m

0.7

0.7

1 .o

-

1.435 1.406 2.351

TABLE VI (Continued) 2

t

fw

0.7

1

-0.4

0

0

c")

Pr

tw

gi(o)

ei(W

I2

I#)

11(2)

5(3)

m

0.5

--

0.7

0.15 0.4 0.6 1.0

0.3031 0.4025 0.4529 0.5258

0.2908 0.3810 0.4265 0.4923

0.4255 0.3252 0.2232 0.0

1.644 1.582 1.535 2.460

1.107 0.7445 0.4798 0.0

1.303 1.241 1.200 2.136

0.9

--

0.7

0.15 0.4 0.6 1 .o

0.2683 0.3469 0.3865 0.4437

0.2653 0.3421 0.3809 0.4368

0.4844 0.3727 0.2553 0.0

1.960 1.918 1.874 2.797

1.257 0.8523 0.5510 0.0

1.478 1.420 1.377 2.309

0.1

1

1

1

0 0.5

1 .o

0.9076 1.189 1.339

0.9076 1.189 1.339

-1.115 -2.871 -4.528

0.9629 0.7798 0.7030

-1.841 -4.278 -6.327

0.6826 0.5481 0.4929

0.1538

1

1

1

0 0.5 1 .o

0.8258 1.075 1.208

0.8258 0.075 1.208

-0.5459 -1.736 -2.885

1.051 0.8575 0.7753

-0.9319 -2.756 -4.265

0.7461 0.6034 0.5442

0.3333

1

1

1

0 0.5 1 .o

0.7063 0.9028 1.010

0.7063 0.9028 1.010

0.0364 -0.5265 -1.104

1.210 1.006 0.9161

0.1645 -0.9472 -1.832

0.8615 0.7098 0.6445

0.625

1

1

1

0.6326 0.7915 0.8793

0.6326 0.7915 0.8793

0.2647 -0.0202 -0.3317

1.331 1.130 1.037

0.7220 -9.0382 -0.6223

0.9504 0.7994 0.7317

1

1

--

0 0.5 1 .o

1

0 0.5 1.0

0.5898 0.7230 0.7979

0.5898 0.7230 0.7979

0.3567 0.1965 0.0

1.412 1.220 1.128

1.010 0.4327 0.0

1.010 0.8654 0.7979

1

SOLUTIONS OF BOUNDARY-LAYER EQUATIONS TABLE VII SOLUTION OF THE INNERLIMITEQUATIONS FOR p +

~~~~

~

V

0.5

a (*I

0.2025 0.4050 0.4920 1.0000

0

1.1547 1.1672 1.1816 1.1885 1.2533

0.75

0 0.2025 0.4050 0.4920 1 .oooo

1.1547 1.1609 1.1679 1.1712 1.1991

1 .o

all

1.1547

*

441

C. FORBES DEWEY, JR.,

442

AND JOSEPH

F. GROSS

SYMBOLS A,, ..., Ah constants defined in Eqs. (106), (116), and (121) function defined in Eqs. (110), ( 154) function defined in Eq. (86)

Chapman-Rubesin constant limit of f ( a , t ) as t + m local skin friction coefficient ;

c,

=

2r/pm Urn2

local heat transfer coefficient ; qwlpm Um(Haw - Hw)

mass concentration of ith component specific heat of the ith component binary diffusion coefficient function as defined in Eq. (103) numerical difference function defined in Eq. (82) function as defined in Eq. (86) similarity function defined in Eq. (23) transformed fluid velocity as defined in Eq. (12) wall constant defined by Eq. (22) function defined in Eq. (83) dimensionless enthalpy function,

(HIH,)

transverse-velocity function, w / w , numerical difference function defined in Eq. (84) total enthalpy of mixture,

h + t(U2

+ d)

numerical difference function defined in Eq. (85) chemical enthalpy of ith component I,, Z2 integrals defined by Eqs. (75) and (76) 11(1),Zl(2), ZL(3) integrals defined by

(77)-(79)

,Jn(t)

j

k

function defined as the nth integral of the error function geometrical index in boundary layer equations ; also Reynolds analogy ratio 2C,,/C, thermal conductivity of the mixture

Lewis number pDI2cp/k Mach number m constant exponent defined in Eq. (34) P7 Prandtl number, c P p / k P fluid pressure 4 heat flow R gas constant r radial coordinate defined in Fig. 1 S Sutherland constant s c Schmidt number, p/pD12

Le

M

S

SIT0

transformed velocity function defined in Eq. (1 50) T temperature of the fluid TO free-stream stagnation temperature t transformed similarity variable defined in Eq. (150); transformedvariable defined by Eq. (109) taw dimensionless adiabatic wall temperature, t = t,, for #(Of = 0 Eckert reference temperature defined in Eq. (53c) dimensionless sweep parameter defined by Eq. (32), 1 ( Urn2/2H,)sin2 A t I" dimensionless wall enthalpy ratio defined by Eq. (31), T,,./To u r n free-stream velocity II flow velocity in the x direction flow velocity in the y direction V W constant defined in Eq. (141) W flow velocity in the z direction (transverse velocity) X coordinate in direction of flow Y coordinate normal to the surface Zi dimensionless concentration function defined in Eq. (15) z dimensionless function, (1 - 2,); also, coordinate transverse to the flow direction pressure gradient parameter defined in Eq. (34) Y adiabatic constant, c,/c,. 6* boundary layer displacement thickness defined by Eq. (80) SO

SOLUTIONS OF BOUNDARY-LAYER EQUATIONS E

7

0 0 K

A h p

5

p u uI u2

asymptotic parameter defined in Eq. (98) similarity variable defined in Eq. (11) boundary-layer momentum thickness defined in Eq. (81) dimensionless enthalpy function, ( H - W / ( H e - Hw) trial function for solution of firstorder equation defined by Eq. (123) sweep angle dimensionless density-viscosity product, (l/C) ( p d p e iue) viscosity transformed x coordinate defined in Eq. (10) fluid density hypersonic parameter ( U , 2/2He) modified hypersonic parameter, (Um2/2He)(Uehrn)’ modified hypersonic parameter, (Um2/2He)[(ue/um)2COG A sin2 A]

+

4.43

total skin friction skin friction in spanwise direction skin friction in x direction T, CJ function defined in Eq. (106) x variable defined by Eq. (141) x(a) function defined in Eq. (112) xo constant defined in Eq. (141) Y ( x , y ) stream fun,ction defined by Eq. (9) ‘PI,. ..,Y, functionsdefined in Eqs. (1 18), (119), and (122) w exponent in the viscosity-temperature law, p TW w, constant defined in Eq. (53b) T

rZ

-

SUBSCRIPTS ( )e function at the edge of the boundary layer ( ) w function at the wall ( )rn function evaluated in the free stream

ACKNOWLEDGMENT T h e authors wish to express their gratitude to Susan Fredlund and Jeannine McGannLamar for their expert assistance in performing the numerical calculations of this chapter. The original numerical procedures were developed by Kathrine Purdom, presently with Lockheed Missiles and Space Division, Sunnyvale, California and James I. Carlstadt of The RAND Corporation. This research is supported and monitored by the Advanced Research Projects Agency under Contract No. SD-79. Any views or conclusions contained in this work should not be interpreted as representing the official opinion or policy of ARPA. One of us (C. F. Dewey, Jr.) was the recipient of a University of Colorado Faculty Fellowship during preparation of this manuscript. We have been privileged to receive a review of this manuscript by Nelson H. Kemp of the AVCO Everett Research Laboratory, Everett, Massachusetts.

REFERENCES 1 . G . Kuerti, T h e laminar boundary layer in compressible flow, Advan. Appl. Mech. 2 , 23-92 (1951). 2. W. D. Hayes and R. F. Probstein, “Hypersonic Flow Theory,” 1st ed., 292-312. Academic Press, New York, 1959. 3. I. E. Beckwith, Similar solutions for the compressible boundary layer on a yawed cylinder with transpiration cooling. NA SA Tech. Rept. TR-R-42(1959). 4. J. F. Gross and C. F. Dewey, Jr., Similar solutions of the laminar boundary-layer equations with variable fluid properties, in “Fluid Dynamic Transactions” (W. Fizdon, ed.) Vol. 2, pp. 529-548. Pergamon Press, New York, 1965.

444

C. FORBES DEWEY, JR.,

AND JOSEPH

F. GROSS

5. N. H. Kemp, Approximate analytical solution of similarity boundary layer equations with variable fluid properties. Publi. No. 64-6. Muss. Inst. Technol. Fluid Mech. Lab., Cambridge, Massachusetts, September 1964. 6. T. Y. Li and H. T. Nagamatsu, Shock-wave effects on the laminar skin friction of an insulated flat plate at hypersonic speeds. J . Aerospace Sci. 20, 345-355 (1953). 7. C. B. Cohen and E. Reshotko, Similar solutions for the compressible laminar boundary layer with heat transfer and arbitrary pressure gradient. N A C A Rept. 1293 (1956). 8. E. Reshotko and I. E. Beckwith, Compressible laminar boundary layer over a yawed infinite cylinder with heat transfer and arbitrary Prandtl number. N A C A Rept. 1379 (1958). 9. I. E. Beckwith and N. B. Cohen, Application of similar solutions to calculation of laminar heat transfer on bodies with yaw and large pressure gradient in high-speed flow. N A S A Tech. Note TN-D-625 (1961). 10. C. F. Dewey, Jr., Use of local similarity concepts in hypersonic viscous interaction problems. A I A A J . 1, 20-33 (1963). 11. A. M. 0. Smith, Improved solutions of the Falkner and Skan boundary-layer equation. Inst. Aerospace. Sci. Sherman M. Fairchild Fund Paper FF-10, 1954. See also: A. M. 0. Smith and D. W. Clutter, Solution of the incompressibk laminar boundarylayer equations, A I A A J . 1, 2062-2071 (1963). 12. A. Busemann, in “Handbuch der Experimental Physik,” Vol. 4, Pt. 1, p. 336. Akad. Verlagsges., Leipzig, 1931. 13. Th. Von Karman and H. S. Tsien, Boundary layer in compressible fluids. .I. Aerospace Sci. 5 , 227 (1938). 14. L. Crocco, The laminar boundary layer in gases. Monogra5c Sci. Aeronaut. No. 3, December 1946 (Translated as North Am. Rept. APL/JHU CF-1038, 1948). 15. G. B. W. Young, and E. Janssen, The compressible boundary layer. J . Aerospace Sci. 19, 229 (1 952). 16. E. R. Van Driest, Investigation of the laminar boundary layer in compressible fluids using the Crocco method. N A C A Tech. Note TN 2597 (1952). 17. H. W. Emmons and D. Leigh, Tabulation of the Blasius function with blowing and suction. Rept. 15966. Fluid Motion Sub-Comm., Aeronaut Res. Council. 1953. 18. D. Meksyn, “New Methods in Laminar Boundary-Layer Theory.” Pergamon Press, Oxford, 1961. 19. L. Rosenhead (ed.), “Laminar Boundary Layers.” Oxford Univ. Press (Clarendon), London and New York, 1963. 20. K. Stewartson, “The Theory of Laminar Boundary Layers in Compressible Fluids”. Oxford Univ. Press (Clarendon), London and New York, 1964. 21. F. K. Moore (ed.), “Theory of Laminar Flows,” Vol. 4, “High-speed Aerodynamics and Jet Propulsion.” Princeton Univ Press, Princeton, New Jersey, 1964. 22. D. B. Spalding and H. L. Evans, Mass transfer through laminar boundary layers: 2, Auxiliary functions for the velocity boundary layer. Intern. ,I. Heat Mass Transjer 2, 199-221. 1961. 23. J. Pretsch, Die laminar Grenzschicht bei Starkem Absaugen und Ausblasen. Untersuch. Mitt. Deut. Luftfahrtforsch., No. 3091 (1944). 24. E. J. Watson, The asymptotic theory of boundary layer flow with suction. Brit. Aeronaut Rex. Council R and M No. 2619 (1952). 25. E. R. G. Eckert, P. L. Donoughe, and B. J. Moore, Velocity and friction characteristics of laminar viscous boundary layer and channel flow with ejection or suction N A C A Tech. Note TN 4102, 1957. 26. B. Thwaites, The development of the laminar boundary layer under conditions of

.

SOLUTIONS OF BOUNDARY-LAYER EQUATIONS

445

continuous suction: I, O n similar profiles Rept. 11,830. Brit. Aeronaut. Res. Council, 1948. 27. H. Schlichting and K. Bussman, Exakte Losungen fur die Laminare Grenzschicht rnit Absaugung und Ausblasen. Schriften Deut. Akad. Luftfahrtforsch. 76, No. 2 (1943). 28. W. Mangler, Laminare Grenzschicht mit Absaugen und Ausblasen. Untersucht. Mitt. Deut. Luftfahrtforsch. No. 3087 (1944). 29. H. Schaefer, Laminare Grenzschicht zur Potentialstromung U = u1 x“‘ mit Absaugung und Ausblasen. Deut. Luftfahrtforsch No. 2043 (1944). 30. J. C. Y. Koh and J. P. Hartnett, Skin friction and heat transfer for incompressible laminar flow over porous wedges. Intern. J . Heat Mass Transfer (Corrigendum) 5 , 593 (1962). 31. K. Stewartson, Further solutions of the Falkner-Skan equation. Proc. Cambridge Phil. SOC.50,454-465 (1954). 32. D. R. Hartree, On an equation occurring in Falkner and Skan’s approximate treatment of the equations of the boundary layer. Proc. Cambridge Phil. Sac. 33, 223-239 (1937). 33. J. H. Hufen and W. Wuest, Ahnliche Losungen bei kompressiblen Grenzschichten mit Warmeubergang und Absaugung oder Ausblasen. 2. Angew. Math. Phys. 17, 385-390 (1966). 34. I. Tani, On the approximate solution of the laminar boundary-layer equations. J . Aerospace Sci. 21 487-504 (1954). 35. F. B. Hanson and P. D. Richardson, Use of a transcendental approximation in laminar boundary layer analysis. J . Mech. Eng. Sci. 7, 131-137 (1965). 36. K.-T. Yang, An improved integral procedure for compressible laminar boundary-layer analysis. J . Appl. Mech. 28, 9-20 (1961). 37. A. Pallone, Nonsimilar solutions of the compressible laminar boundary-layer equations with application to the upstream-transpiration cooling problem. J . Aerospace Sci. 28, 449-456 (1961). 38. 0. M . Belotserkovski and P. I. Chuskin, T h e numerical solution of problems in gas dynamics. In “Basic Developments in Fluid Mechanics” (M. Holt, ed.), p. 1. Academic Press, New York, 1965. 39. A. M . 0. Smith and D. W. Clutter, Solution of the incompressible laminar boundarylayer equations. AZAA ./. 1, 2062-2071 (1963). 40. I. Flugge-Lotz and D. C. Baxter, Computation of the compressible laminar boundarylayer flow including displacement-thickness interaction using finite-difference methods. Tech. Rept. No. 131 (AFOSR2206). Stanford Univ., Stanford, California, January 1962. 41. D. C. Baxter and I. Flugge-Lotz,Z. Angew. Math. Phys. 9b,81 (1958). 42. L. Lees, Laminar heat transfer over blunt-nosed bodies at hypersonic flight speeds. ,let Propulsion, 26, 259-269 (1956). 43. A. M. 0. Smith, Rapid laminar boundary-layer calculations by piecewise application of similar solutions. J . Aerospace Sci. 23, 901-912 (1956). 44. N. H. Kemp, P. H. Rose, and R. W. Detra, Laminar heat transfer around blunt bodies in dissociated air. .I. Aerospace Sci. 26, 421-430 (1959). 45. B. Thwaites, Approximate calculation of the laminar boundary layer. Aeroit. Quart. 1, 245-280 (1949). 46. L. Lees and B. L. Reeves, Supersonic separated and reattaching laminar flows A I A A J . 2, 1907-1920 (1964). 47. B. L. Reeves and L. Lees, Theory of laminar near wake of blunt bodies in hypersonic HOW. AZAA J. 3, 2061-2074 (1965).

446

C. FORBES DEWEY, JR.,

AND JOSEPH

F. GROSS

48. H. J. Merk, Rapid calculations for boundary-layer transfer using wedge solutions and asymptotic expansions. J . Fluid Mech. 5 , 460-480 (1959). 49. W. B. Bush, Local similarity expansions of the boundary-layer equations. AIAA J . 2, 1857-1858 (1964). 50. V. M . Falkner and S. W. Skan, Solutions of the boundary-layer equations. Phil. Mag. 12,865-896 1931. 51. W. B. Bush, A method of obtaining an approximate solution of the laminar boundarylayer equations. J . Aerospace Sci. 28, 350-351 (1961). 52. D. E. Coles, T h e laminar boundary layer near a sonic throat. Proc. Heat Trans. and Fluid Mech. Inst., pp. 119-137. Stanford Univ. Press, Stanford, California, 1957. 53. P. A. Lagerstrom, Laminar flow theory. In “High Speed Aerodynamics and Jet Propulsion,” Vol. 4 : “Theory of Laminar Flows” (F. K. Moore, ed.), pp. 125-129. Princeton Univ. Press, Princeton, New Jersey, 1964. 54. S. Kaplun and P. A. Lagerstrom, Asymptotic expansions of Navier-Stokes solutions for small Reynolds numbers; Low Reynolds number flow past a circular cylinder; Note on the preceding two papers. J . Math. Mech. 6 , 585-606 (1957). 55. P. A. Lagerstrom and J. D. Cole, Examples illustrating expansion procedures for the Navier-Stokes equations. J . Ratl. Mech. Anal. 4, 817-882 (1955). 56. M. Van Dyke, “Perturbation Methods in Fluid Mechanics.” Academic Press, New York, 1964. 57. T. Y. Li and J. F. Gross, Hypersonic strong viscous interaction on a flat plate with mass transfer. Heat Transfer and Fluid Mechanics Institute, 1961. Stanford Univ. Press, Stanford, California, 1961. 58. R. J. Whalen, Boundary-layer interaction of a yawed infinite wing in hypersonic flow. J . Aeron. Sci. 26, 839-851 (1959). 59. L. H. Back and A. B. Witte, Prediction of heat transfer from laminar boundary layers, with emphasis on large free-stream velocity gradients and highly cooled walls. /. Heat Transfer 88, 249-256 (1966). 60. R. Narasimha and S. S. Vasantha, Laminar boundary layer on a flat plate at high Prandtl number. Z . Angew. Math. Phys. 17, 585-592 (1966).

Author Index Numbers in parentheses are reference numbers and indicate that an author’s work is referred to although - his name is not cited in the text. Numbers in italic show the page on which the complete reference is listed. Beckman, E. L., 90(27), 91,92,93,94(27),

A Abdalla, K. L., 228 Abramson, P., 213(98), 214(98), 227 Acrivos, A., 33,63 Adams, C. C., 98,239 Adams, D.E., 42,64 Adams, J. M., 206(85), 226 Adelberg, M., 157,158, 201,224,226 Advani, G . H., 90(28), 91(28), 92(28), 95

(28),109(28), 110(28), 238 T., 206(84), 226

95(33), 96,138

Beckmann, W., 5, 15,62 Beckwith, I. E., 318,325, 330,331, 334,

341 342,343,344,443,444 I

Belding, H. S., 93,238 Belotserkovski, 0. M., 334,445 Benedikt, E. T., 147(4), 222 Benzinger, T. H., 120(93), 240 Berenson, P. J., 98(41, 42), lOO(41, 43),

239,171,225

Alad’ev, I.

Albers,J.A.,210,211,212(96),213(96),227 Bevans, R. S., 108(53), 239 Blalock, A., 118(77), 240 Allingham, W.D., 206,227 Blankenship, V. D., 48,49(83), 64 Altman, P. L., 242

Altmos, D.A,, 228 Amenitskii, A. N., 206(84), 226 Anateg, R.,82(20), 238 Anderson, A. D., 35,63 Anderson, J. E., 269(32), 284(32),285,286,

287,288,289,324

Andracchio, C. R.,228 Arnett, C. D., 150,223 Arpaci, V. S., 49(85),64 Aydelott, J. C., 182(66), 183(66), 186(66),

195(66), 196(66), 225,227,228

B Bach, G. R., 241 (7),242(7),323 Back, L. H., 352,446 Bailey, R.V., 191 (69), 225 Baker, M.J., 50, 64 Barzelay, M.E., 238,323 Batchelor, G. K., 152,223 Bates, D.R.,251,252(15), 323 Baxter, D.C., 334,J45 Bayley, F. J., 56,64 Bazett, H.C . , 88(25), 91(25), 238

Blatt, T., 205(80), 206(80), 226 Blatz, W.J., 114(67), 240 Bobco, R.P., 25,63 Bond, G., 109(55), 110(55), 139 Bonilla, C. F., 166,224 Booda, L. L., 109(56,57),110(56), 239 Boinjakovid, F., 259(23), 260(24), 262(23),

263(23), 264(24), 313

Braginskii, S. I., 268,314 Brett, J. R., 95,138 Briggs, D.G . , 38,39,64 Brindley, J., 10,62 Brogan, T.R.,310(69),326 Bromley, L. A,, 173,174,225 Brown, D.D., 202(77), 203(77),226 Buhler, R.D., 283(48), 325 Bunche, C. M., 116(71), 1-10 Bundy, R.D., 206(82), 226 Burriss, W.L., 98,100,239 Burton, A. C., 88(25), 91(25), 238 Bush, W. B., 335,337,446 Buskirk, E. R.,120,I40 Bussman, A., 331,443 Bussman, K., 331,4J5

447

AUTHORINDEX

448 C

Cambel, A. B., 257(22), 258(22), 259(22), 260(22), 261 (22), 262, 265(22), 266 (22), 313 Campbell, J. B., 115(70), 140 Cann, G. L., 283(48), 315 Carlson, L. D., 92(30), 138 Carlson, W. O., 152, 223 Carne, J. B., 21, 63 Cess, R. D., 28, 63, 224, 228, 313 Chapman, S., 273(35), 314 Chen, F. F., 256(21), 313 Chianta, M. A., 129(110), 130(111), 135 (110), 141 Chiladakis, C., 170(53), 225 Chin, J. H., 153, 223 Ching, P. M., 35, 63, 230, 273,313 Choi, H. Y., 207(91), 227 Chu, N. C., 152, 223 Churchill, S. W., 36, 37, 63 Chuskin, P. I., 334, 445 Cichelli, M. T., 166, 224 Clark, J. A., 47,48,49(83, 85), 64,161,162, 169(39), 170(71), 171, 172(41), 173, 174, 206(88), 225, 226, 227 Clodfelter, R. G., 162, 163, 168, 191, 210, 224,227 Clutter, D. W., 331, 334, 444, 445 Cobine, J. D., 248(20), 255(20), 313 Cochran, T. H., 182, 183, 186, 195, 196, 225 Cohen, C. B., 330, 331, 444 Cohen, N. B., 279,314, 325, 330, 334, 341, 342, 343, 344,444 Colburn, A. P., 54, 64 Cole, J. D., 341, 446 Coles, D. E., 341, 342, 345, 446 Colombo, G. V., 228 Congelliere, J. T., 146, 222, 228 Corpas, E. L., 150(7), 222, 227, 228 Costello, C. P., 206(85), 226, 227 Cousteau, J. Y., 109(58), 139 Covino, B. G., 120(86), 140 Cowling, T. G., 273(35), 314 Crabs, C. C., 150(6), 211(96a), 222,227 Cramer, K. R., 97(38), 100(38), 139 Crandall, J. H., 53, 64 Cremers, C. J., 248(13), 303(65), 305(65), 313,315

Crocco, L., 331,444 Cummings, R. L., 210,214, 227

D D’Angelo, N., 251 (17), 313 Davis, A. H., 50, 54, 55, 59, 64 Day, R., 93(31), 94(31), 96(31), 138 Debolt, H., 283(49), 315 Delattre, M., 146, 222 Denington, R. J., 214, 227 Detra, R. W., 334(44), 445 Dewey, C. F., Jr., 318, 326(4), 330, 345, 354(4), 443, 444 Dillon, R. C., 228 Dittmer, D. S., 141 Donaldson, J. O., 153(24), 223 Donoughe, P. L., 331 (25), 444 Drake, R. M., Jr., 153,223, 270(33), 314 Drawin, H. W., 245, 251 (18), 313 Drufes, H., 284(52), 315 DuBois, E. F., 67(6, 8), 137 Dufton, A. F., 67, 137

E Ebaugh, F. G., Jr., 121(101), 140 Eberhart, R. C., 295, 315 Ebert, W. A , , 274(37), 314 Ecker, G., 295(62), 315 Eckert, E. R. G., 9, 16, 17, 31, 41, 42, 44, 51, 55, 59, 62, 63, 64, 152, 153, 157, 223, 224, 269(32), 270(33), 274, 284 (32), 285, 286, 287, 288, 289, 293(58), 294(58), 295, 296(64), 300(64), 301 (64), 303(64), 304(66), 315, 331, 444 Ede, A. J., 18, 33, 61, 63, 64 Edholm, 0. G., 94(34), 138 Ehlers, R. C., 206(86), 226 Eichhorn, R., 13, 15, 26, 27, 37, 62, 63 Eigenson, L. S., 21(31, 32), 59(32), 63 Eisenberg, M., 16(25), 63 Elenbaas, W., 17, 20, 63 Elsasser, W. M., 67, 138 Elsner, R. W., 92(30), 138 Elwert, G., 250, 259, 313 Emery, A., 152, 223 Emmons, H. W., 331, 444 Ende, W., 178,225 Enders, J. H., 150(6a), 222

AUTHORINDEX Erk, S., 152(14), 208(14), 223 Eshghy, S., 49, 64 Evans, D. G., 200, 201, 205, 226 Evans, H . L., 331,444

F Faber, 0. C., Jr., 164,165,224 Falkner, V. M., 336, 446 Fay, J. A., 271,278, 279, 281,314 Feider, M., 116(71), 140 Feldmanis, C. J., 205, 226 Felenbok, P., 245, 313 Feneberg, W., 268, 314 Feyerherm, A. M., 104(50), 106(50), 139 Finkelnburg, W., 232(3), 233(3), 234(3), 237(4), 240(4), 245, 248(3), 260, 293 (31, 313 Finston, M., 22, 59, 63, 64 Flage, R. A., 228 Flugge-Lotz, I., 334, 445 Foote, J. R., 23, 63 Forslund, R. P., 213(98), 214(98), 227 Forster, H. K., 178, 225 Forstrom, R. J., 274(37), 314 Fox, R. H., 94(34), 138 Frank-Kamenicky, D. A., 54, 64 Frea, W. J., 206, 227 Frederking, T. H. K., 173, 174, 225 Fritsch, C. A,, 30, 63 Fritz, W., 178, 181, 225 Frysinger, T. C., 182(66), 183(66), 186 (66), 195(66), 196(66), 225, 228 Fujii, T., 10, 16, 51, 57,61,62,64 Fullington, F., 92(30), 138

G Gagge, A. P., 67(9), 83, 88(25), 91 ( 2 9 , 104(47), 110, 111(63), 118(78), 138, 139,140 Gallagher, L. W., 153(24), 223 Gambill, W. R., 206, 226 Gartrell, H. E., 51 (92), 64 Gebhart, B., 32, 39, 40, 41, 42, 63, 64 Ginwala, K., 205, 206, 226 Goldman, K., 293(55), 315 Goldman, R. F., 93(33), 94(33), 95(33), 96(33), 138, 139 Goldstein, R. J., 16, 17, 38, 39, 41, 42, 63, 64, 157, 223

449

Goodell, H., 121(103), 141 Gorland, S. H., 211 (96a), 227 Graham, R. W., 201,206(86), 226 Greenberg, L., 67(5), 137 Greene, L. C., 119(81), 121(81), 123(81), 124(81), 125(81, 108), 127(81), 140, 141 Greene, N. D., 206, 226 Gregg, J. L., 12, 15, 17, 21, 23, 29, 42(18), 62,63 Grevstad, P. E., 210(93), 214(93), 227 Griem, H. R., 234(5), 242(5), 245, 313 Griffith, M. V., 135, 136,142 Griffiths, E., 50, 54, 55, 59, 64 Grigull, U., 152(14), 208(14), 223 Grober, H., 152,208,223 Grosh, R. J., 30, 63 Gross, J. F., 318, 326(4), 347, 354(4), 443, 446 Gruber, R. P., 227, 228 Gruszezynski, J. S., 280, 314 Guillemin, V., Jr., 75, 138 Guyton, Arthur C., 141

H Habip, L. M., 146, 148, 222 Haise, F. W., Jr., 150(6a), 222 Hale, F. J., 307, 309, 310, 316 Hall, A. L., 215, 218, 227 Hammel, R. L., 228 Hammond, W. H., 94(34), 138 Hanson, F. B., 334,445 Hara, T., 29, 63 Harder, R. L., 283(48), 315 Hardy, J. D., 67(6, 8), 69(12, 13), 70, 71 (13), 75(12, 17), 81(18), 82(19), 104 (47), 116(73), 117(73), 119(80, 83), 121(97, 99), 123(99), 125(108), 137, 138, 139, 140, 141 Harmathy, T. Z., 190, 225 Harper, E. Y., 153(24), 223 Hartnett, J. P., 331, 445 Hartree, D. R., 331,445 Haurwitz, B., 67, 138 Hausen, A. G., 33, 63 Hayes, W. D., 318,433 Hedgepeth, L. M., 164, 165, 224 Heilmann, R. H., 54, 64 Hellums. J. D., 36, 37, 63

AUTHORINDEX

450

Hendler, E., 121(97), 140 Hendricks, R. C., 206(86), 226 Henriques, F. C., Jr., 119(82), 125,140,141 Hermann, M., 283 (49), 315 Hermann, R., 51,64 Herrington, L. P., 67(9), 111(63), 118(78), 138, 139, 140 Hershey, F. B., 125(106), 141 Hervey, G. R., 94(34), 138 Hill, T., 67, 75, 137 Holm, F. W . , 90(28), 91(28), 92(28), 95 (28), 109(28), 110(28), 138 Holman, J. P., 51, 64 Hong, S. K.,90(26), 92(26,36), 120(86,87), 138, 140 Horton, G. K., 135, 136, 141 Hoshigaki, H., 278, 279,314 Hougen, 0. A , , 54, 64 Houghten, F. C., 104(46), 139 Howell, B. J., 120(86), 140 Howell, J. R., 169, 225 Hsieh, A. C . L., 92(30), 138 Hsu, Y. Y., 173, 201, 225,226 Hufen, J. H., 331, 445 Humphreys, C. M . , 104(49), 105(49), 139 Hunt, H., 90(27), 94(27), 138 Hurd, S. E., 153(24), 223

I Illingworth, C. R., 33, 63 Irvine, T. F., 97(38), 100(38), 139 Isoda, H., 215, 216, 217,227 Ivey, H. J., 206(83), 226

J Jackson, R. G., 228 Jackson, T . , 51, 55, 59, 64 Jakob, M., 54, 64, 155, 156, 223 Jansen, W., 205(80), 206(80), 226 Janssen, J. E., 113(64), 139, 331, 444 Jeffreys, H., 152, 223 Jennings, B. H., 104(49), 105(49), 139 Jerger, E. W., 44,46, 64 John, R. R., 283 (49), 315 Johnson, H. A., 43 (76), 64 Jones, M. C., 170(53), 225 June, R. R., 50, 64

K Kang, B. S., 90(26), 92(26), 95(26, 36), 120(86), 138, 140 Kang, D. H., 90(26), 92(26), 95(26, 361, 138 Kaplun, S., 341, 446 Kemp, N. H., 278, 279, 281,314, 318, 334, 444,445 Kenshalo, D. R., 121(102), 123(102), 141 Kerrebrock, J. L., 290, 305, 306, 307, 309, 310, 315, 316 Keshock, E. G., 145(5), 149(5), 162, 164, 175, 176, 179, 180, 181, 182, 183(60), 184, 185, 186,190, 199,222,225 Khabbag, G., 205(80), 206(80), 226 Kim, P. K., 90(26), 92(26), 95(26), 138 Kimzey, J. H., 217, 218, 227 Kincaide, W. C., 103, 139 King, C. D., 114(65), 139 Kingston, A. E., 251(14, 15, 16), 252(15), 313 Kinsey, J. L., 110(60), 139 Kirk, D. A., 151, 223 Kirkpatrick, M. E., 150, 223 Kitzinger, C., 120(93), 140 Knight, B. A , , 165, 166, 224 Knoche, K. F., 259(23), 262(23), 263(23), 313 Knoll, R. H., 228 Koch, W. B., 21, 63, 104(49), 105(49), 139 Koestel, A., 214(98a), 227 Koh, J. C . Y., 331,445 Kosky, P. G., 170(53), 225 Krantz, P., 111, 139 Kraus, W., 15, 63 Kreider, M. B., 123(104), 141 Krey, R. U . , 241 (7), 242(7), 313 Kudo, M., 26(44), 63 Kuerti, G., 318, 443 Kurnagai, S., 215, 216, 217, 227 Kurtz, E. F., 7(8), 53, 54(99), 62, 64 Kusko, A , , 283 (49), 315 Kutateladge, S . S., 167, 224

L Lagerstrorn, P. A , , 341, 446 Lancet, R. T., 213, 214, 227 Langmuir, I., 17, 14, 63

45 1

AUTHOR INDEX Lanphear, R., 109(55), 110(55), 139 Lee, D. Y., 120, 140 Lee, M. M., 119(85), 140 Lee, P. H., 120(87), 140 Lees, L., 334, 335, 347,445 Le Fevre, E. J., 7, 18, 56, 6.2, 63, 64 Lehninger, A. L., 141 Leigh, D., 331, 444 Lemlich, R., 26, 58,63 Lepper, R., 147(4), 222 Levy, S. E., 118(77), 140 Lewis, E. W., 170, 171, 172, 173, 174, 224 Li, T. Y., 330, 347,444,446 Licht, S., 141 Liebermann, R. W., 241(7), 242(7), 283 (49), 313,315 Lienhard, J. H., 167, 171, 199, 224, 225 Lin, S. H., 98(41,42), lOO(41, 43), 139 Linke, W., 54, 64 Lipkin, M., 119(80), 140 Lorenz, L., 5, 62 Low, A. R., 152,223 Lurie, H., 43(76), 64 Lyon, D. N., 170,225

Matsumoto, R., 26(44), 63 Meigs, P., 82(20), 138 Meksyn, D., 331, 335,444 Menold, E. R., 38, 63 Merk, H. J., 9, 18, 33, 62, 63, 335, 446 Merte, H., Jr., 161, 162, 169(39), 170(41), 171(41), 172(41), 173(41), 174(41), 206(88), 224, 226 Meryrnan, H. T., 123(105), 141 Metz, B., 120(92), 140 Michiyoski, I., 7, 26(44), 33, 62, 63 Mills, E. S., 228 Millsaps, K., 24, 63 Moncrief, J. A., 120(89), 140 Montagna, William, 141 Moore, B. J., 331 (25), 444 Moore, F. K., 331,334, 444 Moore, R. A,, 283 (48), 325 Morozkin, V. I., 206(84), 226 Morris, J. C., 241, 242(7), 313 Morrison, J. F., 120(90), 140 Moritz, A. R., 119(82), 125(82), 140, 141 Munro, A., 120(91, 92), 140 Munroe, L. R., 129(110), 135(110), 141 Murgatroyd, D., 121(97), 140 Murphy, D. W., 191(69), 225

M McAdams, W. H., 155, 156, 223 McArdle, J. G., 150(7), 222, 228 Macbeth, R. V., 204,226 McCook, R. D., 120(96), 140 McEntire, J. A., 206, 227 McGrew, J. L., 191, 225 Mackey, C. O., 67(4), 137 Macosko, R. P., 210, 21 1, 227 McPherson, R. K., 120(91), 140 McWhirter, R. W. P.,251(14, 15), 252(15), 313 Madden, A. J., 44, 61 Mader, P. P., 228 Maecker, H., 232(3), 233(3), 234(3), 243, 245, 248(3), 260, 262(25), 264, 293 (3, 56), 313, 311 Mahn, C., 265(10), 268(10), 313 Mahony, J. J., 156, 221 Malkus, W. V. R., 152, 223 Mangler, W., 331, 115 Mannes, R. L., 162, 164, 191, 194,221 Martin, J. H., 43(75), 61

N Na, T. Y., 33, 63 Nachtsheim, P. R., 53, 64 Nafe, J. P., 121(102), 123(102), 141 Nagamatsu, H. T., 330,444 Narnkoong, D., Jr., 210, 211, 212(96), 213 (96), 227 Nanda, R. J., 35, 63 Narasirnha, R., 355, 146 Nein, M. E., 150, 223 Nestor, 0. H., 295, 315 Neswald, R. G., 115(69), 110 Nevins, R. G., 90(28), 91(28), 92, 95, 104 ( S O ) , 106(50), 109(28), 110, 138, 139 Newburgh, L. H., I11 Ng, C. K., 119(85), 130 Niuman, F., 23, 63 Noyes, R. C., 167,221 Nunamaker, R. R., 150(7), 222, 228 Nusselt, W., 54, 61 Nussle, R. C., 146(2), 150(8), 222, 223

AUTHORINDEX

452 0

Oberbeck, A., 4, 62 Odum, T., 109(55), 110(55), 139 O’Neal, H. A., 109(55), 110(55), 139 Ostrach, S., 1, 6, 14, 15, 62, 64, 152, 223 Otto, E. W., 146(2), 150(8), 222, 223

P Pallone, A., 279, 314, 334, 445 Pannett, R. F., 114(67), 140 Papell, S. S., 164, 165, 202, 203, 204, 224, 226 Parczewski, K. I., 108(53), 139 Park, C., 280, 282,314 Patt, H. J., 284(52), 315 Pellew, A., 152, 223 Penski, K., 283,314 Peters, Th., 237(4), 240(4), 313 Petrash, D. A., 146(2), 150(8), 222, 223 Pfender, E., 248(13), 296(64), 300(64), 301 (64), 303 (64, 65), 304, 305 (65), 313,315 Piret, E. L., 44, 64 Plapp, J. E., 53, 64 Plesset, M. S., 178, 225 Pohlhausen, K., 23, 24, 63 Pomerantz, M. L., 206(87), 226 Pretsch, J., 331, 444 Prins, J. A., 9, 18, 62, 63 Probstein, R. F., 318, 443 Pugh, L. G. C. E., 94(34), 138 Pytte, A., 291, 315

R Rahn, H., 95(36), 138 Raithby, G., 304(66), 315 Randall, W. C., 118(96), 120(96), 140 Rapp, G. M., 104(47), 239 Reeves, B. L., 335, 445 Reeves, E., 90(27), 93(33), 94(27, 33), 95 (33), 96(33), 138 Rehm, T. R., 195, 196, 225 Reitz, J . G., 210(93), 214(93), 227 Rennie, D. W., 120, 140 Reshotko, E., 330, 331,444 Rex, J., 165, 166, 224 Richards, C. H., 69(12), 75(12, 17), 81 (18), 138

Richardson, D. L., 97, 98, 100, 130(112), 139,141 Richardson, P. D., 334,445 Riddell, F. R., 271, 314 Ringler, H., 265(10), 268(10), 313 Ritter, G. L., 170(53), 225 Robinson, J. M., 228 Rohles, F. H., 104(50), 106(50), 139 Rohsenow, W. M., 159, 189, 207(91), 224, 225 Romig, M. R., 28, 63, 230, 313 Rose, P. H., 281, 334(44), 314,445 Rosenhead, L., 331,444 Rosenthal, D., 296(63), 315

S Salgers, E. L., 114(67), 140 Satterlee, H. M., 153(24), 223 Saule, A. V., 214(98a), 227 Saunders, 0. A., 6, 15, 44, 59, 62, 63, 64, 152,223 Scala, S. M., 280, 314 Schaefer, H., 331,445 Scherberg, M. G., 46,47, 64 Schetz, J. A., 26, 27, 37, 63 Schlichting, H., 331, 445 Schmidt, E. H. W., 5, 15, 62, 155, 223 Schmidt, R. J., 152, 223 Schmitz, G., 284(52), 315 Schoeck, P. A., 293(57, 58, 59), 294(57, 58, 59), 295, 315 Schoenhals, R. J., 47, 64 Schuk, H., 6 , 62 Schwartz, S. H., 157, 158, 162, 164, 191, 194,224,228 Scott, C. J., 274(37), 314 Seckendorff, R., 118(75), I40 Senftleben, H., 156,223 Sharma, V. P., 35, 6.3 Shenkman, S., 205(80), 206(80), 226 Shepard, C. E., 283(46, 47), 314 Sherley, J. E., 161, 167, 224 Shure, R. I., 214(98a), 227 Siegel, R., 33, 42(601, 58, 63, 64, 147(5), 149(5), 157, 160(37), 162, 164, 167, 169, 175, 176, 179, 180., 181., 182 183 (60), 184, 185, 186, 190, 191, 209, 222, 224.225.227 .~ Simon, F. F., 202,226

AUTHOR INDEX Simon, H. A., 31, 63 Simoneau, R. J., 202(77), 203(77), 226 Singer, R. M., 228 Siple, P. A., 93(31), 94(31), 96(31), 138 Sissenwine, N., 82(20), 138 Skan, S. W., 336,446 Sliff, H. T . , 228 Smith, A . M. O., 331, 334,444,445 Smolak, G. R., 228 Soderstrom, G . F., 67(6), 116(73), 117(73), 137,140 Soehngen, E. E., 51, 64 Song, S. H., 90(26), 92(26), 95(26, 36), 120(86), 138, 140 Southwell, R. V., 152, 223 Spalding, D. B., 331, 444 Sparrow, E. M., 7,12, 15, 17,21,23, 25,29, 35,42(18), 44, 62, 63,64,209,227,274, 314 Spector, W. S., 118(76), 140 Spitzer, L., 264(26), 314 Springer, W., 104(50), 106(50), 139, 259 (23), 262(23), 263 (23), 313 Squire, H. B., 9, 62 Stambler, I., 114(66), 139 Stankevics, J. O., 281, 314 Steinle, H. F., 167, 225 Stenuit, R., 109, 139 Stevens, G . T . , 214(98a), 227 Stewartson, K., 331, 335, 444, 445 Stine, H. A., 283(45, 47), 314, 315 Stol1,A. M.,69, 71(13), 75(12, 17), 81(18), 82(19), 83, 117(72), 118(72), 119(81), 121(81, loo), 123(81, loo), 124(81), 125(81, 108), 127(81), 129(109, 110), 130(111), 135(110), 138, 140, 141 Stout, E., 51 (93), 64 Stuart, J. T . , 152,223 Sturas, J. I., 211,227 Sugawara, S., 7, 9, 24, 26, 33, 62, 63 Szewczyk, A. A., 16, 51, 52, 53, 63, 64

T Tani, I., 334, 445 Tanner, J. M., 94(34), 138 Taylor, R. B., 214(98a), 227 Thauer, R . , 121 (101), 140 Thompson, R. H., 120(88), 140 Thompson, W. D., 255(19), ,313

453

Thwaites, B., 331, 334, 444, 445 Tobias, C. W., 16(25), 63 Tribus, M., 25, 63 Tritton, D. J., 52, 64 Trusela, R. A., 210, 227 Tsien, H. S., 331, 444 Tsou, F. K., 7(8), 54(99), 62, 64

U Unsold, A., 245, 313 Unterberg, W., 146,222,228 Useller, J. W., 150(6a), 222 Usiskin, C . M., 160(37), 167, 169, 191, 224

v van Dilla, M., 93, 94, 96(31), 138 Van Driest, E. R., 331, 444 van Dyke, M., 341, 446 Van Tassell, W., 279, 314 Vardi, J., 26, 58, 63 Vasantha, S. S., 355,446 Veghte, J. H., 103, 139 Vendrik, A. J. H., 118(79), 140 Vernon, H. M., 67, 137 von Karman, T . , 8, 62, 331,444 Vos, J. J., 118(79), 140

W Wansbrough, R. W., 206(82), 226 Warren, W. R., 280(41), 314 Watanabe, K., 167, 171, 224 Watson, E. J., 331, 444 Watson, V. R., 283 (45, 46, 37), 284(51), 314,315 Webb, P., 116(74), 140 Weber, G . J., 114(67), 140 Weber, H. E., 283(50), 315 Westwater, J. W., 159, 173, 224, 225 Whalen, R. J., 349, 446 Whedon, G. D., 120(88), 140 Whitehouse, R. H., 94(34), 138 Wienecke, R . , 265(10), 267, 268(10, 31), 313,314 Wilke, C. R., 16, 63 Williams, A. R., 291, 315 Williams, C. G., 120(90), 140

454

AUTHOR INDEX

Wilson, J. S., 120(89), 140 Winslow, C.-E. A., 67(5, 9), 111(63), 118 (78), 137, 138, 139, 140 Witkowski, S., 265(10), 267(28), 268(10), 313,314 Witte, A. B., 352, 446 Wolff, H. G., 121 (103), 141 Wolff, H. S., 94(34), 138 Wong, P. T. Y., 199,225 Woodcock, A. H., 87, 138 Wright, L. T., 67(4), 137 Wuest, W., 331, 445 Wurster, R. D., 120(96), 140 Wyndham, C. H., 120(95), 140

Y Yaglou, C. P. J., 104(46, 48), 139 Yang, K. T., 24, 36, 38, 44, 46, 63, 64, 334, 445 Yos, J. M., 241(7), 242(7), 313 Young, G. B. W., 331,444

Z Zankl, G., 265(10), 268(10), 313 Zara, E. A., 164, 165, 224, 228 Zuber, N., 167, 178, 181,224, 225 Zwick, E. B., 228 Zwick, S . A , , 178, 225

Subject Index A

Acoustic wave effects, 47ff Airplane trajectory method of producing reduced gravity, 150 Arcs cathode axis parallel to a plane anode, 295 ff cylinder geometry with annular anode, 320ff free-hurning with plane anodes, 293 ff Asymptotic expansion of local similarity equation, 335ff

B

Bioclimate, 67 Bioclimatology, 67 Bioclimatological data, 84f Biotechnology, heat transfer in, 65ff artificial environments, 103ff indoors, 103 ff space vehicles, 111 ff underwater vessels, 109ff natural environments, 66ff extraterrestrial, 97 ff terraqueous, 90ff, 98f terrestrial, 66ff, 98f skin, its role in heat transfer, 115 ff characteristics, 115 ff functions, 115ff injury, 123ff protection, 128ff thermal sensation, 121 ff Blackbody radiation in a plasma, 238ff Blowing parameter, 322, 326, 330f Boiling in reduced gravity, see Reduced gravity Boltzmann distribution, 235, 245 Bound-bound radiation in a Plasma, 234f Bremsstrahlung in a plasma, 237ff Bubble growth, 174ff

C

Chapman-Ruhesin constant, 324 Clo, 91

Combustion in reduced gravity, see Reduced gravity Condensation in reduced gravity t see Reduced gravity Conservation of energy 4* 2853 308* 319 Conservation of mass equation, 3, 276, 285, 319 Conservation of momentum equations, 3f, 276f, 285, 319 “Critical thermal load”, 124 2779

D Debye length, 254 Dense plasmas, 242ff Displacement thickness, 329, 354 Diver’s wet suit, insulating properties of, 94 ff Drop tower method of producing reduced gravity, 146ff

E Eckert reference temperature, 326 Eckert-Schneider condition, 322 Effective temperature, 104ff Einstein relation, 239 Electrical conductivity, 231 Elenhaas-Heller differential equation, 263 Environmental temperature, 107 Equilibrium in a plasma, see Plasma

F

Fabrics, protective, 129ff Falkner-Skan parameter, 323 Free-hound radiation in a plasma, 235ff Free convection in reduced gravity, see Reduced gravity Free convection on vertical surfaces, 1 ff laminar flow, 4ff circular cylinder, 17 ff flat plate, 4ff differential equation method, 5ff experimental work, 11 ff integral equation method, 8ff 45 5

45 6

SUBJECT INDEX

Free convection on vertical surfaces-cont. laminar flow-cont. Grashof number, very low, 43 ff nonsteady conditions, 33ff experimental work, 41 ff negligible thermal capacity, 33 ff significant thermal capacity, 39ff physical properties effects, 28 ff large temperature differences, 29 f near-critical conditions, 30ff viscous dissipation, 32f radiative effects, 28 surface temperature effects, 22ff differential equation method, 22ff experimental work, 26 integral equation method, 24ff vibration, effects of, 47ff laminar flow instability, 51 ff experimental work, 51 f theoretical work, 52f turbulence, 5Off experimental work, 58 ff theoretical work, 54ff Free-free radiation in a plasma, 237f Fritz-Ende relation, 178f Froude number, 53

G Geometric scale factor, 171ff Grashof number, 3ff, 151

H Hall parameter, 308 Heat conductivity, 233

I Injection parameter, see Blowing parameter Inner limit similarity equations, 344f solutions, 441 Insulation. 87

J

Just noticeable difference (JND), 121

L Lewis number, 270f, 320 frozen, 275

Line radiation in a plasma, 234f Local similarity, 333 ff Local thermal equilibrium (LTE), 246ff

M Magnetic fields in plasmas, see Plasma Magnetic forces used for producing reduced gravity effects, 150f Mass-transfer parameter, see Blowing parameter Maxwell-Boltzmann distribution, 245 Melting, 33 Momentum thickness, 329

N Newton-Raphson scheme, 332 Non-Newtonian fluid, 33 Nusselt number, 3ff, 154ff, 270

0 Oberbeck equations, 5 Operative temperature, 83 Outer limit similarity equations, 341 ff solution, 435ff

P Panradiometer, 71 ff Peclet number, 296, 301 Planck function, 239, 245f Plasma, 229ff characteristic properties, 231 ff blackbody radiation, 238ff bremsstrahlung, 237f line radiation, 234f recombination radiation, 235 ff composition, 256ff heat transfer, 270ff in absence of an externally applied electric or magnetic field, 270ff basic transport equations, 272ff laminar boundary layer equations, 276ff results of reentry studies, 278ff in presence of a magnetic field, 304ff electrically insulating surface, 307ff electrode heat transfer, 305 ff

SUBJECT INDEX

' Reduced gravity-cont. Plasma-cont. --c condensation, forced flow, 209ff, 220f heat transfer-cont. flow behavior, 210f in presence of an electric current, 282ff noncondensable gas, 214 electrically conducting surface, 289ff pressure drop, 21 1ff experimental studies 292ff 'vapor-liquid interface, 213 f cathode axis parallel to a plane 2 condensation without forced flow, 206ff, anode, 295ff 220 cylinder geometry with annular ' laminar film condensation on a vertical anode, 302ff surface, 207f free-burning arcs with plane 4 laminar-to-turbulent transition, 208f anodes, 293ff '1, transient time to establish laminar electrically insulating surface, 282ff condensate film, 209 constricted arc, 283ff / experimental production of, 146 ff influence of on heat transfer, 231 ff 1 ) airplane trajectory, 150 plasma-wall boundaries, 252 ff ' drop tower, 146ff sheaths, 252ff : magnetic forces, 150f steady, dense plasmas, 242ff rockets, 150 thermodynamic equilibrium 244ff ' satellites, 150 local thermal equilibrium, 246ff .' forced convection boiling, 200 f excitation equilibrium, 249 .(,designs involving substitute body ionization equilibrium, 250ff forced, 205f kinetic equilibrium, 248f perfect thermodynamic equilibrium ' 5 two-phase heat transfer, 201 ff I free convection, elimination, 145 245 f .I free convection in, 151 ff, 218f temperature increase of plasma, 262ff fluid flow, 152ff reduction of heat conductivity by a magnetic field, 267ff I i boundary layer theory, 152 f reduction of wall heat fluxes by tran-'boundary layer transition, 153 spiration cooling, 269f Rayleigh numbers encountered in thermodynamic properties, 256ff low-gravity, 153f transport properties, 260ff u threshold of convective motion, 152 Pohlhausen transformation, 22 7 heat transfer, 154ff Pool boiling in reduced gravity, see Reduced :transient development times of boundgravity ary layers, 156ff gravity as an independent parameter, Prandtl number, 3ff, 1 5 1 , 271, 320 equilibrium, 275f 144f .> pool boiling, lSSff, 219f frozen, 275 f ibubble dynamics in saturated nucleate boiling, 174ff R diameter at departure, 181 f Radiant heat load, 83 experimental results, 181f Radiation effects, 67ff theoretical relations, 181 Rayleigh number, 3ff, 151 f forces acting during growth, 182ff modified, 151 ff growth rates, 177ff Recombination radiation in a plasma, 235ff 5 experimental results, 178ff Reduced gravity, 144ff , J theoretical relations, 178 i combustion, 215ff, 221 'i higher heat flux effects, 191ff . candle flame, 215 f nucleation cycle and coalescence, fuel droplets, 21 5 ff 175ff solid fuels, 217f " rise of detached bubbles, 190f

'

J

'

457

,

.

SUBJECT INDEX

45 8 1

Reduced gravity-cont. -2 pool boiling-coat. I bubble dynamics in subcooled boiling, ’ 194ff bubble growth, 195f L, forces acting on bubbles, 196ff .-: critical heat flux, 166ff experimental behavior, 167ff 14 theory, 167 ’; film, boiling, 173f L experimental results, 173 f :i theoretical relations, 173 minimum heat flux between transition boiling and film boiling, 171 f nucleate pool boiling, 158ff experimental results, 159 ff theory, 158f transition region for pool boiling, 171 7 space applications, 144 Reentry studies, 278ff Reynolds analogy, 346ff Reynolds number, 54ff, 208, 345 Rockets, their use in producing reduced gravity, 150 Runge-Kutta method, 331 f I

S Saha-Eggert equation, 238 Satellites, their use in producing reduced gravity, 150 Schmidt number, 16, 320 Self-emitted radiation, 234 Sheaths, plasma, 252ff Similar solutions, 356ff, 389ff, 401 ff Similar solutions for a power-law viscosity relation, 370ff Similar solution for a Sutherland viscositytemperature relation, 426ff Similarity for laminar boundary layers, 319ff

Similarity for laminar boundary layers -cont. conditions for, 322ff equations for solution, 324ff large values of the pressure-gradient parameter, 341 ff inner limit equations, 344f outer limit equations, 341 ff local similarity, 333ff asymptotic expansion of equations, 335ff determination of f, by successive approximations, 337ff numerical integration procedure, 33 1 ff numerical results, 330f range of solutions and parameters, 330 special classes of reduced equations, 327ff computation of boundary layer properties, 328 no mass transfer, 327f Prandtl number equal to unity, 328 pressure-gradient parameter equal to zero, 328 two-dimensional flow, 327 viscosity proportional to temperature, 328 Skin, properties of, 116ff Solar radiation measurements, 77ff Spitzer’s formula, 264 Stream function, 320 Sutherland constant, 326 Sutherland viscosity law, 326

T Thermal thickness, 329 Thermal variability factors (TVF), 107f Thermistor radiometer, 69 f Thermodynamic equilibrium in a plasma, see Plasma Transpiration cooling, 269 f