Some Extensions in the Mathematics of Hydromechanics

VOL,. 9, 1923

13

MA THEMA TICS: R. S. WOOD WARD

SOME EXTENSIONS IN THE MA THEMA TICS OF HYDROMECHANICS By ROBERT S. WOODWARD CARNIGIE INSTITUTION OF WASHINGTON, D. C.

Read before the Academy, April 25, 1922

Present Status and Needs of Hydrokinetics.-Precise description of fluid motion when viscosity is taken into account is a matter of much complexity. No less than twenty symbols and fifteen equations are required for a complete specification of the linear and angular movements of a fluid element, of its expansion and contraction, and of the changes in pressure and density to which it is subject. The formation of what Poinsot called an "image sensible" of these characteristics is a matter of corresponding difficulty which has taxed the resources of the ablest analysts from the time of Euler down to the present day. The nature of the difficulty seems to lie in an inadequacy of our mathematical machinery rather than in defects of our physical concepts. The following abstract, therefore, aims to suggest some extensions and improvements in that machinery, with the hope that greater generality and uniformity may be ultimately realized in the treatment of concrete problems in this branch of physics. The general equations of fluid motion are commonly written in the following simplest forms, namely:'

Pdtu

+ p-

dt

= bx

PX +13U- .ab

+ +

A

A

A2U

1 (66 dv 6p P dt + a- = pY + 3- y '- + ,U 2v2, by by

p

dw ap -wdt + ?az -p

=

1 aaO

pZ + S -0 + 3 ?z 1

A

(1)

A2W.

In these x, y, z are the rectangular coordinates of any element of the fluid at the time t; u, v, w are the linear velocity components of the element; p is the internal stress and p is the density of the fluid at x, y, z, at the time t; X, Y, Z are the force components per unit mass of the fluid at x, y, z; ,u is the coefficient of viscosity of the fluid; and 0 is the time rate of expansion of the element at x, y, z at the time t. This rate is given by the equation expressing the conservation of the mass of the element, that is, by +

?x

v w i+ bz by

_ d log p. dt

(2)

14

'

PRoC. N. A. S.

MA THEMA TICS: R. S. WOOD WARD

The symbol A2 which appears in the second members of (1) is defined, for example, by

2262+2U

62U

by2

az2'

bx2

and the operation here indicated is now commonly called the Laplacian of u. Along with the equations (1) and (2) three others are needed to express the relations between the linear velocity components u, v, w and the angular velocity components (or spin components) of the fluid element. These latter components with reference to the axes of X, Y, Z, respectively, are given by av au 2, = auz- awVxf ay- CJv 2t = aw 2t = __ _

z,

and these are subject to the obvious condition at

ax

+

+ ar by az 71

=

O.

(4)

Moreover, three additional equations, introducing four new functions (potentials), are required to express u, v, w in forms to meet the requirements of the conditions (2) and (4) These equations are u=-

?ax

av +-aw z by

afP + av_ aw by

ao

?z 6v

az

ax

(5)

bx

av by

in which qp is the velocity potential, representing the irrotational part of the motion, and U, V, W may be called the spin potentials, since they represent the rotational part of the motion, these two parts being independent of one another. Finally the conditions (2) and (4) lead to the further relations2

A2V =-2t,

A2p = 0, A2V = - 2n,

A2W

2t,

(6)

- + w = 0. )z ax + by

Much embarrassment hinges on the application of these equations (6), which seem to have figured- thus far chiefly in the r6le of "existence the-

15

MA THEMA TICS: .R. S. WOODWARD

Voi. 9, 1923

orems;" and it is one of the objects of the following paragraphs to indicate how these relations may be utilized in the solution of concrete problems. Extension of Relations between u, v, w and i, n, t.-Equations (3) suggest the following obvious relations derived from (3) and (2): _

2

_a

=

az _ ax

=

ax

2(

az

2(7 ( _ t) ax

0

_

ax y

28

(7)

_ - AW2W

6x

Operating on (7) in a similar manner there result

bA2v -= 2A2t, az

bA2w ay

=

___

~a2v ?x

2A2t,18)

~a^2u =2A2r. by

Again, there follow from (8) and (2) 2

2 2

f{A2¢

by

bA2O bA2\ 7 2 -2_ =u

az

~A2~~\ /~~A2~~

_

x _ A2O A2A2v, = - a2

(9)

/6A2~ 6A2~\I -_A2O A2A2W. =

ax

by

az

The equation (3) shows that when = = = 00, or when there is no rotation, the well-known velocity potential exists, or that u =

ao/bX,

v

-. a/by, A2p= 0.

w =

ao/6Z,

(10)

Similarly, equations (7) show that when 0 is constant and when u, v, w 0 are each harmonic,3 a potential F exists, such that 2 =

bF/lx,

2n = 6F/by, A2F = 0.

2r = bF/lz,

(11)

PROC. N. A. S.

MA THEMA TICS: R. S. WOODWARD

16

Likewise, equations (8) show that when {, , 1 are harmonic, or when &21 = A2r = 0 a generating potential, G, obtains so that

2t=

A2u

=

bG/lx,

A2v

=

A2G

aG/Iy, =

A2w

bG/lz,

=

A20.

(12)

Further, when A20 iS constant and when A2u, A2v, Aw are harmonic (or when u, v, w are preharmonic as defined below) equations (9) show that A2%, M2q, A2r are harmonic and derivable from a function H, such that

2t= bH/bX

2 = H/by, 2 M2H = 0.

A2r

=

bH/lz,

(13)

Harmonic and Preharmonic Functions.-The functions which appear under the symbol A2 in equations (6) to (13) are of surpassing importance and interest in mathematical physics. Functions Q, say, satisfying the equation of Laplace, namely, A2Q = 0, have been very extensively studied, especially during the past half century. They are now generally called harmonic functions, or, briefly, harmonics, following the exposition of their properties and uses given by Thomson and Tait in their Natural Philosophy,4 which is probably still the richest mine in this domain, though somewhat difficult of access. What is given below is only an extension of the suggestions found in their great work. A harmonic function, Q, is any homogeneous function of x, y, z satisfying the equation MQ = 0. From the indications of Thomson and Tait in the work cited it follows that the harmonics of negative integral degree are given, omitting possible constant factors, by the formulas r

Q- (i+1) =

(4

axibykIzi

(14)

r2= X2 + y2 + z2 = j + k + z = 0, 1, 2, ...

Thus, for

*

i=0,

Q-1-

i=1,

Q-2=-r3- r8

r x

y

11 r3

3X22 r'

z

r3;

VOiL. 9, 1923

MATHEMATICS: R. S. WOODWARD

17

Similarly, the harmonics of positive integral degree are given by the formulas I Q, = r2i+ rXbyk ziP

2= i =

(15)

X2

+ y2 + z2 + k + z = O, 1, 2, ...

Thus, for i =O, i =1, i = 2,

Qo + 1; Qi=-x,-y,-z; Q2 =+ 3xy, 3yz, 3zx, - (r2

-

3X2),

-

(r2 - 3y2) - (r2

-

3Z2).

Corresponding to every harmonic there exists another function of nearly equal importance, namely, a function whose Laplacian is the given harmonic. If the latter is denoted by Q, as above, and if P is the corresponding function, A2P = Q,andA2A2P = A2Q = 0. (16) I have ventured to call the function P, defined by (16), a preharmonic, although the term biharmonic has been used by European mathematicians to designate such functions, without special reference, however, so far as I am aware, to their correspondence with harmonics. The preharmonics corresponding to the harmonics given by (14) are readily found. For, since 1 =2A2r,, 2 r 1

(16) gives ~

A2P

bi,&2r

1

2

(Xi lyk bzi'

whence

P-r(i._1)

1

ir

ki + H,

i =j+ k +z

=0,1,

(17)

2,.

wherein H is any harmonic or the sum of any number of such. In each case the degree of the preharmonic is higher by 2 than the degree of the harmonic. Thus, to illustrate, for

18

MA THEMA TICS: R. S. WOOD WA RD

PROC. N. A. S.

i= 0, Q-i = +-,andP+ = -r +H; r

~~~2

i= 1, Q-2 i =2

lx

x

-,andPo =- + H;

= -

11X2\ 3X2\ 11 Q-3 =-y- 6 andP-1= 2- 4 +H.

Similarly, the preharmonics corresponding to the harmonics of positive integral degree, or to those defined by (15), are given by the solution of

&2Pi + 2

= Qi,

and this is found to be

=2(2it2Qi+ 3)

pi+

+ Hp,(8 +

(18)

in which H is any harmonic or the sum of such. To illustrate, by a single example drawn from each of the groups for which i = 0, 1, 2, respectively, and omitting constant factors which would enter in any applications, when i 0,

Qo= + 1, and P2= + r2 + H; 6

i = 1,

Ql =-x,

i=2,

Q2

=

*1 and P3 =- r2x + H; 10

+ 3xy, and P4 =+ 3r2Xy 3x + H. 14

It is seen that by aid of equations (17) and (18) the troublesome triple integrals by which sp, U, V, W of (6) are commonly expressed may be avoided whenever 0, {, , r are known harmonics; and when these latter quantities are known preharmonics, the relations A2A2p = A20, etc., become available. In general it will be found that the relations between the velocity and the spin components, and their Laplacians, given by equations (7) to (13), along with the harmonics and preharmonics given by (14) to (18), serve to remove many of the difficulties of this complex branch of mathematical

physics. See, for example, Lamb's Hydrodynamics, edition of 1895, p. 515. First given in the classic memoir of Helmholtz of 1858, Crelle's Journal, Vol. LV. 3 As explained below, a harmonic function, Q say, is one whose Laplacian is zero, oz briefly A2Q =0. 4 Treatise on Natural Philosophy, by Thomson and Tait, Cambridge, 1879-1883, Vol. I, Part I, Appendix B, pp. 171-218. I

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