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3B2v8:06a=w ðDec 5 2003Þ:51c þ model

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Physica A ] (]]]]) ]]]–]]] www.elsevier.com/locate/physa

5 7

Fluid mechanics revisited

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Howard Brenner

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Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139-4307, USA

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Received 12 November 2005; received in revised form 21 February 2006

15 Abstract

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O¨ttinger’s recent nontraditional incorporation of ﬂuctuations into the formulation of the friction matrix appearing in the phenomenological GENERIC theory of nonequilibrium irreversible processes is shown to furnish transport equations for single-component gases and liquids undergoing heat transfer which support the view that revisions to the Navier–Stokes–Fourier (N–S–F) momentum/energy equation set are necessary, as empirically proposed by the author on the basis of an experimentally supported theory of diffuse volume transport. The hypothesis that the conventional N–S–F equations prevail without modiﬁcation only in the case of ‘‘incompressible’’ ﬂuids, where the density r of the ﬂuid is uniform throughout, serves to determine the new phenomenological parameter a0 appearing in the GENERIC friction matrix. In the case of ideal gases the consequences of this constitutive hypothesis are shown to yield results identical to those derived theoretically by O¨ttinger on the basis of a ‘‘proper’’ coarse-graining of Boltzmann’s kinetic equation. A major consequence of the present work is that the ﬂuid’s speciﬁc momentum density v is equal to its volume velocity vv , rather than to its mass velocity vm , contrary to current views dating back 250 years to Euler. In the case of rareﬁed gases the proposed modiﬁcations are also observed to agree with those resulting from Klimontovich’s molecularly based, albeit ad hoc, self-diffusion addendum to Boltzmann’s collision integral. Despite the differences in their respective physical models—molecular vs. phenomenological—the role played by Klimontovich’s collisional addition to Boltzmann’s equation in modifying the N–S–F equations is noted to constitute a molecular counterpart of O¨ttinger’s phenomenological ﬂuctuation addition to the GENERIC friction matrix. Together, these two theories collectively recognize the need to address multiple- rather than single-encounter collisions between a test molecule and its neighbors when formulating physically satisfactory statistical–mechanical theories of irreversible transport processes in gases. Overall, the results of the present work implicitly support the unorthodox view, implicit in the GENERIC scheme, that the translation of Newton’s discrete mass-point molecular mechanics into continuum mechanics, the latter as embodied in the Cauchy linear momentum equation of ﬂuid mechanics, cannot be correctly effected independently of the laws of thermodynamics. While O¨ttinger’s modiﬁcation of GENERIC necessitates fundamental changes in the foundations of ﬂuid mechanics in regard to momentum transport, no basic changes are required in the foundations of linear irreversible thermodynamics (LIT) beyond recognizing the need to add volume to the usual list of extensive physical properties undergoing transport in singlespecies ﬂuid continua, namely mass, momentum and energy. An alternative, nonGENERICally based approach to LIT, derived from our ﬁndings, is outlined at the conclusion of the paper. Finally, our proposed modiﬁcations of both Cauchy’s linear momentum equation and Newton’s rheological constitutive law for ﬂuid-phase continua are noted to be mirrored by

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Tel.: +1 617 253 6687; fax: +1 617 258 8224.

E-mail address: [email protected] 0378-4371/$ - see front matter r 2006 Published by Elsevier B.V. doi:10.1016/j.physa.2006.03.066

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counterparts in the literature for solid-phase continua dating back to the classical interdiffusion experiments of Kirkendall and their subsequent interpretation by Darken in terms of diffuse volume transport. r 2006 Published by Elsevier B.V. Keywords: ’; ’; ’

7 1. Introduction 9 1.1. Preface 11

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Previous contributions [1–3] relevant to the issues discussed here in have suggested, based primarily upon conﬂicts noted between tracer- and hypothetical mass–velocity experiments, that revisions were needed to the Navier–Stokes–Fourier (N–S–F) equations of viscous ﬂuid mechanics—certainly in the case of singlecomponent ﬂuids. Explicitly, in several key constitutive relations appearing in the N–S–F equation set it was proposed, inter alia, that the ﬂuid’s mass velocity vm appearing therein be replaced by the ﬂuid’s volume velocity vv [4]. The arguments underlying the proposed revisions were based, in part, upon our unorthodox interpretation [5] of experimental data pertaining to thermophoretic particle motion in gases, when considered in conjunction with Kogan et al.’s [6–9] and Bobylev’s [10] re-ordering of the Maxwell [11]—Burnett [12] thermal stress contribution to the viscous stress tensor at small Knudsen numbers, Kn51. These revealed such stresses to be of the same order in Kn as the classical Navier–Stokes stress contributions, at least for Reynolds numbers of order one and Mach numbers small compared with unity. Prior to this reassignment, thermal stresses were regarded as being of a noncontinuum nature in the Chapman–Enskog [13] hierarchical expansion of solutions of the Boltzmann equation in powers of Kn. Our original proposal [1–3] emphasized gases. This owed to a lack at the time of convincing experimental evidence sufﬁcient to demonstrate the applicability of the theory to liquids. Subsequently, however, we were able to offer an interpretation of experimental data involving thermal diffusion in binary liquid mixtures [14,15] that supported the applicability of our proposed N–S–F revisions to liquids as well. It proved unnecessary in our original N–S–F modiﬁcation proposal to address the obvious question of a further revision to these equations regarding the viability of Euler’s constitutive equation for the ﬂuid’s speciﬁc momentum density v [16], including related kinetic energy density and ‘‘work-velocity’’ issues [3]. (The symbol v refers here to the velocity appearing in both the local and convective inertial terms in the Navier–Stokes equations.) That this was unnecessary owed to the smallness of the Reynolds numbers [17] encountered in the thermophoretic and thermal diffusion experiments cited above, as well as in the Knudsen number re-ordering of the thermal stress contributions [6–10]. It was, however, pointed out in Appendix B of Ref. [3] that if one accepts as being correct the standard balance equations for momentum and energy found in textbooks [18–21], while concurrently adopting the unconventional constitutive equations proposed in Ref. [3] for the revised forms of Fourier’s law of heat conduction and Newton’s law of viscosity, the speciﬁc momentum density v would necessarily have to be equal to the ﬂuid’s mass velocity vm , the latter being the velocity appearing in the continuity equation (cf. Eq. (1)). Although not explicitly pointed out in connection with the ‘‘proof’’ set forth in Ref. [3] of the apparent viability of the standard Eulerian constitutive relation v ¼ vm for the ﬂuid’s speciﬁc momentum density, it is implicit in the concomitant use of the nonstandard form of Newton’s viscosity law (cf. the union of Eq. (44) with (56)) (in which the gradient rvv of the ﬂuid’s volume velocity [4] now appears in place of its standard mass velocity counterpart rvm Þ that nonpositive-deﬁnite dissipative terms, namely the ‘‘mixed’’ terms 2Zrvm : rvv and ZB ðr.vm Þðr.vv Þ [22], will appear in the Clausius–Duhem inequality (cf. Eq. (13)) for the local rate of irreversible entropy production. Here, Z and ZB are the respective shear and bulk viscosities. The consequent conﬂict between the apparent dictates of rational mechanics and irreversible thermodynamics manifested by this lack of deﬁnitive positivity leads us to continue to regard the speciﬁc momentum density issue as an open question despite the purported ‘‘proof’’ to the contrary offered in Ref. [3]. As a result of this fundamental dichotomy between the respective predictions of continuum mechanics and the second law of thermo-

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dynamics, the possibility must be entertained that, to quote O¨ttinger [23]: ‘‘Something is missing’’ from the currently accepted energy and entropy transport equations of nonequilibrium irreversible thermodynamics [19–21]; that is, in order to heal the breach engendered by this lack of positivity, additional terms, presumably currently missing from these equations, need to be present in the constitutive expressions for the diffusive ﬂuxes of internal energy and entropy appearing therein. In this context it is pertinent to note that prior to the emergence of this core conﬂict we had earlier [1] opined, so to speak, in favor of thermodynamics over mechanics [24] in resolving apparent conﬂicts between the two, by supposing that v ¼ vv rather than v ¼ vm so as to render the irreversible entropy production rate nonnegative. However, substantive theoretical justiﬁcation for that fundamental change in the constitutive expression governing the speciﬁc momentum density appearing in the Cauchy linear momentum equation was lacking at the time (although we nevertheless regarded our Lagrangian tracer velocity interpretation [1–3] of thermophoretic particle motion as offering convincing experimental evidence in support of the possible change in v from vm to vv Þ. This background sets the stage for the subsequent theoretical analysis, which purports to resolve the momentum density issue in favor of the ‘‘mechanico-thermodynamic’’ relation v ¼ vv . This is accomplished by adding fundamental ﬂuctuational contributions [23] to the standard [19–21] energetic and entropic balance equations of nonequilibrium irreversible thermodynamics, following a recent proposal to this effect by O¨ttinger. In his book [23], ‘‘Beyond Equilibrium Thermodynamics’’ (hereafter referred to as BET), O¨ttinger proposed incorporating ﬂuctuations into the friction matrix appearing in GENERIC theory. This latter nontraditional feature, manifested by the appearance of a new phenomenological coefﬁcient a0 therein, was not explicitly included in earlier versions [25,26] of that theory (although neither was it explicitly ruled out). It is this inclusion in the GENERIC friction matrix M of O¨ttinger’s previously ‘‘Something is missing’’ ﬂuctuation terms that resolves the conﬂict, described earlier, between continuum mechanics and thermodynamics, and which will be seen to lead rationally to the relation v ¼ vv . Explicitly, O¨ttinger’s extension of GENERIC theory harmoniously unites our respective views of momentum density (while also conﬁrming other elements of our theory). A related paper by the present author [27] reinforces this uniﬁcation of our view of momentum transport with that of O¨ttinger by offering an independent argument, due to Klimontovich [28–30], with respect to the viability of the constitutive relation v ¼ vv . In particular, Klimontovich’s unorthodox modiﬁcation of the collisional term in Boltzmann’s equation [13], involving the addition thereto of a physicalspace self-diffusion-like contribution, is counterpart to Ottinger’s ﬂuctuational addition to GENERIC. Both additions, in turn, are formally equivalent to our addition of diffuse volume transport [1–4] to conventional theories of transport phenomena [18]. The independent theories of O¨ttinger [23] and Klimontovich [28–30], which lead to identical results in their common domain of validity (namely, rareﬁed gases), were created to rectify what each regarded as the fundamental failure of existing continuum-mechanical theories—especially those derived from the Boltzmann equation [13]—to incorporate single-component ﬂuctuations and, hence, the notion of Brownian motion into their respective foundations [31]. (Further details with respect to this point of view are discussed in Sections 8.1 and 8.2.) However, the arguments underlying their proposed changes were purely philosophical, in the sense that neither explicitly invoked experimental data, nor appealed to already existing theories to support their claims with respect to the perceived incompleteness of current theories of irreversible thermodynamics and Boltzmann-based gas-kinetic statistical mechanics. In that respect, our conclusions, if accepted as being correct, offer experimental and theoretical evidence in support of both of their theories, whose common results for rareﬁed gases coincide. Conversely, subject to this same caveat, their theories should be regarded as supporting our earlier empirical proposal [1–3] for modiﬁcations in the N–S–F equation set, especially including the speciﬁc momentum density constitutive equation v ¼ vv , as well as the energetic and entropic consequences stemming therefrom. Issues of entropy production and irreversibility are a matter of concern when attempting, as did Boltzmann [13], to translate discrete Newtonian mass-point microscopic molecular mechanics into a rational macroscopic continuum-mechanical analog thereof, namely the Cauchy linear momentum equation of ﬂuid mechanics [18–21]. However, according to the generally accepted view of this microscopic ! macroscopic transition scheme [13], concern about its consistency with the notion of irreversibility, as embodied in the second law of thermodynamics, currently surfaces only after the fact. This secondary ‘‘after the fact’’ role ascribed to the

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1.2. Outline of the paper

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Section 2 reviews the pertinent equations of O¨ttinger’s extended GENERIC theory [23] for the present single-component ﬂuid case, without invoking any constitutive relations for the various physical properties appearing therein. These equations embody the balance equations for mass, linear momentum, energy and entropy as posed by the GENERIC scheme. They differ in substance from the standard balance equations of linear irreversible thermodynamics (LIT) found in textbooks [19–21], to which the former reduce in circumstances where the difference v vm between the ﬂuid’s speciﬁc momentum density v and its mass velocity vm vanishes. It is not that LIT is conceptually wrong, but rather that the present formulation of the subject fails to recognize the role played by the diffuse transport of volume. Section 3 sets forth the general

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second law is evidenced by the subsequent appearance, only after Boltzmann’s transport equation was already fully formulated, of his celebrated H-theorem, aimed at reconciling the consistency of his purely mechanically derived reversible formula for calculating the molecular distribution function with the thermodynamic notion of irreversibility. In short, current thinking holds that while the continuum version of Newtonian mechanics impacts upon the foundations of irreversible thermodynamics, the reverse is not true; that is, the ﬁeld of irreversible thermodynamics is generally (although not universally [32]) believed to be without impact upon ‘‘pre-constitutive’’ Newtonian dynamics-based continuum mechanics—explicitly upon the Cauchy linear momentum equation prior to contemplating the form of the constitutive equation for the stress tensor appearing therein. (Irreversible thermodynamics does, however, impact upon ‘‘post-constitutive’’ continuum mechanics—for example, as a consequence of restrictions imposed by the Clausius–Duhem entropy production inequality [19–21] upon the allowable class of constitutive expressions for the form ascribed to the deviatoric portion of the stress tensor, the latter representing the diffuse momentum ﬂux.) As regards our ‘‘pre’’ and ‘‘post’’ terminology, we are referring here to the accepted view, dating back 250 years to Euler [33–35], that the identiﬁcation v ¼ vm is not a constitutive relation but is, rather, a fundamental physical truth emanating directly from the implicit a priori assumption that Newton’s laws of motion, known to be valid for a discrete ponderable body of mass M, can unquestionably also be applied to a so-called material ﬂuid domain of mass M moving within a ﬂuid continuum—at least in circumstances where M is differential in magnitude. Indeed, we are aware of only a very few circumstances [36, p. 196 (but see also p. 28), ? 37] in which this apparent ‘‘fact,’’ namely that of the viability of the relation v ¼ vm , has been questioned. However, because a material domain does not generally consist permanently of the same molecules, but only of the same net amount of mass [38,39], discrete-body mechanics [40] based upon regarding individual molecules as point masses subject to elementary action–reaction mechanical Newtonian laws cannot, indiscriminately, be applied to such domains. Of course Euler [33], the father of rational ﬂuid mechanics, was unaware at the time of his foundational work in 1755 of the existence of molecules—either static, as in Dalton’s subsequent chemical theory, or mobile, as in Clausius–Maxwell–Boltzmann’s century-later gaskinetic theory. Hence, he was obviously unaware of the hidden constitutive assumption implicit in the relation v ¼ vm . As such, a material domain must necessarily have been viewed by Euler as a continuous body of permanent material integrity insofar as its underlying constitution was concerned. Today we know better. Accordingly, as regards continuum mechanics, and in particular the correctness of the Cauchy linear momentum equation, the relation v ¼ vm needs to be regarded as a tentative constitutive assumption, rather than as an identity, thereby subject to empirical (experimental) veriﬁcation and, if possible, theoretical justiﬁcation. Given these remarks, it remains to be established whether the seemingly unequivocal separation advocated in the literature between the statistical mechanics and thermodynamics of continuum models of large multiparticle systems composed of materially identical particles is viable. GENERIC theory argues forcefully against such separability. The current nontraditional version of GENERIC theory appearing in O¨ttinger’s book [23] represents an attempt at a systematic synthesis of the two ﬁelds. It constitutes a broad general theory, equally applicable to both single-component gases and liquids. In what follows, we point out that O¨ttinger’s theory provides a rational basis for our proposed diffuse volume-based modiﬁcations [1–3] of the N–S–F equations, including, most prominently, changes to the accepted constitutive formula for the speciﬁc momentum [41].

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constitutive equation emanating from GENERIC for the difference v vm in velocities, while temporarily leaving open issues pertaining to the constitutive equations for the heat ﬂux vector and deviatoric stress dyadic. Based solely upon our fundamental hypothesis that v ¼ vm if, and only if, the ﬂuid is ‘‘incompressible,’’ i.e., r ¼ const. throughout the ﬂuid, it is demonstrated that this is tantamount to requiring that the velocity difference be of the constitutive form v vm ¼ Kr ln r, where the coefﬁcient K is nonnegative-deﬁnite. Since this unorthodox momentum density constitutive result holds independently of the constitutive equations governing the heat ﬂux and stress tensor, it necessarily applies to any ﬂuid continuum, gaseous or liquid, irrespective of the ﬂuid’s rheological and thermal properties. Section 4 discusses the thermodynamic consequences stemming from this constitutive relation, including the fact that the diffusive heat ﬂux generally differs from the diffusive internal energy ﬂux, and that the entropy ﬂux is not simply equal to the heat ﬂux divided by the thermodynamic temperature T in single-component ﬂuids. Both conclusions are at odds with the current tenets of (linear) irreversible thermodynamics. Section 5 uses molecularly derived results emanating from Burnett’s solution of the Boltzmann equation to establish that the velocity-difference coefﬁcient K is equal to the gas’s thermometric diffusivity a in the case of rareﬁed gases. In turn, in Section 6, comparison of the preceding dynamical result, namely v ¼ vm þ ar ln r, with the known, purely kinematical, formula [4] for the volume velocity of a liquid or gas, vv ¼ vm þ ar ln r, leads to the conclusion that the ﬂuid’s speciﬁc momentum density v is identical to its volume velocity vv . Since the ﬂuid’s volume velocity has already been shown elsewhere [2,3,15] to be equal to its tracer or Lagrangian velocity, this equality serves, in turn, to demonstrate equality of the ﬂuid’s speciﬁc momentum with its tracer velocity, certainly in the case of rareﬁed gases. Section 7 offers justiﬁcation for three key constitutive assumptions which it was necessary for us to make in the course of evaluating the two undetermined parameters, a0 and D0 , appearing in Ottinger’s nontraditional version of GENERIC [23]. Identifying these two parameters was necessary in order to obtain deﬁnitive results, against which experimental data could be compared. Section 8 broadly discusses the implications of the present theory for LIT as a whole, while stressing the fact that the consequences of the nontraditional GENERIC addition to the friction matrix appearing therein are explicitly manifested in the Onsager force–ﬂux LIT scheme for single-component ﬂuids by the addition thereto of a volume ﬂux, above and beyond the usual heat and momentum ﬂuxes. In turn, the force–ﬂux basis of LIT suggests a simple physical origin (i.e., one independent of the formal constitutive friction matrix approach of GENERIC) for the apparently universal relation v ¼ vm þ Kr ln r, which is believed to be applicable to all gases and liquids, with the value of the phenomenological coefﬁcient K depending upon the particular physical application being addressed. This applicability extends to multicomponent ﬂuids [4], in which circumstance K ¼ D, with D the binary diffusion coefﬁcient D for isothermal two-component ﬂuid mixtures. It is further pointed out (Section 8.4) that, based on the recent ‘‘proper coarse-graining’’ work of O¨ttinger, an alternative may exist to our fundamental incompressibility hypothesis in Section 3, although in the special, but important, case of ideal gases, the conclusions derived therefrom are indistinguishable from those obtained from our hypothesis. Also noted (Section 8.5) is the fact that issues of momentum transport and volume diffusion-induced stress, comparable to those issues identiﬁed here for gases and liquids, also exist in the case of solids. Explicitly, based on experimental data dating back to the classical interdiffusion experiments of Kirkendall and their subsequent interpretation by Darken, it is pointed out that ‘‘two-velocity’’ modiﬁcations of the Cauchy linear momentum equation as well as the Hooke’s law-type stress–strain constitutive equation appearing therein for solid solutions (alloys) have been proposed in the literature. Just as in the case of ﬂuid phases, these unorthodox dynamical solid-phase notions are attributed to diffuse volume transport. Such transport in solids is ascribed largely to atom–vacancy exchanges occurring within the lattice during the diffusional process, and, to a lesser extent, by comparable atom–atom lattice-point interchanges when the respective molecular masses of the interdiffusing species differ. Finally, Section 9 offers a summary and overview of the essence of our ﬁndings.

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2. Review of O¨ttinger’s nontraditional GENERIC theory

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GENERIC (general equation for the non-equilibrium reversible–irreversible coupling) theory, as set forth in BET [23], offers a derivation of the following nontraditional trio of mass, linear momentum and energy

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transport equations for the case of single-component ﬂuids (both gases and liquids) in the absence of external body-force ﬁelds (Eqs. (2.78)–(2.80) of BET): (i) Mass transport (continuity equation): qr þ r.ðrvm Þ ¼ 0. qt

(1)

(ii) Momentum transport (Cauchy linear momentum equation): qðrvÞ þ r.ðrvm vÞ ¼ r.P, qt

(2)

where (3)

P ¼ Ip þ T

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qu þ r.ðvm uÞ ¼ r.½jq ðv vm Þðra0 uÞ þ pu , (4) qt where u is the volumetric (i.e., per unit volume) internal energy density (designated by the symbol in BET), jq is the heat ﬂux vector, and pu ¼ P : ðrvÞT

(5)

is the temporal rate of production of internal energy per unit volume, wherein the superscript T denotes the transpose operator. In addition, a0 is an unconstrained phenomenological parameter appearing in extended GENERIC theory (for which the symbol a is used in BET). This parameter is to be determined constitutively by some scheme, be it theory, simulation or otherwise, but ultimately subject to conﬁrmation by experiment. Determination of a0 , which will be seen to physically represent an Onsager coefﬁcient serving to couple together the respective processes of diffuse internal energy and volume transport, is one of the central goals of the present analysis. In addition to the preceding equations, we shall later require the nontraditional, ﬂuctuation-based GENERIC entropy transport equation, given in BET as Eq. (2.83), namely: (iv) Entropy transport: q qs j p þ u ra0 þ r.ðvm sÞ ¼ r. þ ðv vm Þ (6) þ ps , qt T T

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is the pressure tensor (using the usual ﬂuid-mechanical sign convention for stress, rather than the opposite convention used in BET), with I the dyadic idemfactor and T the symmetric deviatoric or viscous stress tensor. ^ [16] by the The ‘‘velocity’’ v appearing in Eq. (2) is deﬁned in terms of the ﬂuid’s speciﬁc momentum density m ^ relation v:¼m. (BET writes Eq. (2) in a form involving the momentum per unit volume, the latter identiﬁed therein by the symbol M, equivalent to our rv.) (iii) Energy transport:

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r2 jv vm j2 1 1 þ jq .r þ T : ðrvÞT . (7) T T D0 Here, s and ps are, respectively, the volumetric entropy density and temporal rate of irreversible entropy production. It is important to note that neither the symbols pu and ps , respectively, appearing in Eqs. (4)–(5) and (6)–(7) nor the respective volumetric production-rate interpretations that we have assigned to them appear in the original BET equations cited. As such, justiﬁcation for these interpretations is required. This is presented in Section 7 at the conclusion of the analysis. For the time being we simply pursue the consequences of these interpretations (cf. Eqs. (11) and (12)). It will prove convenient in what follows to deﬁne the velocity difference vector:

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ps ¼

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j:¼v vm .

(8)

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qc þ r . n c ¼ pc , (9) qt where c ¼ C=V is the volumetric density of the property, nc is the Eulerian ﬂux density of the property and pc is the volumetric rate of production of the property. The (total) ﬂux density nc ¼ vm c þ jc

appearing above consists, respectively, of a convective portion vm c carried by the moving ﬂuid mass and a diffusive or nonconvective portion jc . In terms of this notation, the continuity equation retains its usual form (1) since the law of conservation of mass together with the deﬁnition vm :¼nm =r of the mass velocity requires that the diffuse mass ﬂux density be zero: jm ¼ 0. On the other hand, the internal energy and entropy transport equations adopt the respective forms

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(11)

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qu þ r.ðvm uÞ ¼ r.ju þ pu qt

qs þ r.ðvm sÞ ¼ r.js þ ps . (12) qt In the latter equation the second law of thermodynamics requires satisfaction of the generic Clausius–Duhem inequality [19–21],

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In the next section we will propose a constitutive equation for j. The system of equations described in the preceding paragraphs represents the entire set of single-component GENERIC equations of which we will avail ourselves in what follows. Of course, in order to understand the origin of these equations one needs to master the basic reversible–irreversible model of transport processes underlying the GENERIC scheme, as set forth in BET. This differs from the more common convective–diffusive model of such phenomena [18]. Before proceeding to a discussion of the constitutive equations required to complete the governing set of equations (1)–(7), it will prove convenient to re-express the preceding energy and entropy transport equations in more conventional terminology. In terms of the general notation set out in Ref. [4] consider the transport of any extensive physical property, with C, say, the amount of the property contained in a volume V. The Eulerian balance equation governing transport of this extensive property is then

ps X0.

(13)

Comparison of Eq. (11) with (4) furnishes the relation ju ¼ jq jðra0 uÞ.

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Similarly, comparison of Eq. (12) with (6) gives 1 q ½j þ jðp þ u ra0 Þ. T Elimination of jq between the latter pair of equations yields the expression

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(15)

(16)

It is important to note that this relation is strictly a consequence of the principles of GENERIC theory, independent of any constitutive relations (although more general nonlinear and nonlocal possibilities than (16) exist within the GENERIC framework). However, its validity hinges critically upon the interpretations we have assigned to the symbols pu and ps appearing in Eqs. (4)–(5) and (6)–(7), since it was those interpretations which resulted in Eqs. (14) and (15), and which thereby led to Eq. (16). The signiﬁcance of Eq. (16) will subsequently be discussed in the context of the combined ﬁrst and second laws of thermodynamics,

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dU ¼ T dS p dV ,

(17)

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1 3 5 7 9 11 13 15 17

for a system of ﬁxed mass. Explicitly, the obvious analogy existing between Eqs. (16) and (17) suggests that a subscript v should be afﬁxed to the symbol j in Eq. (16), with the resulting quantity jv identiﬁed as being the diffusive ﬂux density of volume [4]. The latter is a physically deﬁned, purely kinematical quantity, related to the ﬂuid’s volume velocity vv discussed in the Introduction through the relation vv vm ¼ jv . In turn, comparison of the latter with Eq. (8) would then give rise immediately to the constitutive relation v ¼ vv , thus identifying the ﬂuid’s speciﬁc momentum density with its volume velocity. It is this fact, namely that j ¼ jv , which we propose to formally demonstrate, or at least make plausible, in what is to follow. Before leaving this section, we note that all of the above BET transport equations are deliberately expressed in a space-ﬁxed, Eulerian, format rather than a (moving) ‘‘body-ﬁxed’’ material format. This avoids the ambiguous notion of so-called ‘‘material differentiation following the motion’’ of a material body, such ambiguity arising from the issue of whether the ﬂuid’s mass velocity vm , which is a (normalized) ﬂux, is indeed synonymous with the notion of the ﬂuid’s Lagrangian or tracer ‘‘motion’’ through space [2,3]. The two classes of description, Eulerian and material, are connected through the relation qc Dm c^ , þ r.ðvm cÞ ¼ r Dt qt ^ ¼ c=r is the speciﬁc density of the extensive property C, and where c

(18)

23

3. Constitutive relation for the speciﬁc momentum density v

25

According to O¨ttinger’s ﬂuctuation-based GENERIC scheme [23], the most general possible constitutive equation for the velocity difference (8), applicable to both single-component gases and liquids, is (BET, Eq. (2.82)) D0 p 1 0 j¼ 2 r ðra uÞr . (20) T T r

39 41 43 45

49 51

O O

PR

D

ra0 u ¼ p gT,

(22)

so as to furnish the relation j¼

47

TE

EC

appearing above quantiﬁes the intensity of the ﬂuid’s molecularly based ﬂuctuations [42]. Apart from having to satisfy the preceding inequality, D0 , like a0 , represents an otherwise unconstrained phenomenological parameter, to be chosen in a manner such as to match experimental data and/or accommodate detailed theories, molecular or otherwise, of the pertinent phenomena. Such phenomenological choices, of course, include the trivial possibility that D0 ¼ 0, and hence j ¼ 0, corresponding as a consequence of Eq. (8) to Euler’s conventional momentum density constitutive relation v ¼ vm . Eq. (20) can be rearranged so as to express j in terms of rp and rr as follows: in place of the parameter a0 it proves useful to introduce another parameter, g, the latter deﬁned by the expression

R

37

(21)

R

35

D0 X0

O

33

The nonnegative-deﬁnite diffusion-like phenomenological coefﬁcient [23]

C

31

N

29

(19)

D0 ðrp grTÞ. r2 T

U

27

F

21

Dm q ¼ þ vm . r Dt qt is the material derivative.

19

But, from the single-component ﬂuid equation of state relating T, p and r, we have that rT ¼ ðqT=qpÞr rp þ ðqT=qrÞp rr, whence the above becomes (" ) # D0 qT 1 1g j¼ 2 rp þ g r ln r , qp r b r T

PHYSA : 9965 ARTICLE IN PRESS H. Brenner / Physica A ] (]]]]) ]]]–]]]

23 25 27 29 31 33 35 37 39 41 43 45 47 49 51

F

O

We now introduce what appears to us to be the physically reasonable hypothesis that when the ﬂuid is ‘‘incompressible,’’ namely when r ¼ const, and only then, the two velocities, v and vm , appearing in the basic BET transport equations of Section 2 should coalesce into a single entity, so that v ¼ vm . From Eq. (8) this is equivalent to the relation

O

21

PR

19

j ¼ 0 when r ¼ const ði.e., when rr ¼ 0Þ.

(27)

The latter relation leads to the dual requirements that: (i) the bracketed pressure-gradient coefﬁcient in Eq. (25) needs to be identically zero; and (ii) the diffuse ﬂux j must obey the constitutive relation

D

17

Thus far, no assumptions have been made beyond the basic nontraditional GENERIC transport relations set forth in Section 2, including the general constitutive assumption for j embodied in Eq. (20). For later reference we note that we have, in effect, merely replaced the original parameter a0 appearing in BET by the new parameter K, the relationship between them being br2 T 1 0 . (26) a ¼ uþpK D0 r

j ¼ Kr ln r.

(28)

In the subsequent discussion of Section 7 it will be pointed out that the further, second law-based, Clausius–Duhem requirement, KX0, in the above relation (see Eq. (46)) is formally equivalent to requiring that Fourier’s law (cf. Eq. (42)) be satisﬁed. This surprising linkage of Eq. (28), the latter involving a density gradient, with Fourier’s law, the latter involving an independent temperature gradient, appears surprising. This connection will be seen (in Section 7) to arise from the fact that a0 represents a coupling coefﬁcient which, as a consequence of Onsager’s reciprocal theorem, cannot be arbitrarily chosen if one insists that Fourier’s law of heat conduction must be satisﬁed under all circumstances (i.e., independently of the density gradient, and hence of the pressure gradient in single-component systems). By the chain rule for partial differentiation [43] in conjunction with deﬁnition (23), we have that bðqT=qpÞr ¼ kT , where 1 qr kT ¼ (29) r qp T

TE

15

(25)

EC

13

(24)

R

11

br2 T , D0 whereupon it follows that " # D0 qT j ¼ 2 Kb rp þ Kr ln r. qp r r T g¼K

R

9

In place of g it now proves convenient to instead introduce a related parameter K, deﬁned by the expression

O

7

(23)

C

5

is the coefﬁcient of isothermal compressibility, assumed to be nonnegative for both liquids and gases:

N

3

where b is the coefﬁcient of thermal expansion: 1 qr b¼ . r qT p

kT X0.

U

1

9

(30)

The latter inequality constitutes a requirement for stability of the ﬂuid continuum (as discussed in Section 7.6; cf. Eq. (101)). With use of the preceding relations, the vanishing of the bracketed term in Eq. (25) necessitates that D0 ¼ Kr2 TkT .

(31)

This expression serves to relate D0 to the phenomenological coefﬁcient K appearing in Eq. (28). In turn, from Eq. (24), this makes

PHYSA : 9965 ARTICLE IN PRESS H. Brenner / Physica A ] (]]]]) ]]]–]]]

10

13 15 17 19 21 23 25 27 29

(32)

where we have again used the chain rule [43]. Upon inserting Eq. (31) into (26), subsequent use of Eq. (32) furnishes the relation " # 1 qp 0 uþpT a ¼ . (33) r qT r The latter is, in effect, a constitutive expression serving to relate the GENERIC parameter a0 to the ﬂuid’s local equilibrium thermodynamic (i.e., nontransport) properties. For the moment we leave open the choice of the phenomenological coefﬁcient K appearing in the constitutive equation (28) for the velocity difference j. This same coefﬁcient serves to determine the parameter D0 from Eq. (31). Hence, as a consequence of the hypothesis embodied in Eq. (28), of the original two phenomenological parameters D0 and a0 appearing in the fundamental GENERIC constitutive equation (20), only the parameter D0 remains yet to be determined, as presently manifested in the coefﬁcient K. Of course, as earlier mentioned, this includes the possibility that K ¼ 0, and hence D0 ¼ 0, corresponding to Euler’s traditional momentum density relation, v ¼ vm . It needs to be stressed that Eq. (33), expressing O¨ttinger’s GENERIC ﬂuctuation parameter in terms of the fundamental equilibrium properties of the ﬂuid, derives directly from the constitutive assumption (27), whose validity can be conﬁrmed only by demonstrating that the physical results issuing therefrom accord either directly with experiment or else with an accepted theory. As such, from this point on, the validity of all subsequent relations in this paper derived indirectly from this assumption is to be regarded as tentative, subject to veriﬁcation. This cautionary emphasis is recapitulated and reviewed near the conclusion of the paper, in Section 8.4. However, was Eq. (27) to be proved inconsistent with experiment, the subsequent effort expended in our paper would, nevertheless, not have been fruitless. Rather, all that would be required to rectify the situation would be to simply carry a0 along as a free parameter, beginning with Eq. (20), and with a0 thus appearing explicitly in the subsequent LIT equations of Section 7. This issue is brieﬂy discussed in Section 8.4.

F

11

, r

O

9

O

7

qp qT

PR

5

D

3

b g¼ ¼ kT

TE

1

4. Consequences of the ‘‘incompressibility’’ hypothesis

33

4.1. Internal energy flux

43 45 47 49 51

R

R

O

41

This expression can be re-formulated in more physical terms by noting from the ﬁrst and second laws of thermodynamics that for a single-component ﬂuid [44] qp p d^v. (35) du^ ¼ c^v dT þ T qT v^

C

39

(34)

N

37

Introduction of Eq. (33) into (14) gives, for the internal energy ﬂux, " # qp q ju ¼ j j p T . qT r

The caret atop a symbol denotes a speciﬁc (i.e., per unit mass) density, so that, for example, u^ ¼ u=r denotes the speciﬁc internal energy and v^ ¼ 1=r the speciﬁc volume. Consequently, we have the thermodynamic identity qp qu^ qU pT ¼ , (36) qT v^ q^v T qV T;M

U

35

EC

31

where, with M ¼ rV the mass contained in a volume V ¼ M v^, the extensive internal energy contained in V is ^ It follows that Eq. (34) is equivalent to the relation U ¼ M u.

PHYSA : 9965 ARTICLE IN PRESS H. Brenner / Physica A ] (]]]]) ]]]–]]]

ju ¼ jq þ jðqU=qV ÞT;M .

1 3 5

(37)

In the context of the possibility, discussed following Eq. (17), that j ¼ jv , the second term of Eq. (37) will, in Section 7.2, be seen to possess a well-deﬁned physical signiﬁcance. Eq. (36) shows the internal energy of an ideal gas to be independent of its volume, i.e., ^ vÞT ¼ 0. In such circumstances it follows from Eq. (37) that ðqU=qV ÞT;M ðqu=q^ ju ¼ jq

7 9

11

(ideal gases).

(38)

This relation also holds for any single-component ﬂuid, either gas or liquid, whose speciﬁc internal energy is a function only of its temperature.

11 4.2. Entropy flux 13

23

it follows that js ¼

25 27

jq qS þj . qV T;M T

45 47 49 51

F

R

(42)

R

with kX0

(43)

C

O

the thermal conductivity. Indeed, rather than simply assuming the validity of (42), we will, in Section 7.5, subsequently derive this equation as an immediate consequence of the second law of thermodynamics when the latter is considered in conjunction with Onsager’s reciprocal relation. We also suppose that the deviatoric stress T is given, for example, by the usual rheological constitutive equation for a viscous Newtonian ﬂuid, namely

N

43

jq ¼ krT,

U

41

EC

In order for the present theory to be viable, the constitutive relations entering into the entropy production rate (7) must be of such a nature that ps satisﬁes the Clausius–Duhem inequality (13). Following BET, we suppose that the heat ﬂux vector jq appearing in Eq. (7) obeys Fourier’s law of heat conduction,

35

39

(41)

TE

4.3. Entropy production 31

37

(40)

In the context of the possibility, discussed following Eq. (17), that j ¼ jv , the second term of Eq. (41) will later be seen to possess a well-deﬁned physical signiﬁcance.

29

33

(39)

O

21

Alternatively, with use of the thermodynamic identity [44] qp q^s qS ¼ , qT v^ q^v T qV T;M

O

19

PR

17

From Eqs. (33) and (15) one obtains the following expression for the entropy ﬂux: jq qp js ¼ þ j . qT v^ T

D

15

T ¼ 2Zrv þ ZB Ir.v,

(44)

where ZX0 and ZB X0 are the ﬂuid’s respective shear and bulk viscosities. In as much as T represents the diffuse momentum ﬂux density, it appears appropriate to suppose that the symbol v appearing in the preceding expression refers to the ﬂuid’s speciﬁc momentum density [45] rather than to, say, its mass velocity vm . Upon introducing the above pair of constitutive equations jointly with Eqs. (28) and (31) into (7) we ﬁnd that

PHYSA : 9965 ARTICLE IN PRESS H. Brenner / Physica A ] (]]]]) ]]]–]]]

12

1 3 5

1 K k 2 2 2 . ps ¼ ðrrÞ þ ðrTÞ þ 2Zrv : rv þ ZB ðr vÞ . T kT r2 T

(45)

Given the nonnegative-deﬁnite algebraic signs of kT , k, Z and ZB , Eq. (45) shows that the Clausius–Duhem inequality (13) will be satisﬁed provided that KX0.

7

(46)

23

5. Conﬁrmation of Eq. (28) and identiﬁcation of the phenomenological coefﬁcient K

25

Apart from our acceptance of the nontraditional GENERIC formulation of irreversible thermodynamics set forth in BET [23] as being physically correct, together with our subsequent constitutive assumptions implicit in the production terms (5) and (7), the only additional assumption we have made thus far is explicitly embodied in the hypothesis (27), namely the assumption that v ¼ vm when the ﬂuid is ‘‘incompressible.’’ (Of course, we have also supposed the applicability of Fourier’s law and Newton’s rheological law, Eqs. (42) and (44), although the latter equation is not critical, whereas the former is.) There remains only the task of relating the phenomenological coefﬁcient K appearing in Eq. (28) to the physical properties of the ﬂuid, and of subsequently conﬁrming that the resulting expression for K satisﬁes inequality (46). In the absence of a microscopic theory of the pertinent phenomena this task is normally assigned to experiment, namely that of empirically conﬁrming the physical viability of the union of Eqs. (8) and (28), and, concomitantly, establishing the functional dependence of K upon the system’s parameters. In the case of liquids, for which no fully accepted theory yet exists, one has no recourse other than to revert to experiment in pursuit of these goals. Fortunately, however, in the case of rareﬁed gases an accepted molecular theory already exists, one that will be seen to sufﬁce for these purposes. Explicitly, we can avail ourselves of Burnett’s extension [12] to higher Knudsen numbers of the Chapman–Enskog scheme [13] for solving Boltzmann’s transport equation in the rareﬁed gas regime. Chapman and Enskog’s calculations [13] theoretically predict, inter alia, at least for rareﬁed gases, that the Fourier and Newtonian rheological law constitutive relations (42) and (44), previously regarded as empirical experimental laws valid for continua, are indeed applicable in the so-called ‘‘near-continuum,’’ OðKnÞ, region of small Knudsen numbers, Kn51, with the Oð1Þ terms represented by the inviscid, ideal ﬂuid Euler equations [36]. Moreover, Chapman and Enskog’s theoretical perturbation scheme concomitantly furnishes the values of the phenomenological coefﬁcients k, Z and ZB appearing therein, at least for particular intermolecular collision model choices (e.g., rigid–elastic spheres, Maxwell molecules, Lennard-Jones potentials, etc.). In what follows, Burnett’s [12] Knudsen number extension of the Chapman–Enskog theory [13] will be seen to provide conﬁrmation of the constitutive equation (28) for j, at least for the case of monatomic Maxwell molecules [13,47], while at the same time conﬁrming the inequality (46) by showing that

31 33 35 37 39 41 43 45 47 49 51

O

O

PR

D

TE

29

EC

27

R

19

R

17

O

15

C

13

N

11

U

9

F

21

In view of Eq. (31), the latter inequality is equivalent to that requiring satisfaction of the previously stipulated inequalities (21) and (30). We have referred to the Newtonian rheological constitutive expression (44) as being ‘‘standard.’’ However, this terminology is somewhat ambiguous since the velocity v appearing therein is often implicitly understood in the literature to be the mass velocity vm appearing in the continuity equation (1), rather than representing the speciﬁc momentum density v. (This implicit vm assumption stems from the fact that the symbol for velocity ﬁrst arises in courses in ﬂuid mechanics in the context of deriving the continuity equation. Only later is this mass-based symbol identiﬁed with the Lagrangian notion of the movement of an ‘‘object’’ through space in a trajectory sense, the latter being precursive to the association of this symbol with Newtonian dynamics, and hence its role as a momentum density.) With few exceptions [36, p. 196 (but see also p. 28), 37], the possibility that a difference might exist between the ﬂuid’s momentum density v and its mass velocity vm has not generally been recognized. In a similar vein, referring to Fourier’s law (42) as ‘‘standard’’ is equally ambiguous, since in the past it has not generally been recognized, certainly not in single-component systems [46], that a difference might exist between the diffuse internal energy ﬂux ju and the heat ﬂux jq . Nevertheless, according to Eq. (34), a difference does exist generally, except in the case of ideal gases, where Eq. (38) applies.

K ¼ a,

(47)

PHYSA : 9965 ARTICLE IN PRESS H. Brenner / Physica A ] (]]]]) ]]]–]]]

13 15 17 19 21

F

11

is the thermometric diffusivity, in which c^p is the isobaric speciﬁc heat. As heralded above, the proof to be offered of Eq. (28), wherein K is given by Eq. (47) (thus satisfying inequality (46)), derives from Burnett’s extension of Chapman and Enskog’s expansion scheme for solving the Boltzmann equation to OðKn2 Þ, namely beyond the so-called ‘‘near continuum,’’ OðKnÞ, N–S–F level. In that context, Kogan et al. [6–9] and Bobylev [10] have shown that at small Mach numbers ðMa51Þ and for Reynolds numbers of order unity ðRe ¼ Oð1ÞÞ, the so-called ‘‘thermal stress’’ terms appearing in Burnett’s [12,13] expression for the deviatoric stress T, rather than being of OðKn2 Þ as originally supposed, are, in fact, of the same OðKnÞ order as are the N–S–F equations themselves, at least in the case of so-called SNIF ﬂows (slow nonisothermal ﬂows) [9,10]. The scaling argument used to rationalize this Knudsen number re-ordering derives, in part, from the fact that Kn ¼ Ma=Re, so that a small Knudsen number can be achieved as indicated above, rather than by the classically assumed circumstances [13] where Ma ¼ Oð1Þ and Reb1. The SNIF limit [10] especially includes the limiting case where vm ¼ 0 throughout the ﬂuid, such as would be encountered, for example, in the elementary case of one-dimensional steady-state heat conduction through a gravity-free gas conﬁned between parallel, laterally unbounded, walls permanently maintained at different temperatures, where (from a macroscopic viewpoint) the pressure is sensibly uniform throughout the ﬂuid, except perhaps in thin Knudsen boundary layers existing proximate to the wall [48]. In such circumstances, the Burnett thermal stresses (to which we have referred elsewhere [3] as being the Maxwell–Burnett stresses [49]) are of the form [13, p. 286]

O

9

23 T¼ 25

37 39 41 43 45 47 49 51

D

3Z rðZrTÞ. rT

EC

Now, Z ¼ ru, where u is the kinematic viscosity. Additionally, for a single-component ideal gas, the relation between density and temperature at constant pressure is such that rT ¼ const., whence it readily follows for the present one-dimensional, steady-state, isobaric heat conduction case under consideration that T ¼ 3Zrður ln rÞ.

However, for an ideal monatomic gas, the Prandtl number Pr ¼ u=a has the value preceding relation becomes

R

35

T¼

O

33

where K 1 and K 2 are Oð1Þ nondimensional constants, whose respective numerical values depend slightly upon the particular molecular collision model adopted [13]. For monatomic Maxwellian molecules the values of the two constants appearing above are, respectively, K 1 ¼ 3 and K 2 ¼ 3 d ln Z=d ln T [13, pp. 288–289]. Accordingly, in that case Eq. (49) adopts the form

T ¼ 2Zrðar ln rÞ.

2 3

[18]. Accordingly, the (50)

C

31

(49)

In addition to the latter relation, it is also known theoretically in the case of ideal monatomic gases (which includes Maxwell molecules) that the bulk viscosity is zero [13,47]. Hence, in addition to (50) it is also true in present circumstances that

N

29

Z2 ðK 1 rrT þ K 2 T 1 rTrTÞ, rT

U

27

(48)

O

7

k r^cp

PR

5

a¼

TE

3

where

R

1

13

ZB ¼ 0.

(51)

Upon comparing the union of Eqs. (50) and (51) with that of Eqs. (8) and (44) in the light of the fact that vm ¼ 0 in present circumstances (thus making v ¼ j, and hence T ¼ 2ZrjÞ, it follows that j ¼ ar ln r.

(52)

Comparison of the latter with (28) serves to conﬁrm the viability of hypothesis (27). While we have derived Eq. (52) by considering only a rather restricted set of circumstances, namely, vm ¼ 0 (and rp ¼ 0Þ, the conclusion

PHYSA : 9965 ARTICLE IN PRESS H. Brenner / Physica A ] (]]]]) ]]]–]]]

14

9 11 13 15 17 19 21 23

F

7

O

5

O

3

that K ¼ a bears no such restriction because Eq. (28) is true independently of such restrictions. In particular, as discussed in the derivation of that equation, pressure gradients play no role in its validity. While it is true that Eq. (47) has only been conﬁrmed to hold for ideal monatomic gases composed of Maxwell molecules, on the basis of Eqs. (8) and (28) this would nevertheless appear sufﬁcient to conclude that, in general, there does exist a fundamental difference between the gas’s speciﬁc momentum density v and its mass velocity vm (provided, of course, that we accept the Boltzmann equation and the Burnett–Chapman–Enskog Knudsen number expansion scheme as being valid, at least at small Reynolds numbers, where the momentum density terms in the Navier–Stokes equations are small—a position accepted by most researchers). More importantly, the inequality vavm depends critically upon the assumption that the present ﬂuctuationbased GENERIC equations [23] provide a proper physical foundation for irreversible (gas) transport processes. As such, the experimental veriﬁcation of Eq. (52) for gases (and liquids) would serve to provide a rigorous test of the viability of O¨ttinger’s nontraditional GENERIC scheme [23]. In the next section we discuss such tests. Before doing so, however, we brieﬂy note below the existence of a second, equally unorthodox, theory of irreversible processes for single-component ideal gases due to Klimontovich [28–30]—a theory based directly upon molecular-level arguments rather than upon the present GENERIC macroscopic phenomenological arguments—and which serves to independently conﬁrm Eq. (52). Explicitly, Klimontovich argues in favor of the relation v ¼ vm þ Dr ln r, analogous to Eqs. (8) and (28), in which the symbols v and vm have the same physical signiﬁcance as described above, and in which the symbol D denotes Klimontovich’s physical-space single-component self-diffusion coefﬁcient. Furthermore, he argues that D is equal to the gas’s thermometric diffusivity a, thereby furnishing the same K ¼ a relation (47) as derived earlier from Burnett’s Boltzmann equation-based molecular calculations. In fairness, however, it needs to be emphasized that Klimontovich’s nontraditional scheme has not yet been subjected to the same searching theoretical scrutiny as either Burnett’s original, more traditional, scheme or Kogan et al.’s [6–9] and Bobylev’s [10] re-ordering of the thermal stress portion of the Burnett terms.

PR

1

25

31

Eqs. (8) and (52) or, more precisely, the similar-appearing pair, Eqs. (53) and (54), set forth below (wherein the subscript ‘‘v’’ appears), have a pre-history which is completely independent of the present GENERIC scheme. In particular, in earlier work [4] we derived the purely kinematical relation

D

29

6. Identiﬁcation of the speciﬁc momentum density v as the volume velocity vv

TE

27

vv ¼ vm þ j v ,

41 43 45

EC

R

(54)

R

The pair of relations displayed above, each derived theoretically, have been conﬁrmed indirectly by comparison of the physical consequences stemming therefrom with experimental data for both gases and liquids [2–5,14,50]—without, however, supposing the symbol vv appearing in Eq. (53) to be equal to the ﬂuid’s speciﬁc momentum density v (since inertial effects, wherein v would otherwise have proved pertinent, were negligible in all of the experiments to which Eqs. (53) and (54) pertain [51]). Given that the respective righthand sides of Eqs. (52) and (54) are identical, it follows that

O

39

jv ¼ ar ln r.

C

37

N

35

where vv is the volume velocity (which is equivalent to the Eulerian ﬂux density of volume nv —cf. Eq. (10) wherein c ¼ 1 for the case of volume) and jv is, by deﬁnition, the diffusive ﬂux density of volume. The latter is given constitutively for single-component gases or liquids undergoing heat transfer by the relation [4]

j ¼ jv .

U

33

(53)

(55)

Comparison of Eq. (8) with (53) in the light of the latter identity reveals that 47 49 51

v ¼ vv .

(56)

It has been argued [1–3,14,15] that vv is identical to the ﬂuid’s ‘‘tracer’’ or ‘‘Lagrangian’’ velocity, say vl , the latter representing the actual physical velocity of an object through space (as opposed to the ﬂuid’s mass or volume velocities, both of which are ﬂux densities in disguise [2]). Thus, the physical essence of Eq. (56) is embodied in the relation

PHYSA : 9965 ARTICLE IN PRESS H. Brenner / Physica A ] (]]]]) ]]]–]]]

1 3 5

15

v ¼ vl .

(57)

In as much as momentum also involves the notion of an object moving through space, Eq. (57) constitutes one of the key results of our analysis, namely the conclusion that the ﬂuid’s speciﬁc momentum density is equal to its Lagrangian velocity rather than to its mass velocity. And, to the extent that our hypothesis (27) is valid, it is only in the case where the ﬂuid is ‘‘incompressible’’ that the two results coincide.

7 7. Onsager-based irreversible, thermodynamic justiﬁcation of our post-GENERIC assumptions 9 7.1. Prelude

29 31 33 35 37 39 41 43 45 47 49 51

F

O

O

PR

D

27

TE

25

EC

23

R

21

R

19

O

17

C

15

N

13

The present paper has built upon O¨ttinger’s [23] incorporation of ﬂuctuations into the GENERIC friction matrix. This involved the introduction of two new parameters, a0 and D0 , into the scheme. Except for the required algebraic sign of D0 as set forth in Eq. (21), both parameters are left constitutively undetermined in O¨ttinger’s original BET development (see, however, Section 8.4). Accordingly, bringing his theory to fruition, namely to the stage where its predictions can be compared with experiment, necessitates specifying constitutive expressions for these two key phenomenological coefﬁcients in terms of physically measurable properties. It is this key step to which the present paper has largely been devoted. The critical assumptions that we made enroute to the goal of establishing plausible constitutive relations for a0 and D0 involved our identiﬁcation of the internal energy and entropy production rates, pu and ps , indicated in Eqs. (5) and (7), respectively, together with the adoption of the ‘‘incompressibility’’ hypothesis (27). These led to our constitutive determination of the parameter a0 , Eq. (33), following which subsequent determination of the remaining parameter D0 (cf. Eqs. (31) and (47)) was straightforward. Accordingly, we need focus here only on the three key hypotheses cited above, leading to our eventual determination of a0 , as set forth in Eq. (33). The justiﬁcation offered below for these constitutive assumptions is based, ultimately, upon the agreement of the predictions of the resulting theory with experiment—which is as it should be. Because the theory is, at this stage, limited to single-component ﬂuids, the conﬁrming experiments to which we refer are necessarily limited to such systems. The key experimental laws in this connection are: (i) Fourier’s law of heat conduction, Eq. (42), which refers to diffuse energy transport arising exclusively from a temperature gradient; (ii) the Clausius–Duhem inequality, Eq. (13); (iii) Onsager’s reciprocal theorem, to be discussed, involving coupling between the independent ﬂuxes appearing in the ‘‘force–ﬂux’’ relations appearing therein; and (iv) Curie’s law, which denies the possibility of coupling in linear isotropic systems between ﬂuxes whose respective tensorial orders differ by an odd integer. Together, these laws provide a test of the legitimacy of any theory of linear irreversible processes that purports to be generally applicable. Viewed alternatively, the a priori acceptance of the general correctness of LIT provides a test of the internal consistency of our theory. The critical importance of the coupling issue here resides in our claim that the diffuse ﬂux of volume jv represents an independent ﬂux, over and above the traditional diffuse ﬂuxes of internal energy ju (or heat jq Þ and momentum T. Owing to Curie’s law, the diffuse second-rank tensor momentum ﬂux is necessarily uncoupled from the diffuse vector internal energy (or heat) ﬂux. With diffuse volume accepted as representing yet another independent vector ﬂux, as is advocated here, it too would be uncoupled from the momentum ﬂux. On the other hand diffuse volume would be coupled to the internal energy ﬂux, and thus subject to the restrictions imposed by Onsager reciprocity, requiring equality of the cross coefﬁcients in the ﬂux vs. driving force reciprocal relations. In regard to coupling, the application of the Clausius–Duhem inequality to single-component ﬂuids had, in the past, where the issue of diffuse volume transport had not yet arisen, been essentially trivial. In those conventional circumstances, standard LIT arguments [19–21] resulted in the simple conclusion that 1 1 ps ¼ ju .r (58) þ T : ðrvÞT , T T

U

11

there being no Onsager coupling of the internal energy ﬂux to any other independent ﬂux. As such, as a consequence of Eqs. (42) and (44) together with the traditional assumption that ju jq ¼ krT, the inequality ps X0 was trivially satisﬁed (with v appearing therein identiﬁed as being the ﬂuid’s mass velocity vm Þ.

PHYSA : 9965 ARTICLE IN PRESS H. Brenner / Physica A ] (]]]]) ]]]–]]]

16

23 25 27 29 31 33 35 37 39 41 43 45 47 49 51

F

O

O

21

PR

19

D

17

TE

15

EC

13

R

11

R

9

O

7

C

5

N

3

However, given the present conception of an independent diffuse volume ﬂux jv , the latter dependent solely on the density gradient rr as in Eq. (54), there then arises the possibility of coupling between the vector ﬂuxes jq and jv , subject to the restrictions imposed thereon by both Onsager reciprocity and the Cauchy–Duhem inequality (13). It is the consequences stemming from these requirements that we examine below. It is in this restricted coupling context that our trio of post-GENERIC constitutive assumptions embodied in Eqs. (5), (7) and (27) receives theoretical justiﬁcation. However, the issue is complicated by the fact that there exists no unequivocal deﬁnition of the ‘‘heat ﬂux’’ in the literature. This stems from the fact that since heat is not an extensive property of a system, the notion of a heat ﬂux is essentially ambiguous. Only extensive physical properties such mass, internal energy, momentum, volume, entropy, electric charge, etc. can give rise to ﬂuxes, much less strictly diffusive (i.e., nonconvective) ﬂuxes, such as is implied by the notion of heat conduction. Stated explicitly, there exists no extensive quantity Q to which the symbol jq can be assigned as representing its (diffusive) ﬂux. This contrasts with the fact that extensive quantities like internal energy U can ^ ﬂuxes, with nm the mass ﬂux density. This ambiguity in the possess diffusive (ju Þ as well as convective (nm uÞ interpretation of the heat ﬂux permeates the literature of LIT. One might believe that the problem is at least partially alleviated by invoking Fourier’s law, Eq. (42), in the sense of the latter serving to ‘‘deﬁne’’ the heat ﬂux jq as being that portion of the diffuse (internal) energy ﬂux which vanishes when rT ¼ 0. However, that would be analogous, for example, to deﬁning the concept of a force, say F, in the Newton–Euler point-mass rigid-body law, F ¼ ma [40], by regarding a as the purely kinematically deﬁned acceleration that it indeed is, and then simply deﬁning the force as being the quantity which vanishes when a ¼ 0 (with m the proportionality coefﬁcient). The point here is that, objectively, a relation only achieves acceptance as a physical law of nature when quantitative and independent deﬁnitions of the respective variables appearing in that law (not including the phenomenological proportionality coefﬁcient) have already been set forth prior to proposing that the relation in question be elevated to the status of a bona ﬁde experimentally based law. Accordingly, returning to the Fourier law issue, the heat ﬂux jq must be deﬁned without any reference whatsoever to its possible relationship to a temperature gradient (cf. Eq. (63)), although it is permissible to use the notion of the temperature itself in its deﬁnition. The Dufour effect [19, p. 274], said to result in a ‘‘heat ﬂow’’ in an isothermal mixture undergoing diffusion, is a case in point. The notion of a heat ﬂux in the absence of a temperature gradient strains credulity, as would surely have been true of Fourier. What is almost certainly being referred to in this Dufour context is a diffuse internal energy ﬂow. It is with this lengthy preamble in mind that we now turn to the issue of justifying the trio of assumptions resulting in our constitutive expression for a0 , Eq. (33). In this context we begin by merely summarizing those formulas which have been derived on the basis of those assumptions, and then using this information to suggest a deﬁnition for the heat ﬂux jq that meets the criteria that we have speciﬁed (at least in singlecomponent systems). These criteria involve showing that Fourier’s law (42) is indeed satisﬁed by this deﬁnition of jq , as too are the laws of Onsager and Clausius–Duhem. In this context it is interesting to note that insofar as we are aware there exists no theoretical proof of Fourier’s law in the general case of continua [52] (i.e., involving both liquids and gases, the latter not necessarily rareﬁed). While one might believe that the Boltzmann equation offers the basis for such a proof, at least in the case of single-component rareﬁed gases, such a ‘‘proof’’ involves the implicit assumptions that: (i) the heat ﬂux is identical to the (diffuse) internal energy ﬂux; and (ii) there exist no other independent vector ﬂuxes, such as jv , to which the Boltzmann-based heat ﬂux vector might otherwise couple. However, as can be seen from Eq. (34), according to our present arguments, jq and ju are the same only in the special case of ideal gases; moreover, even in that case a Boltzmann-based proof cannot be accepted as complete without ﬁrst addressing the coupling issue, and subsequently showing that the Clausius–Duhem second law inequality, now possibly including such coupling, is indeed satisﬁed. (Of course, such coupling issues had not even been recognized at the time that Boltzmann and, later, Chapman and Enskog [13] did their foundational work.)

U

1

7.2. Definition of the heat flux jq As a consequence of Eq. (55), Eq. (37) adopts the form ju ¼ jq þ jv ðqU=qV ÞT;M .

(59)

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23 25 27 29 31 33 35 37 39 41 43 45 47 49 51

F

O

ju ¼ Tjs pjv .

O

21

(61)

This result could also have been obtained directly from Eq. (16) upon invoking Eq. (55). This expression obviously constitutes the transport (i.e., ﬂux) counterpart of the thermodynamic relation (17). While Eq. (61) derives from nontraditional GENERIC theory, its validity is nevertheless intimately linked to our further constitutive assumptions, as embodied in Eqs. (5) and (7) (although not Eq. (27)), from which Eqs. (59) and (60) have evolved. Eq. (16) constitutes a powerful incentive to conclude that Eq. (55) must be correct in order to fulﬁll the analogy with Eq. (17). Eq. (59) or its equivalent entropic counterpart, Eq. (60), offers a deﬁnition of the heat ﬂux in terms of welldeﬁned physical properties. Moreover, this deﬁnition of jq is independent of any (subsequent) association with the temperature gradient, such as in Fourier’s law, Eq. (42). A fundamental point here when proposing Eq. (59) as the deﬁnition of the heat ﬂux vector (at least in single-component systems) is that, rationally speaking, only extensive physical properties can possess a ﬂux, as earlier noted. To repeat what was said there: since heat is not an extensive property, in the sense that a system cannot be said to possess an ‘‘amount’’ of heat Q, the concept of a heat ﬂux cannot be a primitive concept in any theory of irreversible processes; rather, it must be a defined quantity. And if heat ﬂux cannot then serve as a primitive concept in nonequilibrium thermodynamics, then heat itself cannot serve as a primitive concept in equilibrium thermodynamics. In effect, we are proposing here that instead of deﬁning the notion of internal energy U in terms of heat Q and work W—with W representing a well-deﬁned, strictly mechanical or electromechanical, concept (and thus able to serve as a primitive concept)—we reverse the scheme by deﬁning heat in terms of internal energy, with the latter now serving as the basic primitive notion. In effect, we define the quantity dQ arising during an inﬁnitesimal, generally irreversible, process involving a change in thermodynamic state as

PR

19

D

17

This expression differs from the classical irreversible thermodynamic result [19–21] for single-component ﬂuids, namely js ¼ jq =T. The extra nontraditional term appearing above represents an ‘‘isothermal’’ ﬂux of entropy accompanying the diffusive volume ﬂow. Note that in contrast with its corresponding energy ﬂux counterpart in Eq. (59), this unorthodox additional term does not vanish for ideal gases since, according to a Maxwell relation, ðqS=qV ÞT;M ¼ ðqp=qTÞV ;M a0 in general. Upon eliminating jq between Eqs. (59) and (60), subsequent use of Eq. (17) in the resulting expression yields the following relation upon rearrangement:

TE

15

EC

13

R

11

R

9

O

7

C

5

N

3

In a single-component isothermal system the quantity ðqU=qV ÞT;M would represent the internal energy per unit volume, namely u. Since jv is the diffuse volume ﬂux density, it follows that the term jv ðqU=qV ÞT;M appearing above represents a nonconvective internal energy ﬂux contribution that would exist in the absence of a temperature gradient (thereby arising from a density gradient). As such, this term constitutes a diffuse ‘‘isothermal’’ ﬂux of internal energy accompanying the diffuse volume ﬂow jv in a piggy-back mode. On the other hand, the Fourier heat ﬂux vector jq represents the nonconvective transport of internal energy across a surface due exclusively to a temperature gradient. As such, it is quite reasonable to suppose that the sum of the two terms appearing on the right-hand side of Eq. (59) should constitute the total nonconvective or diffuse internal energy ﬂux, ju . Issues similar to those discussed above for energy transport also arise in the comparable entropy transport case. In particular, in light of relation (55), Eq. (41) now becomes jq qS js ¼ þ jv . (60) qV T;M T

d=Q:¼dU d=W ,

U

1

17

(62)

where dU is an exact differential. The inexact differential d=W is the (generally path-dependent) work done during the process, the latter presumably obtainable via quantitative dynamical analyses embodying purely mechanical or magneto-electromechanical concepts, even for inherently irreversible processes. Of course, in the case of so-called reversible processes in single-component ﬂuids, we have that d=W rev ¼ p dV . Consistent with this philosophy of having internal energy rather than heat serve as the primitive concept in equilibrium thermodynamics, with the heat transfer d=Q then deﬁned in terms of the change dU in the latter (together with the work d=W done), on the basis of Eq. (59), we instead propose the following deﬁnition of the

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18

23 25 27 29 31 33 35 37 39 41 43 45 47 49 51

F

O

O

21

PR

19

D

17

TE

15

EC

13

R

11

R

9

Of course, this deﬁnition is understood to apply only in the case of single-component ﬂuids. Its extension to more general cases is obvious, and is brieﬂy discussed later in Section 8.6. Use of the above in conjunction with the fundamental proposition (61) and, hence, (60), all again valid only for the single-component case, will be seen to lead to a systematic approach to the subject of nonequilibrium thermodynamics which, like GENERIC theory itself, is not necessarily limited to linear processes. In effect, instead of regarding Eqs. (63) and (61) as having been derived beginning with single-component GENERIC theory (including the demonstration that v ¼ vm þ jv Þ, we propose elevating these equations to the level of fundamental postulates. Together with Eq. (42)—the latter now regarded as an exact relation valid for all temperature gradients, not necessarily small—the proposed scheme collectively serves to set forth the basic equations governing irreversible thermodynamics. It needs to be stressed, however, that we are not advocating basic changes in the fundamental structures of either GENERIC or LIT, both of whose foundations are sufﬁciently robust to, respectively, accommodate therein the existence of newly recognized physical phenomena. Rather, in the case of LIT, our proposed scheme simply entails adding volume to the present list of independent diffuse ﬂuxes appearing in that theory. Equivalently, in the case of GENERIC, following O¨ttinger [23], molecularly based ﬂuctuations are incorporated into the friction matrix appearing in that theory. In the ﬁnal analysis these are the only substantive additions advocated. After all, given that it was GENERIC that led to the precise set of equations which we have proposed, and given the consistency of GENERIC with LIT, it could not seriously be argued that any conﬂict existed between our proposal and either of these two structures. Our proposed reinterpretation of the heat ﬂux does not appear to impact on either of these basic structures, but rather only in the constitutive manner in which they are to be applied. These issues will be discussed in a broader context in subsequent papers, where, in a systematic, formal and axiomatic manner, we propose to go beyond the simple single-component systems studied here. Inasmuch as the concept of heat, and hence of heat transfer, makes its initial entre´e into the realm of thermodynamics in connection with its role in the ﬁrst law, logic would appear to demand that the deﬁnition of the heat ﬂux involve, at most, only ﬁrst law of thermodynamic concepts. This would include the internal energy U (as well as the nonthermodynamic notions of volume V and mass MÞ. Eq. (63) would appear, superﬁcially, to violate this concept since the thermodynamic absolute temperature T appears as one of the variables invoked in the partial derivative. And T is a strictly second law concept. This brings about the recognition that the ‘‘temperature’’ which enters into the deﬁnition of the heat ﬂux need not be formally identiﬁed with the symbol T. Rather, temperature may, in a broader sense, be thought of as a strictly primitive empirical concept—as indeed it was viewed during the reign of the caloric theory of heat, prior to the axiomatic work of Carnot, Joule, Kelvin and Clausius systematizing the foundations of thermodynamics. At that pre-thermodynamic time, the notion of temperature was unrelated to the formal deﬁnition of the symbol T appearing as the ‘‘integrating factor’’ in Clausius’s deﬁnition, dS:¼d=Qrev =T, of the entropy change accompanying a reversible ﬂow of heat (the latter heat ﬂow a strictly ﬁrst law concept); that is, following Fourier and others of that pre-thermodynamic era, one may regard temperature as a primitive quantity, represented, say, by the symbol y, in which case Eq. (63) would be then replaced by the expression

O

7

(63)

jq :¼ju jv ðqU=qV Þy;M ,

C

5

jq :¼ju jv ðqU=qV ÞT;M .

^ v; yÞ. Fourier’s law would where, for a single-component ﬂuid, one has the functional relationship U ¼ M uð^ then read

N

3

heat ﬂux in nonequilibrium thermodynamics:

U

1

jq ¼ kry,

rather than being given by Eq. (42). In the context of the preceding discussion, it is illuminating to read that portion of Fourier’s classical book, ‘‘The Analytical Theory of Heat,’’ concerned with attempting to explain both ‘‘temperature’’ and the ‘‘communication of heat’’ between bodies in different thermal states. Viewed alternatively, the single-component relation js ¼ jq =T appearing in textbooks (cf. [19, Eq. (20), p. 24])—which relation, incidentally, we believe to be incorrect on the basis of Eq. (60)—could not, even were it ? to be correct, be viewed in reverse as the definition, jq :¼ Tjs , of the heat ﬂux. The latter view would violate the

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1 3 5 7 9 11 13

19

required sequential ordering of the ﬁrst and second laws. Without entering into further details, this digression sufﬁces to identify the deep philosophical issues which surround both the deﬁnition of the heat ﬂux and Fourier’s law, especially when embedded in the context of Lebowitz’s oft-repeated view [52] that no formal (statistical–mechanical) proof of Fourier’s law exists despite the almost 200 years that have elapsed since its introduction into physics. Lastly, we note that temperature, y, as a physical concept, may, by analogy with mechanics, be regarded literally as constituting a ‘‘potential’’ for the movement of (internal) energy through space—an interpretation which is consistent with its appearance as a gradient in Fourier’s law. We now resume the discussion that preceded the digression of the preceding two paragraphs. In order that the program proposed herein, involving the deﬁnition of the heat ﬂux, be deemed acceptable, it remains to show that our deﬁnition is internally consistent, in the sense of satisfying the four basic criteria outlined above. We pursue this agenda by ﬁrst identifying the driving forces conjugate to the diffusive ﬂuxes with respect to Onsager’s reciprocity law. This is achieved by expressing the entropy production rate ps in terms of the independent ﬂuxes of the pertinent extensive properties involved in the analysis. It is important to note that this identiﬁcation does not involve the concept of the heat ﬂux jq .

15

From Eqs. (12) and (19) the entropy production rate can be expressed as ps ¼ r

21

F

19

7.3. Identification of the driving forces conjugate to the diffusive fluxes

Dm s^ þ r .js . Dt

O

17

29

where we have deﬁned D¼r

45 47 49 51

pv ¼ r

Dm v^ þ r.jv . Dt

and

(68)

PR

C

(69)

The intensive form of the extensive combined ﬁrst and second laws (Eq. (17)) for single-component systems is du^ ¼ T d^s p d^v, which we rewrite as

N

43

Dm u^ þ r .ju Dt

1 p du^ þ d^v. (70) T T We assume, as is also assumed in the case of irreversible thermodynamics [19–21], that the preceding equation remains valid in the material form

U

41

pu ¼ r

R

37 39

(67)

R

35

(66)

In the following paragraph we demonstrate that D ¼ T 1 T : rv. The internal energy- and volume-production rate analogs of Eq. (64) are, respectively,

O

33

Dm s^ 1 p þ r.ju þ r.jv . T T Dt

(65)

EC

31

D

27

1 p j þ j. T u T v Accordingly, Eq. (64) adopts the form p 1 . þ jv .r þ D, ps ¼ ju r T T js ¼

TE

25

O

However, from Eq. (61), 23

(64)

d^s ¼

1 p Dm u^ þ Dm v^, T T allowing us to write Dm s^ ¼

PHYSA : 9965 ARTICLE IN PRESS H. Brenner / Physica A ] (]]]]) ]]]–]]]

20

1 3 5

Dm s^ 1 Dm u^ p Dm v^ ¼ þ . (71) T Dt T Dt Dt Adoption of this relation is tantamount to supposing the local equilibrium postulate to be valid [19–21]. Hence, with use of Eqs. (68) and (69) the above becomes Dm s^ 1 p ¼ ðpu r.ju Þ þ ðpv r.jv Þ. T T Dt Substitution of the above into (67) gives r

7 9

1 ðpu þ ppv Þ. (72) T However, in the Eulerian form (9) of the volume-transport equation, where C ¼ V , we have that c ¼ 1. Moreover, it follows from (10) and (53) that nv ¼ vv , whence we ﬁnd that [4] D¼

11 13

pv ¼ r.vv . In addition, from Eqs. (5) and (3), pu ¼ pr.v þ T : rv.

17 19

(73)

(74)

Consequently, Eq. (72) becomes

F

15

1 ½pr.ðv vv Þ þ T : rv. T Finally, then, with use of Eq. (56), we obtain

23

1 T : rv. T Substitution of the latter into Eq. (66) makes p 1 1 . ps ¼ T : rv þ ju r þ jv .r . T T T

O

21

31

41

EC

R

R

where we have noted that jM ¼ T. As a consequence of Curie’s law, the Clausius–Duhem inequality (13) applied to Eq. (77) requires separate satisfaction of each of the following inequalities:

O

39

ps ðstressesÞ:¼

43

U

and 45

1 T : rvX0 T

C

37

ps ðvector fluxesÞ Ps :¼ju .r

47

(78)

From (77), the driving forces X c conjugate to the ﬂuxes j c of linear momentum, internal energy and volume are, respectively, p 1 1 X M ¼ rv; X u ¼ r and X v ¼ r , (79) T T T

N

35

(77)

This expression for the entropy production rate possesses the classic ‘‘ﬂux/driving force’’ summation-matrix format of LIT [19–21], namely ps ¼ Sc ðj c X c Þ.

33

(76)

D

29

TE

27

PR

D¼

25

(75)

O

D¼

(80) p 1 þ jv .r X0. T T

(81)

49

7.4. Onsager reciprocity

51

The preceding identiﬁcation of the respective conjugate driving force for each independent ﬂux enables us to explicitly address the restrictions imposed by Onsager’s reciprocal theorem upon our theory. By Curie’s

PHYSA : 9965 ARTICLE IN PRESS H. Brenner / Physica A ] (]]]]) ]]]–]]]

5 7

p 1 jv ¼ Lvu r þ Lvv r . T T

9 11 13 15

This pair of relations may be written alternatively in the matrix form ! " #( ) rð1=TÞ ju Luu Luv ¼ , rðp=TÞ jv Lvu Lvv

Luv ¼ Lvu ,

(85)

as well as that of the trio of inequalities Luu X0;

21

Lvv X0

37 39 41 43

PR

D

TE

EC

Consequently, using Eq. (36), one obtains p qU 1 1 qp r r rv^. ¼ þ T qV T T T q^v T

R

35

Substitute the latter into Eqs. (82) and (83) so as to obtain qU 1 1 qp rv^ þ Luv r ju ¼ Luu Luv qV T T T q^v T and

47

qU 1 1 qp jv ¼ Lvu Lvv rv^. þ Lvv r qV T T T q^v T

U

45

(87)

R

33

we have that qp qp 1 qp 2 qp rT þ rv^ ¼ T r rv^. rp ¼ þ qT v^ q^v T qT v^ T q^v T

O

31

(86c)

It proves convenient to re-express Eqs. (82) and (83) in terms of an alternative set of driving forces, namely frð1=TÞ; rv^g, in place of the previous set, frð1=TÞ; rðp=TÞg. To do so, we note in the single-component case that p ¼ pðT; v^Þ. Thus, in the identity p rp 1 r þ pr ¼ , T T T

C

29

Luu Lvv Luv Lvu X0.

(86a,b)

(88)

N

27

O

and 23 25

(84)

wherein the square ½L matrix appearing in the preceding is both symmetric and nonnegative-deﬁnite in order that inequality (81) be satisﬁed. Thus, we require satisfaction of the equality

17 19

(83)

F

3

theorem, the diffuse momentum ﬂux is uncoupled from those of internal energy and volume. Accordingly, Onsager’s theory together with Eq. (77) requires that the following general constitutive relations apply to the respective ﬂuxes of internal energy and volume: p 1 ju ¼ Luu r , (82) þ Luv r T T

O

1

21

(89)

49

7.5. Fourier’s law

51

Introduction of Eq. (63) into (88) followed by the use of (89) furnishes the following expression for the heat ﬂux:

PHYSA : 9965 ARTICLE IN PRESS H. Brenner / Physica A ] (]]]]) ]]]–]]]

22

"

2 # qU qU 1 qU 1 qp j ¼ Luu ðLuv þ Lvu Þ þ Lvv rv^. þ Luv Lvv r qV T qV T T qV T T q^v T

1

q

3

(90)

Analogous to (27), we now wish to assign to the heat ﬂux jq the universal property that 5

23 25 27 29 31 33 35 37 39

F

1 qp rv^. T q^v T

(94)

O

21

jv ¼ Lvv

Eq. (93) is, of course, Fourier’s law, namely Eq. (42), which we repeat here: jq ¼ krT, wherein kX0 as in Eq. (43). Introduction of (95) into (93) gives 2 qU þ kT 2 . Luu ¼ Lvv qV T

O

19

and

PR

17

(93)

(95)

(96)

It follows from this that if Lvv X0, as required by Eq. (86b), then, as a consequence of (43), it will also be true that Luu X0, in accord with (86a). With use of the constitutive expressions (93) and (94) in Eq. (81) we ﬁnd that " 2 # 2 2 qU 1 1 qp Ps ¼ Luu Lvv þ Lvv ðrv^Þ2 . (97) r qV T T T q^v T

D

15

where we have also taken note of Eq. (85). Substitution of (92) into both (90) and (89) gives " " 2 # 2 # qU 1 1 qU q j ¼ Luu Lvv 2 Luu Lvv r rT qV T T qV T T

TE

13

irrespective of the values of the other independent variables, say v^ (or, equivalently, pÞ and vm , entering into the problem. In turn, this necessitates that the bracketed expression appearing in the last term of Eq. (90) vanish, thus requiring that qU Luv ¼ Lvv ¼ Lvu , (92) qV T

EC

11

(91)

As such, Eq. (81) leads to the requirement that 2 qU . Luu XLvv qV T

R

9

in isothermal systems, i:e:, when rT ¼ 0,

(98)

R

7

jq ¼ 0

47

7.6. Constitutive equation for the diffuse flux of volume

49 51

C

N

43

U

41

O

45

From Eq. (96) it is seen that this condition is satisﬁed by the fact that the thermal conductivity is nonnegative. Given the unequivocal statement embodied in Eq. (91) it is tempting to consider the possibility that Fourier’s law, Eq. (95), may be valid under more general circumstances than would normally be expected, namely the regime beyond the small temperature gradient case that would sufﬁce to assure linearity of the ﬂux/ driving force relation explicit in Fourier’s law. The issue of possible limitations, or lack thereof, on its realm of applicability remains open as of this writing. Explicitly, we are unaware of any experimental data or theory [52] that points to any limits.

In as much as v^ ¼ 1=r, it follows that Eq. (94) is equivalent to the expression jv ¼ Lvv

1 r ln r, TkT

(99)

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23 25 27 29 31 33 35 37 39 41 43 45

F

O

O

21

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where kb is Boltzmann’s constant. The angular brackets denote thermal averaging. In the above, hri ¼ M=V is the mean density, while hðdrÞ2 i is the mean-squared density ﬂuctuation occurring within an open domain of ﬁxed volume V as a result of ﬂuctuations in the instantaneous ﬂuid mass M contained within that volume, owing to the ability of individual molecules to freely enter and leave that domain through its surface. Given the nonnegativity of all of the other parameters appearing in Eq. (101) it is obvious that the theory of ﬂuctuations requires satisfaction of the inequality kT 40. This inequality is also related to the purely thermodynamic fact that stability of the ﬂuid phase requires that ðq2 A=qV 2 ÞT;M X0, wherein A is the extensive Helmholtz free energy. In view of the thermodynamic identity [54, p. 123] ðq2 A=qV 2 ÞT;M ¼ 1=V kT , it is evident that stability demands that the isothermal compressibility be positive. Fluctuations constitute a necessary ingredient when rationalizing the physical role played by O¨ttinger’s positive-deﬁnite material property coefﬁcient D0 in relation to the diffusional contribution M diff to the GENERIC friction matrix M (cf. Ref. [23, Eq. (2.77)]); that is, were the ﬂuid to be truly ‘‘incompressible,’’ in the sense that kT ¼ 0 identically, there would and could be no ﬂuctuations in density, presumably requiring that D0 ¼ 0 too. This is consistent with our basic hypothesis (27), according to which it is only in the case of incompressible ﬂuids that Euler’s speciﬁc momentum relation v ¼ vm holds. Among other things, these remarks point up the (thermodynamically) singular nature of the notion of ﬂuid incompressibility [55], a simpliﬁcation lying at the heart of most contemporary ﬂuid-mechanical applications, especially in the case of liquids. Strict incompressibility corresponds to the case where ðqr=qpÞT ¼ 0, or, equivalently, kT ¼ 0. From Eq. (101) such incompressibility rules out the possibility of ﬂuctuations. At the same time, as evidenced by Eq. (99), this leads to an obvious singularity with regard to the existence of a diffuse volume ﬂux. If nothing else, this interplay between kT and jv shows clearly the intimate relationship of ﬂuctuations to the existence of a diffuse volume ﬂux. In many practical situations this thermodynamic incompressibility singularity, whether in the case of liquids or effectively isobaric gas transport processes, is without appreciable effect on the accuracy of the ﬂuid-mechanical predictions derived from solutions of the classical Navier–Stokes and Fourier equations in this limit. In such circumstances the existence of the singularity may be ignored with impunity. On the other hand, there exist a few key novel fundamental phenomena—for example, thermophoresis [5] and thermal transpiration [50] in single-component gases, and thermal diffusion-based Soret separation phenomena in multicomponent liquid mixtures [14,15]—whereby ignoring the existence of this singularity would lead to fundamentally incorrect predictions, negating the very existence of these physical phenomena.

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(101)

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hðdrÞ2 i kb TkT , ¼ V hri2

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where the thermometric diffusivity a, Eq. (48), is a positive quantity. Consequently, the requirement that Lvv X0 demands that the algebraic sign of kT be nonnegative, as already stipulated in connection with Eq. (30). The required nonnegativity of the isothermal compressibility kT appearing in Eq. (100) is key to O¨ttinger’s [23] central argument that his proposed extension of the GENERIC friction matrix is necessitated by the existence of ﬂuctuations. The intimate relationship of kT , especially its algebraic sign, to the theory of ﬂuctuations lies in its appearance in the theory of equilibrium ﬂuctuations [54, p. 123],

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(100)

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Lvv ¼ aTkT ,

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with kT is the coefﬁcient of isothermal compressibility, deﬁned in Eq. (29). In the light of Eq. (28) the above is equivalent to the relation Lvv ¼ KTkT [53]. Given our identiﬁcation of K with a in Eq. (47), together with the subsequent argument that this relation applies equally to liquids, it follows that, in general,

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8. Discussion 47 49 51

8.1. Brownian motion as the source of the deviation of the specific momentum from Euler’s constitutive hypothesis The issue of Brownian motion bears directly upon the fundamental question of whether or not the Cauchy linear momentum equation (2) follows as an immediate consequence of Newton’s mechanics applied to mass-

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point molecular models, or whether second law of thermodynamic principles is needed in order to correctly effect the transition from discrete to continuum mechanics. In the present single-component context the phrase ‘‘Brownian motion’’ refers to the manifest consequences of multiple collisions occurring between a single tagged molecule of the ﬂuid and other molecules present therein. Explicitly, attention focuses on collisionally induced changes occurring in the momentum and position of a tagged molecule over the long term. As regards its role in transport phenomena, the emphasis here thus focuses upon the consequences of multiple collisions, rather than upon the short-term consequences of single collisions (the latter as already currently embodied in Boltzmann’s collisional integral in the particular case of rareﬁed gases). The founders of gas-kinetic theory, including Clausius, Maxwell and Boltzmann, did not themselves identify Brownian motion as a distinct macroscopic phenomenon arising from collisions of molecules [31], leaving it to Einstein [56] and Smoluchowski [57] to later do so. This failure to incorporate the phenomenon into their collisional model is, in our opinion, responsible for the presently held belief that Brownian motion is irrelevant to the formulation of the subject of Boltzmann-based gas-kinetic theory, an attitude to which neither O¨ttinger [23], Klimontovich [28–30], nor we, among others [58–60], subscribe. The collectively uniform attitude of this latter group toward the issue can be gleaned from explicit remarks made by each on the role of Brownian motion (respectively, focused on: ﬂuctuations, self-diffusion and diffuse volume transport) in singlecomponent ﬂuids during the course of translating Newton’s and Euler’s mass-point rigid-body mechanics into continuum mechanics. This includes: (i) O¨ttinger’s ‘‘Something is missing’’ ﬂuctuation addendum [23] to earlier versions [25,26] of GENERIC theory; (ii) Klimontovich’s self-diffusion physical-space add-on [28–30] to Boltzmann’s collision integral for the purpose of introducing irreversibility directly into mechanics; and (iii) Brenner’s [4] recognition of the phenomenon of diffuse volume transport—the latter, like entropy, a statistical rather than practical concept—over and above the previous, strictly convective, view that in single-component ﬂuids volume could be conveyed through space solely in the company of mass. The seeming irrelevance of Brownian motion with regard to the foundations of gas-kinetic theory is rendered transparent by the obvious fact that in current rareﬁed gas theories [13,61] Brownian statistics do not enter into the calculation of the singlet distribution function f ðx; p; tÞ (with x and p ¼ m dx=dt the respective position and momentum vectors of a mass-point molecule of mass mÞ. Explicitly, the theory of Brownian motion per se does not contribute directly to solving the Boltzmann equation, although the notion of Brownian motion is implicit in the solutions thereof. This observation, in turn, indicates that the multicollision processes, which underlie the phenomenon of Brownian motion, play no role in the basic physics quantifying the macroscopic manifestation of molecular transport phenomena. Thus, philosophically speaking, contemporary thinking argues that the notion of Brownian motion merely enriches the subject of gas-kinetic theory without impacting directly upon its foundations. It is this short-term collisional perspective which O¨ttinger, Klimontovich and we challenge (see the discussion of O¨ttinger’s recent multicollisional model in Section 8.4). The failure of Brownian motion to impact upon the mechanical foundations of statistical mechanics—with such motion viewed merely representing a completely predictable consequence thereof—should appear strange to any unbiased observer unfamiliar with the apparently mechanically reversible treatment of the collisional term in Boltzmann’s theory. That is, macroscopic experience teaches that collisions occurring among a conﬁned and isolated discrete collection of objects (‘‘molecules’’) separated by a vacuum are inherently irreversible, ultimately causing such an isolated system to eventually come to a macroscopic state of rest via ‘‘friction’’, such as certainly occurs in granular gas models [62] lacking a continuous external supply of energy (momentum). That this fate does not befall the molecules in Boltzmann’s collisional model must surely be attributed to the phenomenon of Brownian motion, which should be viewed as the root cause making possible the perpetual motion of (molecular) objects. This suggests that Brownian motion (namely self-diffusion and ﬂuctuations) should be regarded as an essential and heretofore overlooked contribution to kinetic theory, rather than simply constituting a predictable consequence thereof. This attitude with regard to the role of Brownian motion forms the basis of the implicit belief lying at the foundation of ﬂuctuation-based GENERIC theory [23] that the translation of point-mass Newtonian mechanics into continuum mechanics cannot be correctly effected without explicitly incorporating entropy and its statistical-molecular foundations into the translation scheme.

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The agreement of rareﬁed gas solutions of the Boltzmann equation with the N–S–F equations in the socalled near-continuum small Knudsen number regime, such as in the well-known perturbation solutions thereof due to Chapman and Enskog [13], is often cited as convincing evidence of the success of these perturbation schemes and, indeed, of the fundamental correctness of the Boltzmann equation itself. However, such schemes are implicitly based upon the fact that rareﬁed gases obey the ideal gas law. But, according to the theory advanced here, as embodied in the union of Eqs. (59) and (36), an ideal gas constitutes a highly singular case owing to the fact that the contribution of the diffuse volume ﬂux jv to the internal energy or heat ﬂux vanishes identically for such gases (although the corresponding diffuse volume contribution to the entropy ﬂux in Eq. (60) is not correspondingly singular for ideal gases). As such, the fact that our modiﬁed N–S–F equations might appear to be in conﬂict with Chapman–Enskog theory needs to be placed in context. This issue is already implicit in the role of the Maxwell–Burnett thermal stresses, discussed in connection with Eq. (49). In effect, it is the failure of theories of the Boltzmann equation [13] to unambiguously distinguish between the heat ﬂux jq and the diffuse internal energy ﬂux ju , especially in single-component gases, that constitutes the source of the problem. The problem is exacerbated in multicomponent gas mixtures, where, for example, Chapman and Cowling [65] refer to ‘‘. . . the ordinary [my emphasis] ﬂow of heat resulting from inequalities of temperature in the gas,’’ while concomitantly speaking of an additional heat ﬂow due to diffusion (the Dufour effect). Reciprocally, this heretofore unresolved heat ﬂux ambiguity has resulted in the failure of gas-kinetic theory to recognize the fundamental role played by the diffuse volume ﬂux in properly interpreting the hierarchical ordering of the sequential Knudsen number-based terms arising in perturbation solutions of the Boltzmann equation for rareﬁed gases. This is not to state that the Boltzmann equation itself is in error, but rather that one must rigorously avoid supposing that the heat ﬂux and diffuse internal energy ﬂux are synonymous if, at the same time, Fourier’s law, Eq. (42), is to be accepted as generally valid.

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8.2. Limitations of contemporary perturbation solutions of the Boltzmann equation for small Knudsen numbers

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In this latter context it is interesting to contrast the rather different attitude displayed toward Brownian motion when formulating the statistical foundations of quantum mechanics as a ﬁeld theory. There, Nelson [63] and others [64] have invoked fundamental ideas underlying the notion of Brownian motion in a ‘‘hiddenvariable,’’ Bohmian-like, attempt to show that quantum mechanics can be derived directly from little more than Brownian motion concepts when combined with those of Hamilton–Jacobi dynamics.

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8.3. Thermodynamic singularities in fluid mechanics 35

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Thermodynamics and mechanics have traditionally been regarded as essentially separate and distinct ﬁelds of inquiry, except in relation to the second law of thermodynamics, although GENERIC [23] as well as ‘‘extended thermodynamics’’ [66] among other nonequilibrium schemes represent attempts at their uniﬁcation. As regards the second law, the question of the irreversible nature of thermodynamics in contrast to the seemingly reversible nature of Newtonian dynamics has long both intrigued and confounded physicists as well as other scientists interested in the fundamentals of their disciplines, with attempts at the resolution of this seeming paradox as a favorite philosophical topic. As in the case of GENERIC [23], the present paper raises questions about whether continuum mechanics and continuum thermodynamics can be truly separated into distinct branches of physics. Resolution of the question leads, inter alia, to the surprising conclusion that the diffuse ﬂux of volume renders continuum ﬂuid dynamics a branch of irreversible thermodynamics rather than of Newtonian dynamics (a conclusion which will be evidenced more forcefully in subsequent installments in this series). The point to be made is that while Newtonian mechanics is indeed applicable to molecules, and thus subject to the laws of dynamics, the Cauchy linear momentum equation lacks a dynamical (i.e., molecular) basis owing to the presence therein of the stress tensor, a strictly continuum concept. As such, the present series of papers will advance the view that the Cauchy linear momentum equation is, in fact, an irreversible thermodynamic relation rather than a Newton’s law-based dynamical relation. While the case for this unusual perspective may not seem wholly convincing in

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the present paper, it is believed that subsequent papers in the series will bring the issue home more pointedly when we divorce the proposed diffuse volume-based addition to LIT from its present GENERIC ancestry. The fundamental role played by thermodynamics in the present analysis of the Cauchy speciﬁc momentum density v can be seen, for example, in the important role assigned in the present work to Onsager’s reciprocal relation. Prior to this, to the best of the author’s knowledge, Onsager’s theorem had never been cited as being relevant to any ﬂuid-mechanical issue of a dynamical nature, especially that pertaining to the issue of momentum. The linkage here between ﬂuid mechanics and thermodynamics is subtle, as can be seen most clearly by considering the case where the ﬂuid is isothermal. In that case, in particular in the present class of singlecomponent ﬂuids being investigated, only the respective continuity and Cauchy linear momentum equations, (1) and (2), enter the analysis. As such, only the purely ‘‘mechanical’’ variables p, r and vm would normally be expected to arise in traditional views pertaining to this isothermal case. Accordingly, lacking the need for an energy equation, whose presence would otherwise serve to couple ﬂuid mechanics to thermodynamics in an obvious way, one would not normally envision the existence of any common ground between these two ﬁelds of study in the present isothermal instance, except perhaps in the seemingly minor context of the second law of thermodynamics—the latter as embodied in the Clausius–Duhem inequality serving to demonstrate the nonnegativity of the respective shear and bulk viscosity coefﬁcients for rheologically Newtonian ﬂuids (cf. Eqs. (80) and (44)). Yet, despite this belief, these two ﬁelds remain inseparably linked in the isothermal case through the diffuse volume ﬂux jv , as given constitutively by Eq. (54). This ﬂux enters the ﬂuid-mechanical portion of the analysis through the union of the Cauchy linear momentum equation (2) with Eqs. (56) and (53). At the same time, the diffuse volume ﬂux enters the thermodynamic aspect of the analysis through its appearance in Eqs. (59)–(61). It is through this common presence, which transcends the issue of isothermality, that these two ﬁelds are permanently linked despite the ﬂuid being isothermal. The sole exception occurs in the ‘‘incompressible’’ ﬂuid case, where the uniformity of the density results in the fact that jv ¼ 0. Here, however, were the isothermal ﬂuid to be truly incompressible, the pressure would have to be uniform throughout the ﬂuid in accordance with the single-component equation of state, p ¼ pðT; rÞ, since T and r are both constant. This, however, ﬂies in the face of the fact that a pressure gradient rp normally exists in isothermal ﬂow situations involving incompressible ﬂuids. Moreover, since only the pressure gradient, rather than the pressure itself, appears in the equations of ﬂuid mechanics, incompressible ﬂuid mechanics, by itself, can establish the prevailing pressure at a point of the ﬂuid only to within an arbitrary additive, generally timedependent, function. Obviously, one is dealing here with a highly singular situation [55]. This point clearly comes to the fore in the person of the diffuse volume ﬂux, as can be seen from the role played by jv when addressing the Onsager coupling issue during the course of attempting to determine the heat ﬂux jq (even when the latter is identically zero, as in the isothermal case). The role of the diffuse ﬂux of volume, especially in relation to the precise deﬁnition of the heat ﬂux offered here—thereby contributing to the clariﬁcation of this reversibility–irreversibility paradox—has remained hidden until now. In retrospect, the reasons for this failure to recognize the existence of diffuse volume transport, much less its major unifying role, are now obvious, although these reasons are different in the respective gas and liquid cases. Though different in detail, the reason in both cases can be traced to the singular nature of the respective (perturbation) approximations normally made in the literature of gases and liquids. In the case of gases, the Boltzmann equation, with the available solutions thereof largely focused on rareﬁed gases (these obeying the ideal gas law), has, due to this focus, implicitly eliminated the need to clearly distinguish between heat ﬂow and (diffuse) energy ﬂow. This can be seen from Eq. (63) where the distinction disappears owing to the fact that the internal energy of an ideal gas is independent of its volume. As pointed out in Section 8.1, we now recognize the latter condition as a singular limit, in the sense that in such circumstances jv is no longer available to stimulate discussion of a possible distinction existing between jq and ju . Yet a profound philosophical difference exists between the two, since, as earlier stressed, ju represents the ﬂux of an extensive physical property, namely the internal energy U, whereas there exists no extensive physical property, namely heat Q, of which jq could rationally be called its ﬂux (nor does there exist a volumetric density, say cq , of heat). As such, the heat ﬂux must be regarded as a slack variable, as in Eq. (63), namely the

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residue remaining after subtracting from the diffuse internal energy ﬂux all other sources that entrain internal energy (e.g., diffuse volume) other than in the form of temperature (see Eq. (111) below). In the cases of liquids it is the ambiguous notion of ‘‘incompressibility’’ (and the ensuing uncertainty in regard to the notion of pressure) which is the source of the singularity. This is immediately apparent from Eq. (99), where strict incompressibility would require that both r ln r and kT be zero, resulting in a Leibnitz-like mathematical indeterminacy with regard to whether jv was, or was not, zero. This in turn reﬂects upon the latter’s role in connection with Onsager’s reciprocal theorem and, hence, upon the same heat ﬂow/diffuse internal energy ﬂow conundrum as cited above in the case of gases.

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(103)

Thus, were the parameter K to be identiﬁed as being equal to the gas’s thermometric diffusivity a, as in Eq. (47), the constitutive relation thereby obtained for the velocity difference j would have been exactly what we obtained for jv based on our fundamental hypothesis (27) (upon bearing in mind that kT ¼ M w R=rT for ideal gases). Obviously, the agreement between O¨ttinger’s Eq. (102) and our Eq. (28) is, despite their very different origins, a consequence of the fortuitous cancellation of the last two bracketed terms in Eq. (33) in the case of ideal gases. Experimental data involving thermal diffusion in liquids [14,15] support the hypothesis that Eq. (27), and hence (28), including Eq. (33), is not limited to gases, but applies equally well to liquids. However, the limited availability of critical data pertinent to the issue renders conﬁrmation of the general applicability of these relations to liquids somewhat tenuous. As yet, the possible applicability of Eq. (102) to liquids has not been tested. This owes to the absence of experimental data sufﬁcient to the task, e.g., data in which static pressure is imposed externally on a liquid undergoing steady-state heat conduction. Even were such data available, lack of independent knowledge as to the possible effect of pressure on the phenomenological coefﬁcient D0 for liquids would appear to render the interpretation of such data equivocal. As such, it is not yet possible to distinguish among the two possibilities for a0 , namely Eq. (33) vs. a0 ¼ u=r. In this context it needs to be kept in mind that were the incompressibility hypothesis (27) leading to Eq. (33) to prove wrong, the present analysis would, because it is based on the general principles of GENERIC, nevertheless remain intact in broad outline, although not in ﬁne detail. For example, were the constitutive expression for a0 to be kept open throughout the entire development, we would, more generally, in place of Eq. (63), propose the following deﬁnition for the ^ v . Thus, were it to prove true that a0 ¼ u=r, ^ heat ﬂux: jq :¼ju þ ða0 r uÞj the following would obtain for the proposed heat ﬂux deﬁnition: jq :¼ju .This is, of course, the usually assumed constitutive relation (or deﬁnition) of jq for single-component ﬂuids.

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a formula derived theoretically by O¨ttinger [67] based on a ‘‘proper cross graining’’ of the Boltzmann equation. However, since p=T ¼ rR=M w for ideal gases (with R the universal gas constant and M w the molecular weight) the above equation is seen to be constitutively identical to our Eq. (28), wherein

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The basis for the LIT analysis of Section 7 derives largely from our identiﬁcation of O¨ttinger’s GENERIC ﬂuctuation-based phenomenological coefﬁcient a0 as being expressed in terms of the physical properties of the ﬂuid by Eq. (33). This relation, in turn, arose by applying our fundamental hypothesis (27) to Eq. (20), the latter relating the velocity difference, v vm j, to a0 . Other seemingly plausible hypotheses might have led to alternative expressions for a0 . Interestingly, one of these, due to O¨ttinger [67], leads in the special but important case of ideal gases to a formula for a0 which is constitutively identical to our Eq. (28). Explicitly, were it to have been supposed in Eq. (20) that a0 ¼ u=r, this would have led to the relation

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Our analysis has focused exclusively on transport processes occurring in ﬂuids—gases as well as liquids. However, the basic precepts of LIT and GENERIC apply to matter generally, irrespective of its macroscopic physical state. As such, there is every reason to suppose that fundamental ﬂuctuation-based issues similar to those discussed above in the context of ﬂuids also arise in the case of solids, particularly solid solutions (alloys). This belief is supported by the work of Danielewski and his co-workers [68,69], who have repeatedly expressed the view, as explicitly quantiﬁed in their analyses, that the Cauchy linear momentum equation necessitates a two-velocity formulation when applied to solids undergoing interdiffusion, just as in the case of ﬂuids. Concomitantly, they point to the need to modify the classical model of strain-induced stress by adding volume transport to simple strain-based displacement when formulating constitutive equations for stresses in solids arising from diffusion (so-called diffusion-induced stresses). Explicitly, similar to the views expressed here, Danielewski et al. [68,69] argue in favor of the existence of a second fundamental velocity, a so-called ‘‘drift’’ velocity, different from the mass velocity of the solid, and appearing (together with the mass velocity) in the momentum conservation and stress constitutive equations for solids. Moreover, just as in our case, their drift velocity arises as a direct consequence of volume transport, the latter occurring in solids during the course of ‘‘atom–vacancy’’ exchanges within the solid lattice and/or via atom–atom exchanges when the diffusing species possess different molar volumes [70]. (Readers not familiar with the exhaustively detailed terminology employed in connection with atomic transport in solids may ﬁnd it useful to refer to the IUPAC-recommended publication: ‘‘Deﬁnition of terms for diffusion in the solid state’’ [71].) Experimental justiﬁcation for their nontraditional momentum transport model is based upon the widely accepted Darken [72]–Kirkendall [73] notion of diffusion-induced stress [74–76] resulting from volume transport accompanying interdiffusion in multispecies solids [77]. Being based, more or less, exclusively on macroscopic experimental phenomenology, their volume-transport stress model, due to Stephenson [74], lacks the fundamental molecular foundation that we have earlier provided for ﬂuids, at least for single-component gases, and manifesting itself in the notion of temperature-induced stresses. Nevertheless, despite being less well grounded theoretically than in the case of ﬂuids, Danielewski et al.’s [68,69] two-velocity momentum transport model for solids appears to be well-supported macroscopically. Indeed, there exists a vast body of literature concerned with the role of ‘‘volume diffusion’’ [70] in rationalizing the phenomenon of diffusion-induced stress [75,76], the latter manifested explicitly by the permanent deformation of solids noted at the conclusion of the diffusion process.

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Eqs. (59)–(61) were derived on the basis of O¨ttinger’s version [23] of GENERIC supplemented by several key constitutive assumptions. However, once derived, this equation set is seen to collectively possess an obvious physical interpretation in its own right, independent of its GENERIC origin. As such, these fundamental equations, together with the implications that follow therefrom, stand or fall on their own respective merits. Subsequent papers in this series will build upon generalizations stemming from the foundations laid by this trio of equations. Eqs. (59)–(61) are obviously valid only for simple, single-component systems. However, given their structure and interrelationships it is not difﬁcult to speculate on how this trio of equations might be generalized so as to be applicable in more complex circumstances. Such speculations will be conﬁrmed in subsequent papers appearing in this series. These generalizations will be seen to be wholly independent of the GENERIC theory [23] that spawned them. The material which follows immediately below is designed to provide a brief preview of the scheme underlying these proposed generalizations. The previous single-component case dealt with circumstances in which, for a ﬁxed mass M, the thermodynamic state of the system could be described exclusively in terms of the extensive variable set ðU; S; V Þ. This allowed the combined ﬁrst and second laws to be expressed entirely in terms of these three variables as

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representing the extension of Eq. (63) to the present more general situation, is expected to result in the fact that jq vanishes when the temperature T is uniform throughout the ﬂuid, so that Fourier’s law, Eq. (42), will ^ ¼ C=MÞ, continue to prevail irrespective of whether the other intensive variables, namely v^ and c^ (where c vary throughout the ﬂuid. It is this fact, among many, which will be demonstrated in subsequent contributions.

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According to the 250-year old view of Euler [33,34], the momentum velocity v appearing in the Cauchy linear momentum equation (2) is equal to the ﬂuid’s mass velocity vm , the latter being the velocity appearing in the continuity equation (1). And Cauchy’s equation is precursor to the Navier–Stokes equations (as well as the Fourier-based energy equation owing to the dissipative terms appearing therein arising from the stress tensor in Cauchy’s equation). The goal of our paper was to demonstrate on theoretical grounds, using the basic principles of LIT, especially as embodied in GENERIC [23], that Euler’s constitutive equation, v ¼ vm , is incorrect in circumstances where density gradients exist in the ﬂuid. This surprising result is implicit in the unorthodox set of statistical–mechanically based continuum-level transport equations derived for singlecomponent ideal gases by the late Klimontovich [28–30] (see also [27]), although he never explicitly

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emphasized his disagreement with existing theory. A decade later, unaware of this prior work, the present writer arrived, independently, at a set of equations [1–3] identical to those of Klimontovich by a very different route, one based empirically on philosophical arguments stemming from the recognition that volume could be transported purely diffusively in ﬂuids [4]. The resulting unorthodox Cauchy momentum equation (2), in which vavm in circumstances wherein the ﬂuid density r is nonuniform (owing primarily to the presence of temperature gradients), is implicitly supported, albeit somewhat tenuously, by purely macroscopic experiments focused on the phenomenon of thermophoresis [5,14,15], where the issue of slip boundary conditions at solid surfaces, dating back to Maxwell [11] in 1879, complicates the interpretation of the experimental results. Whereas Klimontovich’s theory of ﬂuid mechanics and heat transfer was limited in scope to single-component (ideal) gases by the nature of the assumptions made in its derivation, the present writer’s theory extended to liquids as well, including inhomogeneous ‘‘compressible’’ binary ﬂuid mixtures embodying density gradients, rra0, arising from spatial variations in composition. Focusing solely on the single-component case common to both theories, the velocity difference was predicted to be of the form v vm :¼j, in which j was equivalent to the diffuse volume ﬂux, j ¼ jv , with the latter deﬁned, physically, by the relation [4] jv ¼ nv nm =r, in which nv and nm rvm denote the respective Eulerian ﬂuxes of volume and mass through a surface element dS ﬁxed in space (through which the ﬂuid is ﬂowing). The diffuse volume ﬂux was given constitutively by the expression [4] jv ¼ ar ln r, with a ¼ k=r^cp the thermometric diffusivity, in which k is the thermal conductivity and c^p the isobaric speciﬁc heat. (In the work of Klimontovich [27–30] the velocity disparity j was not identiﬁed in physical terms as being the diffuse volume ﬂux jv , although his constitutive formula for j is the same as that for our jv .) To the extent that the Klimontovich/Brenner diffuse volume interpretation of the momentum–mass velocity difference proves to be correct, their work provides a complete theory of single-component ﬂuid mechanics and heat transfer, albeit different from the orthodox Navier–Stokes–Fourier (N–S–F) versions accepted in the literature [18–21]. The most stroking difference lies in the fact that a single velocity is no longer generally sufﬁcient to characterize the kinematics, dynamics and energetics underlying irreversible thermodynamics. Accord between these respective orthodox and unorthodox views appears to exist only for ‘‘incompressible’’ ﬂuids, r ¼ const: Euler’s proposed constitutive formula, v ¼ vm , for the speciﬁc momentum density predates, by about a century, recognition of the existence of (mobile) molecules, as well as the codiﬁcation of the ﬁrst and second laws of thermodynamics in the mid-1800s. Cauchy’s (1827) pre-molecular, pre-thermodynamic incorporation of Euler’s relation into the linear momentum equation was predicated entirely on the basis of extending Newton’s laws of discrete rigid-body mechanics to ﬂuid continua through the introduction of Cauchy’s stress tensor (3), a continuum concept appearing in place of real, externally imposed, body forces exerted collectively on the contents of a material domain. It is more or less universally believed that Boltzmann’s statistical mechanics has long since resolved any possible doubts in the matter of the constitutive equation for the momentum density v in favor of Euler’s belief that it is the velocity vm of mass. However, objectively speaking, the only extensive experimental justiﬁcation of Boltzmann’s six-dimensional kinetic transport equation lies in the apparent agreement of its small Knudsen number perturbation solutions [13] for rareﬁed gases with the three-dimensional physical-space N–S–F hydrodynamic equations. But the most accurately executed and extensive ﬂuid mechanical experiments to date in support of the conventional form of the N–S–F equations have involved incompressible and/or isothermal liquids, rather than the rareﬁed gases to which Boltzmann’s theory applies. Moreover, and perhaps equally importantly in view of Maxwell’s thermal creep slip condition [11] is the fact that virtually all experiments have involved the use of no-slip boundary conditions imposed upon vm [78], whereas the view we have advanced elsewhere [1–3] with regard to possible critical experiments [5,14,50] involving ﬂuids in which density gradients exist is that the no-slip boundary condition should be imposed, instead, upon the volume vv , the latter being identical to the (total) volume ﬂux nv . And it is only in the uninteresting case of incompressible ﬂuids that vv and vm are the same. Thus, despite the passage of many years in which the N–S–F equations appear to have stood the test of time, and despite the virtually unanimous belief that the Boltzmann equation unequivocally demonstrates the correctness of these equations (especially Euler’s view that v ¼ vm Þ, the doubts raised here and elsewhere suggest the need for a careful reappraisal of the facts. O¨ttinger, on becoming aware of these issues several years ago through reading the manuscript of a then, as yet, unpublished version of Ref. [3] by the writer, recognized that ‘‘Something was missing’’ from earlier

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(the latter derived from Eqs. (31) and (47)), with kT the isothermal compressibility, Eq. (29). By these means we explicitly demonstrated the compatibility of our equations with O¨ttinger’s ﬂuctuation model, without, however, excluding other possibilities. Indeed, as discussed in Section 8.4, O¨ttinger has, himself, on theoretical statistical–mechanical grounds, pointed to another possibility. Were that all, the paper could effectively have concluded without the appearance of Section 7. However, the exercise of establishing the indicated compatibility unearthed an important philosophical fact, one otherwise hidden in the formal, strictly mathematical, aspects of our calculations. This refers to the fundamental reason as to why seemingly minute ﬂuctuations, which would normally be expected to average-out statistically, should have such a profound effect as to set aside 250 years of unquestioned acceptance of Euler’s view? Physically, the answer to this rhetorical question lies in the fact that the ﬂuctuations of the molecules about their average positions, acting in concert with their inhomogeneous spatial distribution, create a macroscopic bias [15]. In effect, the phenomenon is a consequence of the coupling of these two attributes, namely ﬂuctuations and inhomogeneities, the latter as manifested in the molecular number density gradient (which, in single-component systems, translates into a mass density gradient). Their union underlies the inseparable coupling that exists between dynamics and thermodynamics, whose composite nature lies at the very heart of GENERIC. Indeed, the analysis of Section 7 reveals the hidden role that Onsager’s celebrated coupling theorem [79] plays in understanding why Euler’s view was not tenable once the molecular nature of mobile ﬂuid matter was recognized, since Onsager reciprocity theorem brings together the microscopic or molecular, and the macroscopic or continuum. This ﬂuctuation/inhomogeneity argument as the root cause of the breakdown of Euler’s hypothesis is essentially physical in nature, and hence intuitive and informal. In what follows, we show mathematically (and thus formally) why and how Onsager coupling undermines the possibility of there being but a single velocity in ﬂuid mechanics.

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versions of GENERIC [23], the latter but one of several competitive schemes [66] aimed, inter alia, at a rational approach to the physics of non-equilibrium thermodynamics. What he identiﬁed as being ‘‘missing’’ from the basic notions of irreversible thermodynamics was the physical manifestation of molecularly based ﬂuctuations upon the N–S–F equations, more generally of the basic equations of transport processes [18–21]. He proposed to rectify this omission by incorporating into the friction matrix appearing in GENERIC a rational model quantifying these ﬂuctuations. This led him, in effect, to pose Eq. (20) as a general constitutive relation for j based on the principles embodied in GENERIC. Two unknown parameters, D0 and a0 , appear in this expression. Establishing the values of these two parameters, such as to render O¨ttinger’s ﬂuctuation model consistent with the work of Klimontovich and the present writer, formed the heart of the present paper. In this context it must be borne in mind that the constitutive formula j ¼ jv ¼ ar ln r by which we determined D0 and a0 in the present paper has not been independently veriﬁed by others, so that outstanding issues remain. The key to reconciling O¨ttinger’s model with our formulas, namely that appearing in the preceding sentence in conjunction with the deﬁnition jv :¼vv vm of the diffusive volume current, lay in the ‘‘incompressibility hypothesis,’’ Eq. (27), the latter based on our belief that the N–S–F equations are likely to be valid for ﬂuids of uniform density (at least for single-component ﬂuids). This belief derived from a number of sources—experimental, theoretical and philosophical—as already outlined in our earlier papers [1–3], again subject to the caveat of there yet being no independent veriﬁcation of our notions. In any event, the hypothesis embodied in Eq. (27) eventually led to the respective expressions for a0 in Eq. (33) and

9.2. On the mathematical impossibility of Euler’s relation v ¼ vm being valid when viewed in the light of Onsager coupling: thermodynamics vs. mechanics

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From a strictly mathematical viewpoint the necessity for the presence of two velocities rather than one in the Cauchy linear momentum equation (2) can be regarded as formally arising from the need to increase the number of independent vector variables appearing therein in the formulation of the overall ﬂuid-mechanical problem in order to accommodate the additional restriction imposed by Onsager coupling (with the new independent variable j, denoting the difference v vm between these two velocity ﬁelds, as in Eq. (8)). This restriction arose from the presence in the set of independent LIT-based ‘‘force–ﬂux’’ Onsager relations, Eqs.

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(82) and (83), of an additional vector ﬂux, namely the diffuse volume ﬂux jv j (given constitutively by the relation jv ¼ ar ln r, with a the thermometric diffusivity), above and beyond the usual internal energy (or heat) ﬂux vector—the need for which had not been previously recognized. Were but a single velocity to have appeared in Cauchy’s equation, the restriction imposed by the Onsager coupling relation (85) would have resulted in the overall problem being over-determined owing to the resulting disparity between the number of independent variables and the number of independent relations existing among them. As a further consequence, the deﬁnition, jv :¼vv vm , of the diffuse volume ﬂux (with vv the volume velocity) then led to the constitutive relation v ¼ vv for the momentum velocity or speciﬁc momentum density. This general Onsager-based inequality vavm , impacting on Euler’s hypothesis, holds independently of the correctness of the relation j ¼ jv , whose validity has not yet been unequivocally conﬁrmed [67], as discussed in Section 8.4. What appears to be certain, however, contrary to popular belief dating back to Euler, is that the Cauchy linear momentum equation cannot be derived solely by dynamical arguments based simply upon applying Newton’s laws of motion to a ﬁxed mass of ﬂuid viewed as a continuous body (a so-called material domain) moving through space. Rather, because of ﬂuctuations in the instantaneous contents of such a domain, stemming from the molecular constitution of matter, non-equilibrium macroscopic thermodynamic principles deriving from the classical statistical–mechanical work of Onsager necessarily enters into consideration. This perspective serves to inseparably link together continuum mechanics and non-equilibrium thermodynamics in a manner that has not previously been explicitly recognized in LIT [19–21], although already implicit therein, as well as in the more broadly based structure of GENERIC [23]. In turn, this inseparable linkage of mechanics to thermodynamics, with the notion of mechanical work common to both ﬁelds, led naturally in our paper to fundamental questions about the deﬁnition of heat— questions which, in our view, had not previously been satisfactorily addressed, much less answered. Intimately related thereto was the issue of the limits of applicability, if any, to the range of temperature gradients over which Fourier’s law of heat conduction would be valid. Heretofore, the existence of deﬁnitive limits of applicability had been implied, despite the apparent lack of experimental data or rational theoretical argument indicating any such limitation [52]. These issues, which might appear to border on the strictly philosophical, will be discussed in subsequent papers aimed at attempting to extend the thermodynamic analysis of Section 7 to include inhomogeneous multicomponent ﬂuid mixtures as well as more complex single-species ﬂuids than those discussed here?.

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This work is the outgrowth of a dialog begun several years ago with Prof. Hans Christian O¨ttinger of the Department of Materials, Institute of Polymers, of the Swiss Federal Institute of Technology (ETH) in Zu¨rich, following his receipt of a preliminary draft of the work cited in Ref. [3] questioning Euler’s constitutive expression for the speciﬁc momentum. As clearly set forth in the ‘‘Something is missing’’ addendum to his recent book [23], it was he who ﬁrst recognized the commonality of our respective unorthodox views of the current status of transport processes, while implicitly encouraging me to remain ﬁrm in my beliefs when faced with the discouraging views of disbelieving referees. Our common views were recently brought jointly to fruition during a collaborative visit to his Institute in June 2005. It was H.C.O.’s insight, as recorded in his book [23], that provided the theoretical framework resulting in this paper, enabling the constitutive relation for the speciﬁc momentum density embodied in Eq. (56) to be rationalized—a relation in whose correctness I believed deeply intuitively at the outset [1] of my research on the role of diffuse volume transport [4] in ﬂuid mechanics and thermodynamics. Also sharing in the initial phases of the Clausius–Duhem/momentum density issue was my former student and collaborator, Dr. ‘‘Jim’’ Bielenberg, who was a signiﬁcant contributor to the evolution of the thinking reﬂected in the present paper on issues of volume transport and the consistency thereof with the second law.

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[1] H. Brenner, Unsolved problems in ﬂuid mechanics: on the historical misconception of ﬂuid velocity as mass motion, rather than volume motion, in: L.-S. Fan, M. Feinberg, G. Hulse, T.L. Sweeney, J.L. Zakin (Eds.), Unsolved Problems in Chemical Engineering, Proceedings of the Ohio State University, Department of Chemical Engineering Centennial Symposium, April 24–25, 2003, pp. 31–39. This reference is available on-line as hhttp://www.che.eng.ohio-state.edu/centennial/brenner.pdfi, while the lecture on which the reference is based can be seen on-line as a streaming video presentation at hhttp://www.chbmeng.ohio-state.edu/centennial/i. [2] H. Brenner, Phys. Rev. E 70 (2004) 061201 H. Brenner, Inequality of the tracer and mass velocities of physicochemically-inhomogeneous ﬂuid continua, J. Fluid Mech., 2006, submitted for publication. [3] H. Brenner, Physica A 349 (2005) 60. [4] H. Brenner, Physica A 349 (2005) 10. [5] H. Brenner, J.R. Bielenberg, Physica A 355 (2005) 251. [6] M.N. Kogan, Ann. Rev. Fluid Mech. 5 (1973) 383. [7] V.S. Galkin, M.N. Kogan, O.G. Fridlander, Izv. AN SSSR, Mekh. Zhidk. Gaza 5 (1970) 13; M.N. Kogan, V.S. Galkin, O.G. Fridlander, Sov. Phys. Usp. 19 (1976) 420. [8] M.N. Kogan, Non-Navier–Stokes gas dynamics and thermal-stress phenomena, in: Rareﬁed Gas Dynamics, vol. 15, 1986, p. 15. [9] M.N. Kogan, Some solved and unsolved problems in kinetic theory, in: A.D. Ketsdever, E.P. Muntz (Eds.), Rareﬁed Gas Dynamics, 23rd International Symposium, American Institute of Physics, 2003. [10] A.V. Bobylev, J. Stat. Phys. 80 (1995) 1063. [11] J.C. Maxwell, Phil. Trans. R. Soc. London A 170 (1879) 231 reprinted in: W.D. Niven (Ed.), The Scientiﬁc Papers of James Clerk Maxwell, vol. 2, Cambridge University Press, Cambridge, 1890, p. 681. [12] D. Burnett, Proc. London Math. Soc. 39 (1935) 385; D. Burnett, Proc. London Math. Soc. 40 (1936) 382. [13] S. Chapman, T.G. Cowling, The Mathematical Theory of Non-Uniform Gases, third ed., Cambridge University Press, Cambridge, 1970. [14] J.R. Bielenberg, H. Brenner, Physica A 356 (2005) 279. [15] H. Brenner, Phys. Rev. E 72 (2005) 061201. ^ for the quantity here denoted by v. [16] In earlier publications [2,3] we used the symbol m [17] J. Happel, H. Brenner, Low Reynolds Number Hydrodynamics, Prentice-Hall, Englewood Cliffs, NJ, 1965. [18] R.B. Bird, W.E. Stewart, E.N. Lightfoot, Transport Phenomena, second ed., Wiley, New York, 2002. [19] S.R. De Groot, P. Mazur, Non-Equilibrium Thermodynamics, North-Holland, Amsterdam, 1962. [20] R. Haase, Thermodynamics of Irreversible Processes, Dover, New York, 1990 (reprint). [21] G.D.C. Kuiken, Thermodynamics of Irreversible Processes: Applications to Diffusion and Rheology, Wiley, New York, 1994. [22] The overbar operator is deﬁned such that for any dyadic D, the dyadic D:¼ð12ÞðD þ DT Þ ð13ÞI : D represents its symmetric traceless counterpart. [23] H.C. O¨ttinger, Beyond Equilibrium Thermodynamics, Wiley, Hoboken, New Jersey, 2005 The apt phrase, ‘‘Something is missing,’’ which neatly encapsulates the momentum density problem, is taken verbatim from p. 61 of this reference. [24] ‘‘If your theory is found to be against the second law of thermodynamics, I give you no hope; there is nothing for it [your theory] but to collapse in the deepest humiliation,’’ in A.S. Eddington, The Nature of the Physical World, Macmillan, New York, 1928, p. 74; ‘‘[Thermodynamics] is the only physical theory of universal content which I am convinced that within the framework of applicability of its basic concepts will never be overthrown.’’ (A. Einstein), quoted in M.J. Klein, Science 157 (1967) 509. [25] M. Grmela, H.C. O¨ttinger, Phys. Rev E 56 (1997) 6620. [26] H.C. O¨ttinger, J. Non-Equilib. Thermodyn. 57 (1997) 386; H.C. O¨ttinger, Phys. Rev. E 57 (1998) 1416; H.C. O¨ttinger, J. Non-Equilib. Thermodyn. 27 (2002) 105. [27] H. Brenner, originally submitted to Phys. Rev. E (March 2003) under the title ‘‘A molecular basis for the Euler/Lagrange velocity disparity. The demise of the Navier–Stokes paradigm.’’ Revised version to be submitted to Physica A (2005) under the title: ‘‘On Klimontovich’s proposed modiﬁcation of the Boltzmann equation and its consequences for the Navier–Stokes–Fourier equations.’’ [28] Yu.L. Klimontovich, Theor. Math. Phys. 92 (1992) 909. [29] Yu.L. Klimontovich, Theor. Math. Phys. 96 (1993) 1035. [30] Yu.L. Klimontovich, Statistical Theory of Open Systems, Volume 1: A Uniﬁed Approach to Kinetic Descriptions of Processes in Active Systems, Kluwer Academic Publishers, Dordrecht, 1995. [31] Mazo, in the historical background to his book on Brownian motion (R.M. Mazo, Brownian Motion: Fluctuations, Dynamics and Applications, Clarendon Press, Oxford, 2002) discussing the period between Robert Brown’s observations in 1829 and Einstein’s 1905 publication, states that (p. 3): ‘‘It is striking, however, that the founders and main developers of kinetic theory, Maxwell, Boltzmann and Clausius, never published anything on Brownian motion.’’ [32] D. Straub, Alternative Mathematical Theory of Non-Equilibrium Phenomena, Academic Press, San Diego, 1997. [33] L. Euler, Me´m. Acad. Sci. Berlin 11 (1755) 274 (reproduced in Leonhardi Euleri Opera Omnia, Series II, vol. 12, p. 54 (Fu¨ssli, Zu¨rich, 1954)). Additional historical information can be found in the ‘‘Editor’s Introduction’’ to the latter volume by C. Truesdell, ’’Rational

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. [44] J.G. Kirkwood, I. Oppenheim, Chemical Thermodynamics, McGraw-Hill, New York, 1961. [45] Explicitly, the vector v is the momentum density per unit mass appearing in the total Eulerian momentum current dyadic, say nM ¼ nm v T, implicit in the Cauchy linear momentum (2) (rewritten as qðrvÞ=qt þ r.nM ¼ rpÞ, of which nm v and T are its respective convective and diffusive portions (with nm :¼rvm the Eulerian mass current). However, only the total Eulerian ﬂux of an extensive property is physically objective, and not the separate convective and diffusive portions into which it is eventually decomposed [4]. Armed with this fact, consider the particular case where the momentum density v is homogeneous, so that rv ¼ 0. In such circumstances there would no diffusive momentum ﬂux only a convective portion, despite the fact that no restriction is imposed upon vm , thus allowing the possibility that rvm a0. As such, it is evident that the velocity appearing in the constitutive equation (44) for the diffusive momentum current must, in general, represent the ﬂuid’s momentum density v rather than its mass velocity vm . [46] However, as discussed in the most recent edition of Ref. [18, cf. Table 19.2-4], such differences have recently been recognized in multicomponent ﬂuids. [47] C. Truesdell, R.G. Muncaster, Fundamentals of Maxwell’s Kinetic Theory of a Simple Monatomic Gas, Academic Press, New York, 1980. [48] D.W. Mackowski, D.H. Papadopoulos, D.E. Rosner, Phys. Fluids 11 (1999) 2106. [49] The leading term of Eq. (49) was originally derived by Maxwell [11], including the value K 1 ¼ 3 for Maxwell molecules. [50] J.R. Bielenberg, H. Brenner, A continuum model of thermal transpiration, J. Fluid Mech. 246 (2006) 1. [51] Although not relevant to the present arguments, it should be noted that Eq. (54) was originally viewed [4] as being valid only in circumstances where pressure effects on the ﬂuid’s density were small compared with temperature effects. However, according to our present GENERIC derivation, no such restriction appears to exist, either for gases or liquids. Furthermore, it should be noted that while the traditional form of the diffuse internal energy/heat ﬂux relation, namely ju ¼ jq , rather than the GENERIC form (34), was used in the original derivation [4] of Eq. (54), the two forms coincide in the case of ideal gases since ju ¼ jq ¼ krT in that case, as follows from Eq. (34). Accordingly, the derivation of (54) is consistent with that of (52), at least in the case of gases. [52] F. Bonetto, J.J. Lebowitz, L. Rey-Bellet, Fourier’s law: a challenge to theorists, in: A. Fokas, A. Grigoryan, T. Kibble, B. Zegarlinski (Eds.), Mathematical Physics 2000, Imperial College Press, London, 2000, p. 128. [53] We note upon comparing the latter relation with Eq. (31) that D0 ¼ r2 Lvv . This relation is consistent with the requirement (21) that D0 X0. [54] P.G. Debenedetti, Metastable liquids, Princeton University Press, Princeton, NJ, 1996. [55] S. Ansumali, I.V. Karlin, H.C. O¨ttinger, Phys. Rev. Lett. 94 (2005) 080602.

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ﬂuid mechanics, 1687–1765,’’ pp. VII–CXXV; L. Euler, Hist. Acad. Berlin 1755 (1757) 316. A concise history of the conceptual foundations of ﬂuid mechanics, from the time of Newton’s Principia in 1687 up to the deﬁnitive work of Stokes in 1845 [35], can be found in the following article: C. Truesdell, Amer. Math. Monthly 60 (1953) 445; O. Darrigol, Arch. Hist. Exact Sci. 56 (2002) 95; O. Darrigol, Worlds of Flow, Oxford University Press, Oxford, 2005. G.G. Stokes, Trans. Camb. Phil. Soc. 8 (1845) 287 reprinted in: Mathematical and Physical Papers, vol. 1, Cambridge University Press, Cambridge, 1901, p. 75 An account of pre-1845 work on the Navier–Stokes equations can be found in G.G. Stokes, British Assoc. Advance. Sci. (1846) 1; reprinted in Mathematical and Physical Papers, vol. 1, p. 157, Cambridge University Press, Cambridge, 1901. This refers to earlier works, as follows: C.L.M.H. Navier, Me´m. Acad. R. Sci. Paris 6 (1823) 389; S.D. Poisson, J. E´cole Polytech. Paris 13 (1831) 139; and B. Saint-Venant, C. R. Acad. Sci. Paris 17 (1843) 1240. L.D. Landau, E.M. Lifshitz, Fluid Mechanics, second ed., Butterworth-Heinemann, Oxford, 1987. P. Kosta¨dt, M. Liu, Phys. Rev. E 58 (1998) 5535. Recall that a material domain is a hypothetical continuous body of ﬂuid, each of whose surface points moves with the local mass velocity vm [39]. This fact in conjunction with the continuity equation assures that no net mass crosses the surface of such a body at any point. However, this is a strictly macroscopic statement in the sense that individual molecules are nevertheless free to cross a material surface in either direction without violating this no-ﬂux condition, provided only that in some time-average sense there are no large-scale variations in the number of molecules contained within the domain. Clearly, the material view has nothing to say about the role of such ﬂuctuations in regard to their possible impact upon Newton’s laws of motion, which apply strictly only to permanent collections of molecules. J.C. Slattery, Momentum, Energy, and Mass Transfer in Continua, McGraw Hill, New York, 1972. H. Goldstein, Classical Mechanics, Addison-Wesley, Reading, MA, 1950, p. 4; L.D. Landau, E.M. Lifshitz, Mechanics, Addison-Wesley, Reading, MA, 1960. However, neither his work nor that of Klimontovich [28–30] touches upon the volume velocity-based no-slip boundary condition issue, which plays a prominent role in our unorthodox interpretation of the thermophoretic motion of macroscopic particles [1–3,5]. E.M. Lifshitz, L.P. Pitaevskii, Statistical Physics, Part 2, Pergamon Press, Oxford, 1980, p. 369. According to the chain rule, for any function f ðx; y; zÞ ¼ 0,

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[56] A. Einstein, Ann. Phys. 17 (1905) 549 translated in: R. Fu¨rth, (Ed.), A. Einstein, Investigations on the Theory of the Brownian Movement, Dover, New York, 1956 (reprint). [57] M.R. von Smoluchowski, Ann. Phys. 21 (1906) 756. [58] R.F. Streater, Proc. R. Soc. London A 456 (2000) 205. [59] In this connection, see also R.F. Streater, Statistical Dynamics, Imperial College Press, London, 1995; R.F. Streater, J. Math. Phys. 38 (1997) 4570; R.F. Streater, Rep. Math. Phys. 40 (1997) 557; R.F. Streater, J. Stat. Phys. 88 (1997) 447; R.F. Streater, Open Syst. Inf. Dyn. 10 (2003) 3. [60] M.R. Grasselli, R.F. Streater, Rep. Math. Phys. 50 (2002) 13. [61] J.O. Hirschfelder, C.F. Curtiss, R.B. Bird, Molecular Theory of Gases and Liquids, Wiley, New York, 1954; H. Grad, Rariﬁed Gas Dynamics, Pergamon, London, 1960; M.N. Kogan, Rariﬁed Gas Dynamics, Plenum Press, New York, 1969; C. Cercignani, Mathematical Methods in Kinetic Theory, second ed., Plenum Press, New York, 1990; Y. Sone, Kinetic Theory and Fluid Dynamics, Birkha¨user, Boston, 2002. [62] N.V. Brilliantov, T. Po¨schel, Kinetic Theory of Granular Gases, Oxford University Press, Oxford, 2004; J. Javier Brey, M.J. Ruiz-Montero, R. Garcia-Rojo, Phys. Rev. E 60 (1999) 7174. [63] E. Nelson, Dynamical Theories of Brownian Motion, Princeton University Press, Princeton, NJ, 1967 see also the revised online second edition (2001) of this book at hhttp://www.math.princeton.edu/nelson/books/bmotion.pdfi; E. Nelson, Quantum Fluctuations, Princeton University Press Princeton, NJ, 1985 see also hhttp://www.math.princeton.edu/nelson/ books/Quantum_Fluctuations_1985.pdfi. [64] P. Garbaczewski, J.P. Vigier, Phys. Rev. A 46 (1992) 4634. [65] Indeed, their need for the elaborate footnote appearing on p. 143 of their book [13] to explain the unexpected appearance of a factor of 52 in place of 32 appears to be a manifestation of this omission, since the existence of a volume velocity vv , and, hence, a nonzero diffuse volume ﬂux jv in inhomogeneous multicomponent ﬂuid mixtures, is well known in the irreversible thermodynamics literature [19,20]. [66] D. Jou, J. Casas-Va´zquez, G. Lebon, Extended Irreversible Thermodynamics, third ed., Springer, Berlin, 2001; Muller, T. Ruggeri, Rational Extended Thermodynamics, second ed., Springer, New York, 1997; D. Jou, J. Casas-Va´zquez, G. Lebon, Rep. Prog. Phys. 62 (1999) 1035 see also hhttp://telemaco.uab.es/eit/home/principal.phpi. [67] H.C. O¨ttinger, ‘‘Hydrodynamics from Boltzmann’s kinetic equation after proper coarse-graining,’’ Phys. Rev. Lett., 2005, submitted for publication. [68] M. Danielewski, Defect Diffusion Forum 95–98 (1993) 125; M. Danielewski, Defect Diffusion Forum 95–98 (1993) 673; M. Danielewski, Solid. State Phen. 41 (1995) 63. [69] M. Danielewski, W. Krzyzanski, Phys. Status. Solidi. 145 (1994) 351; M. Danielewski, B. Wierzba, J. Phase Equil. Diffusion 26 (2005) 573. [70] Y.C. Chen, Y.G. Zhang, C.Q. Chen, Mater. Sci. Eng. A 368 (2004) 1; S.M. Schwarz, B.W. Kempshall, L.A. Giannuzzi, Acta Materialia 51 (2003) 2765. [71] M. Kizilyalli, J. Corish, R. Metselaar, Pure Appl. Chem. 71 (1999) 1307. [72] L.S. Darken, Trans. AIME 175 (1948) 184. [73] A.D. Smigelskas, E.O. Kirkendall, Trans. AIME 171 (1947) 130. [74] G.B. Stephenson, Acta Metall. 36 (1988) 2663. [75] D.L. Beke, I.A. Szabo´, Z. Erde´lyi, G. Opposits, Mater. Sci. Eng. A 387–389 (2004) 4. [76] W.J. Boettinger, G.B. McFadden, S.R. Coriell, R.F. Sekerka, J.A. Warren, Acta Materialia 53 (2005) 1995. [77] J. Philibert, Atomic Movements, Diffusion and Mass Transport in Solids. Translated from the French by S.J. Rothman, Les Editions de Physique, Les Ulis, Paris, 1991; A.R. Allnatt, A.B. Lidiard, Atomic Transport in Solids, Cambridge, 1993. [78] E. Lauga, M.P. Brenner, H.A. Stone, Dummy, in: J. Foss, C. Tropea, A. Yarin (Eds.), Handbook of Experimental Fluid Dynamics, Springer, New York, 2005 (Chapter 15); C. Neto, D.R. Evans, E. Bonaccurso, H.-J. Butt, V.S.J. Craig, Rep. Prog. Phys. 68 (2005) 2859. [79] L. Onsager, Phys. Rev. 37 (1931) 405; L. Onsager, Phys. Rev. 38 (1931) 2265.

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Fluid mechanics revisited

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Howard Brenner

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Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139-4307, USA

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Received 12 November 2005; received in revised form 21 February 2006

15 Abstract

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O¨ttinger’s recent nontraditional incorporation of ﬂuctuations into the formulation of the friction matrix appearing in the phenomenological GENERIC theory of nonequilibrium irreversible processes is shown to furnish transport equations for single-component gases and liquids undergoing heat transfer which support the view that revisions to the Navier–Stokes–Fourier (N–S–F) momentum/energy equation set are necessary, as empirically proposed by the author on the basis of an experimentally supported theory of diffuse volume transport. The hypothesis that the conventional N–S–F equations prevail without modiﬁcation only in the case of ‘‘incompressible’’ ﬂuids, where the density r of the ﬂuid is uniform throughout, serves to determine the new phenomenological parameter a0 appearing in the GENERIC friction matrix. In the case of ideal gases the consequences of this constitutive hypothesis are shown to yield results identical to those derived theoretically by O¨ttinger on the basis of a ‘‘proper’’ coarse-graining of Boltzmann’s kinetic equation. A major consequence of the present work is that the ﬂuid’s speciﬁc momentum density v is equal to its volume velocity vv , rather than to its mass velocity vm , contrary to current views dating back 250 years to Euler. In the case of rareﬁed gases the proposed modiﬁcations are also observed to agree with those resulting from Klimontovich’s molecularly based, albeit ad hoc, self-diffusion addendum to Boltzmann’s collision integral. Despite the differences in their respective physical models—molecular vs. phenomenological—the role played by Klimontovich’s collisional addition to Boltzmann’s equation in modifying the N–S–F equations is noted to constitute a molecular counterpart of O¨ttinger’s phenomenological ﬂuctuation addition to the GENERIC friction matrix. Together, these two theories collectively recognize the need to address multiple- rather than single-encounter collisions between a test molecule and its neighbors when formulating physically satisfactory statistical–mechanical theories of irreversible transport processes in gases. Overall, the results of the present work implicitly support the unorthodox view, implicit in the GENERIC scheme, that the translation of Newton’s discrete mass-point molecular mechanics into continuum mechanics, the latter as embodied in the Cauchy linear momentum equation of ﬂuid mechanics, cannot be correctly effected independently of the laws of thermodynamics. While O¨ttinger’s modiﬁcation of GENERIC necessitates fundamental changes in the foundations of ﬂuid mechanics in regard to momentum transport, no basic changes are required in the foundations of linear irreversible thermodynamics (LIT) beyond recognizing the need to add volume to the usual list of extensive physical properties undergoing transport in singlespecies ﬂuid continua, namely mass, momentum and energy. An alternative, nonGENERICally based approach to LIT, derived from our ﬁndings, is outlined at the conclusion of the paper. Finally, our proposed modiﬁcations of both Cauchy’s linear momentum equation and Newton’s rheological constitutive law for ﬂuid-phase continua are noted to be mirrored by

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Tel.: +1 617 253 6687; fax: +1 617 258 8224.

E-mail address: [email protected] 0378-4371/$ - see front matter r 2006 Published by Elsevier B.V. doi:10.1016/j.physa.2006.03.066

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counterparts in the literature for solid-phase continua dating back to the classical interdiffusion experiments of Kirkendall and their subsequent interpretation by Darken in terms of diffuse volume transport. r 2006 Published by Elsevier B.V. Keywords: ’; ’; ’

7 1. Introduction 9 1.1. Preface 11

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Previous contributions [1–3] relevant to the issues discussed here in have suggested, based primarily upon conﬂicts noted between tracer- and hypothetical mass–velocity experiments, that revisions were needed to the Navier–Stokes–Fourier (N–S–F) equations of viscous ﬂuid mechanics—certainly in the case of singlecomponent ﬂuids. Explicitly, in several key constitutive relations appearing in the N–S–F equation set it was proposed, inter alia, that the ﬂuid’s mass velocity vm appearing therein be replaced by the ﬂuid’s volume velocity vv [4]. The arguments underlying the proposed revisions were based, in part, upon our unorthodox interpretation [5] of experimental data pertaining to thermophoretic particle motion in gases, when considered in conjunction with Kogan et al.’s [6–9] and Bobylev’s [10] re-ordering of the Maxwell [11]—Burnett [12] thermal stress contribution to the viscous stress tensor at small Knudsen numbers, Kn51. These revealed such stresses to be of the same order in Kn as the classical Navier–Stokes stress contributions, at least for Reynolds numbers of order one and Mach numbers small compared with unity. Prior to this reassignment, thermal stresses were regarded as being of a noncontinuum nature in the Chapman–Enskog [13] hierarchical expansion of solutions of the Boltzmann equation in powers of Kn. Our original proposal [1–3] emphasized gases. This owed to a lack at the time of convincing experimental evidence sufﬁcient to demonstrate the applicability of the theory to liquids. Subsequently, however, we were able to offer an interpretation of experimental data involving thermal diffusion in binary liquid mixtures [14,15] that supported the applicability of our proposed N–S–F revisions to liquids as well. It proved unnecessary in our original N–S–F modiﬁcation proposal to address the obvious question of a further revision to these equations regarding the viability of Euler’s constitutive equation for the ﬂuid’s speciﬁc momentum density v [16], including related kinetic energy density and ‘‘work-velocity’’ issues [3]. (The symbol v refers here to the velocity appearing in both the local and convective inertial terms in the Navier–Stokes equations.) That this was unnecessary owed to the smallness of the Reynolds numbers [17] encountered in the thermophoretic and thermal diffusion experiments cited above, as well as in the Knudsen number re-ordering of the thermal stress contributions [6–10]. It was, however, pointed out in Appendix B of Ref. [3] that if one accepts as being correct the standard balance equations for momentum and energy found in textbooks [18–21], while concurrently adopting the unconventional constitutive equations proposed in Ref. [3] for the revised forms of Fourier’s law of heat conduction and Newton’s law of viscosity, the speciﬁc momentum density v would necessarily have to be equal to the ﬂuid’s mass velocity vm , the latter being the velocity appearing in the continuity equation (cf. Eq. (1)). Although not explicitly pointed out in connection with the ‘‘proof’’ set forth in Ref. [3] of the apparent viability of the standard Eulerian constitutive relation v ¼ vm for the ﬂuid’s speciﬁc momentum density, it is implicit in the concomitant use of the nonstandard form of Newton’s viscosity law (cf. the union of Eq. (44) with (56)) (in which the gradient rvv of the ﬂuid’s volume velocity [4] now appears in place of its standard mass velocity counterpart rvm Þ that nonpositive-deﬁnite dissipative terms, namely the ‘‘mixed’’ terms 2Zrvm : rvv and ZB ðr.vm Þðr.vv Þ [22], will appear in the Clausius–Duhem inequality (cf. Eq. (13)) for the local rate of irreversible entropy production. Here, Z and ZB are the respective shear and bulk viscosities. The consequent conﬂict between the apparent dictates of rational mechanics and irreversible thermodynamics manifested by this lack of deﬁnitive positivity leads us to continue to regard the speciﬁc momentum density issue as an open question despite the purported ‘‘proof’’ to the contrary offered in Ref. [3]. As a result of this fundamental dichotomy between the respective predictions of continuum mechanics and the second law of thermo-

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dynamics, the possibility must be entertained that, to quote O¨ttinger [23]: ‘‘Something is missing’’ from the currently accepted energy and entropy transport equations of nonequilibrium irreversible thermodynamics [19–21]; that is, in order to heal the breach engendered by this lack of positivity, additional terms, presumably currently missing from these equations, need to be present in the constitutive expressions for the diffusive ﬂuxes of internal energy and entropy appearing therein. In this context it is pertinent to note that prior to the emergence of this core conﬂict we had earlier [1] opined, so to speak, in favor of thermodynamics over mechanics [24] in resolving apparent conﬂicts between the two, by supposing that v ¼ vv rather than v ¼ vm so as to render the irreversible entropy production rate nonnegative. However, substantive theoretical justiﬁcation for that fundamental change in the constitutive expression governing the speciﬁc momentum density appearing in the Cauchy linear momentum equation was lacking at the time (although we nevertheless regarded our Lagrangian tracer velocity interpretation [1–3] of thermophoretic particle motion as offering convincing experimental evidence in support of the possible change in v from vm to vv Þ. This background sets the stage for the subsequent theoretical analysis, which purports to resolve the momentum density issue in favor of the ‘‘mechanico-thermodynamic’’ relation v ¼ vv . This is accomplished by adding fundamental ﬂuctuational contributions [23] to the standard [19–21] energetic and entropic balance equations of nonequilibrium irreversible thermodynamics, following a recent proposal to this effect by O¨ttinger. In his book [23], ‘‘Beyond Equilibrium Thermodynamics’’ (hereafter referred to as BET), O¨ttinger proposed incorporating ﬂuctuations into the friction matrix appearing in GENERIC theory. This latter nontraditional feature, manifested by the appearance of a new phenomenological coefﬁcient a0 therein, was not explicitly included in earlier versions [25,26] of that theory (although neither was it explicitly ruled out). It is this inclusion in the GENERIC friction matrix M of O¨ttinger’s previously ‘‘Something is missing’’ ﬂuctuation terms that resolves the conﬂict, described earlier, between continuum mechanics and thermodynamics, and which will be seen to lead rationally to the relation v ¼ vv . Explicitly, O¨ttinger’s extension of GENERIC theory harmoniously unites our respective views of momentum density (while also conﬁrming other elements of our theory). A related paper by the present author [27] reinforces this uniﬁcation of our view of momentum transport with that of O¨ttinger by offering an independent argument, due to Klimontovich [28–30], with respect to the viability of the constitutive relation v ¼ vv . In particular, Klimontovich’s unorthodox modiﬁcation of the collisional term in Boltzmann’s equation [13], involving the addition thereto of a physicalspace self-diffusion-like contribution, is counterpart to Ottinger’s ﬂuctuational addition to GENERIC. Both additions, in turn, are formally equivalent to our addition of diffuse volume transport [1–4] to conventional theories of transport phenomena [18]. The independent theories of O¨ttinger [23] and Klimontovich [28–30], which lead to identical results in their common domain of validity (namely, rareﬁed gases), were created to rectify what each regarded as the fundamental failure of existing continuum-mechanical theories—especially those derived from the Boltzmann equation [13]—to incorporate single-component ﬂuctuations and, hence, the notion of Brownian motion into their respective foundations [31]. (Further details with respect to this point of view are discussed in Sections 8.1 and 8.2.) However, the arguments underlying their proposed changes were purely philosophical, in the sense that neither explicitly invoked experimental data, nor appealed to already existing theories to support their claims with respect to the perceived incompleteness of current theories of irreversible thermodynamics and Boltzmann-based gas-kinetic statistical mechanics. In that respect, our conclusions, if accepted as being correct, offer experimental and theoretical evidence in support of both of their theories, whose common results for rareﬁed gases coincide. Conversely, subject to this same caveat, their theories should be regarded as supporting our earlier empirical proposal [1–3] for modiﬁcations in the N–S–F equation set, especially including the speciﬁc momentum density constitutive equation v ¼ vv , as well as the energetic and entropic consequences stemming therefrom. Issues of entropy production and irreversibility are a matter of concern when attempting, as did Boltzmann [13], to translate discrete Newtonian mass-point microscopic molecular mechanics into a rational macroscopic continuum-mechanical analog thereof, namely the Cauchy linear momentum equation of ﬂuid mechanics [18–21]. However, according to the generally accepted view of this microscopic ! macroscopic transition scheme [13], concern about its consistency with the notion of irreversibility, as embodied in the second law of thermodynamics, currently surfaces only after the fact. This secondary ‘‘after the fact’’ role ascribed to the

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1.2. Outline of the paper

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Section 2 reviews the pertinent equations of O¨ttinger’s extended GENERIC theory [23] for the present single-component ﬂuid case, without invoking any constitutive relations for the various physical properties appearing therein. These equations embody the balance equations for mass, linear momentum, energy and entropy as posed by the GENERIC scheme. They differ in substance from the standard balance equations of linear irreversible thermodynamics (LIT) found in textbooks [19–21], to which the former reduce in circumstances where the difference v vm between the ﬂuid’s speciﬁc momentum density v and its mass velocity vm vanishes. It is not that LIT is conceptually wrong, but rather that the present formulation of the subject fails to recognize the role played by the diffuse transport of volume. Section 3 sets forth the general

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second law is evidenced by the subsequent appearance, only after Boltzmann’s transport equation was already fully formulated, of his celebrated H-theorem, aimed at reconciling the consistency of his purely mechanically derived reversible formula for calculating the molecular distribution function with the thermodynamic notion of irreversibility. In short, current thinking holds that while the continuum version of Newtonian mechanics impacts upon the foundations of irreversible thermodynamics, the reverse is not true; that is, the ﬁeld of irreversible thermodynamics is generally (although not universally [32]) believed to be without impact upon ‘‘pre-constitutive’’ Newtonian dynamics-based continuum mechanics—explicitly upon the Cauchy linear momentum equation prior to contemplating the form of the constitutive equation for the stress tensor appearing therein. (Irreversible thermodynamics does, however, impact upon ‘‘post-constitutive’’ continuum mechanics—for example, as a consequence of restrictions imposed by the Clausius–Duhem entropy production inequality [19–21] upon the allowable class of constitutive expressions for the form ascribed to the deviatoric portion of the stress tensor, the latter representing the diffuse momentum ﬂux.) As regards our ‘‘pre’’ and ‘‘post’’ terminology, we are referring here to the accepted view, dating back 250 years to Euler [33–35], that the identiﬁcation v ¼ vm is not a constitutive relation but is, rather, a fundamental physical truth emanating directly from the implicit a priori assumption that Newton’s laws of motion, known to be valid for a discrete ponderable body of mass M, can unquestionably also be applied to a so-called material ﬂuid domain of mass M moving within a ﬂuid continuum—at least in circumstances where M is differential in magnitude. Indeed, we are aware of only a very few circumstances [36, p. 196 (but see also p. 28), ? 37] in which this apparent ‘‘fact,’’ namely that of the viability of the relation v ¼ vm , has been questioned. However, because a material domain does not generally consist permanently of the same molecules, but only of the same net amount of mass [38,39], discrete-body mechanics [40] based upon regarding individual molecules as point masses subject to elementary action–reaction mechanical Newtonian laws cannot, indiscriminately, be applied to such domains. Of course Euler [33], the father of rational ﬂuid mechanics, was unaware at the time of his foundational work in 1755 of the existence of molecules—either static, as in Dalton’s subsequent chemical theory, or mobile, as in Clausius–Maxwell–Boltzmann’s century-later gaskinetic theory. Hence, he was obviously unaware of the hidden constitutive assumption implicit in the relation v ¼ vm . As such, a material domain must necessarily have been viewed by Euler as a continuous body of permanent material integrity insofar as its underlying constitution was concerned. Today we know better. Accordingly, as regards continuum mechanics, and in particular the correctness of the Cauchy linear momentum equation, the relation v ¼ vm needs to be regarded as a tentative constitutive assumption, rather than as an identity, thereby subject to empirical (experimental) veriﬁcation and, if possible, theoretical justiﬁcation. Given these remarks, it remains to be established whether the seemingly unequivocal separation advocated in the literature between the statistical mechanics and thermodynamics of continuum models of large multiparticle systems composed of materially identical particles is viable. GENERIC theory argues forcefully against such separability. The current nontraditional version of GENERIC theory appearing in O¨ttinger’s book [23] represents an attempt at a systematic synthesis of the two ﬁelds. It constitutes a broad general theory, equally applicable to both single-component gases and liquids. In what follows, we point out that O¨ttinger’s theory provides a rational basis for our proposed diffuse volume-based modiﬁcations [1–3] of the N–S–F equations, including, most prominently, changes to the accepted constitutive formula for the speciﬁc momentum [41].

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constitutive equation emanating from GENERIC for the difference v vm in velocities, while temporarily leaving open issues pertaining to the constitutive equations for the heat ﬂux vector and deviatoric stress dyadic. Based solely upon our fundamental hypothesis that v ¼ vm if, and only if, the ﬂuid is ‘‘incompressible,’’ i.e., r ¼ const. throughout the ﬂuid, it is demonstrated that this is tantamount to requiring that the velocity difference be of the constitutive form v vm ¼ Kr ln r, where the coefﬁcient K is nonnegative-deﬁnite. Since this unorthodox momentum density constitutive result holds independently of the constitutive equations governing the heat ﬂux and stress tensor, it necessarily applies to any ﬂuid continuum, gaseous or liquid, irrespective of the ﬂuid’s rheological and thermal properties. Section 4 discusses the thermodynamic consequences stemming from this constitutive relation, including the fact that the diffusive heat ﬂux generally differs from the diffusive internal energy ﬂux, and that the entropy ﬂux is not simply equal to the heat ﬂux divided by the thermodynamic temperature T in single-component ﬂuids. Both conclusions are at odds with the current tenets of (linear) irreversible thermodynamics. Section 5 uses molecularly derived results emanating from Burnett’s solution of the Boltzmann equation to establish that the velocity-difference coefﬁcient K is equal to the gas’s thermometric diffusivity a in the case of rareﬁed gases. In turn, in Section 6, comparison of the preceding dynamical result, namely v ¼ vm þ ar ln r, with the known, purely kinematical, formula [4] for the volume velocity of a liquid or gas, vv ¼ vm þ ar ln r, leads to the conclusion that the ﬂuid’s speciﬁc momentum density v is identical to its volume velocity vv . Since the ﬂuid’s volume velocity has already been shown elsewhere [2,3,15] to be equal to its tracer or Lagrangian velocity, this equality serves, in turn, to demonstrate equality of the ﬂuid’s speciﬁc momentum with its tracer velocity, certainly in the case of rareﬁed gases. Section 7 offers justiﬁcation for three key constitutive assumptions which it was necessary for us to make in the course of evaluating the two undetermined parameters, a0 and D0 , appearing in Ottinger’s nontraditional version of GENERIC [23]. Identifying these two parameters was necessary in order to obtain deﬁnitive results, against which experimental data could be compared. Section 8 broadly discusses the implications of the present theory for LIT as a whole, while stressing the fact that the consequences of the nontraditional GENERIC addition to the friction matrix appearing therein are explicitly manifested in the Onsager force–ﬂux LIT scheme for single-component ﬂuids by the addition thereto of a volume ﬂux, above and beyond the usual heat and momentum ﬂuxes. In turn, the force–ﬂux basis of LIT suggests a simple physical origin (i.e., one independent of the formal constitutive friction matrix approach of GENERIC) for the apparently universal relation v ¼ vm þ Kr ln r, which is believed to be applicable to all gases and liquids, with the value of the phenomenological coefﬁcient K depending upon the particular physical application being addressed. This applicability extends to multicomponent ﬂuids [4], in which circumstance K ¼ D, with D the binary diffusion coefﬁcient D for isothermal two-component ﬂuid mixtures. It is further pointed out (Section 8.4) that, based on the recent ‘‘proper coarse-graining’’ work of O¨ttinger, an alternative may exist to our fundamental incompressibility hypothesis in Section 3, although in the special, but important, case of ideal gases, the conclusions derived therefrom are indistinguishable from those obtained from our hypothesis. Also noted (Section 8.5) is the fact that issues of momentum transport and volume diffusion-induced stress, comparable to those issues identiﬁed here for gases and liquids, also exist in the case of solids. Explicitly, based on experimental data dating back to the classical interdiffusion experiments of Kirkendall and their subsequent interpretation by Darken, it is pointed out that ‘‘two-velocity’’ modiﬁcations of the Cauchy linear momentum equation as well as the Hooke’s law-type stress–strain constitutive equation appearing therein for solid solutions (alloys) have been proposed in the literature. Just as in the case of ﬂuid phases, these unorthodox dynamical solid-phase notions are attributed to diffuse volume transport. Such transport in solids is ascribed largely to atom–vacancy exchanges occurring within the lattice during the diffusional process, and, to a lesser extent, by comparable atom–atom lattice-point interchanges when the respective molecular masses of the interdiffusing species differ. Finally, Section 9 offers a summary and overview of the essence of our ﬁndings.

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2. Review of O¨ttinger’s nontraditional GENERIC theory

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GENERIC (general equation for the non-equilibrium reversible–irreversible coupling) theory, as set forth in BET [23], offers a derivation of the following nontraditional trio of mass, linear momentum and energy

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transport equations for the case of single-component ﬂuids (both gases and liquids) in the absence of external body-force ﬁelds (Eqs. (2.78)–(2.80) of BET): (i) Mass transport (continuity equation): qr þ r.ðrvm Þ ¼ 0. qt

(1)

(ii) Momentum transport (Cauchy linear momentum equation): qðrvÞ þ r.ðrvm vÞ ¼ r.P, qt

(2)

where (3)

P ¼ Ip þ T

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qu þ r.ðvm uÞ ¼ r.½jq ðv vm Þðra0 uÞ þ pu , (4) qt where u is the volumetric (i.e., per unit volume) internal energy density (designated by the symbol in BET), jq is the heat ﬂux vector, and pu ¼ P : ðrvÞT

(5)

is the temporal rate of production of internal energy per unit volume, wherein the superscript T denotes the transpose operator. In addition, a0 is an unconstrained phenomenological parameter appearing in extended GENERIC theory (for which the symbol a is used in BET). This parameter is to be determined constitutively by some scheme, be it theory, simulation or otherwise, but ultimately subject to conﬁrmation by experiment. Determination of a0 , which will be seen to physically represent an Onsager coefﬁcient serving to couple together the respective processes of diffuse internal energy and volume transport, is one of the central goals of the present analysis. In addition to the preceding equations, we shall later require the nontraditional, ﬂuctuation-based GENERIC entropy transport equation, given in BET as Eq. (2.83), namely: (iv) Entropy transport: q qs j p þ u ra0 þ r.ðvm sÞ ¼ r. þ ðv vm Þ (6) þ ps , qt T T

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is the pressure tensor (using the usual ﬂuid-mechanical sign convention for stress, rather than the opposite convention used in BET), with I the dyadic idemfactor and T the symmetric deviatoric or viscous stress tensor. ^ [16] by the The ‘‘velocity’’ v appearing in Eq. (2) is deﬁned in terms of the ﬂuid’s speciﬁc momentum density m ^ relation v:¼m. (BET writes Eq. (2) in a form involving the momentum per unit volume, the latter identiﬁed therein by the symbol M, equivalent to our rv.) (iii) Energy transport:

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where

41

r2 jv vm j2 1 1 þ jq .r þ T : ðrvÞT . (7) T T D0 Here, s and ps are, respectively, the volumetric entropy density and temporal rate of irreversible entropy production. It is important to note that neither the symbols pu and ps , respectively, appearing in Eqs. (4)–(5) and (6)–(7) nor the respective volumetric production-rate interpretations that we have assigned to them appear in the original BET equations cited. As such, justiﬁcation for these interpretations is required. This is presented in Section 7 at the conclusion of the analysis. For the time being we simply pursue the consequences of these interpretations (cf. Eqs. (11) and (12)). It will prove convenient in what follows to deﬁne the velocity difference vector:

47 49 51

C

N

45

U

43

ps ¼

O

39

j:¼v vm .

(8)

PHYSA : 9965 ARTICLE IN PRESS H. Brenner / Physica A ] (]]]]) ]]]–]]]

13 15 17 19 21 23

qc þ r . n c ¼ pc , (9) qt where c ¼ C=V is the volumetric density of the property, nc is the Eulerian ﬂux density of the property and pc is the volumetric rate of production of the property. The (total) ﬂux density nc ¼ vm c þ jc

appearing above consists, respectively, of a convective portion vm c carried by the moving ﬂuid mass and a diffusive or nonconvective portion jc . In terms of this notation, the continuity equation retains its usual form (1) since the law of conservation of mass together with the deﬁnition vm :¼nm =r of the mass velocity requires that the diffuse mass ﬂux density be zero: jm ¼ 0. On the other hand, the internal energy and entropy transport equations adopt the respective forms

25

31 33 35 37

and

(11)

D

29

qu þ r.ðvm uÞ ¼ r.ju þ pu qt

qs þ r.ðvm sÞ ¼ r.js þ ps . (12) qt In the latter equation the second law of thermodynamics requires satisfaction of the generic Clausius–Duhem inequality [19–21],

TE

27

(10)

F

11

O

9

O

7

PR

5

EC

3

In the next section we will propose a constitutive equation for j. The system of equations described in the preceding paragraphs represents the entire set of single-component GENERIC equations of which we will avail ourselves in what follows. Of course, in order to understand the origin of these equations one needs to master the basic reversible–irreversible model of transport processes underlying the GENERIC scheme, as set forth in BET. This differs from the more common convective–diffusive model of such phenomena [18]. Before proceeding to a discussion of the constitutive equations required to complete the governing set of equations (1)–(7), it will prove convenient to re-express the preceding energy and entropy transport equations in more conventional terminology. In terms of the general notation set out in Ref. [4] consider the transport of any extensive physical property, with C, say, the amount of the property contained in a volume V. The Eulerian balance equation governing transport of this extensive property is then

ps X0.

(13)

Comparison of Eq. (11) with (4) furnishes the relation ju ¼ jq jðra0 uÞ.

R

1

7

(14)

39

R

Similarly, comparison of Eq. (12) with (6) gives 1 q ½j þ jðp þ u ra0 Þ. T Elimination of jq between the latter pair of equations yields the expression

45 47 49 51

C

ju ¼ Tjs pj.

N

43

(15)

(16)

It is important to note that this relation is strictly a consequence of the principles of GENERIC theory, independent of any constitutive relations (although more general nonlinear and nonlocal possibilities than (16) exist within the GENERIC framework). However, its validity hinges critically upon the interpretations we have assigned to the symbols pu and ps appearing in Eqs. (4)–(5) and (6)–(7), since it was those interpretations which resulted in Eqs. (14) and (15), and which thereby led to Eq. (16). The signiﬁcance of Eq. (16) will subsequently be discussed in the context of the combined ﬁrst and second laws of thermodynamics,

U

41

O

js ¼

dU ¼ T dS p dV ,

(17)

PHYSA : 9965 ARTICLE IN PRESS H. Brenner / Physica A ] (]]]]) ]]]–]]]

8

1 3 5 7 9 11 13 15 17

for a system of ﬁxed mass. Explicitly, the obvious analogy existing between Eqs. (16) and (17) suggests that a subscript v should be afﬁxed to the symbol j in Eq. (16), with the resulting quantity jv identiﬁed as being the diffusive ﬂux density of volume [4]. The latter is a physically deﬁned, purely kinematical quantity, related to the ﬂuid’s volume velocity vv discussed in the Introduction through the relation vv vm ¼ jv . In turn, comparison of the latter with Eq. (8) would then give rise immediately to the constitutive relation v ¼ vv , thus identifying the ﬂuid’s speciﬁc momentum density with its volume velocity. It is this fact, namely that j ¼ jv , which we propose to formally demonstrate, or at least make plausible, in what is to follow. Before leaving this section, we note that all of the above BET transport equations are deliberately expressed in a space-ﬁxed, Eulerian, format rather than a (moving) ‘‘body-ﬁxed’’ material format. This avoids the ambiguous notion of so-called ‘‘material differentiation following the motion’’ of a material body, such ambiguity arising from the issue of whether the ﬂuid’s mass velocity vm , which is a (normalized) ﬂux, is indeed synonymous with the notion of the ﬂuid’s Lagrangian or tracer ‘‘motion’’ through space [2,3]. The two classes of description, Eulerian and material, are connected through the relation qc Dm c^ , þ r.ðvm cÞ ¼ r Dt qt ^ ¼ c=r is the speciﬁc density of the extensive property C, and where c

(18)

23

3. Constitutive relation for the speciﬁc momentum density v

25

According to O¨ttinger’s ﬂuctuation-based GENERIC scheme [23], the most general possible constitutive equation for the velocity difference (8), applicable to both single-component gases and liquids, is (BET, Eq. (2.82)) D0 p 1 0 j¼ 2 r ðra uÞr . (20) T T r

39 41 43 45

49 51

O O

PR

D

ra0 u ¼ p gT,

(22)

so as to furnish the relation j¼

47

TE

EC

appearing above quantiﬁes the intensity of the ﬂuid’s molecularly based ﬂuctuations [42]. Apart from having to satisfy the preceding inequality, D0 , like a0 , represents an otherwise unconstrained phenomenological parameter, to be chosen in a manner such as to match experimental data and/or accommodate detailed theories, molecular or otherwise, of the pertinent phenomena. Such phenomenological choices, of course, include the trivial possibility that D0 ¼ 0, and hence j ¼ 0, corresponding as a consequence of Eq. (8) to Euler’s conventional momentum density constitutive relation v ¼ vm . Eq. (20) can be rearranged so as to express j in terms of rp and rr as follows: in place of the parameter a0 it proves useful to introduce another parameter, g, the latter deﬁned by the expression

R

37

(21)

R

35

D0 X0

O

33

The nonnegative-deﬁnite diffusion-like phenomenological coefﬁcient [23]

C

31

N

29

(19)

D0 ðrp grTÞ. r2 T

U

27

F

21

Dm q ¼ þ vm . r Dt qt is the material derivative.

19

But, from the single-component ﬂuid equation of state relating T, p and r, we have that rT ¼ ðqT=qpÞr rp þ ðqT=qrÞp rr, whence the above becomes (" ) # D0 qT 1 1g j¼ 2 rp þ g r ln r , qp r b r T

PHYSA : 9965 ARTICLE IN PRESS H. Brenner / Physica A ] (]]]]) ]]]–]]]

23 25 27 29 31 33 35 37 39 41 43 45 47 49 51

F

O

We now introduce what appears to us to be the physically reasonable hypothesis that when the ﬂuid is ‘‘incompressible,’’ namely when r ¼ const, and only then, the two velocities, v and vm , appearing in the basic BET transport equations of Section 2 should coalesce into a single entity, so that v ¼ vm . From Eq. (8) this is equivalent to the relation

O

21

PR

19

j ¼ 0 when r ¼ const ði.e., when rr ¼ 0Þ.

(27)

The latter relation leads to the dual requirements that: (i) the bracketed pressure-gradient coefﬁcient in Eq. (25) needs to be identically zero; and (ii) the diffuse ﬂux j must obey the constitutive relation

D

17

Thus far, no assumptions have been made beyond the basic nontraditional GENERIC transport relations set forth in Section 2, including the general constitutive assumption for j embodied in Eq. (20). For later reference we note that we have, in effect, merely replaced the original parameter a0 appearing in BET by the new parameter K, the relationship between them being br2 T 1 0 . (26) a ¼ uþpK D0 r

j ¼ Kr ln r.

(28)

In the subsequent discussion of Section 7 it will be pointed out that the further, second law-based, Clausius–Duhem requirement, KX0, in the above relation (see Eq. (46)) is formally equivalent to requiring that Fourier’s law (cf. Eq. (42)) be satisﬁed. This surprising linkage of Eq. (28), the latter involving a density gradient, with Fourier’s law, the latter involving an independent temperature gradient, appears surprising. This connection will be seen (in Section 7) to arise from the fact that a0 represents a coupling coefﬁcient which, as a consequence of Onsager’s reciprocal theorem, cannot be arbitrarily chosen if one insists that Fourier’s law of heat conduction must be satisﬁed under all circumstances (i.e., independently of the density gradient, and hence of the pressure gradient in single-component systems). By the chain rule for partial differentiation [43] in conjunction with deﬁnition (23), we have that bðqT=qpÞr ¼ kT , where 1 qr kT ¼ (29) r qp T

TE

15

(25)

EC

13

(24)

R

11

br2 T , D0 whereupon it follows that " # D0 qT j ¼ 2 Kb rp þ Kr ln r. qp r r T g¼K

R

9

In place of g it now proves convenient to instead introduce a related parameter K, deﬁned by the expression

O

7

(23)

C

5

is the coefﬁcient of isothermal compressibility, assumed to be nonnegative for both liquids and gases:

N

3

where b is the coefﬁcient of thermal expansion: 1 qr b¼ . r qT p

kT X0.

U

1

9

(30)

The latter inequality constitutes a requirement for stability of the ﬂuid continuum (as discussed in Section 7.6; cf. Eq. (101)). With use of the preceding relations, the vanishing of the bracketed term in Eq. (25) necessitates that D0 ¼ Kr2 TkT .

(31)

This expression serves to relate D0 to the phenomenological coefﬁcient K appearing in Eq. (28). In turn, from Eq. (24), this makes

PHYSA : 9965 ARTICLE IN PRESS H. Brenner / Physica A ] (]]]]) ]]]–]]]

10

13 15 17 19 21 23 25 27 29

(32)

where we have again used the chain rule [43]. Upon inserting Eq. (31) into (26), subsequent use of Eq. (32) furnishes the relation " # 1 qp 0 uþpT a ¼ . (33) r qT r The latter is, in effect, a constitutive expression serving to relate the GENERIC parameter a0 to the ﬂuid’s local equilibrium thermodynamic (i.e., nontransport) properties. For the moment we leave open the choice of the phenomenological coefﬁcient K appearing in the constitutive equation (28) for the velocity difference j. This same coefﬁcient serves to determine the parameter D0 from Eq. (31). Hence, as a consequence of the hypothesis embodied in Eq. (28), of the original two phenomenological parameters D0 and a0 appearing in the fundamental GENERIC constitutive equation (20), only the parameter D0 remains yet to be determined, as presently manifested in the coefﬁcient K. Of course, as earlier mentioned, this includes the possibility that K ¼ 0, and hence D0 ¼ 0, corresponding to Euler’s traditional momentum density relation, v ¼ vm . It needs to be stressed that Eq. (33), expressing O¨ttinger’s GENERIC ﬂuctuation parameter in terms of the fundamental equilibrium properties of the ﬂuid, derives directly from the constitutive assumption (27), whose validity can be conﬁrmed only by demonstrating that the physical results issuing therefrom accord either directly with experiment or else with an accepted theory. As such, from this point on, the validity of all subsequent relations in this paper derived indirectly from this assumption is to be regarded as tentative, subject to veriﬁcation. This cautionary emphasis is recapitulated and reviewed near the conclusion of the paper, in Section 8.4. However, was Eq. (27) to be proved inconsistent with experiment, the subsequent effort expended in our paper would, nevertheless, not have been fruitless. Rather, all that would be required to rectify the situation would be to simply carry a0 along as a free parameter, beginning with Eq. (20), and with a0 thus appearing explicitly in the subsequent LIT equations of Section 7. This issue is brieﬂy discussed in Section 8.4.

F

11

, r

O

9

O

7

qp qT

PR

5

D

3

b g¼ ¼ kT

TE

1

4. Consequences of the ‘‘incompressibility’’ hypothesis

33

4.1. Internal energy flux

43 45 47 49 51

R

R

O

41

This expression can be re-formulated in more physical terms by noting from the ﬁrst and second laws of thermodynamics that for a single-component ﬂuid [44] qp p d^v. (35) du^ ¼ c^v dT þ T qT v^

C

39

(34)

N

37

Introduction of Eq. (33) into (14) gives, for the internal energy ﬂux, " # qp q ju ¼ j j p T . qT r

The caret atop a symbol denotes a speciﬁc (i.e., per unit mass) density, so that, for example, u^ ¼ u=r denotes the speciﬁc internal energy and v^ ¼ 1=r the speciﬁc volume. Consequently, we have the thermodynamic identity qp qu^ qU pT ¼ , (36) qT v^ q^v T qV T;M

U

35

EC

31

where, with M ¼ rV the mass contained in a volume V ¼ M v^, the extensive internal energy contained in V is ^ It follows that Eq. (34) is equivalent to the relation U ¼ M u.

PHYSA : 9965 ARTICLE IN PRESS H. Brenner / Physica A ] (]]]]) ]]]–]]]

ju ¼ jq þ jðqU=qV ÞT;M .

1 3 5

(37)

In the context of the possibility, discussed following Eq. (17), that j ¼ jv , the second term of Eq. (37) will, in Section 7.2, be seen to possess a well-deﬁned physical signiﬁcance. Eq. (36) shows the internal energy of an ideal gas to be independent of its volume, i.e., ^ vÞT ¼ 0. In such circumstances it follows from Eq. (37) that ðqU=qV ÞT;M ðqu=q^ ju ¼ jq

7 9

11

(ideal gases).

(38)

This relation also holds for any single-component ﬂuid, either gas or liquid, whose speciﬁc internal energy is a function only of its temperature.

11 4.2. Entropy flux 13

23

it follows that js ¼

25 27

jq qS þj . qV T;M T

45 47 49 51

F

R

(42)

R

with kX0

(43)

C

O

the thermal conductivity. Indeed, rather than simply assuming the validity of (42), we will, in Section 7.5, subsequently derive this equation as an immediate consequence of the second law of thermodynamics when the latter is considered in conjunction with Onsager’s reciprocal relation. We also suppose that the deviatoric stress T is given, for example, by the usual rheological constitutive equation for a viscous Newtonian ﬂuid, namely

N

43

jq ¼ krT,

U

41

EC

In order for the present theory to be viable, the constitutive relations entering into the entropy production rate (7) must be of such a nature that ps satisﬁes the Clausius–Duhem inequality (13). Following BET, we suppose that the heat ﬂux vector jq appearing in Eq. (7) obeys Fourier’s law of heat conduction,

35

39

(41)

TE

4.3. Entropy production 31

37

(40)

In the context of the possibility, discussed following Eq. (17), that j ¼ jv , the second term of Eq. (41) will later be seen to possess a well-deﬁned physical signiﬁcance.

29

33

(39)

O

21

Alternatively, with use of the thermodynamic identity [44] qp q^s qS ¼ , qT v^ q^v T qV T;M

O

19

PR

17

From Eqs. (33) and (15) one obtains the following expression for the entropy ﬂux: jq qp js ¼ þ j . qT v^ T

D

15

T ¼ 2Zrv þ ZB Ir.v,

(44)

where ZX0 and ZB X0 are the ﬂuid’s respective shear and bulk viscosities. In as much as T represents the diffuse momentum ﬂux density, it appears appropriate to suppose that the symbol v appearing in the preceding expression refers to the ﬂuid’s speciﬁc momentum density [45] rather than to, say, its mass velocity vm . Upon introducing the above pair of constitutive equations jointly with Eqs. (28) and (31) into (7) we ﬁnd that

PHYSA : 9965 ARTICLE IN PRESS H. Brenner / Physica A ] (]]]]) ]]]–]]]

12

1 3 5

1 K k 2 2 2 . ps ¼ ðrrÞ þ ðrTÞ þ 2Zrv : rv þ ZB ðr vÞ . T kT r2 T

(45)

Given the nonnegative-deﬁnite algebraic signs of kT , k, Z and ZB , Eq. (45) shows that the Clausius–Duhem inequality (13) will be satisﬁed provided that KX0.

7

(46)

23

5. Conﬁrmation of Eq. (28) and identiﬁcation of the phenomenological coefﬁcient K

25

Apart from our acceptance of the nontraditional GENERIC formulation of irreversible thermodynamics set forth in BET [23] as being physically correct, together with our subsequent constitutive assumptions implicit in the production terms (5) and (7), the only additional assumption we have made thus far is explicitly embodied in the hypothesis (27), namely the assumption that v ¼ vm when the ﬂuid is ‘‘incompressible.’’ (Of course, we have also supposed the applicability of Fourier’s law and Newton’s rheological law, Eqs. (42) and (44), although the latter equation is not critical, whereas the former is.) There remains only the task of relating the phenomenological coefﬁcient K appearing in Eq. (28) to the physical properties of the ﬂuid, and of subsequently conﬁrming that the resulting expression for K satisﬁes inequality (46). In the absence of a microscopic theory of the pertinent phenomena this task is normally assigned to experiment, namely that of empirically conﬁrming the physical viability of the union of Eqs. (8) and (28), and, concomitantly, establishing the functional dependence of K upon the system’s parameters. In the case of liquids, for which no fully accepted theory yet exists, one has no recourse other than to revert to experiment in pursuit of these goals. Fortunately, however, in the case of rareﬁed gases an accepted molecular theory already exists, one that will be seen to sufﬁce for these purposes. Explicitly, we can avail ourselves of Burnett’s extension [12] to higher Knudsen numbers of the Chapman–Enskog scheme [13] for solving Boltzmann’s transport equation in the rareﬁed gas regime. Chapman and Enskog’s calculations [13] theoretically predict, inter alia, at least for rareﬁed gases, that the Fourier and Newtonian rheological law constitutive relations (42) and (44), previously regarded as empirical experimental laws valid for continua, are indeed applicable in the so-called ‘‘near-continuum,’’ OðKnÞ, region of small Knudsen numbers, Kn51, with the Oð1Þ terms represented by the inviscid, ideal ﬂuid Euler equations [36]. Moreover, Chapman and Enskog’s theoretical perturbation scheme concomitantly furnishes the values of the phenomenological coefﬁcients k, Z and ZB appearing therein, at least for particular intermolecular collision model choices (e.g., rigid–elastic spheres, Maxwell molecules, Lennard-Jones potentials, etc.). In what follows, Burnett’s [12] Knudsen number extension of the Chapman–Enskog theory [13] will be seen to provide conﬁrmation of the constitutive equation (28) for j, at least for the case of monatomic Maxwell molecules [13,47], while at the same time conﬁrming the inequality (46) by showing that

31 33 35 37 39 41 43 45 47 49 51

O

O

PR

D

TE

29

EC

27

R

19

R

17

O

15

C

13

N

11

U

9

F

21

In view of Eq. (31), the latter inequality is equivalent to that requiring satisfaction of the previously stipulated inequalities (21) and (30). We have referred to the Newtonian rheological constitutive expression (44) as being ‘‘standard.’’ However, this terminology is somewhat ambiguous since the velocity v appearing therein is often implicitly understood in the literature to be the mass velocity vm appearing in the continuity equation (1), rather than representing the speciﬁc momentum density v. (This implicit vm assumption stems from the fact that the symbol for velocity ﬁrst arises in courses in ﬂuid mechanics in the context of deriving the continuity equation. Only later is this mass-based symbol identiﬁed with the Lagrangian notion of the movement of an ‘‘object’’ through space in a trajectory sense, the latter being precursive to the association of this symbol with Newtonian dynamics, and hence its role as a momentum density.) With few exceptions [36, p. 196 (but see also p. 28), 37], the possibility that a difference might exist between the ﬂuid’s momentum density v and its mass velocity vm has not generally been recognized. In a similar vein, referring to Fourier’s law (42) as ‘‘standard’’ is equally ambiguous, since in the past it has not generally been recognized, certainly not in single-component systems [46], that a difference might exist between the diffuse internal energy ﬂux ju and the heat ﬂux jq . Nevertheless, according to Eq. (34), a difference does exist generally, except in the case of ideal gases, where Eq. (38) applies.

K ¼ a,

(47)

PHYSA : 9965 ARTICLE IN PRESS H. Brenner / Physica A ] (]]]]) ]]]–]]]

13 15 17 19 21

F

11

is the thermometric diffusivity, in which c^p is the isobaric speciﬁc heat. As heralded above, the proof to be offered of Eq. (28), wherein K is given by Eq. (47) (thus satisfying inequality (46)), derives from Burnett’s extension of Chapman and Enskog’s expansion scheme for solving the Boltzmann equation to OðKn2 Þ, namely beyond the so-called ‘‘near continuum,’’ OðKnÞ, N–S–F level. In that context, Kogan et al. [6–9] and Bobylev [10] have shown that at small Mach numbers ðMa51Þ and for Reynolds numbers of order unity ðRe ¼ Oð1ÞÞ, the so-called ‘‘thermal stress’’ terms appearing in Burnett’s [12,13] expression for the deviatoric stress T, rather than being of OðKn2 Þ as originally supposed, are, in fact, of the same OðKnÞ order as are the N–S–F equations themselves, at least in the case of so-called SNIF ﬂows (slow nonisothermal ﬂows) [9,10]. The scaling argument used to rationalize this Knudsen number re-ordering derives, in part, from the fact that Kn ¼ Ma=Re, so that a small Knudsen number can be achieved as indicated above, rather than by the classically assumed circumstances [13] where Ma ¼ Oð1Þ and Reb1. The SNIF limit [10] especially includes the limiting case where vm ¼ 0 throughout the ﬂuid, such as would be encountered, for example, in the elementary case of one-dimensional steady-state heat conduction through a gravity-free gas conﬁned between parallel, laterally unbounded, walls permanently maintained at different temperatures, where (from a macroscopic viewpoint) the pressure is sensibly uniform throughout the ﬂuid, except perhaps in thin Knudsen boundary layers existing proximate to the wall [48]. In such circumstances, the Burnett thermal stresses (to which we have referred elsewhere [3] as being the Maxwell–Burnett stresses [49]) are of the form [13, p. 286]

O

9

23 T¼ 25

37 39 41 43 45 47 49 51

D

3Z rðZrTÞ. rT

EC

Now, Z ¼ ru, where u is the kinematic viscosity. Additionally, for a single-component ideal gas, the relation between density and temperature at constant pressure is such that rT ¼ const., whence it readily follows for the present one-dimensional, steady-state, isobaric heat conduction case under consideration that T ¼ 3Zrður ln rÞ.

However, for an ideal monatomic gas, the Prandtl number Pr ¼ u=a has the value preceding relation becomes

R

35

T¼

O

33

where K 1 and K 2 are Oð1Þ nondimensional constants, whose respective numerical values depend slightly upon the particular molecular collision model adopted [13]. For monatomic Maxwellian molecules the values of the two constants appearing above are, respectively, K 1 ¼ 3 and K 2 ¼ 3 d ln Z=d ln T [13, pp. 288–289]. Accordingly, in that case Eq. (49) adopts the form

T ¼ 2Zrðar ln rÞ.

2 3

[18]. Accordingly, the (50)

C

31

(49)

In addition to the latter relation, it is also known theoretically in the case of ideal monatomic gases (which includes Maxwell molecules) that the bulk viscosity is zero [13,47]. Hence, in addition to (50) it is also true in present circumstances that

N

29

Z2 ðK 1 rrT þ K 2 T 1 rTrTÞ, rT

U

27

(48)

O

7

k r^cp

PR

5

a¼

TE

3

where

R

1

13

ZB ¼ 0.

(51)

Upon comparing the union of Eqs. (50) and (51) with that of Eqs. (8) and (44) in the light of the fact that vm ¼ 0 in present circumstances (thus making v ¼ j, and hence T ¼ 2ZrjÞ, it follows that j ¼ ar ln r.

(52)

Comparison of the latter with (28) serves to conﬁrm the viability of hypothesis (27). While we have derived Eq. (52) by considering only a rather restricted set of circumstances, namely, vm ¼ 0 (and rp ¼ 0Þ, the conclusion

PHYSA : 9965 ARTICLE IN PRESS H. Brenner / Physica A ] (]]]]) ]]]–]]]

14

9 11 13 15 17 19 21 23

F

7

O

5

O

3

that K ¼ a bears no such restriction because Eq. (28) is true independently of such restrictions. In particular, as discussed in the derivation of that equation, pressure gradients play no role in its validity. While it is true that Eq. (47) has only been conﬁrmed to hold for ideal monatomic gases composed of Maxwell molecules, on the basis of Eqs. (8) and (28) this would nevertheless appear sufﬁcient to conclude that, in general, there does exist a fundamental difference between the gas’s speciﬁc momentum density v and its mass velocity vm (provided, of course, that we accept the Boltzmann equation and the Burnett–Chapman–Enskog Knudsen number expansion scheme as being valid, at least at small Reynolds numbers, where the momentum density terms in the Navier–Stokes equations are small—a position accepted by most researchers). More importantly, the inequality vavm depends critically upon the assumption that the present ﬂuctuationbased GENERIC equations [23] provide a proper physical foundation for irreversible (gas) transport processes. As such, the experimental veriﬁcation of Eq. (52) for gases (and liquids) would serve to provide a rigorous test of the viability of O¨ttinger’s nontraditional GENERIC scheme [23]. In the next section we discuss such tests. Before doing so, however, we brieﬂy note below the existence of a second, equally unorthodox, theory of irreversible processes for single-component ideal gases due to Klimontovich [28–30]—a theory based directly upon molecular-level arguments rather than upon the present GENERIC macroscopic phenomenological arguments—and which serves to independently conﬁrm Eq. (52). Explicitly, Klimontovich argues in favor of the relation v ¼ vm þ Dr ln r, analogous to Eqs. (8) and (28), in which the symbols v and vm have the same physical signiﬁcance as described above, and in which the symbol D denotes Klimontovich’s physical-space single-component self-diffusion coefﬁcient. Furthermore, he argues that D is equal to the gas’s thermometric diffusivity a, thereby furnishing the same K ¼ a relation (47) as derived earlier from Burnett’s Boltzmann equation-based molecular calculations. In fairness, however, it needs to be emphasized that Klimontovich’s nontraditional scheme has not yet been subjected to the same searching theoretical scrutiny as either Burnett’s original, more traditional, scheme or Kogan et al.’s [6–9] and Bobylev’s [10] re-ordering of the thermal stress portion of the Burnett terms.

PR

1

25

31

Eqs. (8) and (52) or, more precisely, the similar-appearing pair, Eqs. (53) and (54), set forth below (wherein the subscript ‘‘v’’ appears), have a pre-history which is completely independent of the present GENERIC scheme. In particular, in earlier work [4] we derived the purely kinematical relation

D

29

6. Identiﬁcation of the speciﬁc momentum density v as the volume velocity vv

TE

27

vv ¼ vm þ j v ,

41 43 45

EC

R

(54)

R

The pair of relations displayed above, each derived theoretically, have been conﬁrmed indirectly by comparison of the physical consequences stemming therefrom with experimental data for both gases and liquids [2–5,14,50]—without, however, supposing the symbol vv appearing in Eq. (53) to be equal to the ﬂuid’s speciﬁc momentum density v (since inertial effects, wherein v would otherwise have proved pertinent, were negligible in all of the experiments to which Eqs. (53) and (54) pertain [51]). Given that the respective righthand sides of Eqs. (52) and (54) are identical, it follows that

O

39

jv ¼ ar ln r.

C

37

N

35

where vv is the volume velocity (which is equivalent to the Eulerian ﬂux density of volume nv —cf. Eq. (10) wherein c ¼ 1 for the case of volume) and jv is, by deﬁnition, the diffusive ﬂux density of volume. The latter is given constitutively for single-component gases or liquids undergoing heat transfer by the relation [4]

j ¼ jv .

U

33

(53)

(55)

Comparison of Eq. (8) with (53) in the light of the latter identity reveals that 47 49 51

v ¼ vv .

(56)

It has been argued [1–3,14,15] that vv is identical to the ﬂuid’s ‘‘tracer’’ or ‘‘Lagrangian’’ velocity, say vl , the latter representing the actual physical velocity of an object through space (as opposed to the ﬂuid’s mass or volume velocities, both of which are ﬂux densities in disguise [2]). Thus, the physical essence of Eq. (56) is embodied in the relation

PHYSA : 9965 ARTICLE IN PRESS H. Brenner / Physica A ] (]]]]) ]]]–]]]

1 3 5

15

v ¼ vl .

(57)

In as much as momentum also involves the notion of an object moving through space, Eq. (57) constitutes one of the key results of our analysis, namely the conclusion that the ﬂuid’s speciﬁc momentum density is equal to its Lagrangian velocity rather than to its mass velocity. And, to the extent that our hypothesis (27) is valid, it is only in the case where the ﬂuid is ‘‘incompressible’’ that the two results coincide.

7 7. Onsager-based irreversible, thermodynamic justiﬁcation of our post-GENERIC assumptions 9 7.1. Prelude

29 31 33 35 37 39 41 43 45 47 49 51

F

O

O

PR

D

27

TE

25

EC

23

R

21

R

19

O

17

C

15

N

13

The present paper has built upon O¨ttinger’s [23] incorporation of ﬂuctuations into the GENERIC friction matrix. This involved the introduction of two new parameters, a0 and D0 , into the scheme. Except for the required algebraic sign of D0 as set forth in Eq. (21), both parameters are left constitutively undetermined in O¨ttinger’s original BET development (see, however, Section 8.4). Accordingly, bringing his theory to fruition, namely to the stage where its predictions can be compared with experiment, necessitates specifying constitutive expressions for these two key phenomenological coefﬁcients in terms of physically measurable properties. It is this key step to which the present paper has largely been devoted. The critical assumptions that we made enroute to the goal of establishing plausible constitutive relations for a0 and D0 involved our identiﬁcation of the internal energy and entropy production rates, pu and ps , indicated in Eqs. (5) and (7), respectively, together with the adoption of the ‘‘incompressibility’’ hypothesis (27). These led to our constitutive determination of the parameter a0 , Eq. (33), following which subsequent determination of the remaining parameter D0 (cf. Eqs. (31) and (47)) was straightforward. Accordingly, we need focus here only on the three key hypotheses cited above, leading to our eventual determination of a0 , as set forth in Eq. (33). The justiﬁcation offered below for these constitutive assumptions is based, ultimately, upon the agreement of the predictions of the resulting theory with experiment—which is as it should be. Because the theory is, at this stage, limited to single-component ﬂuids, the conﬁrming experiments to which we refer are necessarily limited to such systems. The key experimental laws in this connection are: (i) Fourier’s law of heat conduction, Eq. (42), which refers to diffuse energy transport arising exclusively from a temperature gradient; (ii) the Clausius–Duhem inequality, Eq. (13); (iii) Onsager’s reciprocal theorem, to be discussed, involving coupling between the independent ﬂuxes appearing in the ‘‘force–ﬂux’’ relations appearing therein; and (iv) Curie’s law, which denies the possibility of coupling in linear isotropic systems between ﬂuxes whose respective tensorial orders differ by an odd integer. Together, these laws provide a test of the legitimacy of any theory of linear irreversible processes that purports to be generally applicable. Viewed alternatively, the a priori acceptance of the general correctness of LIT provides a test of the internal consistency of our theory. The critical importance of the coupling issue here resides in our claim that the diffuse ﬂux of volume jv represents an independent ﬂux, over and above the traditional diffuse ﬂuxes of internal energy ju (or heat jq Þ and momentum T. Owing to Curie’s law, the diffuse second-rank tensor momentum ﬂux is necessarily uncoupled from the diffuse vector internal energy (or heat) ﬂux. With diffuse volume accepted as representing yet another independent vector ﬂux, as is advocated here, it too would be uncoupled from the momentum ﬂux. On the other hand diffuse volume would be coupled to the internal energy ﬂux, and thus subject to the restrictions imposed by Onsager reciprocity, requiring equality of the cross coefﬁcients in the ﬂux vs. driving force reciprocal relations. In regard to coupling, the application of the Clausius–Duhem inequality to single-component ﬂuids had, in the past, where the issue of diffuse volume transport had not yet arisen, been essentially trivial. In those conventional circumstances, standard LIT arguments [19–21] resulted in the simple conclusion that 1 1 ps ¼ ju .r (58) þ T : ðrvÞT , T T

U

11

there being no Onsager coupling of the internal energy ﬂux to any other independent ﬂux. As such, as a consequence of Eqs. (42) and (44) together with the traditional assumption that ju jq ¼ krT, the inequality ps X0 was trivially satisﬁed (with v appearing therein identiﬁed as being the ﬂuid’s mass velocity vm Þ.

PHYSA : 9965 ARTICLE IN PRESS H. Brenner / Physica A ] (]]]]) ]]]–]]]

16

23 25 27 29 31 33 35 37 39 41 43 45 47 49 51

F

O

O

21

PR

19

D

17

TE

15

EC

13

R

11

R

9

O

7

C

5

N

3

However, given the present conception of an independent diffuse volume ﬂux jv , the latter dependent solely on the density gradient rr as in Eq. (54), there then arises the possibility of coupling between the vector ﬂuxes jq and jv , subject to the restrictions imposed thereon by both Onsager reciprocity and the Cauchy–Duhem inequality (13). It is the consequences stemming from these requirements that we examine below. It is in this restricted coupling context that our trio of post-GENERIC constitutive assumptions embodied in Eqs. (5), (7) and (27) receives theoretical justiﬁcation. However, the issue is complicated by the fact that there exists no unequivocal deﬁnition of the ‘‘heat ﬂux’’ in the literature. This stems from the fact that since heat is not an extensive property of a system, the notion of a heat ﬂux is essentially ambiguous. Only extensive physical properties such mass, internal energy, momentum, volume, entropy, electric charge, etc. can give rise to ﬂuxes, much less strictly diffusive (i.e., nonconvective) ﬂuxes, such as is implied by the notion of heat conduction. Stated explicitly, there exists no extensive quantity Q to which the symbol jq can be assigned as representing its (diffusive) ﬂux. This contrasts with the fact that extensive quantities like internal energy U can ^ ﬂuxes, with nm the mass ﬂux density. This ambiguity in the possess diffusive (ju Þ as well as convective (nm uÞ interpretation of the heat ﬂux permeates the literature of LIT. One might believe that the problem is at least partially alleviated by invoking Fourier’s law, Eq. (42), in the sense of the latter serving to ‘‘deﬁne’’ the heat ﬂux jq as being that portion of the diffuse (internal) energy ﬂux which vanishes when rT ¼ 0. However, that would be analogous, for example, to deﬁning the concept of a force, say F, in the Newton–Euler point-mass rigid-body law, F ¼ ma [40], by regarding a as the purely kinematically deﬁned acceleration that it indeed is, and then simply deﬁning the force as being the quantity which vanishes when a ¼ 0 (with m the proportionality coefﬁcient). The point here is that, objectively, a relation only achieves acceptance as a physical law of nature when quantitative and independent deﬁnitions of the respective variables appearing in that law (not including the phenomenological proportionality coefﬁcient) have already been set forth prior to proposing that the relation in question be elevated to the status of a bona ﬁde experimentally based law. Accordingly, returning to the Fourier law issue, the heat ﬂux jq must be deﬁned without any reference whatsoever to its possible relationship to a temperature gradient (cf. Eq. (63)), although it is permissible to use the notion of the temperature itself in its deﬁnition. The Dufour effect [19, p. 274], said to result in a ‘‘heat ﬂow’’ in an isothermal mixture undergoing diffusion, is a case in point. The notion of a heat ﬂux in the absence of a temperature gradient strains credulity, as would surely have been true of Fourier. What is almost certainly being referred to in this Dufour context is a diffuse internal energy ﬂow. It is with this lengthy preamble in mind that we now turn to the issue of justifying the trio of assumptions resulting in our constitutive expression for a0 , Eq. (33). In this context we begin by merely summarizing those formulas which have been derived on the basis of those assumptions, and then using this information to suggest a deﬁnition for the heat ﬂux jq that meets the criteria that we have speciﬁed (at least in singlecomponent systems). These criteria involve showing that Fourier’s law (42) is indeed satisﬁed by this deﬁnition of jq , as too are the laws of Onsager and Clausius–Duhem. In this context it is interesting to note that insofar as we are aware there exists no theoretical proof of Fourier’s law in the general case of continua [52] (i.e., involving both liquids and gases, the latter not necessarily rareﬁed). While one might believe that the Boltzmann equation offers the basis for such a proof, at least in the case of single-component rareﬁed gases, such a ‘‘proof’’ involves the implicit assumptions that: (i) the heat ﬂux is identical to the (diffuse) internal energy ﬂux; and (ii) there exist no other independent vector ﬂuxes, such as jv , to which the Boltzmann-based heat ﬂux vector might otherwise couple. However, as can be seen from Eq. (34), according to our present arguments, jq and ju are the same only in the special case of ideal gases; moreover, even in that case a Boltzmann-based proof cannot be accepted as complete without ﬁrst addressing the coupling issue, and subsequently showing that the Clausius–Duhem second law inequality, now possibly including such coupling, is indeed satisﬁed. (Of course, such coupling issues had not even been recognized at the time that Boltzmann and, later, Chapman and Enskog [13] did their foundational work.)

U

1

7.2. Definition of the heat flux jq As a consequence of Eq. (55), Eq. (37) adopts the form ju ¼ jq þ jv ðqU=qV ÞT;M .

(59)

PHYSA : 9965 ARTICLE IN PRESS H. Brenner / Physica A ] (]]]]) ]]]–]]]

23 25 27 29 31 33 35 37 39 41 43 45 47 49 51

F

O

ju ¼ Tjs pjv .

O

21

(61)

This result could also have been obtained directly from Eq. (16) upon invoking Eq. (55). This expression obviously constitutes the transport (i.e., ﬂux) counterpart of the thermodynamic relation (17). While Eq. (61) derives from nontraditional GENERIC theory, its validity is nevertheless intimately linked to our further constitutive assumptions, as embodied in Eqs. (5) and (7) (although not Eq. (27)), from which Eqs. (59) and (60) have evolved. Eq. (16) constitutes a powerful incentive to conclude that Eq. (55) must be correct in order to fulﬁll the analogy with Eq. (17). Eq. (59) or its equivalent entropic counterpart, Eq. (60), offers a deﬁnition of the heat ﬂux in terms of welldeﬁned physical properties. Moreover, this deﬁnition of jq is independent of any (subsequent) association with the temperature gradient, such as in Fourier’s law, Eq. (42). A fundamental point here when proposing Eq. (59) as the deﬁnition of the heat ﬂux vector (at least in single-component systems) is that, rationally speaking, only extensive physical properties can possess a ﬂux, as earlier noted. To repeat what was said there: since heat is not an extensive property, in the sense that a system cannot be said to possess an ‘‘amount’’ of heat Q, the concept of a heat ﬂux cannot be a primitive concept in any theory of irreversible processes; rather, it must be a defined quantity. And if heat ﬂux cannot then serve as a primitive concept in nonequilibrium thermodynamics, then heat itself cannot serve as a primitive concept in equilibrium thermodynamics. In effect, we are proposing here that instead of deﬁning the notion of internal energy U in terms of heat Q and work W—with W representing a well-deﬁned, strictly mechanical or electromechanical, concept (and thus able to serve as a primitive concept)—we reverse the scheme by deﬁning heat in terms of internal energy, with the latter now serving as the basic primitive notion. In effect, we define the quantity dQ arising during an inﬁnitesimal, generally irreversible, process involving a change in thermodynamic state as

PR

19

D

17

This expression differs from the classical irreversible thermodynamic result [19–21] for single-component ﬂuids, namely js ¼ jq =T. The extra nontraditional term appearing above represents an ‘‘isothermal’’ ﬂux of entropy accompanying the diffusive volume ﬂow. Note that in contrast with its corresponding energy ﬂux counterpart in Eq. (59), this unorthodox additional term does not vanish for ideal gases since, according to a Maxwell relation, ðqS=qV ÞT;M ¼ ðqp=qTÞV ;M a0 in general. Upon eliminating jq between Eqs. (59) and (60), subsequent use of Eq. (17) in the resulting expression yields the following relation upon rearrangement:

TE

15

EC

13

R

11

R

9

O

7

C

5

N

3

In a single-component isothermal system the quantity ðqU=qV ÞT;M would represent the internal energy per unit volume, namely u. Since jv is the diffuse volume ﬂux density, it follows that the term jv ðqU=qV ÞT;M appearing above represents a nonconvective internal energy ﬂux contribution that would exist in the absence of a temperature gradient (thereby arising from a density gradient). As such, this term constitutes a diffuse ‘‘isothermal’’ ﬂux of internal energy accompanying the diffuse volume ﬂow jv in a piggy-back mode. On the other hand, the Fourier heat ﬂux vector jq represents the nonconvective transport of internal energy across a surface due exclusively to a temperature gradient. As such, it is quite reasonable to suppose that the sum of the two terms appearing on the right-hand side of Eq. (59) should constitute the total nonconvective or diffuse internal energy ﬂux, ju . Issues similar to those discussed above for energy transport also arise in the comparable entropy transport case. In particular, in light of relation (55), Eq. (41) now becomes jq qS js ¼ þ jv . (60) qV T;M T

d=Q:¼dU d=W ,

U

1

17

(62)

where dU is an exact differential. The inexact differential d=W is the (generally path-dependent) work done during the process, the latter presumably obtainable via quantitative dynamical analyses embodying purely mechanical or magneto-electromechanical concepts, even for inherently irreversible processes. Of course, in the case of so-called reversible processes in single-component ﬂuids, we have that d=W rev ¼ p dV . Consistent with this philosophy of having internal energy rather than heat serve as the primitive concept in equilibrium thermodynamics, with the heat transfer d=Q then deﬁned in terms of the change dU in the latter (together with the work d=W done), on the basis of Eq. (59), we instead propose the following deﬁnition of the

PHYSA : 9965 ARTICLE IN PRESS H. Brenner / Physica A ] (]]]]) ]]]–]]]

18

23 25 27 29 31 33 35 37 39 41 43 45 47 49 51

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O

O

21

PR

19

D

17

TE

15

EC

13

R

11

R

9

Of course, this deﬁnition is understood to apply only in the case of single-component ﬂuids. Its extension to more general cases is obvious, and is brieﬂy discussed later in Section 8.6. Use of the above in conjunction with the fundamental proposition (61) and, hence, (60), all again valid only for the single-component case, will be seen to lead to a systematic approach to the subject of nonequilibrium thermodynamics which, like GENERIC theory itself, is not necessarily limited to linear processes. In effect, instead of regarding Eqs. (63) and (61) as having been derived beginning with single-component GENERIC theory (including the demonstration that v ¼ vm þ jv Þ, we propose elevating these equations to the level of fundamental postulates. Together with Eq. (42)—the latter now regarded as an exact relation valid for all temperature gradients, not necessarily small—the proposed scheme collectively serves to set forth the basic equations governing irreversible thermodynamics. It needs to be stressed, however, that we are not advocating basic changes in the fundamental structures of either GENERIC or LIT, both of whose foundations are sufﬁciently robust to, respectively, accommodate therein the existence of newly recognized physical phenomena. Rather, in the case of LIT, our proposed scheme simply entails adding volume to the present list of independent diffuse ﬂuxes appearing in that theory. Equivalently, in the case of GENERIC, following O¨ttinger [23], molecularly based ﬂuctuations are incorporated into the friction matrix appearing in that theory. In the ﬁnal analysis these are the only substantive additions advocated. After all, given that it was GENERIC that led to the precise set of equations which we have proposed, and given the consistency of GENERIC with LIT, it could not seriously be argued that any conﬂict existed between our proposal and either of these two structures. Our proposed reinterpretation of the heat ﬂux does not appear to impact on either of these basic structures, but rather only in the constitutive manner in which they are to be applied. These issues will be discussed in a broader context in subsequent papers, where, in a systematic, formal and axiomatic manner, we propose to go beyond the simple single-component systems studied here. Inasmuch as the concept of heat, and hence of heat transfer, makes its initial entre´e into the realm of thermodynamics in connection with its role in the ﬁrst law, logic would appear to demand that the deﬁnition of the heat ﬂux involve, at most, only ﬁrst law of thermodynamic concepts. This would include the internal energy U (as well as the nonthermodynamic notions of volume V and mass MÞ. Eq. (63) would appear, superﬁcially, to violate this concept since the thermodynamic absolute temperature T appears as one of the variables invoked in the partial derivative. And T is a strictly second law concept. This brings about the recognition that the ‘‘temperature’’ which enters into the deﬁnition of the heat ﬂux need not be formally identiﬁed with the symbol T. Rather, temperature may, in a broader sense, be thought of as a strictly primitive empirical concept—as indeed it was viewed during the reign of the caloric theory of heat, prior to the axiomatic work of Carnot, Joule, Kelvin and Clausius systematizing the foundations of thermodynamics. At that pre-thermodynamic time, the notion of temperature was unrelated to the formal deﬁnition of the symbol T appearing as the ‘‘integrating factor’’ in Clausius’s deﬁnition, dS:¼d=Qrev =T, of the entropy change accompanying a reversible ﬂow of heat (the latter heat ﬂow a strictly ﬁrst law concept); that is, following Fourier and others of that pre-thermodynamic era, one may regard temperature as a primitive quantity, represented, say, by the symbol y, in which case Eq. (63) would be then replaced by the expression

O

7

(63)

jq :¼ju jv ðqU=qV Þy;M ,

C

5

jq :¼ju jv ðqU=qV ÞT;M .

^ v; yÞ. Fourier’s law would where, for a single-component ﬂuid, one has the functional relationship U ¼ M uð^ then read

N

3

heat ﬂux in nonequilibrium thermodynamics:

U

1

jq ¼ kry,

rather than being given by Eq. (42). In the context of the preceding discussion, it is illuminating to read that portion of Fourier’s classical book, ‘‘The Analytical Theory of Heat,’’ concerned with attempting to explain both ‘‘temperature’’ and the ‘‘communication of heat’’ between bodies in different thermal states. Viewed alternatively, the single-component relation js ¼ jq =T appearing in textbooks (cf. [19, Eq. (20), p. 24])—which relation, incidentally, we believe to be incorrect on the basis of Eq. (60)—could not, even were it ? to be correct, be viewed in reverse as the definition, jq :¼ Tjs , of the heat ﬂux. The latter view would violate the

PHYSA : 9965 ARTICLE IN PRESS H. Brenner / Physica A ] (]]]]) ]]]–]]]

1 3 5 7 9 11 13

19

required sequential ordering of the ﬁrst and second laws. Without entering into further details, this digression sufﬁces to identify the deep philosophical issues which surround both the deﬁnition of the heat ﬂux and Fourier’s law, especially when embedded in the context of Lebowitz’s oft-repeated view [52] that no formal (statistical–mechanical) proof of Fourier’s law exists despite the almost 200 years that have elapsed since its introduction into physics. Lastly, we note that temperature, y, as a physical concept, may, by analogy with mechanics, be regarded literally as constituting a ‘‘potential’’ for the movement of (internal) energy through space—an interpretation which is consistent with its appearance as a gradient in Fourier’s law. We now resume the discussion that preceded the digression of the preceding two paragraphs. In order that the program proposed herein, involving the deﬁnition of the heat ﬂux, be deemed acceptable, it remains to show that our deﬁnition is internally consistent, in the sense of satisfying the four basic criteria outlined above. We pursue this agenda by ﬁrst identifying the driving forces conjugate to the diffusive ﬂuxes with respect to Onsager’s reciprocity law. This is achieved by expressing the entropy production rate ps in terms of the independent ﬂuxes of the pertinent extensive properties involved in the analysis. It is important to note that this identiﬁcation does not involve the concept of the heat ﬂux jq .

15

From Eqs. (12) and (19) the entropy production rate can be expressed as ps ¼ r

21

F

19

7.3. Identification of the driving forces conjugate to the diffusive fluxes

Dm s^ þ r .js . Dt

O

17

29

where we have deﬁned D¼r

45 47 49 51

pv ¼ r

Dm v^ þ r.jv . Dt

and

(68)

PR

C

(69)

The intensive form of the extensive combined ﬁrst and second laws (Eq. (17)) for single-component systems is du^ ¼ T d^s p d^v, which we rewrite as

N

43

Dm u^ þ r .ju Dt

1 p du^ þ d^v. (70) T T We assume, as is also assumed in the case of irreversible thermodynamics [19–21], that the preceding equation remains valid in the material form

U

41

pu ¼ r

R

37 39

(67)

R

35

(66)

In the following paragraph we demonstrate that D ¼ T 1 T : rv. The internal energy- and volume-production rate analogs of Eq. (64) are, respectively,

O

33

Dm s^ 1 p þ r.ju þ r.jv . T T Dt

(65)

EC

31

D

27

1 p j þ j. T u T v Accordingly, Eq. (64) adopts the form p 1 . þ jv .r þ D, ps ¼ ju r T T js ¼

TE

25

O

However, from Eq. (61), 23

(64)

d^s ¼

1 p Dm u^ þ Dm v^, T T allowing us to write Dm s^ ¼

PHYSA : 9965 ARTICLE IN PRESS H. Brenner / Physica A ] (]]]]) ]]]–]]]

20

1 3 5

Dm s^ 1 Dm u^ p Dm v^ ¼ þ . (71) T Dt T Dt Dt Adoption of this relation is tantamount to supposing the local equilibrium postulate to be valid [19–21]. Hence, with use of Eqs. (68) and (69) the above becomes Dm s^ 1 p ¼ ðpu r.ju Þ þ ðpv r.jv Þ. T T Dt Substitution of the above into (67) gives r

7 9

1 ðpu þ ppv Þ. (72) T However, in the Eulerian form (9) of the volume-transport equation, where C ¼ V , we have that c ¼ 1. Moreover, it follows from (10) and (53) that nv ¼ vv , whence we ﬁnd that [4] D¼

11 13

pv ¼ r.vv . In addition, from Eqs. (5) and (3), pu ¼ pr.v þ T : rv.

17 19

(73)

(74)

Consequently, Eq. (72) becomes

F

15

1 ½pr.ðv vv Þ þ T : rv. T Finally, then, with use of Eq. (56), we obtain

23

1 T : rv. T Substitution of the latter into Eq. (66) makes p 1 1 . ps ¼ T : rv þ ju r þ jv .r . T T T

O

21

31

41

EC

R

R

where we have noted that jM ¼ T. As a consequence of Curie’s law, the Clausius–Duhem inequality (13) applied to Eq. (77) requires separate satisfaction of each of the following inequalities:

O

39

ps ðstressesÞ:¼

43

U

and 45

1 T : rvX0 T

C

37

ps ðvector fluxesÞ Ps :¼ju .r

47

(78)

From (77), the driving forces X c conjugate to the ﬂuxes j c of linear momentum, internal energy and volume are, respectively, p 1 1 X M ¼ rv; X u ¼ r and X v ¼ r , (79) T T T

N

35

(77)

This expression for the entropy production rate possesses the classic ‘‘ﬂux/driving force’’ summation-matrix format of LIT [19–21], namely ps ¼ Sc ðj c X c Þ.

33

(76)

D

29

TE

27

PR

D¼

25

(75)

O

D¼

(80) p 1 þ jv .r X0. T T

(81)

49

7.4. Onsager reciprocity

51

The preceding identiﬁcation of the respective conjugate driving force for each independent ﬂux enables us to explicitly address the restrictions imposed by Onsager’s reciprocal theorem upon our theory. By Curie’s

PHYSA : 9965 ARTICLE IN PRESS H. Brenner / Physica A ] (]]]]) ]]]–]]]

5 7

p 1 jv ¼ Lvu r þ Lvv r . T T

9 11 13 15

This pair of relations may be written alternatively in the matrix form ! " #( ) rð1=TÞ ju Luu Luv ¼ , rðp=TÞ jv Lvu Lvv

Luv ¼ Lvu ,

(85)

as well as that of the trio of inequalities Luu X0;

21

Lvv X0

37 39 41 43

PR

D

TE

EC

Consequently, using Eq. (36), one obtains p qU 1 1 qp r r rv^. ¼ þ T qV T T T q^v T

R

35

Substitute the latter into Eqs. (82) and (83) so as to obtain qU 1 1 qp rv^ þ Luv r ju ¼ Luu Luv qV T T T q^v T and

47

qU 1 1 qp jv ¼ Lvu Lvv rv^. þ Lvv r qV T T T q^v T

U

45

(87)

R

33

we have that qp qp 1 qp 2 qp rT þ rv^ ¼ T r rv^. rp ¼ þ qT v^ q^v T qT v^ T q^v T

O

31

(86c)

It proves convenient to re-express Eqs. (82) and (83) in terms of an alternative set of driving forces, namely frð1=TÞ; rv^g, in place of the previous set, frð1=TÞ; rðp=TÞg. To do so, we note in the single-component case that p ¼ pðT; v^Þ. Thus, in the identity p rp 1 r þ pr ¼ , T T T

C

29

Luu Lvv Luv Lvu X0.

(86a,b)

(88)

N

27

O

and 23 25

(84)

wherein the square ½L matrix appearing in the preceding is both symmetric and nonnegative-deﬁnite in order that inequality (81) be satisﬁed. Thus, we require satisfaction of the equality

17 19

(83)

F

3

theorem, the diffuse momentum ﬂux is uncoupled from those of internal energy and volume. Accordingly, Onsager’s theory together with Eq. (77) requires that the following general constitutive relations apply to the respective ﬂuxes of internal energy and volume: p 1 ju ¼ Luu r , (82) þ Luv r T T

O

1

21

(89)

49

7.5. Fourier’s law

51

Introduction of Eq. (63) into (88) followed by the use of (89) furnishes the following expression for the heat ﬂux:

PHYSA : 9965 ARTICLE IN PRESS H. Brenner / Physica A ] (]]]]) ]]]–]]]

22

"

2 # qU qU 1 qU 1 qp j ¼ Luu ðLuv þ Lvu Þ þ Lvv rv^. þ Luv Lvv r qV T qV T T qV T T q^v T

1

q

3

(90)

Analogous to (27), we now wish to assign to the heat ﬂux jq the universal property that 5

23 25 27 29 31 33 35 37 39

F

1 qp rv^. T q^v T

(94)

O

21

jv ¼ Lvv

Eq. (93) is, of course, Fourier’s law, namely Eq. (42), which we repeat here: jq ¼ krT, wherein kX0 as in Eq. (43). Introduction of (95) into (93) gives 2 qU þ kT 2 . Luu ¼ Lvv qV T

O

19

and

PR

17

(93)

(95)

(96)

It follows from this that if Lvv X0, as required by Eq. (86b), then, as a consequence of (43), it will also be true that Luu X0, in accord with (86a). With use of the constitutive expressions (93) and (94) in Eq. (81) we ﬁnd that " 2 # 2 2 qU 1 1 qp Ps ¼ Luu Lvv þ Lvv ðrv^Þ2 . (97) r qV T T T q^v T

D

15

where we have also taken note of Eq. (85). Substitution of (92) into both (90) and (89) gives " " 2 # 2 # qU 1 1 qU q j ¼ Luu Lvv 2 Luu Lvv r rT qV T T qV T T

TE

13

irrespective of the values of the other independent variables, say v^ (or, equivalently, pÞ and vm , entering into the problem. In turn, this necessitates that the bracketed expression appearing in the last term of Eq. (90) vanish, thus requiring that qU Luv ¼ Lvv ¼ Lvu , (92) qV T

EC

11

(91)

As such, Eq. (81) leads to the requirement that 2 qU . Luu XLvv qV T

R

9

in isothermal systems, i:e:, when rT ¼ 0,

(98)

R

7

jq ¼ 0

47

7.6. Constitutive equation for the diffuse flux of volume

49 51

C

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From Eq. (96) it is seen that this condition is satisﬁed by the fact that the thermal conductivity is nonnegative. Given the unequivocal statement embodied in Eq. (91) it is tempting to consider the possibility that Fourier’s law, Eq. (95), may be valid under more general circumstances than would normally be expected, namely the regime beyond the small temperature gradient case that would sufﬁce to assure linearity of the ﬂux/ driving force relation explicit in Fourier’s law. The issue of possible limitations, or lack thereof, on its realm of applicability remains open as of this writing. Explicitly, we are unaware of any experimental data or theory [52] that points to any limits.

In as much as v^ ¼ 1=r, it follows that Eq. (94) is equivalent to the expression jv ¼ Lvv

1 r ln r, TkT

(99)

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where kb is Boltzmann’s constant. The angular brackets denote thermal averaging. In the above, hri ¼ M=V is the mean density, while hðdrÞ2 i is the mean-squared density ﬂuctuation occurring within an open domain of ﬁxed volume V as a result of ﬂuctuations in the instantaneous ﬂuid mass M contained within that volume, owing to the ability of individual molecules to freely enter and leave that domain through its surface. Given the nonnegativity of all of the other parameters appearing in Eq. (101) it is obvious that the theory of ﬂuctuations requires satisfaction of the inequality kT 40. This inequality is also related to the purely thermodynamic fact that stability of the ﬂuid phase requires that ðq2 A=qV 2 ÞT;M X0, wherein A is the extensive Helmholtz free energy. In view of the thermodynamic identity [54, p. 123] ðq2 A=qV 2 ÞT;M ¼ 1=V kT , it is evident that stability demands that the isothermal compressibility be positive. Fluctuations constitute a necessary ingredient when rationalizing the physical role played by O¨ttinger’s positive-deﬁnite material property coefﬁcient D0 in relation to the diffusional contribution M diff to the GENERIC friction matrix M (cf. Ref. [23, Eq. (2.77)]); that is, were the ﬂuid to be truly ‘‘incompressible,’’ in the sense that kT ¼ 0 identically, there would and could be no ﬂuctuations in density, presumably requiring that D0 ¼ 0 too. This is consistent with our basic hypothesis (27), according to which it is only in the case of incompressible ﬂuids that Euler’s speciﬁc momentum relation v ¼ vm holds. Among other things, these remarks point up the (thermodynamically) singular nature of the notion of ﬂuid incompressibility [55], a simpliﬁcation lying at the heart of most contemporary ﬂuid-mechanical applications, especially in the case of liquids. Strict incompressibility corresponds to the case where ðqr=qpÞT ¼ 0, or, equivalently, kT ¼ 0. From Eq. (101) such incompressibility rules out the possibility of ﬂuctuations. At the same time, as evidenced by Eq. (99), this leads to an obvious singularity with regard to the existence of a diffuse volume ﬂux. If nothing else, this interplay between kT and jv shows clearly the intimate relationship of ﬂuctuations to the existence of a diffuse volume ﬂux. In many practical situations this thermodynamic incompressibility singularity, whether in the case of liquids or effectively isobaric gas transport processes, is without appreciable effect on the accuracy of the ﬂuid-mechanical predictions derived from solutions of the classical Navier–Stokes and Fourier equations in this limit. In such circumstances the existence of the singularity may be ignored with impunity. On the other hand, there exist a few key novel fundamental phenomena—for example, thermophoresis [5] and thermal transpiration [50] in single-component gases, and thermal diffusion-based Soret separation phenomena in multicomponent liquid mixtures [14,15]—whereby ignoring the existence of this singularity would lead to fundamentally incorrect predictions, negating the very existence of these physical phenomena.

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(101)

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hðdrÞ2 i kb TkT , ¼ V hri2

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where the thermometric diffusivity a, Eq. (48), is a positive quantity. Consequently, the requirement that Lvv X0 demands that the algebraic sign of kT be nonnegative, as already stipulated in connection with Eq. (30). The required nonnegativity of the isothermal compressibility kT appearing in Eq. (100) is key to O¨ttinger’s [23] central argument that his proposed extension of the GENERIC friction matrix is necessitated by the existence of ﬂuctuations. The intimate relationship of kT , especially its algebraic sign, to the theory of ﬂuctuations lies in its appearance in the theory of equilibrium ﬂuctuations [54, p. 123],

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(100)

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Lvv ¼ aTkT ,

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with kT is the coefﬁcient of isothermal compressibility, deﬁned in Eq. (29). In the light of Eq. (28) the above is equivalent to the relation Lvv ¼ KTkT [53]. Given our identiﬁcation of K with a in Eq. (47), together with the subsequent argument that this relation applies equally to liquids, it follows that, in general,

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8. Discussion 47 49 51

8.1. Brownian motion as the source of the deviation of the specific momentum from Euler’s constitutive hypothesis The issue of Brownian motion bears directly upon the fundamental question of whether or not the Cauchy linear momentum equation (2) follows as an immediate consequence of Newton’s mechanics applied to mass-

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point molecular models, or whether second law of thermodynamic principles is needed in order to correctly effect the transition from discrete to continuum mechanics. In the present single-component context the phrase ‘‘Brownian motion’’ refers to the manifest consequences of multiple collisions occurring between a single tagged molecule of the ﬂuid and other molecules present therein. Explicitly, attention focuses on collisionally induced changes occurring in the momentum and position of a tagged molecule over the long term. As regards its role in transport phenomena, the emphasis here thus focuses upon the consequences of multiple collisions, rather than upon the short-term consequences of single collisions (the latter as already currently embodied in Boltzmann’s collisional integral in the particular case of rareﬁed gases). The founders of gas-kinetic theory, including Clausius, Maxwell and Boltzmann, did not themselves identify Brownian motion as a distinct macroscopic phenomenon arising from collisions of molecules [31], leaving it to Einstein [56] and Smoluchowski [57] to later do so. This failure to incorporate the phenomenon into their collisional model is, in our opinion, responsible for the presently held belief that Brownian motion is irrelevant to the formulation of the subject of Boltzmann-based gas-kinetic theory, an attitude to which neither O¨ttinger [23], Klimontovich [28–30], nor we, among others [58–60], subscribe. The collectively uniform attitude of this latter group toward the issue can be gleaned from explicit remarks made by each on the role of Brownian motion (respectively, focused on: ﬂuctuations, self-diffusion and diffuse volume transport) in singlecomponent ﬂuids during the course of translating Newton’s and Euler’s mass-point rigid-body mechanics into continuum mechanics. This includes: (i) O¨ttinger’s ‘‘Something is missing’’ ﬂuctuation addendum [23] to earlier versions [25,26] of GENERIC theory; (ii) Klimontovich’s self-diffusion physical-space add-on [28–30] to Boltzmann’s collision integral for the purpose of introducing irreversibility directly into mechanics; and (iii) Brenner’s [4] recognition of the phenomenon of diffuse volume transport—the latter, like entropy, a statistical rather than practical concept—over and above the previous, strictly convective, view that in single-component ﬂuids volume could be conveyed through space solely in the company of mass. The seeming irrelevance of Brownian motion with regard to the foundations of gas-kinetic theory is rendered transparent by the obvious fact that in current rareﬁed gas theories [13,61] Brownian statistics do not enter into the calculation of the singlet distribution function f ðx; p; tÞ (with x and p ¼ m dx=dt the respective position and momentum vectors of a mass-point molecule of mass mÞ. Explicitly, the theory of Brownian motion per se does not contribute directly to solving the Boltzmann equation, although the notion of Brownian motion is implicit in the solutions thereof. This observation, in turn, indicates that the multicollision processes, which underlie the phenomenon of Brownian motion, play no role in the basic physics quantifying the macroscopic manifestation of molecular transport phenomena. Thus, philosophically speaking, contemporary thinking argues that the notion of Brownian motion merely enriches the subject of gas-kinetic theory without impacting directly upon its foundations. It is this short-term collisional perspective which O¨ttinger, Klimontovich and we challenge (see the discussion of O¨ttinger’s recent multicollisional model in Section 8.4). The failure of Brownian motion to impact upon the mechanical foundations of statistical mechanics—with such motion viewed merely representing a completely predictable consequence thereof—should appear strange to any unbiased observer unfamiliar with the apparently mechanically reversible treatment of the collisional term in Boltzmann’s theory. That is, macroscopic experience teaches that collisions occurring among a conﬁned and isolated discrete collection of objects (‘‘molecules’’) separated by a vacuum are inherently irreversible, ultimately causing such an isolated system to eventually come to a macroscopic state of rest via ‘‘friction’’, such as certainly occurs in granular gas models [62] lacking a continuous external supply of energy (momentum). That this fate does not befall the molecules in Boltzmann’s collisional model must surely be attributed to the phenomenon of Brownian motion, which should be viewed as the root cause making possible the perpetual motion of (molecular) objects. This suggests that Brownian motion (namely self-diffusion and ﬂuctuations) should be regarded as an essential and heretofore overlooked contribution to kinetic theory, rather than simply constituting a predictable consequence thereof. This attitude with regard to the role of Brownian motion forms the basis of the implicit belief lying at the foundation of ﬂuctuation-based GENERIC theory [23] that the translation of point-mass Newtonian mechanics into continuum mechanics cannot be correctly effected without explicitly incorporating entropy and its statistical-molecular foundations into the translation scheme.

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The agreement of rareﬁed gas solutions of the Boltzmann equation with the N–S–F equations in the socalled near-continuum small Knudsen number regime, such as in the well-known perturbation solutions thereof due to Chapman and Enskog [13], is often cited as convincing evidence of the success of these perturbation schemes and, indeed, of the fundamental correctness of the Boltzmann equation itself. However, such schemes are implicitly based upon the fact that rareﬁed gases obey the ideal gas law. But, according to the theory advanced here, as embodied in the union of Eqs. (59) and (36), an ideal gas constitutes a highly singular case owing to the fact that the contribution of the diffuse volume ﬂux jv to the internal energy or heat ﬂux vanishes identically for such gases (although the corresponding diffuse volume contribution to the entropy ﬂux in Eq. (60) is not correspondingly singular for ideal gases). As such, the fact that our modiﬁed N–S–F equations might appear to be in conﬂict with Chapman–Enskog theory needs to be placed in context. This issue is already implicit in the role of the Maxwell–Burnett thermal stresses, discussed in connection with Eq. (49). In effect, it is the failure of theories of the Boltzmann equation [13] to unambiguously distinguish between the heat ﬂux jq and the diffuse internal energy ﬂux ju , especially in single-component gases, that constitutes the source of the problem. The problem is exacerbated in multicomponent gas mixtures, where, for example, Chapman and Cowling [65] refer to ‘‘. . . the ordinary [my emphasis] ﬂow of heat resulting from inequalities of temperature in the gas,’’ while concomitantly speaking of an additional heat ﬂow due to diffusion (the Dufour effect). Reciprocally, this heretofore unresolved heat ﬂux ambiguity has resulted in the failure of gas-kinetic theory to recognize the fundamental role played by the diffuse volume ﬂux in properly interpreting the hierarchical ordering of the sequential Knudsen number-based terms arising in perturbation solutions of the Boltzmann equation for rareﬁed gases. This is not to state that the Boltzmann equation itself is in error, but rather that one must rigorously avoid supposing that the heat ﬂux and diffuse internal energy ﬂux are synonymous if, at the same time, Fourier’s law, Eq. (42), is to be accepted as generally valid.

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8.2. Limitations of contemporary perturbation solutions of the Boltzmann equation for small Knudsen numbers

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In this latter context it is interesting to contrast the rather different attitude displayed toward Brownian motion when formulating the statistical foundations of quantum mechanics as a ﬁeld theory. There, Nelson [63] and others [64] have invoked fundamental ideas underlying the notion of Brownian motion in a ‘‘hiddenvariable,’’ Bohmian-like, attempt to show that quantum mechanics can be derived directly from little more than Brownian motion concepts when combined with those of Hamilton–Jacobi dynamics.

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8.3. Thermodynamic singularities in fluid mechanics 35

43 45 47 49 51

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Thermodynamics and mechanics have traditionally been regarded as essentially separate and distinct ﬁelds of inquiry, except in relation to the second law of thermodynamics, although GENERIC [23] as well as ‘‘extended thermodynamics’’ [66] among other nonequilibrium schemes represent attempts at their uniﬁcation. As regards the second law, the question of the irreversible nature of thermodynamics in contrast to the seemingly reversible nature of Newtonian dynamics has long both intrigued and confounded physicists as well as other scientists interested in the fundamentals of their disciplines, with attempts at the resolution of this seeming paradox as a favorite philosophical topic. As in the case of GENERIC [23], the present paper raises questions about whether continuum mechanics and continuum thermodynamics can be truly separated into distinct branches of physics. Resolution of the question leads, inter alia, to the surprising conclusion that the diffuse ﬂux of volume renders continuum ﬂuid dynamics a branch of irreversible thermodynamics rather than of Newtonian dynamics (a conclusion which will be evidenced more forcefully in subsequent installments in this series). The point to be made is that while Newtonian mechanics is indeed applicable to molecules, and thus subject to the laws of dynamics, the Cauchy linear momentum equation lacks a dynamical (i.e., molecular) basis owing to the presence therein of the stress tensor, a strictly continuum concept. As such, the present series of papers will advance the view that the Cauchy linear momentum equation is, in fact, an irreversible thermodynamic relation rather than a Newton’s law-based dynamical relation. While the case for this unusual perspective may not seem wholly convincing in

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the present paper, it is believed that subsequent papers in the series will bring the issue home more pointedly when we divorce the proposed diffuse volume-based addition to LIT from its present GENERIC ancestry. The fundamental role played by thermodynamics in the present analysis of the Cauchy speciﬁc momentum density v can be seen, for example, in the important role assigned in the present work to Onsager’s reciprocal relation. Prior to this, to the best of the author’s knowledge, Onsager’s theorem had never been cited as being relevant to any ﬂuid-mechanical issue of a dynamical nature, especially that pertaining to the issue of momentum. The linkage here between ﬂuid mechanics and thermodynamics is subtle, as can be seen most clearly by considering the case where the ﬂuid is isothermal. In that case, in particular in the present class of singlecomponent ﬂuids being investigated, only the respective continuity and Cauchy linear momentum equations, (1) and (2), enter the analysis. As such, only the purely ‘‘mechanical’’ variables p, r and vm would normally be expected to arise in traditional views pertaining to this isothermal case. Accordingly, lacking the need for an energy equation, whose presence would otherwise serve to couple ﬂuid mechanics to thermodynamics in an obvious way, one would not normally envision the existence of any common ground between these two ﬁelds of study in the present isothermal instance, except perhaps in the seemingly minor context of the second law of thermodynamics—the latter as embodied in the Clausius–Duhem inequality serving to demonstrate the nonnegativity of the respective shear and bulk viscosity coefﬁcients for rheologically Newtonian ﬂuids (cf. Eqs. (80) and (44)). Yet, despite this belief, these two ﬁelds remain inseparably linked in the isothermal case through the diffuse volume ﬂux jv , as given constitutively by Eq. (54). This ﬂux enters the ﬂuid-mechanical portion of the analysis through the union of the Cauchy linear momentum equation (2) with Eqs. (56) and (53). At the same time, the diffuse volume ﬂux enters the thermodynamic aspect of the analysis through its appearance in Eqs. (59)–(61). It is through this common presence, which transcends the issue of isothermality, that these two ﬁelds are permanently linked despite the ﬂuid being isothermal. The sole exception occurs in the ‘‘incompressible’’ ﬂuid case, where the uniformity of the density results in the fact that jv ¼ 0. Here, however, were the isothermal ﬂuid to be truly incompressible, the pressure would have to be uniform throughout the ﬂuid in accordance with the single-component equation of state, p ¼ pðT; rÞ, since T and r are both constant. This, however, ﬂies in the face of the fact that a pressure gradient rp normally exists in isothermal ﬂow situations involving incompressible ﬂuids. Moreover, since only the pressure gradient, rather than the pressure itself, appears in the equations of ﬂuid mechanics, incompressible ﬂuid mechanics, by itself, can establish the prevailing pressure at a point of the ﬂuid only to within an arbitrary additive, generally timedependent, function. Obviously, one is dealing here with a highly singular situation [55]. This point clearly comes to the fore in the person of the diffuse volume ﬂux, as can be seen from the role played by jv when addressing the Onsager coupling issue during the course of attempting to determine the heat ﬂux jq (even when the latter is identically zero, as in the isothermal case). The role of the diffuse ﬂux of volume, especially in relation to the precise deﬁnition of the heat ﬂux offered here—thereby contributing to the clariﬁcation of this reversibility–irreversibility paradox—has remained hidden until now. In retrospect, the reasons for this failure to recognize the existence of diffuse volume transport, much less its major unifying role, are now obvious, although these reasons are different in the respective gas and liquid cases. Though different in detail, the reason in both cases can be traced to the singular nature of the respective (perturbation) approximations normally made in the literature of gases and liquids. In the case of gases, the Boltzmann equation, with the available solutions thereof largely focused on rareﬁed gases (these obeying the ideal gas law), has, due to this focus, implicitly eliminated the need to clearly distinguish between heat ﬂow and (diffuse) energy ﬂow. This can be seen from Eq. (63) where the distinction disappears owing to the fact that the internal energy of an ideal gas is independent of its volume. As pointed out in Section 8.1, we now recognize the latter condition as a singular limit, in the sense that in such circumstances jv is no longer available to stimulate discussion of a possible distinction existing between jq and ju . Yet a profound philosophical difference exists between the two, since, as earlier stressed, ju represents the ﬂux of an extensive physical property, namely the internal energy U, whereas there exists no extensive physical property, namely heat Q, of which jq could rationally be called its ﬂux (nor does there exist a volumetric density, say cq , of heat). As such, the heat ﬂux must be regarded as a slack variable, as in Eq. (63), namely the

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27

residue remaining after subtracting from the diffuse internal energy ﬂux all other sources that entrain internal energy (e.g., diffuse volume) other than in the form of temperature (see Eq. (111) below). In the cases of liquids it is the ambiguous notion of ‘‘incompressibility’’ (and the ensuing uncertainty in regard to the notion of pressure) which is the source of the singularity. This is immediately apparent from Eq. (99), where strict incompressibility would require that both r ln r and kT be zero, resulting in a Leibnitz-like mathematical indeterminacy with regard to whether jv was, or was not, zero. This in turn reﬂects upon the latter’s role in connection with Onsager’s reciprocal theorem and, hence, upon the same heat ﬂow/diffuse internal energy ﬂow conundrum as cited above in the case of gases.

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(103)

Thus, were the parameter K to be identiﬁed as being equal to the gas’s thermometric diffusivity a, as in Eq. (47), the constitutive relation thereby obtained for the velocity difference j would have been exactly what we obtained for jv based on our fundamental hypothesis (27) (upon bearing in mind that kT ¼ M w R=rT for ideal gases). Obviously, the agreement between O¨ttinger’s Eq. (102) and our Eq. (28) is, despite their very different origins, a consequence of the fortuitous cancellation of the last two bracketed terms in Eq. (33) in the case of ideal gases. Experimental data involving thermal diffusion in liquids [14,15] support the hypothesis that Eq. (27), and hence (28), including Eq. (33), is not limited to gases, but applies equally well to liquids. However, the limited availability of critical data pertinent to the issue renders conﬁrmation of the general applicability of these relations to liquids somewhat tenuous. As yet, the possible applicability of Eq. (102) to liquids has not been tested. This owes to the absence of experimental data sufﬁcient to the task, e.g., data in which static pressure is imposed externally on a liquid undergoing steady-state heat conduction. Even were such data available, lack of independent knowledge as to the possible effect of pressure on the phenomenological coefﬁcient D0 for liquids would appear to render the interpretation of such data equivocal. As such, it is not yet possible to distinguish among the two possibilities for a0 , namely Eq. (33) vs. a0 ¼ u=r. In this context it needs to be kept in mind that were the incompressibility hypothesis (27) leading to Eq. (33) to prove wrong, the present analysis would, because it is based on the general principles of GENERIC, nevertheless remain intact in broad outline, although not in ﬁne detail. For example, were the constitutive expression for a0 to be kept open throughout the entire development, we would, more generally, in place of Eq. (63), propose the following deﬁnition for the ^ v . Thus, were it to prove true that a0 ¼ u=r, ^ heat ﬂux: jq :¼ju þ ða0 r uÞj the following would obtain for the proposed heat ﬂux deﬁnition: jq :¼ju .This is, of course, the usually assumed constitutive relation (or deﬁnition) of jq for single-component ﬂuids.

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a formula derived theoretically by O¨ttinger [67] based on a ‘‘proper cross graining’’ of the Boltzmann equation. However, since p=T ¼ rR=M w for ideal gases (with R the universal gas constant and M w the molecular weight) the above equation is seen to be constitutively identical to our Eq. (28), wherein

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The basis for the LIT analysis of Section 7 derives largely from our identiﬁcation of O¨ttinger’s GENERIC ﬂuctuation-based phenomenological coefﬁcient a0 as being expressed in terms of the physical properties of the ﬂuid by Eq. (33). This relation, in turn, arose by applying our fundamental hypothesis (27) to Eq. (20), the latter relating the velocity difference, v vm j, to a0 . Other seemingly plausible hypotheses might have led to alternative expressions for a0 . Interestingly, one of these, due to O¨ttinger [67], leads in the special but important case of ideal gases to a formula for a0 which is constitutively identical to our Eq. (28). Explicitly, were it to have been supposed in Eq. (20) that a0 ¼ u=r, this would have led to the relation

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Our analysis has focused exclusively on transport processes occurring in ﬂuids—gases as well as liquids. However, the basic precepts of LIT and GENERIC apply to matter generally, irrespective of its macroscopic physical state. As such, there is every reason to suppose that fundamental ﬂuctuation-based issues similar to those discussed above in the context of ﬂuids also arise in the case of solids, particularly solid solutions (alloys). This belief is supported by the work of Danielewski and his co-workers [68,69], who have repeatedly expressed the view, as explicitly quantiﬁed in their analyses, that the Cauchy linear momentum equation necessitates a two-velocity formulation when applied to solids undergoing interdiffusion, just as in the case of ﬂuids. Concomitantly, they point to the need to modify the classical model of strain-induced stress by adding volume transport to simple strain-based displacement when formulating constitutive equations for stresses in solids arising from diffusion (so-called diffusion-induced stresses). Explicitly, similar to the views expressed here, Danielewski et al. [68,69] argue in favor of the existence of a second fundamental velocity, a so-called ‘‘drift’’ velocity, different from the mass velocity of the solid, and appearing (together with the mass velocity) in the momentum conservation and stress constitutive equations for solids. Moreover, just as in our case, their drift velocity arises as a direct consequence of volume transport, the latter occurring in solids during the course of ‘‘atom–vacancy’’ exchanges within the solid lattice and/or via atom–atom exchanges when the diffusing species possess different molar volumes [70]. (Readers not familiar with the exhaustively detailed terminology employed in connection with atomic transport in solids may ﬁnd it useful to refer to the IUPAC-recommended publication: ‘‘Deﬁnition of terms for diffusion in the solid state’’ [71].) Experimental justiﬁcation for their nontraditional momentum transport model is based upon the widely accepted Darken [72]–Kirkendall [73] notion of diffusion-induced stress [74–76] resulting from volume transport accompanying interdiffusion in multispecies solids [77]. Being based, more or less, exclusively on macroscopic experimental phenomenology, their volume-transport stress model, due to Stephenson [74], lacks the fundamental molecular foundation that we have earlier provided for ﬂuids, at least for single-component gases, and manifesting itself in the notion of temperature-induced stresses. Nevertheless, despite being less well grounded theoretically than in the case of ﬂuids, Danielewski et al.’s [68,69] two-velocity momentum transport model for solids appears to be well-supported macroscopically. Indeed, there exists a vast body of literature concerned with the role of ‘‘volume diffusion’’ [70] in rationalizing the phenomenon of diffusion-induced stress [75,76], the latter manifested explicitly by the permanent deformation of solids noted at the conclusion of the diffusion process.

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8.6. Preview of non-simple and multicomponent fluid systems

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Eqs. (59)–(61) were derived on the basis of O¨ttinger’s version [23] of GENERIC supplemented by several key constitutive assumptions. However, once derived, this equation set is seen to collectively possess an obvious physical interpretation in its own right, independent of its GENERIC origin. As such, these fundamental equations, together with the implications that follow therefrom, stand or fall on their own respective merits. Subsequent papers in this series will build upon generalizations stemming from the foundations laid by this trio of equations. Eqs. (59)–(61) are obviously valid only for simple, single-component systems. However, given their structure and interrelationships it is not difﬁcult to speculate on how this trio of equations might be generalized so as to be applicable in more complex circumstances. Such speculations will be conﬁrmed in subsequent papers appearing in this series. These generalizations will be seen to be wholly independent of the GENERIC theory [23] that spawned them. The material which follows immediately below is designed to provide a brief preview of the scheme underlying these proposed generalizations. The previous single-component case dealt with circumstances in which, for a ﬁxed mass M, the thermodynamic state of the system could be described exclusively in terms of the extensive variable set ðU; S; V Þ. This allowed the combined ﬁrst and second laws to be expressed entirely in terms of these three variables as

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representing the extension of Eq. (63) to the present more general situation, is expected to result in the fact that jq vanishes when the temperature T is uniform throughout the ﬂuid, so that Fourier’s law, Eq. (42), will ^ ¼ C=MÞ, continue to prevail irrespective of whether the other intensive variables, namely v^ and c^ (where c vary throughout the ﬂuid. It is this fact, among many, which will be demonstrated in subsequent contributions.

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The respective generalizations of Eqs. (59)–(61) are then, obviously, qU qU ju ¼ jq þ jv þ jc , qV T;C;M qC T;V ;M

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According to the 250-year old view of Euler [33,34], the momentum velocity v appearing in the Cauchy linear momentum equation (2) is equal to the ﬂuid’s mass velocity vm , the latter being the velocity appearing in the continuity equation (1). And Cauchy’s equation is precursor to the Navier–Stokes equations (as well as the Fourier-based energy equation owing to the dissipative terms appearing therein arising from the stress tensor in Cauchy’s equation). The goal of our paper was to demonstrate on theoretical grounds, using the basic principles of LIT, especially as embodied in GENERIC [23], that Euler’s constitutive equation, v ¼ vm , is incorrect in circumstances where density gradients exist in the ﬂuid. This surprising result is implicit in the unorthodox set of statistical–mechanically based continuum-level transport equations derived for singlecomponent ideal gases by the late Klimontovich [28–30] (see also [27]), although he never explicitly

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emphasized his disagreement with existing theory. A decade later, unaware of this prior work, the present writer arrived, independently, at a set of equations [1–3] identical to those of Klimontovich by a very different route, one based empirically on philosophical arguments stemming from the recognition that volume could be transported purely diffusively in ﬂuids [4]. The resulting unorthodox Cauchy momentum equation (2), in which vavm in circumstances wherein the ﬂuid density r is nonuniform (owing primarily to the presence of temperature gradients), is implicitly supported, albeit somewhat tenuously, by purely macroscopic experiments focused on the phenomenon of thermophoresis [5,14,15], where the issue of slip boundary conditions at solid surfaces, dating back to Maxwell [11] in 1879, complicates the interpretation of the experimental results. Whereas Klimontovich’s theory of ﬂuid mechanics and heat transfer was limited in scope to single-component (ideal) gases by the nature of the assumptions made in its derivation, the present writer’s theory extended to liquids as well, including inhomogeneous ‘‘compressible’’ binary ﬂuid mixtures embodying density gradients, rra0, arising from spatial variations in composition. Focusing solely on the single-component case common to both theories, the velocity difference was predicted to be of the form v vm :¼j, in which j was equivalent to the diffuse volume ﬂux, j ¼ jv , with the latter deﬁned, physically, by the relation [4] jv ¼ nv nm =r, in which nv and nm rvm denote the respective Eulerian ﬂuxes of volume and mass through a surface element dS ﬁxed in space (through which the ﬂuid is ﬂowing). The diffuse volume ﬂux was given constitutively by the expression [4] jv ¼ ar ln r, with a ¼ k=r^cp the thermometric diffusivity, in which k is the thermal conductivity and c^p the isobaric speciﬁc heat. (In the work of Klimontovich [27–30] the velocity disparity j was not identiﬁed in physical terms as being the diffuse volume ﬂux jv , although his constitutive formula for j is the same as that for our jv .) To the extent that the Klimontovich/Brenner diffuse volume interpretation of the momentum–mass velocity difference proves to be correct, their work provides a complete theory of single-component ﬂuid mechanics and heat transfer, albeit different from the orthodox Navier–Stokes–Fourier (N–S–F) versions accepted in the literature [18–21]. The most stroking difference lies in the fact that a single velocity is no longer generally sufﬁcient to characterize the kinematics, dynamics and energetics underlying irreversible thermodynamics. Accord between these respective orthodox and unorthodox views appears to exist only for ‘‘incompressible’’ ﬂuids, r ¼ const: Euler’s proposed constitutive formula, v ¼ vm , for the speciﬁc momentum density predates, by about a century, recognition of the existence of (mobile) molecules, as well as the codiﬁcation of the ﬁrst and second laws of thermodynamics in the mid-1800s. Cauchy’s (1827) pre-molecular, pre-thermodynamic incorporation of Euler’s relation into the linear momentum equation was predicated entirely on the basis of extending Newton’s laws of discrete rigid-body mechanics to ﬂuid continua through the introduction of Cauchy’s stress tensor (3), a continuum concept appearing in place of real, externally imposed, body forces exerted collectively on the contents of a material domain. It is more or less universally believed that Boltzmann’s statistical mechanics has long since resolved any possible doubts in the matter of the constitutive equation for the momentum density v in favor of Euler’s belief that it is the velocity vm of mass. However, objectively speaking, the only extensive experimental justiﬁcation of Boltzmann’s six-dimensional kinetic transport equation lies in the apparent agreement of its small Knudsen number perturbation solutions [13] for rareﬁed gases with the three-dimensional physical-space N–S–F hydrodynamic equations. But the most accurately executed and extensive ﬂuid mechanical experiments to date in support of the conventional form of the N–S–F equations have involved incompressible and/or isothermal liquids, rather than the rareﬁed gases to which Boltzmann’s theory applies. Moreover, and perhaps equally importantly in view of Maxwell’s thermal creep slip condition [11] is the fact that virtually all experiments have involved the use of no-slip boundary conditions imposed upon vm [78], whereas the view we have advanced elsewhere [1–3] with regard to possible critical experiments [5,14,50] involving ﬂuids in which density gradients exist is that the no-slip boundary condition should be imposed, instead, upon the volume vv , the latter being identical to the (total) volume ﬂux nv . And it is only in the uninteresting case of incompressible ﬂuids that vv and vm are the same. Thus, despite the passage of many years in which the N–S–F equations appear to have stood the test of time, and despite the virtually unanimous belief that the Boltzmann equation unequivocally demonstrates the correctness of these equations (especially Euler’s view that v ¼ vm Þ, the doubts raised here and elsewhere suggest the need for a careful reappraisal of the facts. O¨ttinger, on becoming aware of these issues several years ago through reading the manuscript of a then, as yet, unpublished version of Ref. [3] by the writer, recognized that ‘‘Something was missing’’ from earlier

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(the latter derived from Eqs. (31) and (47)), with kT the isothermal compressibility, Eq. (29). By these means we explicitly demonstrated the compatibility of our equations with O¨ttinger’s ﬂuctuation model, without, however, excluding other possibilities. Indeed, as discussed in Section 8.4, O¨ttinger has, himself, on theoretical statistical–mechanical grounds, pointed to another possibility. Were that all, the paper could effectively have concluded without the appearance of Section 7. However, the exercise of establishing the indicated compatibility unearthed an important philosophical fact, one otherwise hidden in the formal, strictly mathematical, aspects of our calculations. This refers to the fundamental reason as to why seemingly minute ﬂuctuations, which would normally be expected to average-out statistically, should have such a profound effect as to set aside 250 years of unquestioned acceptance of Euler’s view? Physically, the answer to this rhetorical question lies in the fact that the ﬂuctuations of the molecules about their average positions, acting in concert with their inhomogeneous spatial distribution, create a macroscopic bias [15]. In effect, the phenomenon is a consequence of the coupling of these two attributes, namely ﬂuctuations and inhomogeneities, the latter as manifested in the molecular number density gradient (which, in single-component systems, translates into a mass density gradient). Their union underlies the inseparable coupling that exists between dynamics and thermodynamics, whose composite nature lies at the very heart of GENERIC. Indeed, the analysis of Section 7 reveals the hidden role that Onsager’s celebrated coupling theorem [79] plays in understanding why Euler’s view was not tenable once the molecular nature of mobile ﬂuid matter was recognized, since Onsager reciprocity theorem brings together the microscopic or molecular, and the macroscopic or continuum. This ﬂuctuation/inhomogeneity argument as the root cause of the breakdown of Euler’s hypothesis is essentially physical in nature, and hence intuitive and informal. In what follows, we show mathematically (and thus formally) why and how Onsager coupling undermines the possibility of there being but a single velocity in ﬂuid mechanics.

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versions of GENERIC [23], the latter but one of several competitive schemes [66] aimed, inter alia, at a rational approach to the physics of non-equilibrium thermodynamics. What he identiﬁed as being ‘‘missing’’ from the basic notions of irreversible thermodynamics was the physical manifestation of molecularly based ﬂuctuations upon the N–S–F equations, more generally of the basic equations of transport processes [18–21]. He proposed to rectify this omission by incorporating into the friction matrix appearing in GENERIC a rational model quantifying these ﬂuctuations. This led him, in effect, to pose Eq. (20) as a general constitutive relation for j based on the principles embodied in GENERIC. Two unknown parameters, D0 and a0 , appear in this expression. Establishing the values of these two parameters, such as to render O¨ttinger’s ﬂuctuation model consistent with the work of Klimontovich and the present writer, formed the heart of the present paper. In this context it must be borne in mind that the constitutive formula j ¼ jv ¼ ar ln r by which we determined D0 and a0 in the present paper has not been independently veriﬁed by others, so that outstanding issues remain. The key to reconciling O¨ttinger’s model with our formulas, namely that appearing in the preceding sentence in conjunction with the deﬁnition jv :¼vv vm of the diffusive volume current, lay in the ‘‘incompressibility hypothesis,’’ Eq. (27), the latter based on our belief that the N–S–F equations are likely to be valid for ﬂuids of uniform density (at least for single-component ﬂuids). This belief derived from a number of sources—experimental, theoretical and philosophical—as already outlined in our earlier papers [1–3], again subject to the caveat of there yet being no independent veriﬁcation of our notions. In any event, the hypothesis embodied in Eq. (27) eventually led to the respective expressions for a0 in Eq. (33) and

9.2. On the mathematical impossibility of Euler’s relation v ¼ vm being valid when viewed in the light of Onsager coupling: thermodynamics vs. mechanics

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From a strictly mathematical viewpoint the necessity for the presence of two velocities rather than one in the Cauchy linear momentum equation (2) can be regarded as formally arising from the need to increase the number of independent vector variables appearing therein in the formulation of the overall ﬂuid-mechanical problem in order to accommodate the additional restriction imposed by Onsager coupling (with the new independent variable j, denoting the difference v vm between these two velocity ﬁelds, as in Eq. (8)). This restriction arose from the presence in the set of independent LIT-based ‘‘force–ﬂux’’ Onsager relations, Eqs.

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(82) and (83), of an additional vector ﬂux, namely the diffuse volume ﬂux jv j (given constitutively by the relation jv ¼ ar ln r, with a the thermometric diffusivity), above and beyond the usual internal energy (or heat) ﬂux vector—the need for which had not been previously recognized. Were but a single velocity to have appeared in Cauchy’s equation, the restriction imposed by the Onsager coupling relation (85) would have resulted in the overall problem being over-determined owing to the resulting disparity between the number of independent variables and the number of independent relations existing among them. As a further consequence, the deﬁnition, jv :¼vv vm , of the diffuse volume ﬂux (with vv the volume velocity) then led to the constitutive relation v ¼ vv for the momentum velocity or speciﬁc momentum density. This general Onsager-based inequality vavm , impacting on Euler’s hypothesis, holds independently of the correctness of the relation j ¼ jv , whose validity has not yet been unequivocally conﬁrmed [67], as discussed in Section 8.4. What appears to be certain, however, contrary to popular belief dating back to Euler, is that the Cauchy linear momentum equation cannot be derived solely by dynamical arguments based simply upon applying Newton’s laws of motion to a ﬁxed mass of ﬂuid viewed as a continuous body (a so-called material domain) moving through space. Rather, because of ﬂuctuations in the instantaneous contents of such a domain, stemming from the molecular constitution of matter, non-equilibrium macroscopic thermodynamic principles deriving from the classical statistical–mechanical work of Onsager necessarily enters into consideration. This perspective serves to inseparably link together continuum mechanics and non-equilibrium thermodynamics in a manner that has not previously been explicitly recognized in LIT [19–21], although already implicit therein, as well as in the more broadly based structure of GENERIC [23]. In turn, this inseparable linkage of mechanics to thermodynamics, with the notion of mechanical work common to both ﬁelds, led naturally in our paper to fundamental questions about the deﬁnition of heat— questions which, in our view, had not previously been satisfactorily addressed, much less answered. Intimately related thereto was the issue of the limits of applicability, if any, to the range of temperature gradients over which Fourier’s law of heat conduction would be valid. Heretofore, the existence of deﬁnitive limits of applicability had been implied, despite the apparent lack of experimental data or rational theoretical argument indicating any such limitation [52]. These issues, which might appear to border on the strictly philosophical, will be discussed in subsequent papers aimed at attempting to extend the thermodynamic analysis of Section 7 to include inhomogeneous multicomponent ﬂuid mixtures as well as more complex single-species ﬂuids than those discussed here?.

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This work is the outgrowth of a dialog begun several years ago with Prof. Hans Christian O¨ttinger of the Department of Materials, Institute of Polymers, of the Swiss Federal Institute of Technology (ETH) in Zu¨rich, following his receipt of a preliminary draft of the work cited in Ref. [3] questioning Euler’s constitutive expression for the speciﬁc momentum. As clearly set forth in the ‘‘Something is missing’’ addendum to his recent book [23], it was he who ﬁrst recognized the commonality of our respective unorthodox views of the current status of transport processes, while implicitly encouraging me to remain ﬁrm in my beliefs when faced with the discouraging views of disbelieving referees. Our common views were recently brought jointly to fruition during a collaborative visit to his Institute in June 2005. It was H.C.O.’s insight, as recorded in his book [23], that provided the theoretical framework resulting in this paper, enabling the constitutive relation for the speciﬁc momentum density embodied in Eq. (56) to be rationalized—a relation in whose correctness I believed deeply intuitively at the outset [1] of my research on the role of diffuse volume transport [4] in ﬂuid mechanics and thermodynamics. Also sharing in the initial phases of the Clausius–Duhem/momentum density issue was my former student and collaborator, Dr. ‘‘Jim’’ Bielenberg, who was a signiﬁcant contributor to the evolution of the thinking reﬂected in the present paper on issues of volume transport and the consistency thereof with the second law.

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[1] H. Brenner, Unsolved problems in ﬂuid mechanics: on the historical misconception of ﬂuid velocity as mass motion, rather than volume motion, in: L.-S. Fan, M. Feinberg, G. Hulse, T.L. Sweeney, J.L. Zakin (Eds.), Unsolved Problems in Chemical Engineering, Proceedings of the Ohio State University, Department of Chemical Engineering Centennial Symposium, April 24–25, 2003, pp. 31–39. This reference is available on-line as hhttp://www.che.eng.ohio-state.edu/centennial/brenner.pdfi, while the lecture on which the reference is based can be seen on-line as a streaming video presentation at hhttp://www.chbmeng.ohio-state.edu/centennial/i. [2] H. Brenner, Phys. Rev. E 70 (2004) 061201 H. Brenner, Inequality of the tracer and mass velocities of physicochemically-inhomogeneous ﬂuid continua, J. Fluid Mech., 2006, submitted for publication. [3] H. Brenner, Physica A 349 (2005) 60. [4] H. Brenner, Physica A 349 (2005) 10. [5] H. Brenner, J.R. Bielenberg, Physica A 355 (2005) 251. [6] M.N. Kogan, Ann. Rev. Fluid Mech. 5 (1973) 383. [7] V.S. Galkin, M.N. Kogan, O.G. Fridlander, Izv. AN SSSR, Mekh. Zhidk. Gaza 5 (1970) 13; M.N. Kogan, V.S. Galkin, O.G. Fridlander, Sov. Phys. Usp. 19 (1976) 420. [8] M.N. Kogan, Non-Navier–Stokes gas dynamics and thermal-stress phenomena, in: Rareﬁed Gas Dynamics, vol. 15, 1986, p. 15. [9] M.N. Kogan, Some solved and unsolved problems in kinetic theory, in: A.D. Ketsdever, E.P. Muntz (Eds.), Rareﬁed Gas Dynamics, 23rd International Symposium, American Institute of Physics, 2003. [10] A.V. Bobylev, J. Stat. Phys. 80 (1995) 1063. [11] J.C. Maxwell, Phil. Trans. R. Soc. London A 170 (1879) 231 reprinted in: W.D. Niven (Ed.), The Scientiﬁc Papers of James Clerk Maxwell, vol. 2, Cambridge University Press, Cambridge, 1890, p. 681. [12] D. Burnett, Proc. London Math. Soc. 39 (1935) 385; D. Burnett, Proc. London Math. Soc. 40 (1936) 382. [13] S. Chapman, T.G. Cowling, The Mathematical Theory of Non-Uniform Gases, third ed., Cambridge University Press, Cambridge, 1970. [14] J.R. Bielenberg, H. Brenner, Physica A 356 (2005) 279. [15] H. Brenner, Phys. Rev. E 72 (2005) 061201. ^ for the quantity here denoted by v. [16] In earlier publications [2,3] we used the symbol m [17] J. Happel, H. Brenner, Low Reynolds Number Hydrodynamics, Prentice-Hall, Englewood Cliffs, NJ, 1965. [18] R.B. Bird, W.E. Stewart, E.N. Lightfoot, Transport Phenomena, second ed., Wiley, New York, 2002. [19] S.R. De Groot, P. Mazur, Non-Equilibrium Thermodynamics, North-Holland, Amsterdam, 1962. [20] R. Haase, Thermodynamics of Irreversible Processes, Dover, New York, 1990 (reprint). [21] G.D.C. Kuiken, Thermodynamics of Irreversible Processes: Applications to Diffusion and Rheology, Wiley, New York, 1994. [22] The overbar operator is deﬁned such that for any dyadic D, the dyadic D:¼ð12ÞðD þ DT Þ ð13ÞI : D represents its symmetric traceless counterpart. [23] H.C. O¨ttinger, Beyond Equilibrium Thermodynamics, Wiley, Hoboken, New Jersey, 2005 The apt phrase, ‘‘Something is missing,’’ which neatly encapsulates the momentum density problem, is taken verbatim from p. 61 of this reference. [24] ‘‘If your theory is found to be against the second law of thermodynamics, I give you no hope; there is nothing for it [your theory] but to collapse in the deepest humiliation,’’ in A.S. Eddington, The Nature of the Physical World, Macmillan, New York, 1928, p. 74; ‘‘[Thermodynamics] is the only physical theory of universal content which I am convinced that within the framework of applicability of its basic concepts will never be overthrown.’’ (A. Einstein), quoted in M.J. Klein, Science 157 (1967) 509. [25] M. Grmela, H.C. O¨ttinger, Phys. Rev E 56 (1997) 6620. [26] H.C. O¨ttinger, J. Non-Equilib. Thermodyn. 57 (1997) 386; H.C. O¨ttinger, Phys. Rev. E 57 (1998) 1416; H.C. O¨ttinger, J. Non-Equilib. Thermodyn. 27 (2002) 105. [27] H. Brenner, originally submitted to Phys. Rev. E (March 2003) under the title ‘‘A molecular basis for the Euler/Lagrange velocity disparity. The demise of the Navier–Stokes paradigm.’’ Revised version to be submitted to Physica A (2005) under the title: ‘‘On Klimontovich’s proposed modiﬁcation of the Boltzmann equation and its consequences for the Navier–Stokes–Fourier equations.’’ [28] Yu.L. Klimontovich, Theor. Math. Phys. 92 (1992) 909. [29] Yu.L. Klimontovich, Theor. Math. Phys. 96 (1993) 1035. [30] Yu.L. Klimontovich, Statistical Theory of Open Systems, Volume 1: A Uniﬁed Approach to Kinetic Descriptions of Processes in Active Systems, Kluwer Academic Publishers, Dordrecht, 1995. [31] Mazo, in the historical background to his book on Brownian motion (R.M. Mazo, Brownian Motion: Fluctuations, Dynamics and Applications, Clarendon Press, Oxford, 2002) discussing the period between Robert Brown’s observations in 1829 and Einstein’s 1905 publication, states that (p. 3): ‘‘It is striking, however, that the founders and main developers of kinetic theory, Maxwell, Boltzmann and Clausius, never published anything on Brownian motion.’’ [32] D. Straub, Alternative Mathematical Theory of Non-Equilibrium Phenomena, Academic Press, San Diego, 1997. [33] L. Euler, Me´m. Acad. Sci. Berlin 11 (1755) 274 (reproduced in Leonhardi Euleri Opera Omnia, Series II, vol. 12, p. 54 (Fu¨ssli, Zu¨rich, 1954)). Additional historical information can be found in the ‘‘Editor’s Introduction’’ to the latter volume by C. Truesdell, ’’Rational

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. [44] J.G. Kirkwood, I. Oppenheim, Chemical Thermodynamics, McGraw-Hill, New York, 1961. [45] Explicitly, the vector v is the momentum density per unit mass appearing in the total Eulerian momentum current dyadic, say nM ¼ nm v T, implicit in the Cauchy linear momentum (2) (rewritten as qðrvÞ=qt þ r.nM ¼ rpÞ, of which nm v and T are its respective convective and diffusive portions (with nm :¼rvm the Eulerian mass current). However, only the total Eulerian ﬂux of an extensive property is physically objective, and not the separate convective and diffusive portions into which it is eventually decomposed [4]. Armed with this fact, consider the particular case where the momentum density v is homogeneous, so that rv ¼ 0. In such circumstances there would no diffusive momentum ﬂux only a convective portion, despite the fact that no restriction is imposed upon vm , thus allowing the possibility that rvm a0. As such, it is evident that the velocity appearing in the constitutive equation (44) for the diffusive momentum current must, in general, represent the ﬂuid’s momentum density v rather than its mass velocity vm . [46] However, as discussed in the most recent edition of Ref. [18, cf. Table 19.2-4], such differences have recently been recognized in multicomponent ﬂuids. [47] C. Truesdell, R.G. Muncaster, Fundamentals of Maxwell’s Kinetic Theory of a Simple Monatomic Gas, Academic Press, New York, 1980. [48] D.W. Mackowski, D.H. Papadopoulos, D.E. Rosner, Phys. Fluids 11 (1999) 2106. [49] The leading term of Eq. (49) was originally derived by Maxwell [11], including the value K 1 ¼ 3 for Maxwell molecules. [50] J.R. Bielenberg, H. Brenner, A continuum model of thermal transpiration, J. Fluid Mech. 246 (2006) 1. [51] Although not relevant to the present arguments, it should be noted that Eq. (54) was originally viewed [4] as being valid only in circumstances where pressure effects on the ﬂuid’s density were small compared with temperature effects. However, according to our present GENERIC derivation, no such restriction appears to exist, either for gases or liquids. Furthermore, it should be noted that while the traditional form of the diffuse internal energy/heat ﬂux relation, namely ju ¼ jq , rather than the GENERIC form (34), was used in the original derivation [4] of Eq. (54), the two forms coincide in the case of ideal gases since ju ¼ jq ¼ krT in that case, as follows from Eq. (34). Accordingly, the derivation of (54) is consistent with that of (52), at least in the case of gases. [52] F. Bonetto, J.J. Lebowitz, L. Rey-Bellet, Fourier’s law: a challenge to theorists, in: A. Fokas, A. Grigoryan, T. Kibble, B. Zegarlinski (Eds.), Mathematical Physics 2000, Imperial College Press, London, 2000, p. 128. [53] We note upon comparing the latter relation with Eq. (31) that D0 ¼ r2 Lvv . This relation is consistent with the requirement (21) that D0 X0. [54] P.G. Debenedetti, Metastable liquids, Princeton University Press, Princeton, NJ, 1996. [55] S. Ansumali, I.V. Karlin, H.C. O¨ttinger, Phys. Rev. Lett. 94 (2005) 080602.

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ﬂuid mechanics, 1687–1765,’’ pp. VII–CXXV; L. Euler, Hist. Acad. Berlin 1755 (1757) 316. A concise history of the conceptual foundations of ﬂuid mechanics, from the time of Newton’s Principia in 1687 up to the deﬁnitive work of Stokes in 1845 [35], can be found in the following article: C. Truesdell, Amer. Math. Monthly 60 (1953) 445; O. Darrigol, Arch. Hist. Exact Sci. 56 (2002) 95; O. Darrigol, Worlds of Flow, Oxford University Press, Oxford, 2005. G.G. Stokes, Trans. Camb. Phil. Soc. 8 (1845) 287 reprinted in: Mathematical and Physical Papers, vol. 1, Cambridge University Press, Cambridge, 1901, p. 75 An account of pre-1845 work on the Navier–Stokes equations can be found in G.G. Stokes, British Assoc. Advance. Sci. (1846) 1; reprinted in Mathematical and Physical Papers, vol. 1, p. 157, Cambridge University Press, Cambridge, 1901. This refers to earlier works, as follows: C.L.M.H. Navier, Me´m. Acad. R. Sci. Paris 6 (1823) 389; S.D. Poisson, J. E´cole Polytech. Paris 13 (1831) 139; and B. Saint-Venant, C. R. Acad. Sci. Paris 17 (1843) 1240. L.D. Landau, E.M. Lifshitz, Fluid Mechanics, second ed., Butterworth-Heinemann, Oxford, 1987. P. Kosta¨dt, M. Liu, Phys. Rev. E 58 (1998) 5535. Recall that a material domain is a hypothetical continuous body of ﬂuid, each of whose surface points moves with the local mass velocity vm [39]. This fact in conjunction with the continuity equation assures that no net mass crosses the surface of such a body at any point. However, this is a strictly macroscopic statement in the sense that individual molecules are nevertheless free to cross a material surface in either direction without violating this no-ﬂux condition, provided only that in some time-average sense there are no large-scale variations in the number of molecules contained within the domain. Clearly, the material view has nothing to say about the role of such ﬂuctuations in regard to their possible impact upon Newton’s laws of motion, which apply strictly only to permanent collections of molecules. J.C. Slattery, Momentum, Energy, and Mass Transfer in Continua, McGraw Hill, New York, 1972. H. Goldstein, Classical Mechanics, Addison-Wesley, Reading, MA, 1950, p. 4; L.D. Landau, E.M. Lifshitz, Mechanics, Addison-Wesley, Reading, MA, 1960. However, neither his work nor that of Klimontovich [28–30] touches upon the volume velocity-based no-slip boundary condition issue, which plays a prominent role in our unorthodox interpretation of the thermophoretic motion of macroscopic particles [1–3,5]. E.M. Lifshitz, L.P. Pitaevskii, Statistical Physics, Part 2, Pergamon Press, Oxford, 1980, p. 369. According to the chain rule, for any function f ðx; y; zÞ ¼ 0,

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