Navier-Stokes revisited

Physica A: Statistical Mechanics and its Applications

Volumn 349, Issue 1-2, 2005, Pages 60-132

  Brenner, Howard ^a  

^a Massachusetts Institute of Technology Cambridge   (United States)

Author keywords

Korteweg stress; Navier Stokes; No slip; Rheology; Thermophoresis

Indexed keywords

BOUNDARY CONDITIONS; CONTINUUM MECHANICS; HEAT TRANSFER; KINETIC THEORY OF GASES; MATHEMATICAL MODELS; PERTURBATION TECHNIQUES; PRESSURE EFFECTS; RHEOLOGY; TENSORS; THERMOPHORESIS; VISCOSITY OF LIQUIDS; VISCOUS FLOW;

CONTINUUM FLUID MECHANICS; KORTEWEG STRESSES; NEWTON'S LAW; NO-SLIP TANGENTIAL VELOCITY;

NAVIER STOKES EQUATIONS;

EID: 12344284937     PISSN: 03784371     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.physa.2004.10.034     Document Type: Article

References (248)

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3. An account of pre-1845 work by others on the Navier-Stokes equations can be found in G.G. Stokes, Report on recent researches in hydrodynamics, British Assoc. Advance. Sci., 1846, pp. 1-20. Reprinted in Mathematical and Physical Papers, vol. 1, Cambridge University Press, Cambridge, 1901, p. 157. 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, can be found in the following articles: C. Truesdell, Am. Math. Monthly 60 (1953) 445
4. An account of pre-1845 work by others on the Navier-Stokes equations can be found in G.G. Stokes, Report on recent researches in hydrodynamics, British Assoc. Advance. Sci., 1846, pp. 1-20. Reprinted in Mathematical and Physical Papers, vol. 1, Cambridge University Press, Cambridge, 1901, p. 157. 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, can be found in the following articles: C. Truesdell, Am. Math. Monthly 60 (1953) 445
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6. Here and throughout, a question mark surmounting an equality sign serves to suggest that the stated equality is, at this point in the manuscript, an as yet unresolved issue, one which will, however, later be resolved against the equality !
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8. The volume velocity v v is, in fact, identical with the volume flux density or volume current n v [4], defined such that with d S a directed element of surface area fixed in space, the scalar d S · n v gives the volume flowing across d S per unit time. This is the analog of the fact that with d S · n m the flux of mass across d S, in which n m is the mass current, the mass velocity, v m := n m / ρ, represents the convective portion of the volume flux and j v the diffusive portion.
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20. Presumably, v m can be independently measured experimentally at a point of the fluid by some colorimetric method, involving the addition of dye to the fluid or, even better, instead of adding a foreign coloring agent (and thereby obfuscating the notion of a single-component fluid) by performing an optical experiment with a single-component fluid whose molecules are photochromic or fluorescent. These latter techniques involve so-called "molecular tagging velocimetry" (MTV) [11], as opposed to "particle-image velocimetry" (PIV) [9], which involves monitoring tracer particles, namely foreign objects deliberately introduced into the fluid.
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38. L.D. Landau, E.M. Lifshitz, Fluid Mechanics, second ed., Butterworth-Heinemann, Oxford, 1987, p. 196; P. Kostädt, and M. Liu Phys. Rev. E 58 1998 5535
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42. H. Brenner, J.R. Bielenberg, A continuum theory of phoretic phenomena: thermophoresis, Physica A (2004) submitted.
43. J.R. Bielenberg, H. Brenner, A continuum model of thermal transpiration, J. Fluid Mech., 2004, submitted.
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45. Indeed, at the hands of D.D. Joseph, his co-workers, and others (see Ref. [127] as well as the extensive references cited in Ref. [4]), Eq. (2.13) is often used in applications to 'compressible' fluids, at least in the case of isothermal binary diffusion problems, where our single-component adiabatically additive volume 'law' based on (∂ v ̂ / ∂ T) p is replaced by its better known (cf. [4]) multicomponent species additive volume 'law' counterpart based on (∂ v ̂ / ∂ w i) p, T, where w i is the mass fraction of species i. In the latter context, Eq. (2.13) is referred to as expressing a condition of "quasi-incompressibility" [27] in circumstances where ρ is not constant throughout the fluid.
46. J. Lowengrub, and L. Truskinovsky Proc. Roy. Soc. (London) A 454 1998 2617
47. This elementary equivalence is true only in circumstances where the no-penetration boundary condition (1.8) imposed upon v m at solid surfaces can be replaced by a comparable condition imposed upon v v ; for in such circumstances Eq. (1.7), in conjunction with the latter condition, then leads to the single vector velocity boundary condition, v v = 0 on ∂ V s. This no-penetration equivalency will obviously obtain in circumstances where n · (v m - v v) = 0 on ∂ V s. Equivalently, from Eq. (1.5) this necessitates that n · j v = 0 on ∂ V s. From (1.6), this latter condition will prevail whenever n · ∇ ρ = 0 on ∂ V s or, equivalently, when n · ∇ v ̂ = 0 on ∂ V s. In the present single-component case, and for the case where the law of adiabatically additive volumes prevails, this requires that n · ∇ T = 0 on ∂ V s and, hence, from Eq. (2.5) that n · q = 0 on ∂ V s. In turn, from Eq. (2.7) this is equivalent to the condition that n · j u = 0 on ∂ V s, which, because it is also true that in these same circumstances that n · j v = 0 on ∂ V s, leads to the observation that in such circumstances it is immaterial whether j u is given constitutively by the classic expression (2.4) or by its nontraditional counterpart (2.6). In summary, the complete vector velocity boundary condition, v v = 0 on ∂ V s, will obtain whenever no diffusive transport of internal energy occurs across the solid-fluid interface, corresponding to the 'insulation' boundary condition, n · j u = 0 on ∂ V s. For nonconducting cases, Eqs. (2.13) and (2.14) together with the boundary conditions (1.7) and (1.8) are indistinguishable from those governing v m in the classical creeping flow case.
48. With regard to use of the term "phoretic" forces to describe particle motion in the presence of gradients, Anderson [30] has inadvertantly sowed some degree confusion owing to his use of terms like thermophoresis and diffusiophoresis, normally reserved for gases [23-25], to describe phenomena that are actually driven by surface-gradient forces in liquids [31,32], see also [132]. The latter category is typified by Marangoni forces resulting from interfacial tension gradient ∇ s γ, caused by a surface temperature gradient ∇ s T along the particle surface, owing to the functional dependence of interfacial tension γ upon T. The resulting Marangoni surface stress causes the particle to move against the temperature gradient. However, the forces associated therewith give rise to a particle velocity U generally dependent upon the size of the particle [32], whereas in non-Brownian thermophoretic experiments [23] U is observed to be independent of particle size, ruling out Marangoni forces as possibly responsible for the observed, size-independent, thermophoretic movement.
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67. Knudsen's work on noncontinuum effects in thermal transpiration flows did not even appear until 1910 [42], whence it is unlikely that the concept of "noncontinuum" behavior would have even arisen in Maxwell's mind in 1879. [Indeed, the fact that Maxwell applied his slip condition to the strictly continuum N-S-F equations supports our belief that he regarded his so-called slip condition to be a continuum effect arising from the surface temperature gradient. In this context it is noteworthy that the adherence of the fluid to a solid surface - so widely accepted today in the case of continua, irrespective of whether or not the surface is isothermal - would, in the case of nonisothermal continua, not likely to have been regarded as sacrosanct during Maxwell's era. After all, very little data pertinent to the issue existed at that time.] Concomitantly, the standard explanation found in textbooks [43] to the effect that the thermophoretic particle motion observed in gases is "molecular" (i.e., noncontinuum) in origin, arising from more energetic particles striking the hotter side of the particle and overcoming the opposing effects of the less energetic particles on the colder side, is untenable in the continuum limit.
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74. Others [32] have suggested that "phoretic motion" in liquids may actually be due to Marangoni-like surface effects [31], wherein the surface is not "passive", as in our model, but rather interacts physicochemically with the fluid. However, as discussed in Section 8 such particle motion requires the action of body forces, which are absent as the animating mechanism underlying Eq. (3.1) for liquids and (3.4) for gases.
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76. J.R. Bielenberg, H. Brenner, A hydrodynamic/Brownian motion model of thermal diffusion in liquids, Phys. Rev. E (2004), to be submitted.
77. The absence of bulk viscosity effects in (4.1) derives from the fact that the volume velocity appearing in Eq. (1.1) is assumed to obey Eq. (2.13), a conclusion consistent with the choice of the constitutive equation (1.6) and valid, for example, in the case of ideal gases.
78. To describe these as being "the" noncontinuum terms, without including the first-order "near-continuum" O (Kn) N-S-F terms in the appellation, is surely confusing, certainly to fluid mechanicians who regard the O (Kn) N-S-F terms, and not the O (Kn 0) Euler terms, as the equations of continuum fluid mechanics; that is, owing to their apparent Knudsen number dependence, the latter classical "near-continuum" first-order N-S-F terms should, for consistency, also be classified as noncontinuum terms, despite their being regarded by fluid mechanicians as strictly continuum-level terms.
79. Even higher-order, O (Kn 3), so-called super-Burnett terms [52] exist. For a contextual evaluation of the Burnett, super-Burnett, and generally higher-order contributions to the linear momentum equation, see Ref. [53].
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84. The comparable Burnett terms for the heat flux do not impact upon whether or not Eq. (2.6) is or is not correct, since gas kinetic theory [6] draws no clear-cut distinction between the heat flux q and the diffuse internal energy current j u.
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91. Although not required for the subsequent calculations, as an aside we note that μ = (const.) T [6] for Maxwell molecules, from which it follows that K 2 = 3 for such molecules.
92. In the latter context, note that Eq. (4.8) is consistent with the fact that κ is known [6] to be identically zero for monatomic ideal gases owing to the assumed spherically symmetric nature of such molecules.
93. A.D. Kovalenko Thermoelasticity 1969 Wolters-Noordhoff Groningen; H. Parkus Thermoelasticity 1976 Springer Wien, New York; D. lesan, and A. Scalia Thermoelastic Deformations 1996 Kluwer Dordrecht; G.A. Maugin, and A. Berezovski J. Thermal Stresses 22 1999 421; N. Noda, R.B. Hetnarski, and Y. Tanigawa Thermal Streses 2002 Taylor and Francis London
94. A.D. Kovalenko Thermoelasticity 1969 Wolters-Noordhoff Groningen; H. Parkus Thermoelasticity 1976 Springer Wien, New York; D. lesan, and A. Scalia Thermoelastic Deformations 1996 Kluwer Dordrecht; G.A. Maugin, and A. Berezovski J. Thermal Stresses 22 1999 421; N. Noda, R.B. Hetnarski, and Y. Tanigawa Thermal Streses 2002 Taylor and Francis London
95. A.D. Kovalenko Thermoelasticity 1969 Wolters-Noordhoff Groningen; H. Parkus Thermoelasticity 1976 Springer Wien, New York; D. lesan, and A. Scalia Thermoelastic Deformations 1996 Kluwer Dordrecht; G.A. Maugin, and A. Berezovski J. Thermal Stresses 22 1999 421; N. Noda, R.B. Hetnarski, and Y. Tanigawa Thermal Streses 2002 Taylor and Francis London
96. A.D. Kovalenko Thermoelasticity 1969 Wolters-Noordhoff Groningen; H. Parkus Thermoelasticity 1976 Springer Wien, New York; D. lesan, and A. Scalia Thermoelastic Deformations 1996 Kluwer Dordrecht; G.A. Maugin, and A. Berezovski J. Thermal Stresses 22 1999 421; N. Noda, R.B. Hetnarski, and Y. Tanigawa Thermal Streses 2002 Taylor and Francis London
97. A.D. Kovalenko Thermoelasticity 1969 Wolters-Noordhoff Groningen; H. Parkus Thermoelasticity 1976 Springer Wien, New York; D. lesan, and A. Scalia Thermoelastic Deformations 1996 Kluwer Dordrecht; G.A. Maugin, and A. Berezovski J. Thermal Stresses 22 1999 421; N. Noda, R.B. Hetnarski, and Y. Tanigawa Thermal Streses 2002 Taylor and Francis London
98. Indeed, in the case of solids, the notion of a "noncontinuum solid" does not even appear to exist, except perhaps in the case of granular materials, although fractures and dislocations, representing isolated singularities, may exist within the solid.
99. D.J. Korteweg Arch. Néerl. Sci. Exactes Naturelles II 6 1901 1
100. C. Cercignani Mathematical Methods in Kinetic Theory second ed. 1990 Plenum Press New York
101. H.A. Kramers, and J. Kistemaker Physica 10 1943 699
102. D.A. Noever Phys. Fluids A 2 1990 858; D.A. Noever Phys. Lett. A 144 1990 253; D.A. Noever Phys. Rev. Lett. 65 1990 1587; D.A. Noever Phys. Rev. A 45 1992 7302
103. D.A. Noever Phys. Fluids A 2 1990 858; D.A. Noever Phys. Lett. A 144 1990 253; D.A. Noever Phys. Rev. Lett. 65 1990 1587; D.A. Noever Phys. Rev. A 45 1992 7302
104. D.A. Noever Phys. Fluids A 2 1990 858; D.A. Noever Phys. Lett. A 144 1990 253; D.A. Noever Phys. Rev. Lett. 65 1990 1587; D.A. Noever Phys. Rev. A 45 1992 7302
105. D.A. Noever Phys. Fluids A 2 1990 858; D.A. Noever Phys. Lett. A 144 1990 253; D.A. Noever Phys. Rev. Lett. 65 1990 1587; D.A. Noever Phys. Rev. A 45 1992 7302
106. W. Crookes Philos. Trans. Roy. Soc. (London) 166 1876 325
107. A detailed and historical discussion of attempts to explain the principles underlying the windmill-like rotation undergone by the rotor in Crookes's radiometer based upon noncontinuum concepts is given in Ref. [44].
108. J.C. Maxwell Philos. Mag. 19 1860 19; J.C. Maxwell Philos. Mag. 20 1860 21
109. J.C. Maxwell Philos. Mag. 19 1860 19; J.C. Maxwell Philos. Mag. 20 1860 21
110. Some of the historical context, chronology, and acrimony in the matter of priority surrounding the competition between Maxwell and Osborne Reynolds [71] to use their respective thermal transpiration models to explain the physical mechanism underlying the working of Crookes's radiometer [67] can be found in the biography by I. Tolstoy, James Clerk Maxwell, University of Chicago Press, Chicago, 1981, pp. 150-151, 166-167; see also Ref. [44].
111. O. Reynolds, Proc. Roy. Soc. London 38 (1879-1880) 300. This paper is only a preliminary abstract of the lengthier paper published some time afterwards as O. Reynolds, Philos. Trans. Roy. Soc. (London) 170 (1879) 727.
112. O. Reynolds, Proc. Roy. Soc. London 38 (1879-1880) 300. This paper is only a preliminary abstract of the lengthier paper published some time afterwards as O. Reynolds, Philos. Trans. Roy. Soc. (London) 170 (1879) 727.
113. The importance of understanding the mechanism behind Crookes's radiometer [67] played a vital, and under-appreciated, role in the history of gas-kinetic theory, in particular in regard to the boundary conditions to be applied to the Boltzmann equation at solid surfaces. After all, an important part of the verification of the validity of the Boltzmann equation necessarily lies in the agreement of its predictions with experiment, for which circumstances the solution of boundary-value problems (either imposed upon the Boltzmann equation itself or upon the coarser-scale transport equations derived therefrom, such as the N-S-F equations) plays a pre-eminent role.
114. For a modem version of the slip boundary condition involving G for gases, see F. Sharpov, D. Kalempa, Phys. Fluids 15 (2003) 1800.
115. Given the interpretation of Maxwell slip as a noncontinuum O (Kn 2) effect owing to its origin in connection with the Burnett terms, Epstein [35] and those who followed should, for mathematical consistency as regards the hierarchical ordering of the Knudsen number terms appearing in their transport equations, have then solved the corresponding noncontinuum O (Kn 2) -level transport equations, rather than the near-continuum O (Kn) N-S-F equations. At a minimum, this would have resulted in adding the Maxwell thermal stress term (5.3) to the O (Kn) viscous Newtonian term (4.2) appearing in the momentum equation. Additionally, because the gas is "compressible" owing to its density varying with temperature, the continuity equation used by Epstein, namely ∇ · v m = 0, is valid only to O (Kn). At O (Kn 2) another term should have appeared in his continuity equation in order that the latter be correct. However, as discussed in Appendix C, owing to a fortuitous combination of circumstances in the present class of phoretic thermal problems [23,24], these additions do not affect the calculation of U.
116. As discussed in connection with Eqs. (5.4) and (5.5), the notion of noncontinuum slip is associated with the parameter G appearing therein, rather than with the last term of Eq. (5.4), which alone governs Maxwell's "slip coefficient", C s. In the literature [73], G is associated with the notion of "velocity slip", a truly noncontinuum phenomenon occurring even in isothermal fluids.
117. L. Euler, Mém. Acad. Sci. Berlin 11 (1755) 274. Reproduced in: Leonhardi Euleri Opera Omnia. Series II, vol. 12, Füssli, Zürich, 1954, p. 54. Additional historical information can be found in the "Editor's Introduction" to the latter volume by C. Truesdell, Rational fluid mechanics, 1687-1765, pp. VII-CXXV; see also L. Euler, Hist. Acad. Berlin 1755 (1757) 316-361.
118. L. Euler, Mém. Acad. Sci. Berlin 11 (1755) 274. Reproduced in: Leonhardi Euleri Opera Omnia. Series II, vol. 12, Füssli, Zürich, 1954, p. 54. Additional historical information can be found in the "Editor's Introduction" to the latter volume by C. Truesdell, Rational fluid mechanics, 1687-1765, pp. VII-CXXV; see also L. Euler, Hist. Acad. Berlin 1755 (1757) 316-361.
119. L. Euler, Mém. Acad. Sci. Berlin 11 (1755) 274. Reproduced in: Leonhardi Euleri Opera Omnia. Series II, vol. 12, Füssli, Zürich, 1954, p. 54. Additional historical information can be found in the "Editor's Introduction" to the latter volume by C. Truesdell, Rational fluid mechanics, 1687-1765, pp. VII-CXXV; see also L. Euler, Hist. Acad. Berlin 1755 (1757) 316-361.
120. L. Euler, Mém. Acad. Sci. Berlin 11 (1755) 274. Reproduced in: Leonhardi Euleri Opera Omnia. Series II, vol. 12, Füssli, Zürich, 1954, p. 54. Additional historical information can be found in the "Editor's Introduction" to the latter volume by C. Truesdell, Rational fluid mechanics, 1687-1765, pp. VII-CXXV; see also L. Euler, Hist. Acad. Berlin 1755 (1757) 316-361.
121. G.K. Batchelor An Introduction to Fluid Dynamics 1967 Cambridge University Press Cambridge
122. By "small" is meant the following: If a is the maximum linear dimension of the particle, it is required that a || ∇ v l || / | v l | ≪ 1, with the modulus bars denoting appropriate norms.
123. C.C. Truesdell, and R.A. Toupin The classical field theories S. Flügge Handbuch der Physik, vol. IIII/1, Principles of Classical Mechanics and Field Theory 1960 Springer Berlin 226; C. Truesdell, and W. Noll The Nonlinear Field Theories of Mechanics S. Flügge Handbuch der Physik vol. III/3 1965 Springer Berlin; W. Noll, R.A. Toupin, and C.C. Wang Continuum Theory of Inhomogeneities in Simple Bodies 1968 Springer Berlin
124. C.C. Truesdell, and R.A. Toupin The classical field theories S. Flügge Handbuch der Physik, vol. IIII/1, Principles of Classical Mechanics and Field Theory 1960 Springer Berlin 226; C. Truesdell, and W. Noll The Nonlinear Field Theories of Mechanics S. Flügge Handbuch der Physik vol. III/3 1965 Springer Berlin; W. Noll, R.A. Toupin, and C.C. Wang Continuum Theory of Inhomogeneities in Simple Bodies 1968 Springer Berlin
125. C.C. Truesdell, and R.A. Toupin The classical field theories S. Flügge Handbuch der Physik, vol. IIII/1, Principles of Classical Mechanics and Field Theory 1960 Springer Berlin 226; C. Truesdell, and W. Noll The Nonlinear Field Theories of Mechanics S. Flügge Handbuch der Physik vol. III/3 1965 Springer Berlin; W. Noll, R.A. Toupin, and C.C. Wang Continuum Theory of Inhomogeneities in Simple Bodies 1968 Springer Berlin
126. Of course, in the case of unsteady flows, the necessity of performing repetitive experiments with different size particles, all at the same instant of time, would, no doubt, pose a daunting challenge to the experimentalist!
127. A perhaps equally remarkable fact about Eq. (3.2), applicable to gases, is that it reveals a totally counter-intuitive fluid-mechanical phenomenon - namely, the larger the viscosity of the gas the faster does the particle move! This fact alone signals the extraordinarily unique nature of thermophoretic motion, since viscosity generally retards rather than enhances relative particle motion through fluids, a fact well known to every low Reynolds number fluid mechanician [82].
128. J. Happel, and H. Brenner Low Reynolds Number Hydrodynamics 1965 Prentice-Hall Englewood Cliffs, NJ
129. P. Goldsmith, and F.G. May Diffusiophoresis and thermophoresis in water vapour systems C.N. Davies Aerosol Science 1966 Academic Press London 163 (see also Ref. [38])
130. In order for an investigator be able to objectively identify his velocity measurements as representative of those of the fluid itself, and not an artifact of the properties of the tracer particle, he needs to assure himself that his experimental tracer particles do not possess any physical attributes that, in the zero-size limit, would distinguish the particle's velocity from that of the fluid itself. It was in order to fulfill this requirement of "passivity" that only (effectively) thermally insulated thermophoretic spheres were selected by us in order to identify the velocity v l of the undisturbed fluid. As revealed by Eq. (3.1), thermophoretically animated spheres possessing a nonzero k s / k ratio move with a velocity that depends significantly upon the magnitude of this conductivity ratio, even in the limit of effectively zero size. As such, (effectively) noninsulated particles may not serve as fluid velocity tracers. It is only to this extent that the experimental fluid mechanician, in deciding upon the choice of appropriate tracer particles with which to conduct his velocity experiments, would have to contemplate the possible complicating effects of temperature gradients. Even were he insufficiently insightful to recognize a priori the need for insulated particles, were he to next perform a sequence of replicate size-varying experiments using a series of particles possessing different thermal conductivities (just as he might do with a series of particles of different densities, so as to assure himself of their zero-size "passivity"), he would presumably soon come to recognize that all low conductivity particles yielded identical extrapolated zero-size velocities. Accordingly, he would presumably then reject all zero-size particle data obtained with his high conductivity particles as failing to fulfill the requirement of "passivity" (even were he unable to identify thermal conductivity as the source of the observed differences in the zero-size velocity measurements).
131. Even were external forces such as gravity to act on the fluid, enabling the particle to sediment relative to the surrounding fluid if its density differed from that of the fluid, such relative motion would vanish in the pointsize tracer-particle limit, thereby having no effect upon the ability of the tracer particle to monitor the fluid velocity that exists in its absence.
132. By the phrase "gas-kinetic molecular interpretation" is meant that the property cannot be derived directly simply by summing each of the three elemental extensive properties of the individual molecules in some small domain of volume V (namely the mass m, kinetic energy mc 2 / 2, and momentum m c of the molecules, with c the molecular velocity) and subsequently dividing by the volume of that domain in order to obtain the corresponding intensive volumetric pointwise mass, kinetic energy, and momentum densities at a point of the continuum.
133. The reason for separating these two items stems from the fact (noted in connection with Table 2 appearing in Appendix C) that it is possible under certain well-defined circumstances for the traditional and modified N-S-F equation set to fortuitously yield identical results, both of which accord with experiment, albeit on the proviso that the correct velocity boundary condition be used (either that of no-slip imposed upon v l or the equivalent Maxwell slip condition imposed upon v m).
134. The isothermal assumption is needed in order to avoid complications associated with thermal diffusion species fluxes, while the isobaric assumption is similarly required to avoid pressure diffusion contributions to the species flux density j i [18].
135. This has the effect of enabling the right-hand side of (7.1) [and, equivalently, that of Eq. (1.6) for the isothermal, isobaric, binary diffusion case] to be re-written in the form j v = D ∇ ln ρ = - D * ∇ v ̂ = - D * (∂ v ̂ / ∂ w 1) p, T ∇ w 1 = (∂ v ̂ / ∂ w 1) p, T j 1.
136. We use the word "semi-empirical" here because there does not appear to exist in the literature a theoretical proof of the concentration-slip boundary condition, derived along the lines laid out by Maxwell [8] in the thermal gradient case, wherein the concentration analog of Eq. (5.4) is derived from the analog of the Maxwell-Burnett thermal stress term (5.3). Rather, owing to this lack, Kramers and Kistemaker [65] adopted their widely-used concentration-slip velocity condition on a different basis, namely a molar rather than mass basis. Explicitly, we are not aware of the existence in the literature of the Burnett extra stress concentration analog of Eq. (4.5), although if our theory is correct it should be given by Eq. (4.6), in which j v = D ∇ ln ρ (see Ref. [89]). According to our theory, the generic no-slip boundary condition should be given by Eq. (5.8), where I s · j v = D ∇ s ln ρ in the present binary mixture case.
137. Reynolds' [71] experiments were actually performed with porous plugs rather than with well-defined capillary tubes
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140. S.E. Vargo, E.P. Muntz, G.R. Shiflett, and W.C. Tang J. Vac. Sci. Technol. A 17 1999 2308; J.P. Hobson, and D.B. Salzman J. Vac. Sci. Technol. A 18 2000 1758; F. Ochoa, C. Eastwood, P.D. Ronney, and B. Dunn Thermal transpiration based microscale propulsion and power generation devices, seventh Intl. Microgravity Combustion Workshop June 2003 Cleveland OH; Y. Sone, and K. Sato Phys. Fluids 12 2000 1864
141. S.E. Vargo, E.P. Muntz, G.R. Shiflett, and W.C. Tang J. Vac. Sci. Technol. A 17 1999 2308; J.P. Hobson, and D.B. Salzman J. Vac. Sci. Technol. A 18 2000 1758; F. Ochoa, C. Eastwood, P.D. Ronney, and B. Dunn Thermal transpiration based microscale propulsion and power generation devices, seventh Intl. Microgravity Combustion Workshop June 2003 Cleveland OH; Y. Sone, and K. Sato Phys. Fluids 12 2000 1864
142. In regard to these experiments, note that according to Eq. (3.3) the slip coefficient is different for monatomic and diatomic gases.
143. H. Brenner, Molecular, Brownian motion confirmation of the tracer/mass velocity disparity, Phys. Rev. E (under revision; originally submitted under the title "Molecular, Brownian motion confirmation of the Euler/Lagrange velocity disparity").
144. E. Nelson Phys. Rev. 150 1966 1079; E. Nelson, Dynamical Theories of Brownian Motion, Princeton Univ. Press, Princeton, 1967; E. Nelson, Connection between Brownian motion and quantum mechanics, in: H. Nelkowski, A. Hermann, H. Poser, R. Schrader, R. Seiler (Eds.), Einstein Symposium, Berlin, Lecture Notes in Physics, vol. 100, Springer, Berlin, 1979, p. 168.
145. E. Nelson Phys. Rev. 150 1966 1079; E. Nelson, Dynamical Theories of Brownian Motion, Princeton Univ. Press, Princeton, 1967; E. Nelson, Connection between Brownian motion and quantum mechanics, in: H. Nelkowski, A. Hermann, H. Poser, R. Schrader, R. Seiler (Eds.), Einstein Symposium, Berlin, Lecture Notes in Physics, vol. 100, Springer, Berlin, 1979, p. 168.
146. E. Nelson Phys. Rev. 150 1966 1079; E. Nelson, Dynamical Theories of Brownian Motion, Princeton Univ. Press, Princeton, 1967; E. Nelson, Connection between Brownian motion and quantum mechanics, in: H. Nelkowski, A. Hermann, H. Poser, R. Schrader, R. Seiler (Eds.), Einstein Symposium, Berlin, Lecture Notes in Physics, vol. 100, Springer, Berlin, 1979, p. 168.
147. K. Itô, Stochastic Calculus, Springer Lecture Notes in Physics, vol. 39, Springer, New York, 1975, p. 218.
148. In probabilistic terminology, v l constitutes Nelson's "drift velocity" (more precisely, his "forward" drift velocity, since a hypothetical mathematically-defined "backwards" drift velocity also appears in Nelson's theory, a fact that need not concern us here). In his notation, Nelson's symbol v [no relation to our v in either Eqs. (1.3) and (1.4)] is equivalent to our mass velocity v m, as is apparent from its appearance in Nelson's continuity equation, analogous to our Eq. (1.2).
149. In our interpretation of Nelson's work, D might better be termed the fluid's "self-diffusion" coefficient, since his analysis appears limited to single-component fluids within which mass density gradients exist; that is, the symbol D appearing in (7.3) is regarded as being a "self-diffusion" coefficient, intrinsic to the single-component fluid itself, rather than arising from the presence a foreign object, namely a colloidal particle, present in the fluid. In this sense, D should be regarded as an isotropic correlation coefficient, I D = (1 / 2) 〈 Δ x Δ x 〉 / Δ t, in which the position vector x = x (x 0, t) represents the statistical location at time t of a "fluid particle" that at time t = 0 was located at the position x 0. The phrase "fluid particle" here refers not to a "material particle" (which is an extensive entity) but rather to a fluid particle (an intensive entity) in the sense implicitly understood in connection with Eq. (1.9), where the tracer fluid field, v l (x 0, t), is regarding as describing the (mean) tracer motion, the so-called "forward motion," of such a hypothetical fluid particle. Mathematically, the symbol D is that appearing in the Markoff process stochastic relation [99] d x (t) = v l [ x (t), t ] d t + 2 D d w (t), in which d x (t) ≡ x - x 0, with d w (t) a normalized Wiener process [100].
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172. The essentially kinematic analyses of Einstein and Smoluchowski preceded the later dynamical theories of Brownian motion phenomena, such as those due to Langevin, Ornstein-Uhlenbeck, Kramers, etc. (cf. Ref. [110]).
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175. Youri L'vovich Klimontovich, late Professor Emeritus in the Physics Department at Moscow State University, died of cancer on November 27, 2002, approximately three months before I learned of his contribution to the subject under discussion.
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179. It is interesting to note that an identical term appears in the well-known book of de Groot and Mazur [13], but only in the context of a class of applications involving what they term "discontinuous" systems [cf. Eqs. (69) and (72) of their Chapter XV]. Indeed, they explicitly identify the term j v appearing in our subsequent Eq. (7.4), which they term the "volume flow".
180. L. Onsager, Ann. Trans. NY Acad. Sci. 46 (1945) 241. It is interesting to note in the present context that Onsager comments as follows when discussing problems of "pure" multicomponent diffusion in liquids, involving what appears to us to be thermodynamically ideal solutions: "...provided only that the volume change due to mixing may be neglected, it is possible to arrange matters such that v = 0 everywhere" (where Onsager's "hydrodynamic" velocity, v, is understood by us to be the volume velocity). Moreover, he goes on later to further state that: "Viscous flow is a relative motion of adjacent portions of a liquid. Diffusion is a relative motion of its different constituents. Strictly speaking, the two are inseparable; for the "hydrodynamic" velocity in a diffusing mixture is merely an average determined by some arbitrary convention".
181. As pointed out by Haase [15, p. 221] and others [116], experimentalists who measure molecular diffusivities usually choose a volume- rather than mass-based reference frame (so as to avoid having to explicitly address what Haase terms "convective velocities", namely nonzero mass-average velocities, v m). In this frame of reference it is supposed: (i) that the volume velocity vanishes everywhere, v v = 0 (corresponding here to v l = 0), despite the fact that v m ≠ 0 ; and (ii) that the diffusional process is unidirectional, However, to the best of our knowledge, it appears never to have been pointed in this connection that the assumption of requiring that v v = 0 everywhere, including on the boundary, is incompatible with the traditional no-slip tangential boundary condition, I s · (v m - U) = 0, imposed upon v m [117]. Equally, the Kramers-Kistemaker [65] species concentration boundary condition, analogous to (5.2), would also be violated unless it was true that, when expressed in appropriate binary diffusion terminology, v v = v m - C s υ ∇ s ln T ; for, in that case, Eq. (3.5) becomes identical with Eq. (5.1), corresponding to no slip of the volume velocity (and hence of the Lagrangian velocity v l).
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215. In the case of gases, kinetic theory [18] suggests that a better assumption would be D v = D v * / ρ, where D v * is a constant, independent of density. This would result in different expressions for the four Korteweg coefficients than those given in Eq. (8.2).
216. In the interest of greater generality we could have added a body force per unit volume, say f, to the right-hand side of (8.3). However, we have refrained from doing so in order to clarify subsequent arguments regarding other classes of phoretic particle motion [30], which, in contrast with the thrust of our work involving circumstances where f = 0, are driven by nonzero body forces.
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243. In a generic context, the diffuse current j ψ, of some extensive property Ψ is defined as the flux density of the property over and above the corresponding convective contribution n m ψ ̂ thereto carried by the mass current n m = ρ v m. Stated more explicitly, the total current n ψ of the extensive property under discussion in a Eulerian space-fixed reference frame is regarded as being of the form n ψ = n m ψ ̂ + j ψ, with ψ ̂ is the amount of the property per unit mass, i.e., the specific density of the property Ψ. The latter density appears in the generic Eulerian transport equation ∂ ψ / ∂ t + ∇ · n ψ = π ψ in which ψ = ρ ψ ̂ and π ψ are, respectively, the amount of the property and temporal rate of production of the property, both on a per unit volume basis. This generic Eulerian transport is formally equivalent to the generic material derivative form, ρ D m ψ ̂ / D t + ∇ · j ψ = π ψ.
244. The latter fact shows, for example, why the thermomolecular pressure difference in the thermal transpiration problem [24] is correctly given by Maxwell's scheme despite the fact that Maxwell's transport equations are inappropriate.
245. Note that in terms of the fundamental "Newtonian" stress issue (1.1), the "extra" deviatoric stress [cf. (4.2)], T γ + = 2 μ ∇ j γ, makes no contribution to the present problems, just as was true in Appendix C, owing to the fact that since ∇ 2 j γ = 0, it follows that ∇ · T γ + = 0.
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