Chapman-Enskog $\Rightarrow$ viscosity-capillarity
Author:
Marshall Slemrod
Journal:
Quart. Appl. Math. 70 (2012), 613-624
MSC (2010):
Primary 35Q53; Secondary 35B20, 35L60, 76P05
DOI:
https://doi.org/10.1090/S0033-569X-2012-01305-1
Published electronically:
May 2, 2012
MathSciNet review:
2986137
Full-text PDF Free Access
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Abstract: This paper reviews earlier work of A. Gorban and I. Karlin for the exact summation of the Chapman-Enskog expansion for the linearized Grad’s 13-moment equations. One consequence of their exact summation, not noted in their papers, is that the exact summation yields a nonlocal version of Korteweg’s theory of capillarity, which has proved to be useful as an admissibility criterion in gas dynamics.
References
- Alexander N. Gorban and Iliya V. Karlin, Structure and approximations of the Chapman-Enskog expansion for the linearized Grad equations, Transport Theory Statist. Phys. 21 (1992), no. 1-2, 101–117. MR 1149364, DOI https://doi.org/10.1080/00411459208203524
- A.N. Gorban and I.V. Karlin, Short-wave limit of hydrodynamics: a soluble model. Physical Review Letters, 77 (1996), 282-285.
- Iliya V. Karlin and Alexander N. Gorban, Hydrodynamics from Grad’s equations: what can we learn from exact solutions?, Ann. Phys. 11 (2002), no. 10-11, 783–833. MR 1957348, DOI https://doi.org/10.1002/1521-3889%28200211%2911%3A10/11%3C783%3A%3AAID-ANDP783%3E3.0.CO%3B2-V
- D.J. Korteweg, Sur la forme que prennent les équations des mouvements des fluides si l’on tient couple des forces capillaires par des variations de densité. Arch. Neerl. Sci. Exactes Nat. Ser. II 6 (1901), 1-24.
- C.A. Truesdell and W. Noll, The Non-linear Theories of Mechanics. S. Flugge, Editor, Encylopedia of Physics, Vol. III (2nd edition), Springer-Verlag, Berlin/New York (1965).
- M. Slemrod, Admissibility criteria for propagating phase boundaries in a van der Waals fluid, Arch. Rational Mech. Anal. 81 (1983), no. 4, 301–315. MR 683192, DOI https://doi.org/10.1007/BF00250857
- M. Slemrod, Dynamic phase transitions in a van der Waals fluid, J. Differential Equations 52 (1984), no. 1, 1–23. MR 737959, DOI https://doi.org/10.1016/0022-0396%2884%2990130-X
- Constantine M. Dafermos, Hyperbolic conservation laws in continuum physics, 3rd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 325, Springer-Verlag, Berlin, 2010. MR 2574377
- Philip Rosenau, Dynamics of dense lattices, Phys. Rev. B (3) 36 (1987), no. 11, 5868–5876. MR 914756, DOI https://doi.org/10.1103/PhysRevB.36.5868
- P. Rosenau, Extension of Landau-Ginzburg free energy functionals to high-gradient domains. Physical Review A 39 (1989), 6614-6617.
- Philip Rosenau, Extending hydrodynamics via the regularization of the Chapman-Enskog expansion, Phys. Rev. A (3) 40 (1989), no. 12, 7193–7196. MR 1031939, DOI https://doi.org/10.1103/PhysRevA.40.7193
- C. Cercignani, The Boltzmann equation and fluid dynamics, Handbook of mathematical fluid dynamics, Vol. I, North-Holland, Amsterdam, 2002, pp. 1–69. MR 1942464, DOI https://doi.org/10.1016/S1874-5792%2802%2980003-9
- A. V. Bobylev, Instabilities in the Chapman-Enskog expansion and hyperbolic Burnett equations, J. Stat. Phys. 124 (2006), no. 2-4, 371–399. MR 2264613, DOI https://doi.org/10.1007/s10955-005-8087-6
References
- A.N. Gorban and I.V. Karlin, Structure and approximation of the Chapman-Enskog expansion for linearized Grad equations. Soviet Physics, JETP 73 (1991), 637-641. MR 1149364 (92m:82117)
- A.N. Gorban and I.V. Karlin, Short-wave limit of hydrodynamics: a soluble model. Physical Review Letters, 77 (1996), 282-285.
- I.V. Karlin and A.N. Gorban, Hydrodynamics from Grad’s equations: What can we learn from exact solutions?. Ann. Physics (Leipzig) 11 (2002), 783-833. MR 1957348 (2004e:82050)
- D.J. Korteweg, Sur la forme que prennent les équations des mouvements des fluides si l’on tient couple des forces capillaires par des variations de densité. Arch. Neerl. Sci. Exactes Nat. Ser. II 6 (1901), 1-24.
- C.A. Truesdell and W. Noll, The Non-linear Theories of Mechanics. S. Flugge, Editor, Encylopedia of Physics, Vol. III (2nd edition), Springer-Verlag, Berlin/New York (1965).
- M. Slemrod, Admissibility criteria for propagating phase boundaries in a van der Waals fluid. Archive for Rational Mechanics and Analysis 81 (1983), 301-315. MR 683192 (84a:76030)
- M. Slemrod, Dynamic phase transitions in a van der Waals fluid. Journal of Differential Equations 52 (1984), 1-23. MR 737959 (85e:76040)
- C.M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics. Third Edition, Springer (2010). MR 2574377 (2011i:35150)
- P. Rosenau, Dynamics of dense lattices. Physical Review B 36 (1987), 5868-5876. MR 914756 (88m:82046)
- P. Rosenau, Extension of Landau-Ginzburg free energy functionals to high-gradient domains. Physical Review A 39 (1989), 6614-6617.
- P. Rosenau, Extending hydrodynamics via the regularization of the Chapman-Enskog expansion. Physical Review A 40 (1989), 7193-7196. MR 1031939 (91b:82047)
- C. Cercignani, The Boltzmann Equation and Fluid Dynamics. Handbook of Mathematical Fluid Dynamics, Vol. 1, North Holland, Amsterdam (2002), 1-69. MR 1942464 (2003k:76119)
- S. Bobylev, Instabilities in the Chapman-Enskog expansion and hyperbolic Burnett equations. J. Statistical Physics 124 (2006), 371-399. MR 2264613 (2007i:82066)
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Additional Information
Marshall Slemrod
Affiliation:
Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
MR Author ID:
163635
Email:
slemrod@math.wisc.edu
Received by editor(s):
December 13, 2011
Published electronically:
May 2, 2012
Dedicated:
Dedicated to Constantine M. Dafermos on the Occasion of his 70$^{\textrm {th}}$ Birthday
Article copyright:
© Copyright 2012
Brown University
The copyright for this article reverts to public domain 28 years after publication.
