In this note we provide proofs of the following statements for a compressible Newtonian fluid: (i) internal energy being a convex function of entropy and specific volume is equivalent to nonnegativity of both specific heat at constant volume and isothermal bulk modulus; (ii) convexity of internal energy together with the second law of thermodynamics imply linear stability of the rest state; and (iii) linear stability of the rest state together with the second law imply convexity of internal energy.
Connections Between Stability, Convexity of Internal Energy, and the Second Law for Compressible Newtonian Fluids
Contributed by the Applied Mechanics Division of THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS for publication in the ASME JOURNAL OF APPLIED MECHANICS. Manuscript received by the ASME Applied Mechanics Division, March 28, 2002, final revision, June 16, 2004. Associate Editor: K. R. Rajagopal.
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Bechtel , S. E., Rooney , F. J., and Forest , M. G. (March 15, 2005). "Connections Between Stability, Convexity of Internal Energy, and the Second Law for Compressible Newtonian Fluids." ASME. J. Appl. Mech. March 2005; 72(2): 299–300. https://doi.org/10.1115/1.1831297
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