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.

Keywords:
flow instability, entropy, density, specific heat, elastic moduli, compressible flow, equations of state
Topics:
Entropy, Fluids, Internal energy (Physics), Stability, Specific heat
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Batchelor, G. K., 1967, An Introduction to Fluid Dynamics, Cambridge University Press, Cambridge.
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Bechtel
,
S. E.
,
Forest
,
M. G.
,
Rooney
,
F. J.
, and
Wang
,
Q.
,
2003
, “
Thermal Expansion Models of Viscous Fluids Based on Limits of Free Energy
,”
Phys. Fluids
,
15
, pp.
2681
–
2693
.
Copyright © 2005
 by ASME
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