Entropy in Fluids

Dear Colleagues,

The entropy of a system provides a measure of missing information (or randomness) in the system. The concept of entropy was introduced by Clausius in 1865 to reformulate the second law of thermodynamics in a more elegant way. In a revolutionary stroke, in 1870 Boltzmann explained how entropy can indeed be used to understand the macroscopic world via the underlying molecular dynamics.

In general, fluid flows are out of equilibrium, so it is not formally possible to ascribe the concept of entropy to fluid flows. However, one typically circumvents this issue by assuming the fluid to be locally close to equilibrium. In the same vein, fully developed turbulence (FDT) in fluids is a dissipative dynamical system with enormous strongly interacting degrees of freedom in a state of strong departure from absolute statistical equilibrium. So, equilibrium statistical mechanics is not formally applicable to FDT, and equilibrium states are not realizable in FDT. Nevertheless, as Kraichnan ingeniously pointed out in 1964, equilibrium states prove to be useful to FDT because they indicate the direction toward which the nonlinear interactions in conjunction with a selective rapid viscous decay of high-wavenumber modes drive the system and produce an energy cascade.

In dealing with the statistical properties of systems of critical points with long-range interactions, Tsallis pointed out perceptively in 1988 that it is useful to generalize the Boltzmann–Gibbs entropy to the non-extensive regime. The Tsallis entropy has been shown to provide alternate perspectives to some aspects of FDT (as well as a wide variety of problems in physics).

The entropy concept has also proved useful when applied to superfluids. In the standard model of superfluid helium II given by Landau in 1941, the superfluid below the lambda point (2.17 K) is taken to be an inviscid, irrotational fluid with thermal excitations modeled by a normal fluid moving on that underlying superfluid. This system supports pressure waves of ordinary sound in which the normal fluid and superfluid components move in phase. On the other hand, this system also supports a new compression wave of entropy called the “second sound” in which the normal fluid and superfluid components move out of phase and the overall mass density remains nearly constant.

Prof. Dr. Bhimsen Shivamoggi
Guest Editor

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