Density-gradient dependence in the thermodynamics of fluids

A generalized thermodynamics of fluids is presented in which the density S and flux F of entropy are allowed to depend on the gradient of the mass density. Equilibrium and non-equilibrium are treated successively. The general forms of S and F are obtained to first order in the departure from equilibrium. It is shown that the non-convective part of F cannot simply be the heat flux divided by the temperature. The simplest consistent form for the constitutive equations is obtained. The junction conditions at an interface with another material are also derived.