Symmetric forms for hyperbolic‐parabolic systems of multi‐gradient fluids

Abstract

AbstractWe consider multi‐gradient fluids endowed with a volume internal‐energy that is a function of mass density, volume entropy and their successive gradients. We obtain a thermodynamic form of the equation of motion and an equation of energy compatible with the two laws of thermodynamics. The equations of multi‐gradient fluids belong to the class of dispersive systems. In the conservative case, we can replace the set of equations by a quasi‐linear system written in a divergence form. Near an equilibrium position, we obtain a Hermitian‐symmetric system written in the form of Godunov's systems. Equilibrium positions are stable when the total volume energy of the fluid is a convex function of the conjugated variables ‐ called the main field ‐ of mass density, volume entropy, their successive gradients and velocity.