Symmetric form for the hyperbolic-parabolic system of fourth-gradient fluid model

Abstract

The fourth-gradient model for fluids—associated with an extended molecular mean-field theory of capillarity—is considered. By producing fluctuations of density near the critical point like in computational molecular dynamics, the model is more realistic and richer than van der Waals’ one and other models associated with a second order expansion. The aim of the paper is to prove—with a fourth-gradient internal energy already obtained by the mean field theory—that the quasi-linear system of conservation laws can be written in an Hermitian symmetric form implying the stability of constant solutions. The result extends the symmetric hyperbolicity property of governing-equations’ systems when an equation of energy associated with high order deformation of a continuum medium is taken into account.