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

TL;DR

This paper demonstrates that the fourth-gradient fluid model's governing equations can be expressed in a symmetric hyperbolic form, ensuring stability and extending the mathematical framework for complex fluid models with high-order effects.

Contribution

It proves the symmetric hyperbolic form of the fourth-gradient fluid system, extending stability analysis to high-order continuum models derived from mean-field theory.

Findings

The system can be written in an Hermitian symmetric form.

Stability of constant solutions is guaranteed.

Extends symmetric hyperbolicity to high-order fluid models.

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.