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

Abstract

We 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.

Augmented skew-symmetric system for shallow-water system with surface tension allowing large gradient of density

“…Note that there exists several interesting papers developing augmented systems such as [12] and [17] for symmetric form for capillarity fluids with a capillarity energy E(h, ∇h) or multi-gradient fluids with a capillarity energy E(h, ∇h, • • • , ∇ n h). See also recently [8] for the defocusing Schrödinger equation which is linked to the quantum-Euler system (E(h, p) = Φ(h) + σ p 2 /h where σ is constant) through the Madelung transform and some numerical simulation.…”

“…It is interesting to note that the augmented system in [12] and [17] is related to the unknowns (h, u, ∇h, • • • , ∇ n h). In [21], the authors developed a similar augmented version in order to deal with internal capillarity energies (1) for numerical purposes namely:…”