Some mathematical properties of a barotropic multiphase flow model

Khaled Saleh,Nicolas Seguin,Philippe Helluy,Jean-Marc Hérard

doi:10.1051/proc/202069070

Abstract

We study a model for compressible multiphase flows involving N non miscible barotopic phases where N is arbitrary. This model boils down to the barotropic Baer-Nunziato model when N = 2. We prove the weak hyperbolicity property, the non-strict convexity of the natural mathematical entropy, and the existence of a symmetric form.

Highlights

The modeling and numerical simulation of multiphase flows is a relevant approach for a detailed investigation of some patterns occurring in many industrial sectors.In [4,8,9,14], some modeling efforts have been provided for the design of compressible multiphase flow models allowing unique jump conditions and for which the initial-value problem is well posed
As in the Baer-Nunziato model, the partial differential equations (PDEs) are composed of a hyperbolic first order convective part consisting in N Euler-like systems coupled through non-conservative terms and zero-th source terms accounting for pressure, velocity and temperature relaxation phenomena between the phases
In [5, 11], two crucial properties have been proven for a class of two phase flow models containing the BaerNunziato model, namely, the convexity of the natural entropy associated with the system, and the existence of a symmetric form

Summary

Introduction

The modeling and numerical simulation of multiphase flows is a relevant approach for a detailed investigation of some patterns occurring in many industrial sectors. The N -phase flow models developed therein consist in an extension to N ≥ 3 phases of the well-known Baer-Nunziato two phase flow model [1]. As in the Baer-Nunziato model, the PDEs are composed of a hyperbolic first order convective part consisting in N Euler-like systems coupled through non-conservative terms and zero-th source terms accounting for pressure, velocity and temperature relaxation phenomena between the phases. In [5, 11], two crucial properties have been proven for a class of two phase flow models containing the BaerNunziato model, namely, the convexity of the natural entropy associated with the system, and the existence of a symmetric form. We prove the convexity of the entropy and the existence of a symmetric form for a barotropic multiphase flow model with N - where N is arbitrarily large - phases. We restrict the study to the case where the interfacial velocity coincides with one of the phasic material velocities

The barotropic multiphase flow model

Eigenstructure of the system

Mathematical Entropy

Symmetrizability

Conclusion