Hyperbolic Relaxation Model for Granular Flow

In this work we describe an efficient model for the simulation of a two-phase flow made of a gas and a granular solid. The starting point is the two-velocity two-pressure model of Baer-Nunziato [1]. The model is supplemented by a relaxation source term in order to take into account the pressure equilibrium between the two phases and the granular stress in the solid phase. We show that the relaxation process can be made thermodynamically coherent with an adequate choice of the granular stress. We then propose a numerical scheme based on splitting approach. Each step of the time marching algorithm is made of two stages. In the first stage, the homogeneous convection equations are solved by a standard finite volume Rusanov scheme. In the second stage, the volume fraction is updated in order to take into account the equilibrium source term. The whole procedure is entropy dissipative. For simplified pressure laws (stiffened gas laws) we are able to prove that the approximated volume fraction satisfies a maximum principle. Introduction We are interested in the numerical modeling of a two-phase (granular-gas) flow with two velocities and two pressures p1 and p2. In one space dimension, the model is made up of seven non-homogeneous partial differential equations: two mass balance laws, two momentum balance laws, two energy balance laws and one volume fraction evolution equation. It is similar to the initial model proposed by Baer-Nunziato [1]. The main feature of this model is that the left hand side of the equations is hyperbolic. This property is very important because it ensures the mathematical stability of the model. However, in many industrial applications it is not realistic to admit two independent pressures. Generally, an algebraic relation between the two pressures is assumed. An example (among many others) of such a modeling in the framework of internal ballistics is given by Gough in [7]. For a general presentation of twophase flow models, we refer to the book of Gidaspow [4]. The relation between the two pressures is classically of the form p2 = p1 +R where R is the granular stress. In the general case, the granular stress depends on all the thermodynamic variables of the two phases. Because of the pressure relation, the system is now overdetermined. The volume fraction equation can be eliminated and a six-equation model is obtained. Unfortunately, the new model has a reduced hyperbolicity domain. The worst situation corresponds to a vanishing granular stress R = 0. In this case, the model is almost never hyperbolic. In the case of a vanishing granular stress, several authors have proposed to relax the algebraic relation p2 = p1 by adding an adequate source term to the volume fraction evolution [12], [3], [8], etc. An important parameter of the source term is the characteristic equilibrium time. When the equilibration time tends to zero, the 1991 Mathematics Subject Classification. 76M12, 65M12.