The dispersive wave dynamics of a two-phase flow relaxation model

¹ Department of Mathematical Sciences, Norwegian University of Science and Technology (NTNU), NO-7491 Trondheim, Norway
susanne.solem@ntnu.no; peder.aursand@math.ntnu.no
² SINTEF Materials and Chemistry, P. O. Box 4760 Sluppen, NO-7465 Trondheim, Norway
Tore.Flatten@sintef.no

Received: 25 October 2013
Revised: 7 May 2014

Abstract

We consider a general Euler-type two-phase flow model with relaxation towards phase equilibrium. We provide a complete description of the transition between the wave dynamics of the homogeneous relaxation system and that of the local equilibrium approximation. In particular, we present generally valid analytical expressions for the amplifications and velocities of each Fourier component. This transitional wave dynamics is fully determined by only two dimensionless parameters; a stiffness parameter and the ratio of the sound velocities in the stiff and non-stiff limits. A direct calculation verifies that the stability criterion is precisely the subcharacteristic condition. We further prove a maximum principle in the transitional regime, similar in spirit to the subcharacteristic condition; the transitional wave speeds can never exceed the largest wave speed of the homogeneous relaxation system. Finally, we identify the existence of a critical region of wave numbers where the sonic waves completely disappear from the system. This region corresponds to the casus irreducibilis of the describing cubic polynomial.