Travelling wave solutions to model equations of van der Waals fluids

Abstract

We consider the existence and uniqueness of travelling wave solutions to the model hydrodynamics equations (without capillarity) obtained from a four-velocity kinetic model of van der Waals fluids. We analyze both the Euler and the Navier-stokes equations. The Euler equations are shown to change their type. The Rankine-Hugoniot conditions are discussed in detail. It is shown that the Hugoniot locus can be disconnected even if the equations are hyperbolic. Using the Navier-stokes equations we show how to modify the Oleinik-Liu conditions of admissibility of shock waves to such situations. The shock-wave structures are found numerically. In particular, the so-called impending shock splitting is obtained.