Shock profiles for hydrodynamic models for fluid-particles flows in the flowing regime

Abstract

We consider systems of conservation laws derived from coupled fluid-kinetic equations intended to describe particle-laden flows. By means of Chapman–Enskog type expansion, we determine second order corrections and we discuss the existence and stability of shock profiles. Entropy plays a central role in this analysis.

This approach is implemented on a simplified model, restricting the fluid description to the Burgers equation, and a more realistic model based on the Euler equations. The comparison between the two systems gives the opportunity to bring out the role of certain structural properties, like the Galilean invariance, which is satisfied only by the Euler-based system.

We justify existence and stability of small amplitude shock profiles for both systems. For the Euler-based model, we also employ a geometric singular perturbation approach in view of passing from small- to large-amplitude shock profiles, considering temperature as small parameter. This program, fully achieved for the zero-temperature regime, is extended on numerical grounds to small positive temperatures.