Toward a mathematical analysis for a model of suspension flowing down an inclined plane

TL;DR

This paper develops a mathematical method to analyze shock and rarefaction waves in suspension flow models, specifically addressing the Riemann problem with variable volume fractions in a fluid flowing down an inclined plane.

Contribution

It introduces a novel approach for solving the Riemann problem in suspension flow equations, extending existing methods to account for spatiotemporal variations in volume fractions.

Findings

Method effectively finds shock and rarefaction waves.

Applicable to suspension flow with variable particle concentration.

Builds on classical Riemann problem techniques.

Abstract

We consider the Riemann problem of the dilute approximation equations with spatiotemporally dependent volume fractions from the full model of suspension, in which the particles settle to the solid substrate and the clear liquid film flows over the sediment [Murisic et al., J. Fluid. Mech. {\bf{717}}, 203--231 (2013)]. We present a method to find shock waves, rarefaction waves for the Riemann problem of this system. Our method is mainly based on [Smoller, Springer-Verlag, New York, second edition, (1994)].