Contact Manifolds in a Hyperbolic System of Two Nonlinear Conservation Laws

TL;DR

This paper investigates a hyperbolic system of two nonlinear conservation laws modeling bidisperse suspensions, focusing on contact manifolds, wave classification, and semi-analytical solutions to Riemann problems.

Contribution

It identifies contact manifolds within the phase space and analyzes their influence on solution structures for a specific hyperbolic system.

Findings

Detection of quasi-umbilic points and a contact manifold in the interior of phase space

Classification of elementary waves originating from the origin

Semi-analytical solutions to prototypic Riemann problems connecting key states

Abstract

This paper deals with a hyperbolic system of two nonlinear conservation laws, where the phase space contains two contact manifolds. The governing equations are modelling bidisperse suspensions, which consist of two types of small particles that are dispersed in a viscous fluid and differ in size and viscosity. For certain parameter choices quasi-umbilic points and a contact manifold in the interior of the phase space are detected. The dependance of the solutions structure on this contact manifold is examined. The elementary waves that start in the origin of the phase space are classified. Prototypic Riemann problems that connect the origin to any point in the state space and that connect any state in the state space to the maximum line are solved semi-analytically.