Eulerian droplet model: Delta-shock waves and solution of the Riemann problem

TL;DR

This paper analyzes an Eulerian droplet model, establishing conditions for solution regularity loss, and constructs solutions to the Riemann problem, including delta-shock waves, supported by numerical examples.

Contribution

It introduces a method to solve the Riemann problem for the Eulerian droplet model, including delta-shock solutions, extending previous work on pressureless gas systems.

Findings

Conditions for regularity loss in the model are established.

Constructive solutions to the Riemann problem are provided.

Existence of delta-shock solutions is proven.

Abstract

We study an Eulerian droplet model which can be seen as the pressureless gas system with a source term, a subsystem of this model and the inviscid Burgers equation with source term. The condition for loss of regularity of a solution to Burgers equation with source term is established. The same condition applies to the Eulerian droplet model and its subsystem. The Riemann problem for the Eulerian droplet model is constructively solved by going through the solution of the Riemann problems for the inviscid Burgers equation with a source term and the subsystem, respectively. Under suitable generalized Rankine-Hugoniot relations and entropy condition, the existence of delta-shock solution is established. The existence of a solution to the generalized Rankine-Hugoniot conditions is proven. Some numerical illustrations are presented.