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

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.

Delta-Shock Solution to the Eulerian Droplet Model by Variable Substitution Method

“…To our knowledge, investigations on the Eulerian droplet model (1) have mostly focused on the numerical level [1] and the practical level [3,4,[15][16][17]. Recently, the theoretical arguments for (1) were completed by Keita and Bourgault [18]. They solved the Riemann problem for the Eulerian droplet model by going through the solution of the Riemann problems for the inviscid Burgers equation with a source term and the subsystem, respectively.…”

“…Particularly, for the delta-shock solution, the generalised Rankine-Hugoniot condition, which is in the form of ordinary differential equations, was proposed. Nevertheless, as was pointed out in [18], "In general, it might be hard to find the analytical solution of this ordinary differential equations". Thus, for the delta-shock solution of the Eulerian droplet model (1), with the help of the Cauchy-Peáno theorem, the Cauchy-Lipschitz existence theorem, and the Arzla-Ascoli theorem, they obtained the existence of a solution to the generalised Rankine-Hugoniot condition satisfying the Lax entropy condition.…”

“…Here, α ± and u ± are constants. As was done in [18], just for the convenience of the study, in the system (1), µ is assumed to be a positive constant, u a is also supposed to be constant, and α > 0 is required. Recall that, in 2016, Shen [19] studied the Riemann problem for the pressureless Euler system with the following source term:…”

“…Then, the existence and uniqueness of the delta-shock solution are solved completely under the entropy condition by studying the function equation, which is one of the outstanding advantages of our approach. Compared to the discussions in [18], this approach avoids using some analytical theorems that are complicated and not easy to understand for hydrodynamicists and engineers. Although we take a different method from [18], the results are just the same.…”

“…Compared to the discussions in [18], this approach avoids using some analytical theorems that are complicated and not easy to understand for hydrodynamicists and engineers. Although we take a different method from [18], the results are just the same.…”