The two-dimensional Riemann problem for isentropic Chaplygin gas dynamic system$^*$

Abstract

The two-dimensional Riemann problem for isentropic Chaplygin gas dynamic system consists of interactions of four planar elementary waves. Different from polytropic gas, all of them are contact discontinuities due to the system is full linear degenerate, i.e., the three eigenvalues of the system are linear degenerate. They include compressive one (S ± ), rarefactive one (R ± ) and slip lines (J ± ). We still call S ± as shock and R ± as rarefaction wave.In this paper, we study the problem systematically. According to different combination of four elementary waves, we deliver a complete classification to the problem. It contains 14 cases in all. The Riemann solutions are self-similar, and the flow is transonic in self-similar plane (x/t, y/t). The boundaries of the interaction domains are obtained. Solutions in supersonic domains are constructed in no J cases. While in the rest cases, the structure of solutions are conjectured except for the case 2J + + 2J − . Especially, delta waves and simple waves appear in some cases. The Dirichlet boundary value problems in subsonic domains or the boundary value problems for transonic flow are formed case by case. The domains are convex for two cases, and non-convex for the rest cases. The boundaries of the domains are composed of sonic curves and/or slip lines.

The Convergence of Riemann Solutions to the Modified Chaplygin Gas Equations with a Coulomb-Like Friction Term as the Pressure Vanishes

“…For the Riemann problem of homogeneous Chaplygin gas equations, there are lots of results. We refer the readers to [17][18][19][20][21][22][23]. For inhomogeneous Chaplygin gas equations, Shen [24] studied Riemann problem by introducing a new velocity:…”

Riemann problem for the isentropic relativistic Chaplygin Euler equations

“…For a Chaplygin gas, Brenier [18] studied system (1.4) and obtained the Riemann solutions with concentration when initial data belong to a certain domain in phase plane. Furthermore, Guo, Sheng, and Zhang [19] abandoned this constrain and obtained the general solutions. In [19], the two-dimensional case is also considered systematically and some conjectures on the structures of solutions were delivered.…”

Riemann problem for a compressible perfect fluid with a constant external force for the Chaplygin gas

“…(1.4) If f = 0, the solutions with concentration to the Riemann problem for (1.4) with the Chaplygin gas were obtained by Brenier [9]. Guo, Sheng and Zhang [10] solved completely this problem, where the delta shock wave solutions were constructed. Roughly speaking, the delta shock wave solution is a solution such that at least one of the variables contains Dirac delta function [11][12][13][14][15][16].…”