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

Abstract

The Riemann problem for a compressible perfect fluid with a constant external force for the Chaplygin gas is considered. We obtain two kinds of exact solutions. The first one consists of contact discontinuities, while the other one involves a delta shock wave in which both density and internal energy contain a Dirac delta function. The position, speed and weights of the delta shock wave are derived from both generalized Rankine-Hugoniot relation and entropy condition, which are established in detail. Moreover, the solutions are no longer self-similar due to the influence of the constant external force.

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

“…First, compared with the method presented in [18], our method seems simpler and has the advantage of being easily workable. Second, the variable substitution (4) introduced in this paper is obviously different from that in previous works [19,[22][23][24][25][26][27][28][29][30]. Third, influenced by the source term, the Riemann solutions for (1) are not selfsimilar any more.…”

“…This method adopted here is easy to understand and workable. Furthermore, the variable substitution introduced here is different from that in [19,[22][23][24][25][26][27][28][29][30], etc. It is also shown that, because of the effect of the external force, the contact discontinuities and delta-shock waves are curved, and then the solutions are not self-similar.…”

“…Now we solve the Riemann problem (5) and (6) when u − > u + . Under the entropy condition (24), the Riemann problem is reduced to solving the ordinary differential equations (23) with the initial data x(0) = 0, w(0) = 0.…”

“…Variable substitution is the common method used to discuss the balance law with the source term. For example, see [22] for the generalised zero-pressure flow model, [23] for the Suliciu relaxation system, and [24] for the perfect fluid model. Besides, one can also refer to [19] and [25][26][27][28][29][30] for the Chaplygin and generalised Chaplygin gas equations with friction.…”

“…Besides, one can also refer to [19] and [25][26][27][28][29][30] for the Chaplygin and generalised Chaplygin gas equations with friction. Motivated by the works in [19] and [22][23][24][25][26][27][28][29][30], here we introduce a new state variable v(x, t) and perform the following variable substitution for (1)…”