The Riemann problem for the one-dimensional isentropic Euler system under the body force with varying gamma law

Abstract

The exact Riemann solutions are presented in fully explicit forms for the one-dimensional isentropic Euler system of gas dynamics with the body force, in which the shock and rarefaction waves are accelerated into the parabola curves with the same degree under the influence of such body force. Moreover, the limit of Riemann solution composed of two shock waves tends to an accelerated delta shock solution as well as the limit of Riemann solution constituted by two rarefaction waves converges to a solution made up of two contact discontinuities along with the vacuum state encompassed by them when the adiabatic exponent tends to one, in which the intrinsic phenomena of concentration and cavitation can be analyzed and observed carefully. It is of interest to notice that the internal states in two rarefaction wave fans are transformed gradually into the corresponding vacuum states under this limiting circumstance, which is distinguished from the previously established result that a whole rarefaction wave is concentrated into only one contact discontinuity.