Isentropic Chaplygin Research Articles

Multi-dimensional optimal system and conservation laws for Chaplygin gas Cargo-LeRoux model

The well-known Cargo-LeRoux model for the isentropic Chaplygin gas is investigated using classical Lie symmetry method. Optimal systems up to six dimensions are constructed using the adjoint transformation and the invariants of the admitted Lie algebras. We develop exact solutions to the Cargo-LeRoux model by using the one-dimensional optimal system and discussed the physical behavior of such solutions graphically. Conservation laws to the governing equations are also obtained by using the nonlinear self-adjointness of the model. The performance of the developed solutions is explored through the evolutionary behavior of a discontinuity wave.

Vanishing Magnetic Field Limits of Solutions to the Isentropic Chaplygin Gas Magnetogasdynamics Equations

Vanishing Magnetic Field Limits of Solutions to the Isentropic Chaplygin Gas Magnetogasdynamics Equations

Delta shocks and vacuum states for the isentropic magnetogasdynamics equations for Chaplygin gas as pressure and magnetic field vanish

This paper is concerned with the Riemann problem for the isentropic Chaplygin gas magnetogasdynamics equations and the formation of $$\delta $$ -shocks and vacuum states as pressure and magnetic field vanish. Firstly, the Riemann problem of the isentropic magnetogasdynamics equations for Chaplygin gas is solved analytically. Secondly, it is rigorously proved that, as both the pressure and the magnetic field vanish, the Riemann solution containing two shocks tends to a $$\delta $$ -shock solution to the transport equations, and the intermediate density between the two shocks tends to a weighted $$\delta $$ -measure which form the $$\delta $$ -shock; while the Riemann solution containing two rarefaction waves tends to a two-contact-discontinuity solution to the transport equations if we do not consider the virtual velocity in the vacuum region, the intermediate state between the two contact discontinuities is a vacuum state. Moreover, we also give some numerical simulations to confirm the theoretical analysis.

Global structure of two dimensional Riemann solutions to the Euler system for isentropic Chaplygin gas

AbstractIn this paper, we study the two dimensional Riemann problem of the Euler system for isentropic Chaplygin gas, and the initial data consist of three pieces constant states divided by an inverted Y‐type curve. The results extended the results in Chen and Qu where the solution of every one dimensional Riemann problem contains no slip plane. We divide the discussion into several cases and for each case, we give the structure of the solution containing slip planes by the method of generalized characteristic analysis.

Steepened wave in two-phase Chaplygin flows comprising a source term

This manuscript brings some qualitative features of steepened wave in isentropic Chaplygin two-phase flows with a non-constant source term via Lie group transformation. The transport equation for steepened wave is determined. The behaviour of amplitude of steepened wave is investigated using the numerical solution of the system. The effects of inclination of the flow on the amplitude of singular surface are also shown.

Global smooth solutions to 3D irrotational Euler equations for Chaplygin gases

We study here the Cauchy problem associated with the isentropic and compressible Euler equations for Chaplygin gases. Based on the new formulation of the compressible Euler equations in J. Luk and J. Speck [The hidden null structure of the compressible Euler equations and a prelude to applications, J. Hyperbolic Differ. Equ. 17 (2020) 1–60] we show that the wave system satisfied by the modified density and the velocity for Chaplygin gases satisfies the weak null condition. We then prove the global existence of smooth solutions to the irrotational and isentropic Chaplygin gases without introducing a potential function, when the initial data are small perturbations to a constant state.

Riemann problem for the isentropic Chaplygin gas Cargo-LeRoux model

The so-called Cargo-LeRoux model is a hyperbolic system of conservation laws derived from the Euler equations with a constant gravity. This paper is concerned with such a model for an isentropic Chaplygin gas. The Riemann problem is solved with four kinds of structures. The first is composed of the contact discontinuities, the second contains an overcompressible δ-wave (δ-shock wave), and the third and fourth involve a contact discontinuity and a kind of non-overcompressible δ-wave. One of the highlights of this paper lies in the introduction of the non-overcompressible δ-waves.

On the delta shockwave interactions for the isentropic Chaplygin gas system consisting of three scalar equations

The Riemann problem for the one-dimensional version of isentropic compressible Euler system for the Chaplygin gas consisting of three scalar equations is considered. It is shown that the Riemann solutions involve only two situations: the combination of three contact discontinuities or a delta shock wave. The generalized Rankine-Hugoniot conditions of delta shock wave are derived and the exact delta shock wave solution including the strength and propagation speed is obtained explicitly. The solutions to the perturbed Riemann problem are constructed globally when the initial data are taken to be the three piecewise constant initial data. The wave interaction problem is extensively investigated and some interesting phenomena are observed. It is shown that the limits of solutions to the perturbed Riemann problem converge to the corresponding ones to the Riemann problem when the perturbation parameter tends to zero.

Stability of the Riemann solutions for a Chaplygin gas

In this paper, we study the structural stability of solutions to the Riemann problem for one-dimensional isentropic Chaplygin gas. We perturb the Riemann initial data by taking three piecewise constant states and construct the global structure. By letting the perturbed parameter ε tend to zero, we prove that the Riemann solutions are stable under the local small perturbations of the Riemann initial data even when the initial perturbed density depending on the parameter but with no mass concentration limit.

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

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 ($S^\pm$), rarefactive one ($R^\pm$) and slip lines ($J^\pm$). We still call $S^\pm$ as shock and $R^\pm$ 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.