Vanishing Pressure Limit Research Articles

The singular limits of the Riemann solutions as pressure vanishes for a reduced two-phase mixtures model with non-isentropic gas state

We study the cavitation and concentration phenomena of the Riemann solutions for a reduced two-phase mixtures model with non-isentropic gas state in vanishing pressure limit. We solve the Riemann problem by constructing the regions in (p, u, s) coordinate system. Then we obtain the limiting behaviors of the Riemann solutions and the formation of δ-shock waves and vacuum as pressure vanishes. We conclude that, as pressure vanishes, the limit of Riemann solutions is the Riemann solutions of the reduced 2-dimensional pressureless gas dynamics model. Finally, we present numerical simulations which are consistent with our theoretical analysis.

The vanishing pressure limits of Riemann solutions for the Aw-Rascle hydrodynamic traffic flow model with the logarithmic equation of state

Two kinds of Riemann solutions for the Aw-Rascle hydrodynamic traffic flow model with the logarithmic equation of state are explicitly obtained by using the combination between 1-rarefaction or 1-shock wave along with 2-contact discontinuity. The formation of vacuum state and delta shock wave is identified and analyzed when the perturbation parameter in the pressure term drops to zero, where the intrinsic cavitation and concentration phenomena are surveyed and explored concretely. Additionally, several numerical results displaying the formation process of vacuum state and delta shock wave are also presented by taking three different perturbation parameters for comparison.

The vanishing pressure limits of Riemann solutions to the isothermal two-phase flow model under the external force

The vanishing pressure limits of Riemann solutions to the isothermal two-phase flow model under the external force

Concentration and cavitation in the vanishing pressure limit of solutions to the relativistic Euler equations with the logarithmic equation of state

In this paper, we constructively solve the Riemann problem for the relativistic Euler equations with the logarithmic equation of state. The concentration and cavitation phenomena are observed and analyzed during the process of vanishing pressure in the Riemann solutions. It is rigorously proved that, as the pressure vanishes, they tend to the two kinds of Riemann solutions to the zero-pressure relativistic Euler equations, which include a delta shock formed by a weighted δ-measure and a vacuum state.

The transition of Riemann solutions with composite waves for the improved Aw–Rascle–Zhang model in dusty gas

We study the cavitation and concentration of the Riemann solutions for the improved Aw–Rascle–Zhang (IARZ) model in dusty gas with a non-genuinely nonlinear field. The Riemann solutions containing composite waves are constructed by Liu-entropy condition first. Second, we investigate the limits of the inflection point and tangent point along the 1-family wave curve and find that the composite waves tend to elementary waves as pressure vanishes. Third, we obtain the limiting behavior of the Riemann solutions and observe the formation of δ-shock wave and vacuum as pressure vanishes. We conclude that the limit of Riemann solutions of the IARZ model is not the Riemann solutions of the limit of the IARZ model. The phenomenon is consistent with the work of C. Shen and M. Sun [“Formation of delta-shocks and vacuum states in the vanishing pressure limit of solutions to the Aw–Rascle model,” J. Differ. Equations 249, 3024–3051 (2010)]. Finally, we perform some numerical simulations to verify our theoretical analysis.

Concentration and cavitation of Riemann solutions to two-phase Chaplygin flows under vanishing pressure and flux approximation

In this manuscript, the intrinsic phenomena of concentration and cavitation are investigated in the Riemann problem to the pressureless two-phase flow using the three parameters flux approximation. The self-similar solution of the Riemann problem for modified Chaplygin two-phase flows is obtained. It is shown that Riemann solutions for the system under consideration consist two shock wave and one contact discontinuity, which converge to a delta shock wave in the vanishing pressure limit. It is also analyzed that Riemann solution consisting 1-rarefaction wave and 3-rarefaction wave of new approximated model converges to the two contact discontinuities and vacuum state, present between them, for without approximation.

Delta Shocks and Vacuum States in the Vanishing Pressure Limit of Riemann Solutions to the Isentropic Euler System for Chaplygin Gas

Delta Shocks and Vacuum States in the Vanishing Pressure Limit of Riemann Solutions to the Isentropic Euler System for Chaplygin Gas

On vanishing pressure limit of continuous solutions to the isentropic Euler equations

The vanishing pressure limit of continuous solutions isentropic Euler equations is analyzed, which is formulated as small parameter [Formula: see text] goes to [Formula: see text]. Due to the characteristics being degenerated in the limiting process, the resonance may cause the mass concentration. It is shown that in the pressure vanishing process, for the isentropic Euler equations, the continuous solutions with compressive initial data converge to the mass concentration solution of pressureless Euler equations, and with rarefaction initial data converge to the continuous solutions globally. It is worth to point out: [Formula: see text] converges in [Formula: see text], while [Formula: see text] converges in [Formula: see text], due to the structure of pressureless Euler equations. To handle the blow-up of density [Formula: see text] and spatial derivatives of velocity [Formula: see text], a new level set argument is introduced. Furthermore, we consider the convergence rate with respect to [Formula: see text], both [Formula: see text] and the area of characteristic triangle are [Formula: see text] order, while the rates of [Formula: see text] and [Formula: see text] depend on the further regularity of the initial data of [Formula: see text].

Concentration and Cavitation in the Vanishing Pressure Limit of Solutions to a 3 × 3 Generalized Chaplygin Gas Equations

The phenomena of concentration and cavitation are identified and analyzed by studying the vanishing pressure limit of solutions to the 3x3 isentropic compressible Euler equations for generalized Chaplygin gas (GCG) with a small parameter. It is rigorously proved that, any Riemann solution containing two shocks and possibly one-contact-discontinuity of the GCG equations converges to a delta-shock solution of the same system as the parameter decreases to a certain critical value. Moreover, as the parameter goes to zero, that is, the pressure vanishes, the limiting solution is just the delta-shock solution of the pressureless gas dynamics (PGD) model, and the intermediate density between the two shocks tends to a weighted δ-measure that forms the delta shock wave; any Riemann solution containing two rarefaction waves and possibly one contact-discontinuity tends to a two-contact-discontinuity solution of the PGD model, and the nonvacuum intermediate state in between tends to a vacuum state. Finally, some numerical results are presented to exhibit the processes of concentration and cavitation as the pressure decreases.

The intrinsic phenomena of cavitation and concentration in Riemann solutions for the isentropic two-phase model with the logarithmic equation of state

The Riemann solutions for a simplified two-phase flow model with the logarithmic equation of state are obtained in fully explicit forms for all four possible different structures. The intrinsic phenomena of cavitation and concentration are identified and analyzed in the Riemann solutions by using the vanishing pressure limit in the mixture momentum equation. It is shown that the Riemann solution consisting of 1-rarefaction wave, 2-contact discontinuity, and 3-rarefaction wave converges to a solution made up of three different vacuum states together with the left and right states separated successively by four contact discontinuities as the perturbed parameter tends to zero. By comparison, the Riemann solution composed of 1-shock wave, 2-contact discontinuity, and 3-shock wave also tends to a solution consisting of a single delta shock wave in the vanishing pressure limit. In addition, it should be pointed out that Dirac delta measures are developed simultaneously in the densities of liquid and gas in the limiting situation.

Delta waves and vacuum states in the vanishing pressure limit of Riemann solutions to Baer-Nunziato two-phase flow model

<p style='text-indent:20px;'>The phenomena of concentration and cavitation for the Riemann problem of the Baer-Nunziato (BN) two-phase flow model has been investigated in this paper. By using the characteristic analysis method, the formation of <inline-formula><tex-math id="M1">\begin{document}$ \delta- $\end{document}</tex-math></inline-formula>waves and vacuum states are obtained as the pressure for both phases vanish in the BN model. The solid contact wave is carefully dealt. The comparison with the solutions of pressureless two-phase model shows that, two shock waves tend to a <inline-formula><tex-math id="M2">\begin{document}$ \delta- $\end{document}</tex-math></inline-formula>shock solution, and two rarefaction waves tend to a two contact discontinuity solution when the solid contact discontinuity is involved. Moreover, the detailed Riemann solutions for two-phase flow model are given as the double pressure parameters vanish. This may contribute to the design of numerical schemes in the future research.</p>

Concentration and Cavitation in the Vanishing Pressure Limit of Solutions to a Simplified Isentropic Relativistic Euler Equations

We identify and analyze the phenomena of concentration and cavitation by studying the vanishing pressure limit of solutions to a simplified isentropic relativistic Euler equations. Firstly, both the explicit expressions and geometric properties of the rarefaction wave curve and shock wave curve based on any left state are given with the help of Lorentz invariance, and the Riemann problem for this system is considered. Then, we rigorously prove that, as pressure vanishes, any two-shock Riemann solution tends to a delta-shock solution of the pressureless relativistic Euler equations, and the intermediate density between the two shocks tends to a weighted $$\delta $$ -measure, which forms the delta shock wave. This describes the phenomenon of mass concentration. On the other hand, any two-rarefaction Riemann solution tends to a two-contact-discontinuity solution of the pressureless relativistic Euler equations and the nonvacuum intermediate state in between tends to a vacuum state, which reveals the phenomenon of cavitation. Both concentration and cavitation are fundamental and physical in fluid dynamics.

The asymptotic limits of Riemann solutions for the isentropic drift‐flux model of compressible two‐phase flows

The formation of vacuum state and delta shock wave are observed and studied in the limits of Riemann solutions for the one‐dimensional isentropic drift‐flux model of compressible two‐phase flows by letting the pressure in the mixture momentum equation tend to zero. It is shown that the Riemann solution containing two rarefaction waves and one contact discontinuity turns out to be the solution containing two contact discontinuities with the vacuum state between them in the limiting situation. By comparison, it is also proved rigorously in the sense of distributions that the Riemann solution containing two shock waves and one contact discontinuity converges to a delta shock wave solution under this vanishing pressure limit.

Concentration in the flux approximation limit of Riemann solutions to the extended Chaplygin gas equations with friction

In this paper, two kinds of occurrence mechanisms on the phenomenon of concentration and the formation of delta shock waves are analyzed and identified in the flux approximation limit of Riemann solutions to the extended Chaplygin gas equations with Coulomb-like friction, whose special case can also be seen as the model of the magnetogasdynamics with Coulomb-like friction. First, by introducing a transformation, the Riemann problem for the extended Chaplygin gas equations with Coulomb-like friction is solved completely. Second, we rigorously show that, as the pressure vanishes, any two-shock Riemann solution to the nonhomogeneous extended Chaplygin gas equations tends to a δ-shock solution to the corresponding nonhomogeneous transportation equations, and the intermediate density between the two shocks tends to a weighted δ-measure that forms the δ-shock; any two-rarefaction-wave Riemann solution to the nonhomogeneous extended Chaplygin gas equations tends to a two-contact-discontinuity solution to the corresponding nonhomogeneous transportation equations, and the nonvacuum intermediate state between the two rarefaction waves tends to a vacuum state. Finally, we also show that, as the pressure approaches the generalized Chaplygin pressure, any two-shock Riemann solution to the nonhomogeneous extended Chaplygin gas equations tends to a delta-shock solution to the corresponding nonhomogeneous generalized Chaplygin gas equations. In a word, we have not only generalized all the results about the vanishing pressure limit now available for homogeneous equations to nonhomogeneous cases but also obtained the stability of the delta shock Riemann solutions to the nonhomogeneous transportation equations and generalized Chaplygin gas equations with respect to flux function perturbation.

Concentration and cavitation in the vanishing pressure limit of solutions to the generalized Chaplygin Euler equations of compressible fluid flow

Abstract The phenomena of concentration and cavitation and the formation of delta shock waves and vacuum states in vanishing pressure limits of solutions to the generalized Chaplygin Euler equations of compressible fluid flow are analyzed. It is proved that, as the pressure vanishes, any two-shock-wave Riemann solution of the generalized Chaplygin Euler equations of compressible fluid flow tends to a delta-shock solution to the transport equations, and the intermediate density between them tends to a weighted δ -measure that forms a delta shock wave; any two-rarefaction-wave solution is shown to tend to two contact discontinuities connecting the constant states and vacuum states, which form a vacuum solution of the transport equations. Moreover, some numerical simulations completely coinciding with the theoretical analysis are presented.

The Vanishing Pressure Limit of Riemann Solutions to the Non-Isentropic Euler Equations for Generalized Chaplygin Gas

We analyze the appearance of delta shock wave and vacuum state in the vanishing pressure limit of Riemann solutions to the non-isentropic generalized Chaplygin gas equations. As the pressure vanishes, the Riemann solution including two shock waves and possible one contact discontinuity converges to a delta shock wave solution. Both the densityρand the internal energyHsimultaneously present a Dirac delta singularity. And the Riemann solution involving two rarefaction waves and possible one contact discontinuity converges to a solution involving vacuum state of the transport equations.

Vanishing pressure limit for compressible Navier–Stokes equations with degenerate viscosities

In this paper we study a vanishing pressure process for highly compressible Navier–Stokes equations as the Mach number tends to infinity. We first prove the global existence of weak solutions for the pressureless system in the framework (Li and Xin 2015 arXiv:1504.06826v2), where the weak solutions are established for compressible Navier–Stokes equations with degenerate viscous coefficients. Furthermore, a rate of convergence of the density in is obtained, in case when the velocity corresponds to the gradient of density at initial time.

The vanishing pressure limits of Riemann solutions to the Chaplygin gas equations with a source term

We study the vanishing pressure limits of Riemann solutions to the Chaplygin gas equations with a source term. The phenomena of concentration and cavitation to Chaplygin gas equations with a friction term are identified and analyzed as the pressure vanishes. Due to the influence of source term, the Riemann solutions are no longer self-similar. When the pressure vanishes, the Riemann solutions to the inhomogeneous Chaplygin gas equations converge to the Riemann solutions to the pressureless gas dynamics model with a friction term.

Concentration in vanishing pressure limit of solutions to the modified Chaplygin gas equations

Two kinds of occurrence mechanism on the phenomenon of concentration and the formation of delta shock wave in vanishing pressure limit of solutions to the modified Chaplygin gas equations are analyzed and identified. The Riemann problem of the modified Chaplygin gas equations is first solved. Then it is shown that, as the pressure vanishes, any two-shock Riemann solution tends to a delta-shock solution to the transport equations, and the intermediate density between the two shocks tends to a weighted δ-measure which forms a delta shock wave; any two-rarefaction-wave Riemann solution tends to a two-contact-discontinuity solution to the transport equations, and the nonvacuum intermediate state in between tends to a vacuum state. It is also shown that, as the pressure approaches the generalized Chaplygin gas pressure, any two-shock Riemann solution tends to a delta-shock solution to the generalized Chaplygin gas equations. Some numerical results are presented to show the formation process of delta shock waves and vacuum states.

Delta shocks and vacuum states in vanishing pressure limits of solutions to the relativistic Euler equations for generalized Chaplygin gas

The Riemann solutions for the relativistic Euler equations for generalized Chaplygin gas are considered. It is rigorously proved that, as the pressure vanishes, they tend to the two kinds of Riemann solutions to the zero-pressure relativistic Euler equations, which include a delta shock formed by a weighted $\delta$-measure and a vacuum state.