Chaplygin Gas Equation Research Articles

Sound speed in extended Chaplygin fluid

We consider an extended Chaplygin gas equation of state which is driven from D-brane action and construct a cosmological model based on this equation of state. In this regard, we compute the scale factor of the model under a certain approximation. The conservation equation of this case is a non-linear differential equation which should solve using the special conditions. We also analyze the stability of the model by using sound speed as well as adiabatic index and discuss certain special cases of the model. We find special equation of state in this model which yields to dynamical and thermodynamical stability. Furthermore, we study the cosmological consequences of this model under certain conditions.

Delta Shock Wave Solution of the Riemann Problem for the Non-homogeneous Modified Chaplygin Gasdynamics

The motivation of the present study is to derive the solution of the Riemann problem for modified Chaplygin gas equations in the presence of constant external force. The analysis leads to the fact that in some special circumstances delta shock appears in the solution of the Riemann problem. Also, the Rankine–Hugoniot relations for delta shock wave which are utilized to determine the strength, position and propagation speed of the delta shocks have been derived. Delta shock wave solution to the Riemann problem for the modified Chaplygin gas equation is obtained. It is found that the external force term, appearing in the governing equations, influences the Riemann solution for the modified Chaplygin gas equation.

Stability of thin shell wormhole in f(R) theory of gravity

Abstract Dynamics of thin-shell wormholes within the context of f ( R ) gravity by joining two identical copies of Reissner–Nordstrom–de Sitter manifold using the cut and paste approach is discussed. The stability of wormholes is analyzed through the radial perturbation about the static solution and the modified generalized Chaplygin gas equation of state (EoS). This formalism gives rise two specific different configurations of f ( R ) gravity, in which quadratic type model and cubic type model. The presence of both stable and unstable regions depending on the appropriate values of different parameters are examined their EoS, with taken into account several f ( R ) gravity models as well as the metric space are illustrated. Also, the stability regions under the radial perturbation depend on the EoS parameter β 2 usually is explained as the subluminal square speed of sound exist. Such a region of stability in the existence of a large value of the charge is significantly is increased and extended by decreasing the value of the cosmological constant The effect of increasing dark sources δ and α parameters leads to an increase in the stable regions is displayed. Stability regions increase for increasing EoS parameters γ , ς , η and also the influence of the curvature parameter R may increase the stability regions.

Stability of asymmetric thin shell wormhole with a variable equation of state

Dynamical equations of asymmetric thin shell wormholes connecting two different spacetimes by applying the cut and paste technique are constructed. The linear stability with variable modified generalized Chaplygin gas (VMGCG) equation of state, where the pressure will be a function of both radius and density, is investigated. This formalism is applied to two specific cases of asymmetric wormhole of a Schwarzschild connected to Reissner–Nordström spacetime and Reissner–Nordström connected to Schwarzschild–De Sitter (SDS) spacetime.

Global smooth solutions for the Chaplygin gas equations with source terms in [formula omitted

This article studies the existence of global smooth solution for the Chaplygin gas equations with source terms in Rd, d≥1. We establish the existence and uniqueness of global smooth solution provided the initial data is sufficiently small, which tells us that the damping terms can prevent the development of singularity in small amplitude.

Time dependent dark energy and the thermodynamics of many-body systems

In this paper, we study the thermodynamics and statistics of the galaxies clustering affected by the dynamical dark energy. We consider two important dark energy models based on time dependent equation of state to evaluate the gravitational partition function. In the first model, we consider barotropic dark energy with time dependent equation of state. However, in the second model, we consider various kinds of Chaplygin gas equation of state which originally introduced by string theory. We calculate, analytically and numerically, the thermodynamic quantities in canonical and grand canonical ensembles. We investigate validity of the second law of thermodynamics for the total system of clustering of galaxies and dynamical dark energy. We finally evaluate the galaxy-galaxy correlation function and compare our model with Peebles's power law and find that the model based on generalized Chaplygin gas may yields to more agreement with observations.

Symmetry analysis and optimal systems of generalized Chaplygin gas equations with a source term

The system of generalized Chaplygin gas equations with a coulomblike friction term has been investigated by using the famous Lie symmetry method. A direct and systematic algorithm based on the adjoint transformation and invariants of the admitted Lie algebras is then used to construct one‐ and two‐dimensional optimal system of the Chaplygin gas equations. Inequivalent classes of group invariant solutions are then obtained using the one‐dimensional optimal system. Further, the evolutionary behaviour of the weak discontinuity wave within the state characterized by one of the group invariant solutions is investigated in detail, and certain observations are noted in respect to their contrasting behaviour.

The generalized Riemann problem for the Chaplygin gas equation

The main motive of the present study is to determine the closed form solution of the generalized Riemann Problem (GRP) for the 1-D Euler’s equation for Chaplygin gas by using the Differential Constraint Method (DCM). The consistency conditions and differential constraint for the governing model is determined. Also, the compatibility condition between differential constraints and Chaplygin gas model is computed here. Further, to obtain an exact (closed form) solution of Generalized Riemann Problem (GRP), we characterize the solution for a smooth arbitrary function as initial data. Ultimately, we solve the GRP with generalized discontinuous initial data.

Concentration in the zero-exponent limit of solutions to the isentropic Euler equations for extended Chaplygin gas

By introducing an isentropic Euler system with a new version of extended Chaplygin gas equation of state, we study two kinds of occurrence mechanism on the phenomenon of concentration and the formation of delta shock waves in the zero-exponent limit of solutions to the extended Chaplygin gas equations as the two exponents tend to zero wholly or partly. The Riemann problem is first solved. Then, we show that, as both the two exponents tend to zero, that is, the extended Chaplygin gas pressure tends to a constant, any two-shock-wave Riemann solution of the extended Chaplygin gas equations converges to a delta-shock solution to the zero-pressure flow system, 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 zero-pressure flow system, and the nonvacuum intermediate state in between tends to a vacuum. It is also shown that, as one of the exponents goes to zero, namely, the extended Chaplygin gas pressure approaches to some special generalized Chaplygin gas pressure, any two-shock-wave Riemann solution tends to a delta-shock solution to the generalized Chaplygin gas equations.

Well-posedness and blow-up criterion for the Chaplygin gas equations in ℝN

We establish a well-posedness theory and a blow-up criterion for the Chaplygin gas equations in [Formula: see text] for any dimension [Formula: see text]. First, given [Formula: see text], [Formula: see text], we prove the well-posedness property for solutions [Formula: see text] in the space [Formula: see text] for the Cauchy problem associated with the Chaplygin gas equations, provided the initial density [Formula: see text] is bounded below. We also prove that the solution of the Chaplygin gas equations depends continuously upon its initial data [Formula: see text] in [Formula: see text] if [Formula: see text], and we state a blow-up criterion for the solutions in the classical BMO space. Finally, using Osgood’s modulus of continuity, we establish a refined blow-up criterion of the solutions.

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.

The Chaplygin gas as a model for modified teleparallel gravity?

This paper explores the possibility of treating the exotic Chaplygin-gas (CG) fluid model as some manifestation of an f(T) gravitation. To this end, we use the different cosmological CG equations of state, compare them with the equation of state for the modified teleparallel gravity and reconstruct the corresponding Lagrangian densities. We then explicitly derive the equation of state parameter of the torsion fluid w_T and study its evolution for vacuum-torsion, radiation-torsion, dust-torsion, stiff fluid-torsion and radiation-dust-torsion multi-fluid systems. The obtained Lagrangians have, in general, matter dependence due to the matter-torsion coupling appearing in the energy density and pressure terms of the modified teleparallel gravity theory. For the simplest CG models, however, it is possibly to reconstruct f(T) Lagrangians that depend explicitly on the torsion scalar T only. The preliminary results show that, in addition to providing Chaplygin-gas-like solutions to the modified teleparallel gravitation, which naturally behave like dark matter and dark energy at early and late times respectively, the technique can be used to overcome some of the challenges attributed to the CG cosmological alternative.

Limits of Solutions to the Relativistic Euler Equations for Modified Chaplygin Gas by Flux Approximation

We study the Riemann problem and flux-approximation limits of solutions to the relativistic Euler equations with the state equation for modified Chaplygin gas. The limits of Riemann solutions for the relativistic modified Chaplygin gas equations and the corresponding flux-approximation system are discussed when the pressure term and flux-perturbation parameters tend to zero. It is rigorously proved that, as the pressure and flux approximations vanish respectively, any two-shock-wave Riemann solution tends to a delta-shock solution to the pressureless relativistic Euler equations, and the intermediate density between them tends to a weighted $\delta $ -measure that forms a delta shock wave. Correspondingly, any two-rarefaction-wave solution becomes two contact discontinuities connecting the constant states and vacuum states, which form a vacuum solution.

Delta Shock Waves as Flux-Approximation Limit of Solutions to the Modified Chaplygin Gas Equations

By introducing a triple parameter flux perturbation including pressure in the modified Chaplygin gas equations, we prove that, as the triple parameter flux perturbation vanishes, any two-shock Riemann solution and any two-rarefaction-wave Riemann solution to the perturbed modified Chaplygin gas equations tend to a delta-shock and a vacuum solution to the transport equations, respectively. Then we show that, as a double parameter flux perturbation vanishes, any two-shock Riemann solution under a certain condition to the perturbed modified Chaplygin gas equations tends to a delta shock wave solution to the generalized Chaplygin gas equations. In addition, we also show that, as a single parameter flux perturbation vanishes, any two-shock solution satisfying certain initial data and any parameterized delta shock wave solution to the perturbed generalized Chaplygin gas equations tend to a delta shock wave solution to the generalized Chaplygin gas equations. Finally, we exhibit some representative numerical simulations to confirm the theoretical analysis.

Invariant Solutions and Nonlinear Self-Adjointness of the Two-Component Chaplygin Gas Equation

In this paper, the Lie group method is performed on a special dark fluid, the Chaplygin gas, which describes both dark matter and dark energy in the present universe. Based on an optimal system of one-dimensional subalgebras, similarity reductions and group invariant solutions are given. Finally, by means of Ibragimov’s method, conservation laws are obtained.

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.

Kaluza Klein Type FRW Cosmological Model With Extended Chaplygin Gas

In this paper we consider the extended Chaplygin gas equation of state as a model of dark energy for which it recovers barotropic fluids with quadratic equation of state. We obtain scale factor dependence energy density and Hubble expansion parameter for particular values of n, m and α. Similarly for arbitrary values of n, m and α we have discuss nature of cosmological parameters such as scale factor, Hubble expansion parameter, time dependent dark energy density and deceleration parameter in the framework of Kaluza Klein type FRW cosmological model. Further, we study stability of the model by using speed of sound and observe that the model is stable at late time.

On sign-changeable interaction in FLRW cosmology

We investigate an interacting two-fluid model in a spatially flat Friedmann–Lemaître–Robertson–Walker (FLRW) Universe, when the energy transfer between these two dark components is produced by a factorizable nonlinear sign-changeable interaction depending linearly on the energy density and quadratically on the deceleration parameter. We solve the source equation and obtain the effective energy densities of the dark sector and their components. We show that the effective equation of state of the dark sector includes some of the several kinds of Chaplygin gas equations of state as well as a generalization of the polytropic equation of state. We use Bayesian statistics methods to constrain free parameters in the models during the most recent evolution considering supernovae type Ia and measurements of the Hubble expansion rate. The resulting constraints provide new information on sign-changeable interactions, its equivalences and compatibility with previous models and novel late time universe dynamics.

Tolman--Bondi--Lema\^\i tre Spacetime with a Generalised Chaplygin Gas

The Tolman-Bondi-Lema\^itre type of inhomogeneous spacetime with generalised Chaplygin gas equation of state given by $p = -\frac{A}{\rho^{\alpha}}$ is investigated where $ \alpha$ is a constant. We get an inhomogeneous spacetime at early stage but at the late stage of universe the inhomogeneity disappear with suitable radial co-ordinate transformation. For the large scale factor our model behaves like $\Lambda$CDM type which is in accord with the recent WMAP studies. We have calculated $\frac{\partial \rho}{\partial r}$ and it is found to be negative for $\alpha > 0$ which is in agreement with the observational analysis. A striking difference with Chaplygin gas ($ \alpha = 1$) lies in the fact that with any suitable co-ordinate transformation our metric cannot be reduced to the Einstein-de Sitter type of homogeneous spacetime as is possible for the Chaplygin gas. We have also studied the effective deceleration parameter and find that the desired feature of \emph{flip} occurs at the late universe. It is seen that the flip time depends explicitly on $\alpha$. We also find that flip is not synchronous occurring earlier at the outer shells, thus offering a natural path against occurrence of wellknown shell crossing singularity. This is unlike the Tolman-Bondi case with perfect gas where one has to impose stringent external conditions to avoid this type of singularity. We further observe that if we adopt separation of variables method to solve the field equations the inhomogeneity in matter distribution disappears. The whole situation is later discussed with the help of Raychaudhury equation and the results compared with previous cases. This work is the generalisation of our previous article where we have taken $\alpha =1$.

Nonlocal symmetries classifications and exact solution of Chaplygin gas equations

In this work, a complete tree of nonlocally related partial differential equations of Chaplygin gas equations is constructed. This tree includes the systems, which are obtained through local conservation laws and the local symmetry based method. We classify the nonlocal symmetries from potential systems as well as inverse potential systems (IPSs). Furthermore, we propose a systematic algorithm for identification of nonlocal symmetries through IPSs by combining the ideas in the studies of Bluman and Yang [J. Math. Phys. 54, 093504 (2013)] and Yang and Cheviakov [J. Math. Phys. 55, 083514 (2014)]. Finally, we obtain a new exact solution through nonlocal symmetry analysis and physical behavior of solution is presented.