Global stability of the rarefaction wave of a one-dimensional model system for compressible viscous gas

This paper is concerned with the asymptotic behavior toward the rarefaction wave of the solution of a one-dimensional barotropic model system for compressible viscous gas. We assume that the initial data tend to constant states atx=±, respectively, and the Riemann problem for the corresponding hyperbolic system admits a weak continuous rarefaction wave. If the adiabatic constant satisfies 12, then the solution is proved to tend to the rarefaction wave ast under no smallness conditions of both the difference of asymptotic values atx=± and the initial data. The proof is given by an elementaryL ²-energy method.     

Pressure based finite volume method for calculation of compressible viscous gas flows

A pressure based, iterative finite volume method is developed for calculation of compressible, viscous, heat conductive gas flows at all speeds. The method does not need the use of under-relaxation coefficient in order to ensure a convergence of the iterative process. The method is derived from a general form of system of equations describing the motion of compressible, viscous gas. An emphasis is done on the calculation of gaseous microfluidic problems. A fast transient process of gas wave propagation in a two-dimensional microchannel is used as a benchmark problem. The results obtained by using the new method are compared with the numerical solution obtained by using SIMPLE (iterative) and PISO (non-iterative) methods. It is shown that the new iterative method is faster than SIMPLE. For the considered problem the new method is slightly faster than PISO as well. Calculated are also some typical microfluidic subsonic and supersonic flows, and the Rayleigh–Bénard convection of a rarefied gas in continuum limit. The numerical results are compared with other analytical and numerical solutions.     

An RKDG finite element method for the one-dimensional inviscid compressible gas dynamics equations in a Lagrangian coordinate

In this paper,Runge-Kutta Discontinuous Galerkin(RKDG) finite element method is presented to solve the onedimensional inviscid compressible gas dynamic equations in a Lagrangian coordinate.The equations are discretized by the DG method in space and the temporal discretization is accomplished by the total variation diminishing Runge-Kutta method.A limiter based on the characteristic field decomposition is applied to maintain stability and non-oscillatory property of the RKDG method.For multi-medium fluid simulation,the two cells adjacent to the interface are treated differently from other cells.At first,a linear Riemann solver is applied to calculate the numerical ?ux at the interface.Numerical examples show that there is some oscillation in the vicinity of the interface.Then a nonlinear Riemann solver based on the characteristic formulation of the equation and the discontinuity relations is adopted to calculate the numerical ?ux at the interface,which suppresses the oscillation successfully.Several single-medium and multi-medium fluid examples are given to demonstrate the reliability and efficiency of the algorithm.     

A RKDG finite element method for the one-dimensional inviscid compressible gas dynamics equations in Lagrangian coordinate

In this paper, Runge-Kutta Discontinuous Galerkin (RKDG) finite element method is presented to solve the one-dimensional inviscid compressible gas dynamic equations in Lagrangian coordinate. The equations are discretized by the DG method in space and the temporal discretization is accomplished by the total variation diminishing Runge-Kutta method. A limiter based on the characteristic field decomposition is applied to maintain stability and non-oscillatory property of the RKDG method. For multi-medium fluid simulation, the two cells adjacent to the interface are treated differently from other cells. At first, a linear Riemann solver is applied to calculate the numerical flux at the interface. Numerical examples show that there is some oscillation in the vicinity of the interface. Then a nonlinear Riemann solver based on the characteristic formulation of the equation and the discontinuity relations is adopted to calculate the numerical flux at the interface, which suppress the oscillation successfully. Several single-medium and multi-medium fluid examples are given to demonstrate the reliability and efficiency of the algorithm.     

Convergence of the viscosity method for isentropic gas dynamics

A convergence theorem for the method of artificial viscosity applied to the isentropic equations of gas dynamics is established. Convergence of a subsequence in the strong topology is proved without uniform estimates on the derivatives using the theory of compensated compactness and an analysis of progressing entropy waves.     

Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion

The asymptotic stability of traveling wave solutions with shock profile is investigated for several systems in gas dynamics. 1) The solution of a scalar conservation law with viscosity approaches the traveling wave solution at the ratet ^– (for some>0) ast, provided that the initial disturbance is small and of integral zero, and in addition decays at an algebraic rate for |x|. 2) The traveling wave solution with Nishida and Smoller's condition of the system of a viscous heat-conductive ideal gas is asymptotically stable, provided the initial disturbance is small and of integral zero. 3) The traveling wave solution with weak shock profile of the Broadwell model system of the Boltzmann equation is asymptotically stable, provided the initial disturbance is small and its hydrodynamical moments are of integral zero. Each proof is given by applying an elementary energy method to the integrated system of the conservation form of the original one. The property of integral zero of the initial disturbance plays a crucial role in this procedure.     

The Fermi gas model of one-dimensional conductors

The Fermi gas model of one-dimensional conductors is reviewed. The exact solutions known for particular values of the coupling constants in a single chain problem (Tomonaga model, Luther-Emery model) are discussed. Renormalization group arguments are used to extend these solutions to arbitrary values of the couplings. The instabilities and possible ground states are studied by investigating the behaviour of the response functions. The relationship between this model and others is discussed and is used to obtain further information about the behaviour of the system. The model is generalized to a set of coupled chains to describe quasi-one-dimensional systems. The crossover from one-dimensional to three-dimensional behaviour and the type of ordering are discussed.     

Analytical solution of navier-stokes flow of a viscous compressible fluid

An analytical procedure is proposed to study the flow of viscous compressible continuous fluids.     

Convergence of the fractional step Lax-Friedrichs scheme and Godunov scheme for the isentropic system of gas dynamics

A convergence theorem of the fractional step Lax-Friedrichs scheme and Godunov scheme for an inhomogeneous system of isentropic gas dynamics (1<5/3) is established by using the framework of compensated compactness. Meanwhile, a corresponding existence theorem of global solutions with large data containing the vacuum is obtained.Partially supported by U.S. NSF Grant # DMS-850403     

On the instability of an isentropic model for a gaseous relativistic star

It is shown that the isentropic perfect fluid subclass of Buchdahl's exact solution for a gaseous relativistic star is unstable.     

Kinetic formulation of the isentropic gas dynamics andp-systems

We consider the 2×2 hyperbolic system of isentropic gas dynamics, in both Eulerian or Lagrangian variables (also called thep-system). We show that they can be reformulated as a kinetic equation, using an additional kinetic variable. Such a formulation was first obtained by the authors in the case of multidimensional scalar conservation laws. A new phenomenon occurs here, namely that the advection velocity is now a combination of the macroscopic and kinetic velocities. Various applications are given: we recover the invariant regions, deduce newL estimates using moments lemma and proveL –w* stability for 3.     

Structure result for steady-state solutions of a one-dimensional Frémond model of SMA

This paper deals with the complete characterization of solutions to an elliptic system of variational inequalities. The latter model arises in the study of the long-time behavior of shape memory materials and is suitable of describing a variety of experimentally observed phenomena.     

The stability of an isentropic model for a gaseous relativistic star

We show that the isentropic subclass of Buchdahl’s exact solution for a gaseous relativistic star is stable and gravitationally bound for all values of the compactness ratio u [≡ (M/R), where M is the total mass and R is the radius of the configuration in geometrized units] in the range, 0 <  u ≤  0.20, corresponding to the regular behaviour of the solution. This result is in agreement with the expectation and opposite to the earlier claim found in the literature.     

Global spherically symmetric solutions to the equations of a viscous polytropic ideal gas in an exterior domain

We consider the equations of a viscous polytropic ideal gas in the domain exterior to a ball in ⁿ (n=2 or 3) and prove the global existence of spherically symmetric smooth solutions for (large) initial data with spherical symmetry. The large-time behavior of the solutions is also discussed. To prove the existence we first study an approximate problem in a bounded annular domain and then obtain a priori estimates independent of the boundedness of the annular domain. Letting the diameter of the annular domain tend to infinity, we get a global spherically symmetric solution as the limit.Dedicated to Professor Rolf Leis on the occasion of his 65th birthdaySupported by the SFB 256 of the Deutsche Forschungsgemeinschaft at the University of Boon.     

Some exact solutions of magnetized viscous model in string cosmology

In this paper, we study anisotropic Bianchi-V Universe with magnetic field and bulk viscous fluid in string cosmology. Exact solutions of the field equations are obtained by using the equation of state (EoS) for a cloud of strings, and a relationship between bulk viscous coefficient and scalar expansion. The bulk viscous coefficient is assumed to be inversely proportional to the expansion scalar. It is interesting to examine the effects of magnetized bulk viscous string model in early and late stages of evolution of the Universe. This paper presents different string models like geometrical (Nambu string), Takabayasi (p-string) and Reddy string models by taking certain physical conditions. We discuss the nature of classical potential for viscous fluid with and without magnetic field. The presence of bulk viscosity stops the Universe from becoming empty in its future evolution. It is observed that the Universe expands with decelerated rate in the presence of viscous fluid with magnetic field whereas, it expands with marginal inflation in the presence of viscous fluid without magnetic field. The other physical and geometrical aspects of each string model are discussed in detail.     

Condensation of a one-dimensional lattice gas

We consider a one-dimensional lattice gas in the canonical ensemble with interaction energy 1/r , 1<2. Using an energy-entropy argument we show that the gas condenses at sufficiently low temperatures meaning that the gas has a non-uniform density in the thermodynamic limit.Research supported by the Swedish research councils NFR and STUF     

All flow-stationary cylindrically symmetric solutions of the Einstein field equations for a rotating isentropic perfect fluid

In previous papers by the author a class of flow-stationary cylindrically symmetric solutions of the Einstein field equations for a rotating isentropic perfect fluid was found. The present paper shows how all such solutions may be obtained by methods very similar to those used previously. The solutions depend on one variable and contain one completely arbitrary function f of that variable. The choice of a definite form of f corresponds to fixing the equation of state. After this is fixed, the enthalpy per unit rest-energy of the fluid, H, is determined by a linear homogeneous differential equation of second order, and all the other components of the metric are algebraically determined in terms of f and H.     

Nonlinear model of a granulated medium containing viscous fluid layers and gas cavities

We analyze nonlinear oscillations and waves in a simple model of a granular medium containing inclusions in the form of fluid layers and gas cavities. We show that in such a medium, the velocity of one of the wave modes is low; therefore, the nonlinearity is high and the effects of interaction are more strongly expressed than usual.