STABILITY OF THE RAREFACTION WAVE IN THE SINGULAR LIMIT OF A SHARP INTERFACE PROBLEM FOR THE COMPRESSIBLE NAVIER-STOKES/ALLEN-CAHN SYSTEM

This paper is concerned with the global well-posedness of the solution to the compressible Navier-Stokes/Allen-Cahn system and its sharp interface limit in one-dimensional space.For the perturbations with small energy but possibly large oscillations of rarefaction wave solutions near phase separation,and where the strength of the initial phase field could be arbitrarily large,we prove that the solution of the Cauchy problem exists for all time,and converges to the centered rarefaction wave solution of the corresponding standard two-phase Euler equation as the viscosity and the thickness of the interface tend to zero.The proof is mainly based on a scaling argument and a basic energy method.

NONLINEAR STABILITY OF RAREFACTION WAVES TO THE COMPRESSIBLE NAVIER-STOKES EQUATIONS FOR A REACTING MIXTURE WITH ZERO HEAT CONDUCTIVITY

In this paper,we study the time-asymptotically nonlinear stability of rarefaction waves for the Cauchy problem of the compressible Navier-Stokes equations for a reacting mixture with zero heat conductivity in one dimension.If the corresponding Riemann problem for the compressible Euler system admits the solutions consisting of rarefaction waves only,it is shown that its Cauchy problem has a unique global solution which tends time-asymptotically towards the rarefaction waves,while the initial perturbation and the strength of rarefaction waves are suitably small.

STABILITY OF THE RAREFACTION WAVE FOR THE GENERALIZED KDV-BURGERS EQUATION

This paper is concerned with the stability of the rarefaction wave for the generalized KdV-Burgers equation [GRAPHICS] Roughly speaking, under the assumption that u(-) < u(+), the solution u(x, t) to Cauchy problem (1) satisfying (sup)(x&ISIN;R)\u(x, t) - u(R)(x/t)\ --> 0 as t --> infinity, where u(R)(x/t) is the rarefaction wave of the non-viscous Burgers equation u(t) + f(u)(x) = 0 with Riemann initial data [GRAPHICS]

ASYMPTOTIC STABILITY OF RAREFACTION WAVE FOR GENERALIZED BURGERS EQUATION

This paper is concerned with the stability of the rarefaction wave for the Burgers equationwhere 0 ≤ a < 1/4p (q is determined by (2.2)). Roughly speaking, under the assumption that u_ < u+, the authors prove the existence of the global smooth solution to the Cauchy problem (I), also find the solution u(x, t) to the Cauchy problem (I) satisfying sup |u(x, t) -uR(x/t)| → 0 as t → ∞, where uR(x/t) is the rarefaction wave of the non-viscous Burgersequation ut + f(u)x = 0 with Riemann initial data u(x, 0) =

Analysis of the stress wave and rarefaction wave produced by hypervelocity impact of sphere onto thin plate

Shock wave is emitted into the plate and sphere when a sphere hypervelocity impacts onto a thin plate.The fragmentation and phase change of the material caused by the propagation and unloading of shock wave could result in the formation of debris cloud eventually.Propagation models are deduced based on one-dimensional shock wave theory and the geometry of sphere,which uses elliptic equations(corresponding to ellipsoid equations in physical space)to describe the propagation of shock wave and the rarefaction wave.The“Effective thickness”is defined as the critical plate thickness that ensures the rarefaction wave overtake the shock wave at the back of the sphere.The“Effective thickness”is directly related to the form of the debris cloud.The relation of the“Effective thickness”and the“Optimum thickness”is also discussed.The impacts of Al spheres onto Al plates are simulated within SPH to verify the propagation models and associated theories.The results show that the wave fronts predicted by the propagation models are closer to the simulation result at higher impact velocity.The curvatures of the wave fronts decrease with the increase of impact velocities.The predicted“Effective thickness”is consistent with the simulation results.The analysis about the shock wave propagation and unloading in this paper can provide a new sight and inspiration for the quantitative study of hypervelocity impact and space debris protection.

Condensation Induced by Rarefaction Waves and Reflected Rarefaction Waves

In this paper,homogeneous condensation induced by unsteady rarefaction waves and reflected rarefaction waves in vapor-gas mixture was investigated experimentally.It is shown that the temperature of condensation onset during very fast unsteady expansion in vapor-gas mixture is much lower than that during equilibrium process in the atmosphere. It is of interest to indicate that the size of droplets approximates a constant,but the number density and the mass density of droplets change rapidly in the region of static flow.

ASYMPTOTIC STABILITY OF RAREFACTION WAVE FOR HYPERBOLIC-ELLIPTIC COUPLED SYSTEM IN RADIATING GAS

In this article, authors study the Cauch problem for a model of hyperbolic-elliptic coupled system derived from the one-dimensional system of the rudiating gas. By considering the initial data as a small disturbances of rarefaction wave of inviscid Burgers equation, the global existence of the solution to the corresponding Cauchy problem and asymptotic stability of rarefaction wave is proved. The analysis is based on a priori estimates and L^2-energy method.

ASYMPTOTIC RAREFACTION WAVES FOR BALANCE LAWS WITH STIFF SOURCES

We study the long time formation of rarefaction waves appearing in balance laws by means of singular perturbation methods. The balance laws are non standard because they contain a variable u that appears only in the flux terms. We present a concrete example occurring in flow of steam, nitrogen and water in porous media and an abstract example for a class of systems of three equations. In the concrete example the zero-order equations resulting from the expansion yield a type of conservation law system called compositional model in Petroleum Engineering. In this work we show how compositional models originate from physically more fundamental systems of balance laws. Under appropriate conditions, we prove that certain solutions of the system of balance laws decay with time to rarefaction wave solutions in the compositional model originating from the system of balance laws.

DECAY RATES OF PLANAR VISCOUS RAREFACTION WAVE FOR MULTI-DIMENSIONAL SCALAR CONSERVATION LAW WITH DEGENERATE VISCOSITY ON HALF SPACE

We investigate the decay rates of the planar viscous rarefaction wave of the initial-boundary value problem to scalar conservation law with degenerate viscosity in several dimensions on the half-line space, where the corresponding one-dimensional problem admits the rarefaction wave as an asymptotic state. The analysis is based on the standard L2-energy method and L1-estimate.

ZERO DISSIPATION LIMIT TO RAREFACTION WAVES FOR THE ONE-DIMENSIONAL COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH SELECTED DENSITY-DEPENDENT VISCOSITY

This paper is devoted to studying the zero dissipation limit problem for the one-dimensional compressible Navier-Stokes equations with selected density-dependent viscosity.In particular,we focus our attention on the viscosity taking the formμ(ρ)=ρ^(ϵ)(ϵ>0).For the selected density-dependent viscosity,it is proved that the solutions of the one-dimensional compressible Navier-Stokes equations with centered rarefaction wave initial data exist for all time,and converge to the centered rarefaction waves as the viscosity vanishes,uniformly away from the initial discontinuities.New and subtle analysis is developed to overcome difficulties due to the selected density-dependent viscosity to derive energy estimates,in addition to the scaling argument and elementary energy analysis.Moreover,our results extend the studies in[Xin Z P.Comm Pure Appl Math,1993,46(5):621-665].

ASYMPTOTIC STABILITY OF A BOUNDARY LAYER AND RAREFACTION WAVE FOR THE OUTFLOW PROBLEM OF THE HEAT-CONDUCTIVE IDEAL GAS WITHOUT VISCOSITY

This article is devoted to studying the initial-boundary value problem for an ideal polytropic model of non-viscous and compressible gas.We focus our attention on the outflow problem when the flow velocity on the boundary is negative and give a rigorous proof of the asymptotic stability of both the degenerate boundary layer and its superposition with the 3-rarefaction wave under some smallness conditions.New weighted energy estimates are introduced,and the trace of the density and velocity on the boundary are handled by some subtle analysis.The decay properties of the boundary layer and the smooth rarefaction wave also play an important role.

NONLINEAR STABILITY OF RAREFACTION WAVES FOR A COMPRESSIBLE MICROPOLAR FLUID MODEL WITH ZERO HEAT CONDUCTIVITY

In 2018,Duan[1]studied the case of zero heat conductivity for a one-dimensional compressible micropolar fluid model.Due to the absence of heat conductivity,it is quite difficult to close the energy estimates.He considered the far-field states of the initial data to be constants;that is,lim x→±∞(v0,u0,w0,θ0)(x)=(1,0,0,1).He proved that the solution tends asymptotically to those constants.In this article,under the same hypothesis that the heat conductivity is zero,we consider the far-field states of the initial data to be different constants-that is,lim x→±∞(v0,u0,w0,θ0)(x)=(v±,u±,o,θ±)-and we prove that if both the initial perturbation and the strength of the rarefaction waves are assumed to be suitably small,the Cauchy problem admits a unique global solution that tends time-asymptotically toward the combination of two rarefaction waves from different families.

STABILITY OF RAREFACTION WAVE FOR A MACROSCOPIC MODEL DERIVED FROM THE VLASOV-MAXWELL-BOLTZMANN SYSTEM

In this article, we are concerned with the nonlinear stability of the rarefaction wave for a one-dimensional macroscopic model derived from the Vlasov-Maxwell-Boltzmann system. The result shows that the large-time behavior of the solutions coincides with the one for both the Navier-Stokes-Poisson system and the Navier-Stokes system. Both the timedecay property of the rarefaction wave profile and the influence of the electromagnetic field play a key role in the analysis.

Firing Stability of Rarefaction Wave Gun Based on Virtual Prototype

The dynamics characteristics of rarefaction wave gun( RAVEN) in launching are studied. Based on its characteristics of structure and interior ballistics, the launch dynamic model and virtual prototype of RAVEN are established. After simulating and solving by ADAMS software, through comparison RAVEN to conventional gun,influences of some structural parameters on firing stability are acquired. Optimization analysis of structural parameters of gun based on analysis of the firing stability is done. It puts forward the theoretic basis for improving firing stability and security.

On the Stability of Rarefaction Wave Solutions for Viscous p-system with Boundary Effect

Abstract The inflow problem in the supersonic case for a one-dimensional compressible viscous gas on the half line (0,+X) is investigated. A stability theorem concerning the long time behavior of rarefaction wave is established.

Convergence to the Rarefaction Wave for a Model of Radiating Gas in One-dimension

In this paper, a sequence of solutions to the one-dimensional motion of a radiating gas are con- structed. Furthermore, when the absorption coefficient a tends to oo, the above solutions converge to the rarefaction wave, which is an elementary wave pattern of gas dynamics, with a convergence rate α -1/3｜lnα｜2.

Zero Dissipation Limit to Rarefaction Waves for the p-System

We study the zero-dissipation problem for a one-dimensional model system for the isentropic flow of a compressible viscous gas, the so-called p-system with viscosity. When the solution of the inviscid problem is a rarefaction wave with finite strength, there exists unique solution to the viscous problem with the same initial data which converges to the given inviscid solution as c goes to zero. The proof consists of a scaling argument and elementary energy analysis, based on the underlying wave structure.

Stability of Rarefaction Wave for Compressible Navier-Stokes Equations on the Half Line

This paper is concerned with the large-time behavior of solutions to an initial-boundary-value problem for full compressible Navier-Stokes equations on the half line（0,∞）,which is named impermeable wall problem.It is shown that the 3-rarefaction wave is stable under partially large initial perturbation if the adiabatic exponent γ is close to 1.Here partially large initial perturbation means that the perturbation of absolute temperature is small,while the perturbations of specific volume and velocity can be large.The proof is given by the elementary energy method.

Development of a rarefaction wave at discharge initiation in a storage silo--DEM simulations

The generation of a rarefaction wave at the initiation of discharge from a storage silo is a phenomenon of scientific and practical interest. The effect, sometimes termed the dynamic pressure switch, may create dangerous pulsations of the storage structure. Owing to the nonlinearity, discontinuity, and heterogeneity of granular systems, the mechanism of generation and propagation of stress waves is complex and not yet completely understood. The present study conducted discrete element simulations to model the formation and propagation of a rarefaction wave in a granular material contained in a silo. Modeling was performed for a flat-bottom cylindrical container with diameter of 0.1 or 0.12 m and height of 0.5 m. The effects of the orifice size and the shape of the initial discharging impulse on the shape and extent of the rarefaction wave were examined. Positions, velocities, and forces of particles were recorded every 10-5 s and used to infer the location of the front of the rarefaction wave and loads on construction members. Discharge through the entire bottom of the bin generates a plane rarefaction wave that may be followed by a compaction wave, depending on the discharge rate. Discharge through the orifice generates a spherical rarefaction wave that, after reflection from the silo wall, travels up the silo as a sequence of rarefaction-compaction cycles with constant wavelength equal to the silo diameter, During the travel of the wave along the bin height, the wave amplitude increases with the distance traveled. Simulations confirmed earlier findings of laboratory and numerical （finite element method） experiments and a theoretical approach, estimating the speed of the front of the rarefaction wave to range from 70 to 80 m/s and the speed of the tail to range from 20 to 60 m/s.

Rarefaction Wave Interaction and Shock-Rarefaction Composite Wave Interaction for a Two-Dimensional Nonlinear Wave System

In order to construct global solutions to two-dimensional(2 D for short)Riemann problems for nonlinear hyperbolic systems of conservation laws,it is important to study various types of wave interactions.This paper deals with two types of wave interactions for a 2 D nonlinear wave system with a nonconvex equation of state:Rarefaction wave interaction and shock-rarefaction composite wave interaction.In order to construct solutions to these wave interactions,the authors consider two types of Goursat problems,including standard Goursat problem and discontinuous Goursat problem,for a 2 D selfsimilar nonlinear wave system.Global classical solutions to these Goursat problems are obtained by the method of characteristics.The solutions constructed in the paper may be used as building blocks of solutions of 2 D Riemann problems.