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.

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]

Rarefaction and Temperature Gradient Effect on the Performance of the Knudsen Pump

The prediction of the multiscale flow in the Knudsen pump is important for understanding its pumping mechanism.However,there is little research on such interesting multiscale phenomenon in the Knudsen pumps.In this paper,a novel numerical analysis method combining the direct simulation Monte Carlo（DSMC） method with the smoothed particle hydrodynamics（SPH） method is presented for simulating the multiscale flow,which is often encountered in the application of the Knudsen pumps.Validity and accuracy of the new method are given by comparing its results with that of the previous research.Using the coupled multiscale approach,the rarefaction and the temperature drive are studied,which are two main factors on the performance of the Knudsen pumps.To investigate the effect of rarefaction on the performance of the Knudsen pump,various pump operation pressures are compared.The flow characteristics and pumping ability at different rarefaction are analyzed,and the phenomenon of the multiscale flow is also discussed.Several cases with different linear or nonlinear temperature gradients are set to investigate the effect of temperature gradient on the performance of the Knudsen pump.The flow characteristics of the Knudsen pump such as the velocity,pressure increase,and the mass flowrate are presented.A unique phenomenon,the reverse transpiration effect caused by the nonlinear temperature gradient is studied,and the reason of the significant pressure increase in the pump channel is also analyzed.Since the multiscale gas flow is widely encountered in the microflow systems,the above method and its results can also be greatly beneficial and provide significant insights for the design of the MEMS devices.

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.

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.

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.

Numerical method for simulating rarefaction shocks in the approximation of phase-flip hydrodynamics

A finite-difference algorithm is proposed for numerical modeling of hydrodynamic flows with rarefaction shocks, in which the fluid undergoes a jump-like liquid-gas phase transition. This new type of flow discontinuity, unexplored so far in computational fluid dynamics, arises in the approximation of phase-flip(PF) hydrodynamics, where a highly dynamic fluid is allowed to reach the innermost limit of metastability at the spinodal, upon which an instantaneous relaxation to the full phase equilibrium(EQ) is assumed. A new element in the proposed method is artificial kinetics of the phase transition, represented by an artificial relaxation term in the energy equation for a "hidden"component of the internal energy, temporarily withdrawn from the fluid at the moment of the PF transition. When combined with an appropriate variant of artificial viscosity in the Lagrangian framework, the latter ensures convergence to exact discontinuous solutions, which is demonstrated with several test cases.

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.

ASYMPTOTIC STABILITY OF THE RAREFACTION WAVE FOR THE NON-VISCOUS AND HEAT-CONDUCTIVE IDEAL GAS IN HALF SPACE

This article is concerned with the impermeable wall problem for an ideal polytropic model of non-viscous and heat-conductive gas in one-dimensional half space. It is shown that the 3-rarefaction wave is stable under some smallness conditions. The proof is given by an elementary energy method and the key point is to do the higher order derivative estimates with respect to t because of the less dissipativity of the system and the higher order derivative boundary terms.

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.

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.

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.

Asymptotic Stability of Combination of Contact Discontinuity with Rarefaction Waves for the One-Dimensional Viscous Micropolar Fluid Model

In this paper, we consider with the large time behavior of solutions of the Cauchy problem to the one-dimensional compressible micropolar fluid model, where the far field states are prescribed. When the corresponding Riemann problem for the Euler system admits the solution consisting of contact discontinuity and rarefaction waves, it is shown that the combination wave corresponding to the contact discontinuity, with rarefaction waves is asymptotically stable provided that the strength of the combination wave and the initial perturbation are suitably small. This result is proved by using elementary L2-energy methods.

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.

Effects of rarefaction on the characteristics of micro gas journal bearings

Given the definition of the reference Knudsen number for micro gas journal bearings,the range in the number is related to the viscosity of air at different temperatures. A modified Reynolds equation for micro gas journal bearings based on Burgdorfer's first-order slip boundary condition is proposed that takes into account the gas rarefaction effect. The finite difference method (FDM) is adopted to solve the modified Reynolds equation to obtain the pressure profiles,load capacities and attitude angles for micro gas journal bearings at different reference Knudsen numbers,bearing numbers and journal eccentricity ratios. Numerical analysis shows that pressure profiles and non-dimensional load capacities decrease markedly as gas rarefaction in-creases. Attitude angles change conversely,and when the eccentricity ratio is less than 0.6,the attitude angles rise slightly and the influence of the reference Knudsen number is not marked. In addition,the effect of gas rarefaction on the non-dimensional load capacity and attitude angle decreases with smaller bearing numbers.