Asymptotic Stability of Rarefaction Wave for the Navier–Stokes Equations for a Compressible Fluid in the Half Space

This paper is concerned with the asymptotic stability towards a rarefaction wave of the solution to an outflow problem for the Navier–Stokes equations in a compressible fluid in the Eulerian coordinate in the half space. This is the second one of our series of papers on this subject. In this paper, firstly we classify completely the time-asymptotic states, according to some parameters, that is the spatial-asymptotic states and boundary conditions, for this initial boundary value problem, and some pictures for the classification of time-asymptotic states are drawn in the state space. In order to prove the stability of the rarefaction wave, we use the solution to Burgers’ equation to construct a suitably smooth approximation of the rarefaction wave and establish some time-decay estimates in L ^p -norm for the smoothed rarefaction wave. We then employ the L ²-energy method to prove that the rarefaction wave is non-linearly stable under a small perturbation, as time goes to infinity. P. Zhu was supported by JSPS postdoctoral fellowship under P99217.     

Asymptotic Stability of Combination of Viscous Contact Wave with Rarefaction Waves for One-Dimensional Compressible Navier–Stokes System

We are concerned with the large-time behavior of solutions of the Cauchy problem to the one-dimensional compressible Navier–Stokes system for ideal polytropic fluids, where the far field states are prescribed. When the corresponding Riemann problem for the compressible Euler system admits the solution consisting of contact discontinuity and rarefaction waves, it is proved that for the one-dimensional compressible Navier–Stokes system, the combination wave of a “viscous contact wave”, which corresponds to the contact discontinuity, with rarefaction waves is asymptotically stable, provided the strength of the combination wave is suitably small. This result is proved by using elementary energy methods.     

Wave interactions and variation estimates for self-similar zero-viscosity limits in systems of conservation laws

We consider the problem of self-similar zero-viscosity limits for systems ofN conservation laws. First, we give general conditions so that the resulting boundary-value problem admits solutions. The obtained existence theory covers a large class of systems, in particular the class of symmetric hyperbolic systems. Second, we show that if the system is strictly hyperbolic and the Riemann data are sufficiently close, then the resulting family of solutions is of uniformly bounded variation and oscillation. Third, we construct solutions of the Riemann problem via self-similar zero-viscosity limits and study the structure of the emerging solution and the relation of self-similar zero-viscosity limits and shock profiles. The emerging solution consists ofN wave fans separated by constant states. Each wave fan is associated with one of the characteristic fields and consists of a rarefaction, a shock, or an alternating sequence of shocks and rarefactions so that each shock adjacent to a rarefaction on one side is a contact discontinuity on that side. At shocks, the solutions of the self-similar zero-viscosity problem have the internal structure of a traveling wave.     

Simple waves in a plasma with anisotropic pressure

Fast and slow simple waves are studied in the framework of the anisotropic magnetohydrodynamics of Chew, Goldberger, and Low [1]. Baranov [2] has constructed fields of integral curves for fast and slow waves and in two special cases has shown that such waves break in the compression section. The possibility of breaking of a slow wave in a rarefaction section was noted by Akhiezer et al. [3]. However, their general relations in simple waves [3] have been shown to be incorrect [2, 4]. In the present paper the nature of the variation of the longitudinal and transverse plasma pressures is determined, and the problem of the breaking of fast and slow waves is completely solved. Conditions under which a slow wave breaks in a rarefaction section are found. A fast wave always breaks in a rarefaction section.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 181–183, July–August, 1988.     

Hyperbolicity singularities in Rarefaction Waves

For mixed-type systems of conservation laws, rarefaction waves may contain states at the boundary of the elliptic region, where two characteristic speeds coincide, and the Lax family of the wave changes. Such contiguous rarefaction waves form a single fan with a continuous profile. Different pairs of families may appear in such rarefactions, giving rise to novel Riemann solution structures. We study the structure of such rarefaction waves near regular and exceptional points of the elliptic boundary and describe their effect on Riemann solutions.     

Solution of Riemann problem for dusty gas flow

A direct approach is used to solve the Riemann problem for a quasilinear hyperbolic system of equations governing the one dimensional unsteady planar flow of an isentropic, inviscid compressible fluid in the presence of dust particles. The elementary wave solutions of the Riemann problem, that is, shock waves, rarefaction waves and contact discontinuities are derived and their properties are discussed for a dusty gas. The generalised Riemann invariants are used to find the solution between rarefaction wave and the contact discontinuity and also inside rarefaction fan. Unlike the ordinary gasdynamic case, the solution inside the rarefaction waves in dusty gas cannot be obtained directly and explicitly; indeed, it requires an extra iteration procedure. Although the case of dusty gas is more complex than the ordinary gas dynamics case, all the parallel results for compressive waves remain identical. We also compare/contrast the nature of the solution in an ordinary gasdynamics and the dusty gas flow case.     

Analysis of some supersonic nonsymmetric nozzles

Supersonic nonsymmetric plane nozzles, which are characterized mainly by centered compression and rarefaction waves, were constructed earlier in [1]. An intensive compression wave is undesirable in a number of cases, because of the possibility of boundary-layer detachment on the nozzle walls, for instance. Hence, some constraints on the pressure gradient, for example, the condition that the pressure does not grow at the walls, must be included in the formulation of the problem. The presence of this last condition distinguishes the results in [1] only by the fact that the centered compression waves in the nozzles are replaced by constant parameter sections. Considered below under identical conditions are nonsymmetric nozzles with compression waves, with constant parameter sections, and, also, known nozzles with a straight lower wall for comparison.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 154–156, May–June, 1976.The author is grateful to A. N. Kraiko for his interest and attention to the research and for useful discussions.     

Pairwise Wave Interactions in Ideal Polytropic Gases

We consider the problem of resolving all pairwise interactions of shock waves, contact waves, and rarefaction waves in the one-dimensional flow of an ideal polytropic gas. Here, resolving an interaction means to determine the types of the three outgoing (backward, contact, and forward) waves in the Riemann problem defined by the extreme left and right states of the two incoming waves, together with possible vacuum formation. This problem has been considered by several authors and turns out to be surprisingly involved. For each type of interaction (head-on, involving a contact, or overtaking) the outcome depends on the strengths of the incoming waves. In the case of overtaking waves the type of the reflected wave also depends on the value of the adiabatic constant. Our analysis provides a complete breakdown and gives the exact outcome of each interaction.     

Vanishing Viscosity Limit of the Compressible Navier–Stokes Equations for Solutions to a Riemann Problem

We study the vanishing viscosity limit of the compressible Navier–Stokes equations to the Riemann solution of the Euler equations that consists of the superposition of a shock wave and a rarefaction wave. In particular, it is shown that there exists a family of smooth solutions to the compressible Navier–Stokes equations that converges to the Riemann solution away from the initial and shock layers at a rate in terms of the viscosity and the heat conductivity coefficients. This gives the first mathematical justification of this limit for the Navier–Stokes equations to the Riemann solution that contains these two typical nonlinear hyperbolic waves.     

Strong explosion in a combustible gas mixture

Assume that a planar, cylindrical, or spherical point explosion takes place in a combustible mixture of gases. As a result of the explosion a strong shock wave develops and triggers chemical reactions with the release of heat. The solution of the problem for the case in which the thickness of the heat release zone is neglected (the infinitely thin detonation wave model) was obtained in [1–3].It was emphasized in [4] that these solutions can be considered only as asymptotic solutions for time and distance scales which are large in comparison with the scales which are characteristic for the chemical reactions, and under the assumption that as the overdriven detonation wave which is formed in the explosion is weakened by the rarefaction waves it does not degenerate into an ordinary compression shock. Here the question remains open of the possibility of obtaining such asymptotic solutions with account for finite chemical-reaction rates.In conclusion the authors wish to thank E. Bishimov for carrying out most of the computations for this study.     

An Exact Riemann Solver for Multicomponent Turbulent Flow

The main objective of this paper is to provide some adequate way to compute the non-conservative hyperbolic system which describes a multicomponent turbulent flow. The model is written for an isentropic gas. The exact solution of the Riemann Problem (RP) associated to the hyperbolic system is exhibited. It is composed of constant states separated by rarefaction waves, or shock waves and a contact discontinuity.

The selection of the admissible part of the shock curve is obtained using an entropy criterion. This entropy is the total energy of the system. Thanks to the latter, one may compute the exact solution of the Riemann problem, assuming genuinely non linear fields contain sufficiently weak shocks.     

Solution to the Riemann problem for drift-flux model with modified Chaplygin two-phase flows

In this paper, we concern about the Riemann problem for compressible no-slip drift-flux model which represents a system of quasi-linear partial differential equations derived by averaging the mass and momentum conservation laws with modified Chaplygin two-phase flows. We obtain the exact solution of Riemann problem by elaborately analyzing characteristic fields and discuss the elementary waves namely, shock wave, rarefaction wave and contact discontinuity wave. By employing the equality of pressure and velocity across the middle characteristic field, two nonlinear algebraic equations with two unknowns as gas density ahead and behind the middle wave are formed. The Newton–Raphson method of two variables is applied to find the unknowns with a series of initial data from the literature. Finally, the exact solution for the physical quantities such as gas density, liquid density, velocity, and pressure are illustrated graphically.     

Dynamics of vapor-gas bubbles

The nonlinear problem of thermal, mass, and dynamic interaction between a vapor-gas bubble and a liquid is considered. The results of numerical solution of the problem of radial motion of the bubble caused by a sudden pressure change in the liquid, illustrating the behavior of vapor-gas bubbles in compression and rarefaction waves, are presented. The corresponding problem for single-component gas and vapor bubbles was considered in [1, 2].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 56–61, November–December. 1976.     

One-dimensional linear waves with axial and central symmetries in saturated porous media

The features of propagation of one-dimensional monochromatic waves and dynamics of weak perturbations with axial and central symmetries in liquid-saturated porous medium are investigated. Non-stationary interaction forces and viscoelastic skeleton characteristics are taken into account. The research is carried out within the two-velocity, two-stress tensor model by applying methods of multiphase media mechanics. The system of equations is solved numerically by applying Fast Fourier Transform (FFT) algorithm. The influence of geometry of the process on wave propagation behavior is studied.It is shown that the initial pressure perturbation splits into two waves: fast (deformational) wave and slow (filtrational) one. Each of them is followed by the balance wave: that is, rarefaction wave after compression wave and compression wave after rarefaction wave; at that slow wave and balance one following fast wave may interfere.     

The Riemann problem for a hyperbolic model of two-phase flow in conservative form

This article is to continue the present author's work (International Journal of Computational Fluid Dynamics (2009) 23 (9), 623–641) on studying the structure of solutions of the Riemann problem for a system of three conservation laws governing two-phase flows. While existing solutions are limited and found quite recently for the Baer and Nunziato equations, this article presents the first instance of an exact solution of the Riemann problem for two-phase flow in gas–liquid mixture. To demonstrate the structure of the solution, we use a hyperbolic conservative model with mechanical equilibrium and without velocity equilibrium. The Riemann problem solution for the model equations comprises a set of elementary waves, rarefaction and discontinuous waves of various types. In particular, such a solution treats both the wave structure and the intermediate states of the two-phase gas–liquid mixture. The resulting exact Riemann solver is fully non-linear, direct and complete. On this basis then, we use locally the exact Riemann solver for the two-phase flow in gas–liquid mixture within the framework of finite volume upwind Godunov methods. In order to demonstrate the effectiveness and accuracy of the proposed solver, we consider a series of test problems selected from the open literature and compare the exact and numerical results by using upwind Godunov methods, showing excellent oscillation-free results in two-phase fluid flow problems.     

WAVE INTERACTIONS IN SULICIU MODEL FOR DYNAMIC PHASE TRANSITIONS

Elementary waves in Suliciu model for dynamic phase transitions are obtained through traveling wave analysis.For any given initial data with two pieces of constant states,the Riemann solutions are constructed as a combination of elementary waves. When the initial profile contains three pieces of constant states,the solution may be constructed from the Riemann solutions,with each two adjacent states connected by elementary waves.A new Riemann problem forms when these two waves collide.Through the exploration of these Riemann problems,the outcome of wave interactions may be classified in a suitable parametric space.     

Interaction of a wave with an obstacle in a multicomponent two-phase medium

The author's model [1] of a multicomponent liquid medium with nonlinear limiting compression diagrams and constant coefficient of viscosity is improved by the introduction of a coefficient of viscosity that varies during the deformation. The new model is used to obtain a numerical solution to the problem of the propagation of a plane wave produced by a shock load and the interaction of the wave with a fixed obstacle. Such a problem was solved earlier [2] in the case of a viscous medium for linear diagrams of static and dynamic compression and constant coefficient of viscosity. It is shown that the nonlinearity of the diagram of static compression leads with increasing pressure first to an increase in the reflection coefficient and then to a decrease of it. If the load has a sufficient duration, the initial section of the obstacle is subject to a succession of several waves, the number of which increases with increasing duration and amplitude of the load. The calculation was made for glycerine with air bubbles. It is assumed that at pressures up to 400·10⁵ N/m² glycerine is a linearly elastic medium In this case, the dynamic compression diagram of the two-component glycerine—gas-bubble medium is linear.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 183–187, May–June, 1981.I thank Yu. A. Sozonenko for discussion and valuable comments.     

An explicit formulation for the evolution of nonlinear surface waves interacting with a submerged body

An explicit formulation to study nonlinear waves interacting with a submerged body in an ideal fluid of infinite depth is presented. The formulation allows one to decompose the nonlinear wave–body interaction problem into body and free‐surface problems. After the decomposition, the body problem satisfies a modified body boundary condition in an unbounded fluid domain, while the free‐surface problem satisfies modified nonlinear free‐surface boundary conditions. It is then shown that the nonlinear free‐surface problem can be further reduced to a closed system of two nonlinear evolution equations expanded in infinite series for the free‐surface elevation and the velocity potential at the free surface. For numerical experiments, the body problem is solved using a distribution of singularities along the body surface and the system of evolution equations, truncated at third order in wave steepness, is then solved using a pseudo‐spectral method based on the fast Fourier transform. A circular cylinder translating steadily near the free surface is considered and it is found that our numerical solutions show excellent agreement with the fully nonlinear solution using a boundary integral method. We further validate our solutions for a submerged circular cylinder oscillating vertically or fixed under incoming nonlinear waves with other analytical and numerical results. Copyright © 2007 John Wiley & Sons, Ltd.     

The Benjamin–Lighthill Conjecture for Steady Water Waves (revisited)

The two-dimensional nonlinear problem of steady gravity waves on water of finite depth is considered. The Benjamin–Lighthill conjecture is proved for these waves provided Bernoulli’s constant attains near-critical values. In fact this is a consequence of the following more general results. If Bernoulli’s constant is near-critical, then all corresponding waves have sufficiently small heights and slopes. Moreover, for every near-critical value of Bernoulli’s constant, there exist only the following waves: a solitary wave and the family of Stokes waves having their crests strictly below the crest of this solitary wave; this family is parametrised by wave heights which increase from zero to the height of the solitary wave. All these waves are unique up to horizontal translations. Most of these results were proved in our previous paper (Kozlov and Kuznetsov in Arch Rational Mech Anal 197, 433–488, 2010), in which it was supposed that wave slopes are bounded a priori. Here we show that the latter condition is superfluous by proving the following theorem. If any steady wave has the free-surface profile of a sufficiently small height, then the slope of this wave is also small.