Decay Of Discontinuity Research Articles

Decay of initial nonlinear Alfv�n wave discontinuities in a collisionless plasma

A system of Whitham equations for the derivative nonlinear Schrodinger equation in the Riemann form was used to analyze possible types of decay of discontinuities which accompany the overturning of the simplest quasilongitudinal nonlinear Alfven wave. A whole class of structures is found which are formed when the discontinuities decay into two simple collisionless shock waves without forming a plateau between them. Different types of decay of the initial discontinuities are considered for cases when the overturning of the Alfven wave is modulationally stable.

The analytical solution of the Riemann problem in relativistic hydrodynamics

We consider the decay of an initial discontinuity in a polytropic gas in a Minkowski space–time (the special relativistic Riemann problem). In order to get a general analytical solution for this problem, we analyse the properties of the relativistic flow across shock waves and rarefactions. As in classical hydrodynamics, the solution of the Riemann problem is found by solving an implicit algebraic equation which gives the pressure in the intermediate states. The solution presented here contains as a particular case the special relativistic shock-tube problem in which the gas is initially at rest. Finally, we discuss the impact of this result on the development of high-resolution shock-capturing numerical codes to solve the equations of relativistic hydrodynamics.

Spherically symmetric escape of a self-gravitating ideal gas into a vacuum

The spherically symmetric flows of an ideal gas are considered assuming that Newtonian gravitation acts on the mass of gas. The problem of the decay of a special discontinuity is investigated by the method of characteristic series, and exact solutions of the initial boundary-value problems of the nonlinear integrodifferential partial-differential system are constructed in the form of converging series. It is proved that the particles of gas on the free gas-vacuum surface move as particles in the field of attraction of a material point situated at the centre of symmetry and having a mass equal to the initial mass of gas. In the case when the gas and the vacuum are continuously adjacent to one another one can prove a theorem of the existence and uniqueness of the solution for only rational adiabatic indices, and one can show that for certain values of the gas-dynamic parameters the gas sphere disperses to infinity, while in other cases the gas-vacuum boundary stops and the mass of gas begins to collapse. Singularities of the solution on this boundary only appear at the instant of focusing, which can be treated as the instant when the whole mass of gas collapses into the centre of gravity.

Growth and decay of sonic waves in thermally radiative magnetogasdynamics

The propagation of weak discontinuities headed by a wavefront and their formation into shock waves are investigated in a thermally-radiative ideal plasma. It is found that all compressive waves grow without bound only if the magnitude of initial discontinuity associated with the wave exceeds a critical value. In particular, the grow and decay of weak discontinuities in plane, cylindrical, and spherical geometries have been discussed and the shock formation distance and time in all these cases have been derived in presence of magnetic field and radiation, separately.

Longitudinal waves in an elastic medium with a piecewise-linear dependence of the stress on the strain

Longitudinal waves in an elastic body in which the derivative of the stress with respect to the strain (the coefficient of elasticity) can undergo a discontinuity are considered. Fronts can be propagated in such a body, in which a jumplike change in the coefficient of elasticity and the characteristic velocities occurs. It is shown that if the front velocity reaches the value of one of the velocities of the characteristics, reconstruction of the motion occurs, where the special case of the problem of the decay of a weak discontinuity of second-order arises, whereupon a change in the front velocity and type occurs. The problem under consideration is very similar to the problem of the behaviour of an elastic-plastic medium with hardening (although mathematically it does not the same as it), for which the front classification and certain solutions with front reconstruction are given in /1–3/. For a medium of different-modulus, a classification of the front is made in /4, 5/. Some of these fronts are identical with fronts in elastic-plastic media. Solutions of problems of the behaviour of weak first-order discontinuities in a different-modulus elastic body were examined in /4–7/. The paper by G.Ya. Galin, “Phase Transformation Waves” presented at the international conference “Modern Mathematical Problems of Mechanics and Their Application” (Moscow, 1987) was also devoted to flows with a change in properties of the medium.

The Evaluation of the Diffusion Activity of Migrating Grain Boundaries by the Discontinuous Decay Kinetics in the Nickel Superalloy

The Evaluation of the Diffusion Activity of Migrating Grain Boundaries by the Discontinuous Decay Kinetics in the Nickel Superalloy

Stability of shock waves

The progress already made in studies of the stability of shock waves and some tasks for the future are reviewed. The following aspects of the problem are discussed: 1) stability of shock waves as hydrodynamic discontinuities irrespective of the events which occur in the relaxation zone of a wave (this aspect includes also the problem of stability of a shock wave sustained by a piston and anomalies in the reaction of a wave to external perturbation, including reflection and refraction of perturbations); 2) stability of flow in the relaxation zone (structural stability) of a shock wave. The stress is on the theoretical side of the problem. However, potential practical realization of the shock-wave instability criteria are also considered. Schemes of decay of unstable shock-wave discontinuities are discussed.

On simple waves in an elastic-ideal plastic medium

Elastic-plastic flow is described by a non-linear hyperbolic /1/ system of equations and an inequality (non-negativity of the factor in the associated law) that is the loading condition. There are overturning waves among the simple waves (SW) of this system of equations. However, only those SW that are not overturning satisfy the loading condition. This fact is known for the case when Hooke's law for elastic deformation and the Mises flow criterion /2/ are assumed; its foundation substantially utilized these particular properties. The absence of SW overturning is established below for a body with an arbitrary smooth flow surface and linear anisotropic elasticity. In this case jumps do not occur from the elastic-plastic SW, which indicates the possibility of solving the problem of the decay of an arbitrary discontinuity without the insertion of jumps of any new kinds that require the search for additional conditions.

The decay of an arbitrary initial discontinuity in an elastic medium

The solution of the problem of the decay of an arbitrary discontinuity in elastic theory is studied. It is assumed that a plane boundary separates an elastic homogeneous, non-heat-conducting medium into two half-spaces with different elastic properties and densities. Each of the media possesses an arbitrary kind of homogneous initial strain (stress) and velocity. In the sequel the stresses and velocities of the media are assumed to be continuous at the boundary. This results in the formation of a system of plane selfsimilar waves (simple and shock), which propagate in each of the half-spaces. The problem is solved under the assumption of weak non-linearity and anisotropy of the materials. This permits an approximate evaluation of the stress and strain at the contact discontinuity. After this the problem on the decay of an arbitrary initial discontinuity is reduced to two problems on the sudden change of load on a half-space boundary, which are solved independently for each of the media.

Numerical solution of the generalized equations of gas dynamics

Numerical algorithms for solving the generalized equations of gas dynamics, obtained from Boltzmann's kinetic equation and conserving second-order small terms, are proposed. Comparison of the algorithms given here with the scheme with counter-flow differences and Gopunov's method in solving the problem of the decay of a strong discontinuity showed their effectiveness and accuracy.

Global existence for 1D, compressible, isentropic Navier-Stokes equations with large initial data

We prove the global existence of weak solutions of the Cauchy problem for the Navier-Stokes equations of compressible, isentropic flow of a polytropic gas in one space dimension. The initial velocity and density are assumed to be in L 2 {L^2} and L 2 ∩ B V {L^2} \cap BV respectively, modulo additive constants. In particular, no smallness assumptions are made about the intial data. In addition, we prove a result concerning the asymptotic decay of discontinuities in the solution when the adiabatic constant exceeds 3 / 2 3/2 .

A monotonic scheme of through computation based on an implicit algorithm of flux correction

A reversible implicit algorithm of flux correction is used to construct a monotonic scheme of through computation for dynamic problems of an inviscid thermally non-conducting gas. Comparison with other schemes of through computation when solving various problems, notably, the problem of the decay of a strong discontinuity, showed the high efficiency and accuracy of the present scheme when studying complex flow.

Spontaneous decay of a tangential discontinuity

Classical reconnection is phrased as initial value problem within the framework of ideal magnetohydrodynamics.

Propagation of weak discontinuities in a vibrationally-excited gas flow

Propagation of weak discontinuities headed by wavefronts of arbitrary shape in three dimensions are studied in vibrationally relaxing gas flow. The transport equations representing the rate of change of discontinuities in the normal derivatives of the flow variables are obtained, and it is found that the nonlinearity in the governing equations does not contribute anything to the vibrationally relaxing gas. An explicit criterion for the growth and decay of weak discontinuities along bi-characteristic curves in the characteristic manifold of the governing differential equations is given. A special case of interest is also discussed.

Discontinuities of the variables that characterize the propagation of solitary waves in a fluid layer

A nonlinear system of equations of hyperbolic type describing the propagation of solitary waves is considered [1]. A solitary wave is characterized in this approximation by two variables — the energy density per unit length measured along its crest, and the direction of the normal to the wave crest. The evolution of a wave described by the system may lead to the appearance of discontinuities, at which there are jumps in the energy density and the direction of the wave crest [2]. To establish the conditions at the discontinuities, a solution describing the interaction of nonparallel solitons [3, 4] is used. The obtained conditions are used to solve the problem of the decay of an arbitrary discontinuity in terms of soliton variables.

Eulerian formulation of transport equations for three-dimensional shock waves in simple elastic solids

The propagation of a three-dimensional shock wave in an elastic solid is studied. The material is assumed to be a simple elastic solid in which the Cauchy stress depends on the deformation gradient only. It is shown that the growth or decay of a discontinuity ψ depends on (i) an unknown quantity φ− behind the shock wave, (ii) the two principal curvatures of the shock surface, (iii) the gradient on the shock surface of the shock wave speeds and (iv) the inhomogeneous term which depends on the motion ahead of the shock surface and vanishes when the motion ahead of the shock surface is uniform. If a proper choice is made of the propagation vectorb along which the growth or decay of the discontinuity is measured, the dependence on item (iii) can be avoided. However,b assumes different directions depending on the choice of discontinuity ψ with which one is concerned and the unknown quantity φ− behind the shock wave on which one chooses to depend. As in the case of one-dimensional shock waves, the growth (or decay) of one discontinuity may not be accompanied by the growth (or decay) of other discontinuities. A universal equation relating the growth or decay of discontinuities in the normal stress, normal velocity and specific volume is also presented.

Growth and decay of sonic discontinuities in non-equilibrium magnetogasdynamics

The phenomenon associated with sonic discontinuities in non-equilibrium magnetogasdynamics has been studied by the use of singular surface theory. The fundamental differential equations for growth and decay of sonic discontinuities have been formulated for general class of relaxing gas applying compatibility conditions for surface of discontinuity in continuum mechanics. This class of equations have been solved completely and for particular case of plane. We have also obtained the critical time at which sonic wave terminates in shock wave.

Time evolution of discontinuities at the wave head in a non-equilibrium two-phase flow of a mixture of a gas and dusty particles

The behavior of weak discontinuities in a non-equilibrium flow of a mixture of a gas and small dusty particles is analysed using the singular surface theory. The propagation velocity of these weak waves, “a”, is determined and is shown to agree with the ordinary gasdynamic results in a limit of vanishingz, the volume fraction of the solid particles in the mixture. The differential equation governing the growth and decay of weak discontinuities is derived and solved both analytically and numerically. Using a representative set of data for the upstream flow, it is shown that all expansive waves are ultimately damped, whereas not all compressive waves grow into shock waves. The effect of the volume fraction of the solid particles and the wave front curvature are to enhance the flattening rate of an expansive wave and to diminish the steeping rate of a compressive wave.

Stability of flows with discontinuity surfaces

A study is made in the linear formulation of flows with homogeneous distribution of the parameters in expanding regions separated by boundaries that are either discontinuity surfaces of an arbitrary nature or surfaces with effective boundary conditions. Examples of such flows are the decay of an arbitrary discontinuity [1] and flow in a tube with a region of heat release [2].

Unidirectional nonstationary flow of instantaneously heated gas from a cylinder for different positions of the heated zone

The finite-difference method and one-dimensional approximation are used to consider the processes of decay of an initial pressure discontinuity for the individual cases of its position in a bounded region.