Initial Discontinuity Research Articles

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.

On the formation of secondary shocks in a self-similar magnetogasdynamic flow

The problem of a self-similar flow behind a cylindrical shock wave, produced by the sudden release of a finite amount of energy per unit length along an axis of infinite extent in a plasma permeated by a transverse magnetic field, has been solved exactly for the specific heat ratio γ = 2. This value for the specific heat ratio γ is known to approximate the plasma state in a uniform magnetic field. We used the singular surface theory to study the propagation of a weak discontinuity in this self-similar flow, and calculated the critical time for a shock formation. We find that the appearance of a secondary shock cannot be ignored if the amplitude of the initial discontinuity satisfies appropriate conditions. The condition of the nonimpact between a weak discontinuity and the primary shock has also been discussed.

Fractal gases of zero momentum with respect to all inertial frames

Unlike gases containing molecules obeying Maxwell–Boltzmann statistics, ‘‘fractal gases’’ obey the fundamental axiom that their linear momentum is zero when resolved in any direction with respect to all inertial frames of reference. In contrast to speculation, the properties of fractal gases obeying Newtonian mechanics, or special or general relativity can be investigated in a way similar to the logic of geometry. They have the following properties: (1) An entire fractal gas cannot decay into a Maxwell–Boltzmann distribution, but is in a state of overall equilibrium. (2) Localities tend toward localized equilibrium, upset later by collision with other localities. (3) Fractal gases contain temperature differences that are a source of temporary order. (4) The appearance of a fractal gas to a hypothetical observer would suggest creation in an initial discontinuity.

Short-wave instabilities and ill-posed initial-value problems

We characterize ill-posed problems as catastrophically (Hadamard) unstable to short waves. The growth rate tends to infinity as the wavelength tends to zero. The mathematical description of ill-posed problems is framed in terms of instability. These problems cannot be integrated numerically; the finer the mesh, the worse is the result. The instability must be regularized. Ill-posed problems which arise in problems involving interfaces, oil recovery, granular media, and viscoelastic fluids are regularized in different ways, by adding effects of surface tension or viscosity or compressibility or by weakening the initial discontinuity. Problems which are stables t → ∞ for any fixed wavelength λ, no matter how small, can be Hadamard unstable with catastrophic instability as λ → 0 for a fixed t, no matter how large. We stress the utility of freezing coefficients in nonlinear and quasilinear systems and prove that in general ill-posed problems cannot be solved unless the initial data is analytic. We show why the shock up of first-order systems which are nonlinear in first derivatives can be expected to lead to discontinuities in second, rather than first, derivatives.

Delayed diffusion due to flux limitation

Consider a flux-limited diffusion process, where ut=[G(ux)]x with G(∞)<∞ and G′(s)⩾0. We show that if the pr ofile u(0, x initially has a sharp front, then the sharp front may not be resolved immediately. Instead, the front may remain perfectly sharp for a finite time, during which the height of the jump decays to zero. In this case the profile near the sharp front takes a self-similar form u=tf(x/t), with the front remaining sharp for a time [h/2f(0)]2, where h is the height of the initial discontinuity . We also determine when an initially sharp jump will remain sharp for a finite time, and when it will be resolved immediately.

Improved initial approximation and intensity-guided discontinuity detection in visible-surface reconstruction

Improved initial approximation and intensity-guided discontinuity detection in visible-surface reconstruction

Do large structures control their own growth in a mixing layer? An assessment

This study makes a specific comparison between two different two-dimensional free shear layers: the T-layer which develops in time from an initial tangential velocity discontinuity separating the two half-spaces; and the S-layer which develops downstream of the origin where two uniform streams of unequal velocity are brought into tangential contact. The method of comparison is to assume that the vorticity of the S-layer is given parabolically by a Galilean mapping of that of the T-layer; to satisfy the appropriate boundary conditions in the S-layer and to compute the velocity induced at any point in the S-layer by its vorticity field; and to compare this velocity to that which can be derived from the velocity of the T-layer at corresponding points by a Galilean transformation of the velocity itself. The purpose of this calculation is to assess approximately how far the flow in the S-layer is from parabolic and, in particular, to what extent the perturbations induced upstream by large concentrations of vorticity found downstream are instrumental in hastening or retarding the subharmonic instability that leads to the formation of these large structures. The calculations suggest that this elliptic influence, or the feedback, in a mixing layer is relatively small, at least for small velocity ratios.

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 BENCHMARK TEST NO. 4.1: SEDIMENTATION

The test problem considered is to simulate the gravitation-driven sedimentation process of a dense phase (ρ1=1.000) that rests on top of a light phase (ρ2=0.999). The gravitational force is assumed not to be normal to the initial discontinuity in volume fraction, which leads to this problem to be two-dimensional. The purpose of this test is to provide a good examination of the numerical method.

On the solution of reaction‐diffusion equations with double diffusivity

In this paper, solution of a pair of Coupled Partial Differential equations is derived. These equations arise in the solution of problems of flow of homogeneous liquids in fissured rocks and heat conduction involving two temperatures. These equations have been considered by Hill and Aifantis, but the technique we use appears to be simpler and more direct, and some new results are derived. Also, discussion about the propagation of initial discontinuities is given and illustrated with graphs of some special cases.

Solutions in the large for certain nonlinear parabolic systems

We prove the global existence of smooth solutions for certain systems of the form ut + f(u)x = Duxx. Here u and f are vectors and D is a constant, positive matrix. We assume that the Cauchy data u0 satisfies ‖u0−u¯‖L∞(ℝ)<r, where ū is a fixed vector and f is defined in an r-ball about ū, and that ‖u0−u¯‖L2(ℝ) is sufficiently small. We show how our results apply to the equations of (nonisentropic) gas dynamics, and we include a result which shows that for the Navier-Stokes equations of compressible flow, smoothing of initial discontinuities must occur for the velocity and energy, but cannot occur for the density.

Device for measuring thermal-inertia index of resistance thermometers

Meters have been developed for determining the index of thermal inertia of thermometers under operating conditions by creating the initial termperature discontinuity in the thermometer itself by the passage of current and recording the cooling process [i, 2]; however, they are either suitable for operation with thermometers that have high sensitivity [2] or do not satisfy the accuracy requirements imposed on small (i00 mm ~ or less) thermometers [i]. In the proposed meter, the measurement accuracy is increased and the performance is improved, owing to use of the circuit described below.

The fatigue crack direction and threshold behaviour of mild steel under mixed mode I and III loading

As part of a programme to investigate the mixed mode fatigue crack growth threshold behaviour of mild steel, tests were carried out on three-point bend specimens with spark machined initial slits inclined to give mixed Mode I and III displacements. Overall the expected tendency to Mode I crack growth showed as an initial directional discontinuity followed by a smooth rotation of the crack front until it was almost perpendicular to the specimen sides. At a smaller scale, initial crack growth was by the formation of Mode I branch cracks which developed into a ‘twist’ fracture surface consisting of narrow Mode I facets separated by cliffs. The facets eventually grew out and the fracture surface became smooth. The result in the initiation it was necessary to distinguish between the threshold conditions which result in the initiation of crack growth, specimen failure and crack arrest. An envelope based on Mode I branch crack growth provides a reasonable lower bound to the results for crack initiation and specimen failure. The crack arrest threshold results and some of the crack growth threshold results could not be analysed in detail because of lack of appropriate stress intensity factors.

A selfsimilar problem on the action of a sudden load on the boundary of an elastic half-space

The solution of the non-linear problem of the action of a constant stress suddenly applied to the plane boundary of an elastic half-space that has homogeneous prestrain is investigated. The problem is selfsimilar, and its solution is constructed from shock and selfsimilar simple waves investigated earlier /1–5/. The problem under consideration is the necessary element that should be contained in solutions of different non-stationary problems, for instance, in the problem of the decay of an arbitrary initial discontinuity. Moreover, the selfsimilar solution constructed below represents the asymptotic form long times of the corresponding non-selfsimilar problems when the stress on the half-space boundary varies from some values to others according to an arbitrary law over a limited time.

On the structure of generalized solutions of the one-dimensional equations of a polytropic viscous gas

A model system of equations that defines the unsteady one-dimensional flow of a visoucs gas is considered on the assumption that the pressure is determined by the adiabatic Poisson law. Generalized solutions are investigated in the class of discontinuous functions, a class of correctness is separated, and the structure of solutions of this class is clarified. It is shown that the initial velocity discontinuities are instantly smoothed out, and from the discontinuity points of the initial density,lines of contact discontinuity are formed. These lines exist for an infinite time, and the pressure jumps on them vanish exponentially.

The Riemann Problem for Nonconvex Scalar Conservation Laws and Hamilton-Jacobi Equations

We present a closed form expression for the viscosity solution to the Riemann problem for any scalar nonconvex conservation law. We then define an analogous problem for any scalar nonconvex Hamilton-Jacobi equation and obtain its (even simpler) solution. Extensions to two (and by inference, higher) space dimensional problems, when the initial discontinuity lies on a hyperplane, are also given.

The Riemann problem for nonconvex scalar conservation laws and Hamilton-Jacobi equations

We present a closed form expression for the viscosity solution to the Riemann problem for any scalar nonconvex conservation law. We then define an analogous problem for any scalar nonconvex Hamilton-Jacobi equation and obtain its (even simpler) solution. Extensions to two (and by inference, higher) space dimensional problems, when the initial discontinuity lies on a hyperplane, are also given.

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.

Nonlinear Breaking of Wave Fronts in a Vibrationally Relaxing Gas

Using the method of characteristics, some explicit criteria for the nonlinear breaking of wave fronts in a plane, cylindrically or spherically symmetric flows of a vibrationally relaxing gas are obtained. It is shown that except a spherically converging compressive wave, all compressive waves break only when the initial discontinuity at the wave-head exceeds a critical value, and the effects of vibrational relaxation and that of the wave front curvature are to increase the time of their breaking. It is found that the stabilizing influences of vibrational relaxation and that of the wave front curvature are not strong enough to overcome the breaking tendency of a spherically converging compressive wave; in fact, in contrast to a cylindrically converging wave, it always breaks down before the formation of a focus, no matter how small be the initial discontinuity at the wave-head.

Studies of moisture effects in simple atmospheric models: The stable case

Abstract The shallow-water equations are supplemented by a moisture equation in order to model effects of latent heat release on the horizontal propagation of disturbances. Since latent heating in regions of upward motion compensates the reduction in perturbation temperature due to lifting, buoyancy effects are reduced and disturbances propagate more slowly. Analytic solutions are obtained to illustrate this effect, a good example being provided by the collapse of an initial horizontal temperature discontinuity. Another interesting example is provided by the case of disturbances propagating through a region from one side. Moist regions are found to have front and rear boundaries which both move faster than rainbands do inside the moist region. If random disturbances are passed through a region which is initially saturated, the relative humidity drops until it reaches a value which depends on the disturbance level.