Anisotropic propagation of weak discontinuities in flows of thermally conducting and dissociating gases

The object of the present investigation is to study the anisotropic propagation of weak discontinuities in flows of thermally conducting and dissociating gases. The velocity of propagation of the wave frcnt is determined. A set of differential equations governing the growth and decay of weak discontinuities are obtained and solved. It is found that if the sonic wave is a compressive wave of order 1, then it terminates into a shock wave after a critical timet _c which has been determined. It is also observed that the effects of heat conduction and dissociation are to decrease the duration of time by which a weak discontinuity will generate into a shock wave.     

Propagation of discontinuities along bicharacteristics in the unsteady flow of a relaxing gas

Growth and decay of weak discontinuities headed by wave front of arbitrary shape in three dimensions are investigated in an unsteady flow of a relaxing gas. 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 plays an important role in the interplay of damping and steepening tendencies of the wave. An explicit criterion for the growth and decay of weak discontinuities along bicharacteristic curves in the characteristic manifold of the governing differential equations is given and special reference is made of diverging and converging waves under different thermodynamical situations. It is shown that there is a special case of a compressive converging wave, irrespective of the thermodynamical state whether weak or strong, in which the effects of thermodynamical influences and that of wave front curvature are unable to overcome the tendency of the wave to grow into a shock.     

Transient thermoelastic response in a cracked strip of functionally graded materials via generalized fractional heat conduction

This work is devoted to analyzing a thermal shock problem of an elastic strip made of functionally graded materials containing a crack parallel to the free surface based on a generalized fractional heat conduction theory. The embedded crack is assumed to be insulated. The Fourier transform and the Laplace transform are employed to solve a mixed initial-boundary value problem associated with a time-fractional partial differential equation. Temperature and thermal stresses in the Laplace transform domain are evaluated by solving a system of singular integral equations. Numerical results of the thermoelastic fields in the time domain are given by applying a numerical inversion of the Laplace transform. The temperature jump between the upper and lower crack faces and the thermal stress intensity factors at the crack tips are illustrated graphically, and phase lags of heat flux, fractional orders, and gradient index play different roles in controlling heat transfer process. A comparison of the temperature jump and thermal stress intensity factors between the non-Fourier model and the classical Fourier model is made. Numerical results show that wave-like behavior and memory effects are two significant features of the fractional Cattaneo heat conduction, which does not occur for the classical Fourier heat conduction.     

A difference scheme with anti-diffusion terms to improve the computation of contact discontinuities and the study of weak discontinuities in compressible flow

The investigation of Mach reflection formed after the impingement of a weak plane shock wave on a wedge with shock Mach number M_s near 1, is still an open problem[12]. It's difficult for shock tube experiments with interferometer to detect contact discontinuities if it is too weak; also difficult to catch with due accuracy the transition condition between Mach reflection and regular reflection. The interest to this phenomenon is continuing, especially for weak shocks, because there was systematic discrepancy between simplified three shock theory of von Neumann [8] and shock tube results [15] which was named by G. Birkhoff as “von Neumann Paradox on three shock theory” [18].In 1972, K.O.Friedrichs called for more computational efforts on this problem. Recently it is known that for weak impinging shocks it's still difficult to get contact discontinuities and curved Mach stem with satisfactory accuracy. Recent numerical computation sometimes even fails to show reflected shock wave[6]. These explain why von Neumann paradox of the three shock theory in case of weak discontinuities is still a problem of interesting [9,12,14]. In this paper, on one hand, we investigate the numerical methods for Euler's equation for compressible inviscid flow, aiming at improving the computation of contact discontinuities, on the other hand, a methodology is suggested to correctly plot flow data from the massive information in storage. On this basis, all the reflected shock wave , contact discontinuities and the curved Mach stem are determined. We get Mach reflection under the condition when over-simplified shock theory predicts no such configuration[5].     

Weak Discontinuities in Solutions of Long‐Wave Equations for Viscous Flow

Nonlinear integrodifferential equations describing the propagation of disturbances in a thin layer of viscous liquid with free surface are studied. These equations admit solutions with weak discontinuities, which are located on the characteristics. The possibility of an unbounded increase in the amplitude of the weak discontinuity and the formation of the shock in the process of flow evolution is established. Differential balance laws approximating the integrodifferential model are proposed. These laws are used to perform numerical simulation of wave propagation in a fluid.     

Quasilinear hyperbolic-hyperbolic singular perturbation problem: Study of shock layer

We present the asymptotic analysis of a quasilinear hyperbolic–hyperbolic singular perturbation problem in one dimension. The leading part of the analysis concerns the construction of some shock layers associated with discontinuities of a hyperbolic problem. This study is a generalization of the case of viscous perturbation for a hyperbolic problem.     

Investigation of the behavior of weak waves in the solution of a vector variational wave equation

We consider nonsmooth solutions of the system of Euler-Lagrange equations corresponding to a variational problem with several unknown functions of several variables and with a quadratic functional. The propagation of weak discontinuities is described by the equations of the method of singular characteristics developed by Melikyan. The onset and interaction of weak discontinuities of the solution caused by nonsmooth initial conditions are studied by numerical-analytic methods. We develop two computer programs for shock-fitting and shock-capturing computations. The approach was earlier applied by the authors to the analysis of a variational wave equation, namely, to the solution of the Euler-Lagrange equation for a variational problem with a single unknown function.     

Exponential stability of a linear viscoelastic bar with thermal memory

In this paper we study a one-dimensional evolution problem arising in the theory of linear thermoviscoelasticity with hereditary heat conduction. Depending on the istantaneous conductivity K₀, both Coleman-Gurtin (K₀>0) and Gurtin-Pipkin (K₀=0) heat flow theories are involved. In any case, the exponential stability of the corresponding semigroup is proved for a class of memory functions including weakly singular kernels. In order to achieve the exponential decay of the energy, we assume that mechanical and thermal memory kernels decay exponentially for large time. Entrata in Redazione il 23 luglio 1998.     

An approximate solution for spherical and cylindrical piston problem

A new theory of shock dynamics (NTSD) has been derived in the form of a finite number of compatibility conditions along shock rays. It has been used to study the growth and decay of shock strengths for spherical and cylindrical pistons starting from a non-zero velocity. Further a weak shock theory has been derived using a simple perturbation method which admits an exact solution and also agrees with the classical decay laws for weak spherical and cylindrical shocks.     

Shock Wave Structure of Multi-Temperature Euler Equations from Kinetic Theory for a Binary Mixture

A multi-temperature hydrodynamic limit of kinetic equations is employed for the analysis of the steady shock problem in a binary mixture. Numerical results for varying parameters indicate possible occurrence of either smooth profiles or of weak solutions with one or two discontinuities.     

Partial regularity of suitable weak solutions to the four‐dimensional incompressible magneto‐hydrodynamic equations

In this paper, we study the partial regularity of suitable weak solutions to the incompressible magneto‐hydrodynamic equations in dimension four by borrowing and improving the arguments given by Caffarelli, Kohn, and Nirenberg for incompressible Navier–Stokes equations. The so‐called ε‐regularity criteria are established for suitable weak solutions. As an application, an estimate on Hausdorff dimension of the possible singular points set for a suitable weak solution is given. Finally, we present further information on distribution of the possible singular points if the given initial data decay sufficiently rapidly or are not too singular at the origin, in some sense. Copyright © 2012 John Wiley & Sons, Ltd.     

Transport equations of higher order discontinuities for quasilinear hyperbolic systems

The transport equations satisfying ordinary linear differential equations of first order which govern the behaviour of higher order discontinuities for quasilinear hyperbolic systems along the rays associated with a singular surface are derived. It is shown that the transport equations depend on the Gaussian curvature of wave front.     

Multi-Dimensional Quadrature of Singular and Discontinuous Functions

In many simulations of physical phenomena, discontinuous material coefficients and singular forces pose severe challenges for the numerical methods. The singularity of the problem can be reduced by using a numerical method based on a weak form of the equations. Such a method, combined with an interface tracking method to track the interfaces to which the discontinuities and singularities are confined, will require numerical quadrature with singular or discontinuous integrands. We introduce a class of numerical integration methods based on a regularization of the integrand. The methods can be of arbitrary high order of accuracy. Moment and regularity conditions control the overall accuracy.     

Propagation of weak discontinuities in binary mixtures of ideal gases

The propagation of the weak discontinuities in binary non-reacting mixtures of classical ideal monoatomic gases is analyzed. The normal speeds of propagation are determined and compared with those of a single fluid. The differential equation governing the growth and the decay of the acceleration waves is obtained and the solutions for plane, cylindrical and spherical waves are shown. The influence of the different atomic masses of the constituents is also investigated.     

Hypersonic flow past a sphere

The effects of dissociation or ionization of air on the analytical solution for hypersonic flow past a sphere are considered here, under certain assumptions. It has been assumed that the shock wave is in the shape of a sphere, that the density ratio across the shock is constant, that the flow behind the shock is at constant density and that dissociation or ionization only occurs behind the shock wave. Thus the effects of the compressibility of the air, variation of density ratio along the shock, and the department of the shock shape from being circular are not taken into account. Here the velocity, pressure, temperature, pressure coefficient and vorticity, etc., at any point between the shock and the surface of the sphere in the presence of dissociation or ionization are obtained. In addition, shock detachment distance, drag coefficient, stagnation point velocity gradient and sonic points on the shock and the surface have also been obtained. The results have been compared with the corresponding results obtained in the case when dissociation or ionization does not occur behind the shock.     

Dissociation effect on the hypersonic flow past a circular cylinder

The effects of dissociation of air on hypersonic flow past a circular cylinder at zero angle of incidence are considered under the assumptions that the shock wave is in the shape of a circular cylinder, the density ratio across the shock is constant, the flow behind the shock is at constant density and dissociation occurs only behind the shock wave. In the present paper, the velocity, pressure and drag coefficients, vorticity, shock detachment distance, stagnation point velocity gradient and sonic points on the shock and the surface have been obtained in the presence of dissociation. The results have been compared with the corresponding results obtained in the case when dissociation dose not occur and the corresponding results in the case of the sphere in the presence of dissociation.     

Singularities, Shocks, and Instabilities in Interface Growth

Two-dimensional interface motion is examined in the setting of geometric crystal growth. We focus on the relationships between local curvature and global shape evolution displaying the dual role of singularities and shocks depending on the parameterization of the curve—the crystal surface. Discontinuities in surface slope accompany regions of asymptotically decreasing curvature during transient growth, whereas an absence of discontinuities preempts such asymptotic curvature evolution. In one parameterization, these discontinuities manifest themselves as a finite-time continuous blowup of curvature, and in another, as a shock and hence a localized divergence of curvature. Previously, it has been conjectured, based on numerical evidence, that the minimum blowup time is preempted by shock formation. We prove this conjecture in the present paper. Additionally we prove that a class of local geometric models preserves the convexity of the surface. These results are connected to experiments on crystal growth.     

On sonic discontinuities in an ideal radiating gas

Following Elcrat the phenomena associated with the sonic discontinuities in an ideal radiating gas are studied here. The differential equations for growth and decay of these discontinuities are formulated. In order to integrate them in full generality they are transformed to an equation along the bicharacteristic curve in the characteristic manifold. The criterion for the decay or blow up of weak discontinuities has been given. Radiation contributes to the fast development of shock.

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Résumé Dans cet article il s'agit de l'étude des phénomènes associés aux discontinuités soniques dans un gaz parfait rayonnant. On formule les équations différentielles gouvernant la croissance et l'amortissement de ces discontinuités. Afin de pouvoir les intégrer en toute généralité, on les transforme en une équation sur la courbe bicaractéristique dans la variété caractéristique. On donne le critère d'amortissement ou d'éclatement des discontinuités faibles. Le rayonnement contribue au développement rapide du choc.

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This author is thankful to CSIR, India for financial assistance for this work.     

Formation and Dynamics of Shock Waves in the Degasperis-Procesi Equation

Solutions of the Degasperis-Procesi nonlinear wave equation may develop discontinuities in finite time. As shown by Coclite and Karlsen, there is a uniquely determined entropy weak solution which provides a natural continuation of the solution past such a point. Here we study this phenomenon in detail for solutions involving interacting peakons and antipeakons. We show that a jump discontinuity forms when a peakon collides with an antipeakon, and that the entropy weak solution in this case is described by a "shockpeakon" ansatz reducing the PDE to a system of ODEs for positions, momenta, and shock strengths.