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.     

Evolution of weak discontinuities in shallow water equations

In this paper, we determine the critical time, when a weak discontinuity in the shallow water equations culminates into a bore. Invariance group properties of the governing system of partial differential equations (PDEs), admitting Lie group of point transformations with commuting infinitesimal operators, are presented. Some appropriate canonical variables are characterized that transform equations at hand to an equivalent form, which admits non-constant solutions. The propagation of weak discontinuities is studied in the medium characterized by the particular solution of the governing system.     

Growth of weak discontinuities in arbitrarily moving gas

The theory of singular surfaces is combined with the ray theory to the study of anisotropic, non-linear wave-propagation in an arbitrarily moving gas. The governing equation for the strength of the wave along the rays is expressed in an integral form. Use of the analysis is made by working out two examples in detail.     

Propagation of non-planar weak and strong shocks in a non-ideal relaxing gas

In this article, the kinematics of one-dimensional motion have been applied to construct evolution equations for non-planar weak and strong shocks propagating into a non-ideal relaxing gas. The approximate value of exponent of shock velocity, at the instant of shock collapse, obtained from systematic approximation method is compared with those obtained from characteristic rule and Guderley’s scheme. Computation of exponent is carried out for different values of van der Waals excluded volume. Effects of non-ideal and relaxation parameters on the wave evolution, governed by the evolution equations, are analyzed.

     

Riemann problem in non-ideal gas dynamics

In this paper we consider the Riemann problem for gas dynamic equations governing a one dimensional flow of van der Waals gases. The existence and uniqueness of shocks, contact discontinuities, simple wave solutions are discussed using R-H conditions and Lax conditions. The explicit form of solutions for shocks, contact discontinuities and simple waves are derived. The effects of van der Waals parameter on the shock and simple waves are studied. A condition is derived on the initial data for the existence of a solution to the Riemann problem. Moreover, a necessary and sufficient condition is derived on the initial data which gives the information about the existence of a shock wave or a simple wave for a 1-family and a 3-family of characteristics in the solution of the Riemann problem.     

Decay of a model system of radiating gas

This paper concerns the Cauchy problem of a model system to the radiating gas in . Large time behaviors of classical solutions to the Cauchy problem are studied without needing the smallness assumption of initial perturbation in L¹‐norm. We obtain the optimal H^N‐norm time‐decay rates of the solutions in with 1 ≤ n ≤ 4 by applying the Fourier splitting method introduced by Schonbek (1980) with a slight modification and an energy method. Furthermore, when initial perturbation is bounded in L^p‐norm (p ∈ (1,2]), optimal L^p‐ L² decay estimates of the derivatives of solutions are shown. Copyright © 2013 John Wiley & Sons, Ltd.     

Expansion of a wedge of non-ideal gas into vacuum

We study the problem of expansion of a wedge of non-ideal gas into vacuum in a two-dimensional bounded domain. The non-ideal gas is characterized by a van der Waals type equation of state. The problem is modeled by standard Euler equations of compressible flow, which are simplified by a transformation to similarity variables and then to hodograph transformation to arrive at a second order quasilinear partial differential equation in phase space; this, using Riemann variants, can be expressed as a non-homogeneous linearly degenerate system provided that the flow is supersonic. For the solution of the governing system, we study the interaction of two-dimensional planar rarefaction waves, which is a two-dimensional Riemann problem with piecewise constant data in the self-similar plane. The real gas effects, which significantly influence the flow regions and boundaries and which do not show-up in the ideal gas model, are elucidated; this aspect of the problem has not been considered until now.     

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.     

The propagation of weak discontinuities in quasi-linear symmetric hyperbolic systems

Zusammenfassung Die vorliegende Abhandlung untersucht die Fortpflanzung kleiner Unstetigkeiten in Systemen von nichtlinearen hyperbolischen Differentialgleichungen. Ein Ausdruck wird abgeleitet, der die Änderung in der Intensität der Unstetigkeit angibt, wenn diese sich entlang eines Strahls des hyperbolischen Gleichungssystems fortbewegt. Schliesslich wird als Beispiel mit Hilfe des angegebenen Verfahrens die Fortpflanzung von Schallwellen behandelt.     

An optically thin radiating gas jet

A radiating gas jet near the optically thin limit is examined. The gas is assumed to be inviscid, optically grey, and in the state of chemical equilibrium or to obey the ideal gas law. The jet is assumed to be plane-symmetric or axi-symmetric and is fully expanded. Under these conditions and by the method of quasi-one-dimensional flow, the problem is analyzed. In addition to this, numerical examples are also included to illustrate the radiative effects.

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Zusammenfassung Es wird ein strahlender Gasstrahl untersucht, nahe zum optisch dünnen Grenzfall. Das Gas wird reibungsfrei, optisch grau und im chemischen Gleichgewicht (oder als ideales Gas) vorausgesetzt. Der Strahl wird eben oder axisymmetrisch angenommen und ist voll expandiert. Unter diesen Voraussetzungen und mit der Methode der fast-eindimensionalen Strömung wird das Problem analysiert. Zudem sind numerische Beispiele angegeben, um den Strahlungseffekt zu veranschaulichen.

     

Conservation laws and growth of discontinuities in a reactive polytropic gas

Summary We shall consider a conservation law associated with a non-equilibrium gas flow with chemical reaction. Some results on the growth of discontinuities and the occurrence of breakdown of the solution are pointed out.

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Sommario Si prende in esame una legge di conservazione che viene associata alle equazioni che governano un gas in non equilibrio in cui avviene una reazione chimica. Si mettono in evidenza alcuni risultati sull'evoluzione delle discontinuita' e sulla eventuale perdita di regolarita' della soluzione.

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This work was partially supported by the G.N.F.M. (Gruppo Nazionale Fisica Matematica) of C.N.R.     

Global well-posedness and relaxation limits of a model for radiating gas

We study the initial value problem for a hyperbolic-elliptic coupled system with arbitrary large discontinuous initial data. We prove existence and uniqueness for that model by means of L¹-contraction and comparison properties. Moreover, after suitable scalings, we study both the hyperbolic-parabolic and the hyperbolic-hyperbolic relaxation limits for that system.     

On weak discontinuities through thermally conducting and dissociating gases

Using singular surface theory, the phenomena associated with the uniform and nonuniform propagation of weak discontinuities through thermally conducting and dissociating gases is studied. The basic differential equations governing the criteria for decay or blow up of these discontinuities is obtained. It turns out that growth and decay of weak discontinuities are derived and solved completely. The the thermal conduction and dissociation allow the existence of a singular surface carrying a weak discontinuity which grows into a shock and the role of dissociation and thermal conduction is to cause rapid damping in the formation of this shock.     

Convergence to the rarefaction wave for a model of radiating gas in one-dimension

In this paper, a sequence of solutions to the one-dimensional motion of a radiating gas are constructed. Furthermore, when the absorption coefficient α tends to ∞, the above solutions converge to the rarefaction wave, which is an elementary wave pattern of gas dynamics, with a convergence rate \(\alpha ^{ - \tfrac{1}{3}} \left| {\ln \alpha } \right|^2\).     

Asymptotic behavior of solutions to a hyperbolic-elliptic coupled system in multi-dimensional radiating gas

This paper is concerned with the asymptotic behavior of solutions to the Cauchy problem of a hyperbolic-elliptic coupled system in the multi-dimensional radiating gas

     

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.     

Compressibility effect on laminar convection to a radiating gas past a vertical plate

Summary Fluid motion generated by a long vertical hot plate in the presence of an externally applied magnetic field is investigated when radiative transfer is important. Even though the velocities involved are small, this paper takes compressibility into effect owing to the extreme heat encountered in radiation problems. Thus the viscosity () and the thermal conductivity (K) are assumed to vary with the temperature (T). In this paper we takeT ⁿ while the Prandtl number (Pr) which is defined in terms of the specific heat at constant pressure (c _p) byPr=c _p/K is constant. Assuming a general differential approximation for the radiative flux, the asymptotic problem, valid far away from the tips of the plate, is reduced to a set of nonlinear coupled ordinary differntial equations. Solution to this problem is discussed quantitatively.

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Zusammenfassung Es wird die Bewegung eines Fluids entlang einer heißen vertikalen Platte untersucht in Gegenwart eines außen angebrachten Magnetfeldes, im Falle daß die Strahlung wichtig ist. Obwohl die Geschwindigkeiten klein sind, werden in dieser Arbeit Kompressibilitätseffekte berücksichtigt, da bei Strahlungsproblemen hohe Temperaturen auftreten. Die Zähigkeit und der WärmeleitungskoeffizientK werden mit der Temperatur veränderlich angenommen, mitT ⁿ, und der Prandtl-ZahlPr=c _p/K konstant. Für den Strahlungsfluß wird eine allgemeine différentielle Form angenommen, und das asymptotische Problem, gültig in einigem Abstand von den Plattenenden, wird auf ein System von nichtlinearen gekoppelten gewöhnlichen Differentialgleichungen reduziert. Die Lösung des Problems wird quantitativ diskutiert.

     

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.     

One-dimensional cylindrical shock waves in non-ideal gas under magnetic field

In the present paper, we analyze the evolutionary behavior of imploding strong shock waves propagating through a non-ideal gas in the presence of axial magnetic field. An evolution equation has been constructed by using the method based on the kinematics of one-dimensional motion of shock waves. The values of similarity exponents have been calculated by using the first order truncation approximation which describes the decay behavior of strong shocks. The approximate values of the similarity exponents are compared with the similarity exponents calculated by the CCW approximation, the exact similarity solution and perturbation technique.

     

The growth and decay of sonic waves in a radiating gas at high temperature

The propagation of a sonic discontinuity in an optically thick gray gas at temperature 10⁵°K or higher has been studied. The effects of radiation pressure and radiation energy density have been taken into account, while the profiles structured by radiant heat transfer are imbedded in the discontinuities under high temperature conditions of an optically thick medium. When the sonic discontinuity is propagating into a gas at rest, its velocity of propagation is found to be a constant which is the effective speed of sound in a radiating gas. The fundamental differential equations governing the growth of the sonic discontinuity are obtained and solved. It is concluded that if the sonic discontinuity is a compressive wave of order 1, then it terminates into a shock wave after a critical timet _c which has been determined. But on the other hand, when the sonic discontinuity is an expansion wave of order 1, then it will decay and will vanish ultimately. Particular cases of interest have been studied in details.