Oblique magnetohydrodynamic shock wave decay

Summary A modified hodograph transformation is used to obtain an exact solution of the equations governing the one-dimensional unsteady flow of an ideal, inviscid, perfectly conducting compressible fluid, subjected to an oblique magnetic field. This solution is used to obtain an approximate representation of the path of an initially uniform shock wave which intersects a centered simple wave. The solutions for the corresponding problems in the conventional, non-magnetic case and for a transverse orientation of the applied magnetic field are contained as special limiting cases of the solutions of the present paper. This provides a valuable check on the theory.

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Résumé Une transformation hodographe modifiée est employée pour obtenir une solution exacte des équations relatives aux écoulements unidimensionnels non-stationnaires et non-isentropiques d'un fluide non visqueux idéal, parfaitement conducteur d'électricité et compressible, soumis à l'action d'un champ magnétique oblique avec deux composantes différentes de zero. On utilise cette solution pour obtenir une représentation approximative de la trajectoire d'une onde de choc magnétohydrodynamique initialement uniforme, rencontrant une onde simple centrée. Les solutions pour le cas non magnétique et pour un champ transversal apparaissent comme des cas limites particuliers de la solution présentée ici, c'est là une confirmation de la validité de la theorie.

     

Magnetohydrodynamic shock wave decay

Summary A modified hodograph transformation is used to obtain an exact solution of the equations governing the one-dimensional unsteady flow of an ideal, inviscid, perfectly conducting compressible fluid, subjected to a transverse magnetic field. This solution is used to obtain an approximate representation of the path of an initially uniform shock wave which intersects a centered simple wave. In the limit of vanishing magnetic field, the solution reduces exactly to the solution of the corresponding problem for conventional gas dynamics.

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Résumé Une transformation hodographe modifiée est employée pour obtenir une solution exacte des équations relatives aux écoulements unidimensionnels non-stationnaires et non-isentropiques d'un fluide non visqueux idéal, parfaitement conducteur d'électricité et compressible, soumis à l'action d'un champ magnétique transversal. On utilise cette solution pour obtenir une représentation approximative de la trajectoire d'une onde de choc magnétohydrodynamique initialement uniforme, rencontrant une onde simple centrée.Dans le cas limite d'un champ magnétique nul, la solution se réduit exactement à celle du problème correspondant de la dynamique classique des gaz. C'est là une confirmation de la validité de la théorie.

     

Optimal decay rates of the compressible magnetohydrodynamic equations

We use a pure energy method recently developed by Guo and Wang to prove the optimal time decay rates of the solutions to the compressible magnetohydrodynamic equations in the whole space. In particular, the optimal decay rates of the higher-order spatial derivatives of solutions are obtained.     

The exponential decay of a system of nonlinear wave equations

The exponential decay of a system of nonlinear wave equations with initial boundary values is considered. We have some sufficient conditions that ensure that the energy admits exponential decay by a compactness uniqueness argument and the energy estimates.     

The motion of a non-uniform hydromagnetic shock wave

Sommaire Nous considérons un écoulement, unidimensionnel, non stationaire, non isentropique, d'un fluide parfait, parfaitement conducteur de l'électricité, soumis à l'action d'un champ magnétique pour une orientation transversale du champ et le mouvement non continu d'une onde de choc magnetohydrodynamique.

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This research was supported by a National Aeronautics and Space Administration general research grant to the University of Wisconsin.     

Spatial decay bounds for time dependent magnetohydrodynamic geophysical flow

This article is concerned with spatial decay bounds for the time dependent magnetohydrodynamic geophysical flow in an infinite pipe when homogeneous lateral surface boundary conditions are applied. Assuming that the entrance velocity and magnetic field data are small enough and the fluid flow converges to laminar flow as the distance down the pipe tends to infinity, we derive a second order differential inequality that leads to an exponential decay estimate for the “energy” associated with the velocity and magnetic field represented by the difference between the entrance flow and fully developed laminar flow. We also show how to establish the explicit decay bounds for the total energy.     

Perturbation of a shock wave

The Cauchy problem is considered for the perturbed Hopf equation u_t+uu_x=εf(u), ε→0. The solution in the continuity domain can be expanded in the standard asymptotic series in integral powers of the small parameter. An asymptotic representation is found for the line of propagation of the shock wave. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 118, No. 3, pp. 462–466, March, 1999.     

Exponential decay for a neutral wave equation

A strongly damped wave equation involving a delay of neutral type in its second order derivative is considered. It is proved that solutions decay to zero exponentially despite the fact that delays are, in general, sources of instability.     

Superexponential decay of solutions of a semilinear wave equation

Consider a smooth solution of $\text{u}{}_{\mathrm{tt}}\text{? Δu + q(x) ¦ u ¦}{}^{\mathrm{p?1}}\text{u = 0 x ? R}{}^3\text{, q ? 0}$ and is C¹, and 1 < p < 5. Assume that the initial data decay sufficiently rapidly at infinity, $\text{q(x) ? a}\text{exp}\text{(?b ¦ x ¦}{}^{\mathrm{c}}\text{), a, b > 0, c > 1}$, and for simplicity, q_r ? 0. Then the local energy decays faster than exponentially.     

Convergence of Dirichlet quotients and selective decay of 2D magnetohydrodynamic flows

The selective decay phenomena have been observed by physicists for many dynamic flows such as Navier-Stokes flows, barotropic geophysical flows, and magnetohydrodynamic (MHD) flows in either actual physical experiments or numerical simulations. In the previous paper (M.-Q. Zhan, 2010 [20]), the author showed the validity of the selective decay principle for the 2D magnetohydrodynamic (MHD) flows in the case of small magnetic Prandtl number. In this paper, we shall show the validity of the selective decay principle for the 2D magnetohydrodynamic (MHD) flows for any magnetic Prandtl number with periodic boundary conditions.     

Exponential decay of solutions of a nonlinearly damped wave equation

The issue of stablity of solutions to nonlinear wave equations has been addressed by many authors. So many results concerning energy decay have been established. Here in this paper we consider the following nonlinearly damped wave equation a, b > 0, in a bounded domain and show that, for suitably chosen initial data, the energy of the solution decays exponentially even if m > 2.     

The dissociation of a pure diatomic gas behind a strong normal shock wave

Zusammenfassung Die Arbeit behandelt das Problem der Dissoziation eines reinen diatomischen Gases durch einen starken geraden Verdichtungsstoss. Mit Hilfe einiger vereinfachender Annahmen und eines theoretischen Ausdruckes für die Dissoziationsgeschwindigkeit wird eine analytische Darstellung der Resultate erzielt. Der Einfluss der Erhitzung und der Kompression durch den Stoss wird eingehend diskutiert.

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The research reported in this document has been sponsored in part by the Air Research and Development Command, United States Air Force, under contract AF 61(514)1124, through the European Office, A.R.D.C.     

The diffraction of a shock wave by an inclined body of revolution

Summary The interaction between a plane normal shock wave of arbitrary strength and a slender body of revolution, held with its axis at a small angle to the direction of propagation of the shock, is considered. The pressure field is determined by using the method of matched asymptotic expansions and the general results are applied to the problem of diffraction by an inclined circular cone.

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Zusammenfassung Es wird die Beeinflussung einer ebenen Stosswelle beliebiger Stärke durch einen schlanken Rotationskörper untersucht, dessen Achse einen kleinen Winkel mit der Ausbreitungsrichtung des Stosses bildet. Das Druckfeld wird mit Hilfe der abgestimmten asymptotischen Entwicklungen bestimmt, und die allgemeinen Resultate werden auf das Problem der Diffraktion durch einen angestellten Kreiskegel angewendet.

     

Formation and decay of electromagnetic shock waves

Résumé Ce traité se rapporte principalement à la dérivation de la loi de détérioration asymptotique pour les ondes de choc électromagnetiques faibles se propageant dans un milieu de type ferromagnétique. On dérive la loi de détérioration pour les paramètres de choc en utilisant une, technique de corrélation entre une des solutions d'une onde simple et les équations d'un choc faible.     

The balance of material momentum at a shock wave propagating in a thermo-elastic material

At a surface of discontinuity, the mechanical balance laws are represented by jump conditions. It is shown how the balance of material momentum at an adiabatic shock propagating in a thermo-elastic material is obtained from the discontinuous balances of physical momentum and energy. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The stability of a metastable shock wave in a viscoelastic medium under two-dimensional perturbations

The unsteady motions of a viscoelastic medium are considered, taking account of a small anisotropy and a small non-linearity. The behaviour of a metastable, quasi-transverse shock wave when it interacts with non-one-dimensional perturbations is investigated numerically. The stability of this wave under non-one-dimensional perturbations of large amplitude is demonstrated.