Self-similar time-varying flows of an ideal gas with a change in the adiabatic exponent in a “reflected” shock wave

Self-similar one-dimensional time-varying problems are considered under the assumption that there is a change in the adiabatic exponent in a shock wave (SW) running (“reflected”) from a centre or axis of symmetry (later from a centre of symmetry, CS) or from a plane. The medium is an ideal (inviscid and non-heat-conducting) perfect gas with constant heat capacities. In problems with strong SW, the change in the adiabatic exponent in a gas approximately simulates physicochemical processes such as dissociation and ionization and, in the problem of the collapse of a spherical cavity in a liquid, the conversion of liquid into vapour. In both cases, the adiabatic exponent decreases on passing across a reflected SW. Problems of the collapse of a spherical cavity, the reflection of a strong SW from a centre of symmetry and a simpler problem with a self-similarity index of one are examined. When it is assumed that there is an increase in the adiabatic exponent, the self-similar solutions of the first two problems are rejected due to the decrease in entropy from the instant when the SW is reflected. When it is assumed that there is a decrease in the adiabatic exponent, the solutions of these problems only become unsuitable after a finite time has elapsed for the same reason. Up to this time when the decrease in the adiabatic exponent has not reached a certain threshold, the structure of the self-similar solution does not undergo qualitative changes. When the above-mentioned threshold is exceeded, a self-similar solution is possible if a cylindrical or spherical piston expands according to a special law from the instant of SW reflection from the CS. When there is no piston, the flow behind the reflected wave becomes non-self- similar. In the case of the deceleration of a plane flow, conditions are possible with the joining of SW from different sides to a centred rarefaction wave.     

Similarity solution for a cylindrical shock wave in a rotational axisymmetric dusty gas with heat conduction and radiation heat flux

The propagation of shock waves in a rotational axisymmetric dusty gas with heat conduction and radiation heat flux, which has a variable azimuthally fluid velocity together with a variable axial fluid velocity, is investigated. The dusty gas is assumed to be a mixture of non-ideal (or perfect) gas and small solid particles, in which solid particles are continuously distributed. It is assumed that the equilibrium flow-condition is maintained and variable energy input is continuously supplied by the piston (or inner expanding surface). The fluid velocities in the ambient medium are assume to be vary and obey power laws. The density of the ambient medium is assumed to be constant, the heat conduction is express in terms of Fourier’s law and the radiation is considered to be of the diffusion type for an optically thick grey gas model. The thermal conductivity K and the absorption coefficient α_R are assumed to vary with temperature and density. In order to obtain the similarity solutions the angular velocity of the ambient medium is assume to be decreasing as the distance from the axis increases. The effects of the variation of the heat transfer parameter and non-idealness of the gas in the mixture are investigated. The effects of an increase in (i) the mass concentration of solid particles in the mixture and (ii) the ratio of the density of solid particles to the initial density of the gas on the flow variables are also investigated.     

The discharge shock wave in a condensed ideally conducting medium with a magnetic field

Using a model equation of state containing a point of inflection, we solve the problem of the formation of a discharge shock wave in an ideally conducting medium with a magnetic field. Dissipative processes are neglected. Translated fromMatematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 41, No. 1, 1998, pp. 51–53.     

Computation of a shock wave structure in monatomic gas with accuracy control

The structure of a shock wave in a monatomic one-component gas was computed by solving the Boltzmann kinetic equation with accuracy controlled with respect to computational parameters. The hard-sphere molecular model and molecules with the Lennard-Jones potential were considered. The computations were performed in a wide range of Mach numbers with the accuracy no less than 3% for the shock front width and 1% for local values of density and temperature. The shock wave structure was studied in terms of macroscopic gas characteristics and in terms of the molecular velocity distribution function.     

Propagation of a shock wave in an isentropic gas with small viscosity

A new method is considered for constructing the front and the dynamics of motion of a shock wave propagating in an isentropic gas with small viscosity with a speed close to the speed of sound. The proposed algorithm of solution makes it possible to considerably raise the accuracy and effectiveness of numerical computations on the computer as compared with known universal numerical methods based on the use of difference schemes.Translated from Itogi Nauki i Tekhniki, Sovremennye Problemy Matematiki, Vol. 8, pp. 273–308, 1977.     

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.     

Thermodynamically compatible conservation laws in the model of heat conducting radiating gas

Thermodynamic compatibility of the mass, momentum, and energy conservation laws that describe the motion of heat conducting gas in the presence of radiation heat exchange is considered. The study is based on the one-velocity two-component mathematical model of continuous compressible medium with the gas and radiation components. The work uses experimental data for radiation and other experimental data of modern physics.     

Dynamics of heat conducting fluids inside heat conducting domains

The existence of a solution is proved for a model of compressible, heat conducting fluid inside a heat conducting domain. Boundary conditions for the temperature involving a radiation term are assumed. Compared to other approaches, the scheme used for the proof requires lower regularity of the domain boundary in the case of its application to barotropic compressible fluids.     

Stability of an imploding spherical shock wave in a van der Waals gas II

The emission of light from a sonoluminescing bubble may dependon whether shock waves launched each acoustic cycle by the implodingsurface of the bubble focus on to a sufficiently small volumeat the bubble centre. This in turn may depend on whether theshock maintains its stability as it travels inwards. With thisapplication in mind, the linear stability of an imploding sphericalshock was studied in Part I, using a van der Waals equationof state for the gas. Conditions for instability were determined,but the subsequent fate of the perturbations of the bubble surfacewas unknown. Would the instabilities grow and persist at finiteamplitude or would they disappear during implosion? The answersto such questions are sought here by integrating the gas dynamicsequations using the finite-difference essentially non-oscillatorymethod of Shu and Osher. The shock is initiated by a nearlyspherical ‘piston’ and its subsequent evolution,including its finite-amplitude deviations from sphericity, isdetermined. Two types of behaviour are found depending on theparameter , where b is thevan der Waals excluded volume and is the initial uncompressed density of gas ahead of the shock.When is sufficiently large, an initially smooth shock front remains smooth as it focusesand, although it is impossible to continue the integrationsup to the moment of implosion, it appears that it will focuson a small volume at the centre of the bubble. This is in sharpcontrast to what happens at smaller values of for which the initial distortion of the shock front,if sufficiently large, becomes and remains polygonal shaped.This is consistent with experimental results for cylindricalimploding shocks as well as with earlier theoretical investigationsof imploding cylindrical and spherical shocks in an ideal gas() that used the Chisnell, Chester and Witham (CCW) approximation or the geometrical shockapproximation of Whitham. Plausibly, the polygonal distortionsreduce the volume on to which the imploding shock in a sonoluminescingbubble focuses.     

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.     

Propagation of shock wave type singularities in equations of two-dimensional zero-pressure gas dynamics

We study a system of equations consisting of the two-dimensional Bürgers equation and the continuity equation. In 1970 such a system was proposed by Ya. B. Zeldovich for describing the formation of the large-scale structure of the Universe. In the present paper, for the divergent form of this system (the zero-pressure gas dynamics system), we rigorously define the notion of its generalized solution (in the sense of distributions) in terms of Radon measures and obtain a generalization of the Rankine-Hugoniot relations. By using these relations, we show that, in general, the variational representation of generalized solutions valid for the one-dimensional system of zero-pressure gas dynamics does not make sense in the two-dimensional case. Translated fromMatematicheskie Zametki, Vol. 66, No. 5, pp. 760–769, November, 1999.     

Group theoretic method for analyzing interaction of a discontinuity wave with a strong shock in an ideal gas

A group theoretic method is used to obtain an exact particular solution to the system of partial differential equations, describing one-dimensional unsteady planar, cylindrically and spherically symmetric motions in an ideal gas, involving shock waves. It is interesting to remark that the exact solution obtained here is precisely the blast wave solution obtained earlier using a different method of approach. Further, the evolution of a discontinuity wave and its interaction with the strong shock are studied within the state characterized by the exact particular solution. The properties of reflected and transmitted waves and the jump in the shock acceleration are completely characterized, and certain observations are noted in respect to their contrasting behavior.     

On a cylindrical blast ware propagating in a conducting gas

Zusammenfassung Die Ähnlichkeitslösung von Lin für starke zylindrische Stösse wird verallgemeinert für den Fall, dass der Stoss in einem Plasma und in einem Magnetfeld parallel zur Zylinderachse sich ausbreitet.     

On the shock wave spectrum for isentropic gas dynamics with capillarity

We consider the stability problem for shock layers in Slemrod's model of an isentropic gas with capillarity. We show that these traveling waves are monotone in the weak capillarity case, and become highly oscillatory as the capillarity strength increases. Using a spectral energy estimate we prove that small-amplitude monotone shocks are spectrally stable. We also show that monotone shocks have no unstable real spectrum regardless of amplitude; this implies that any instabilities of these monotone traveling waves, if they exist, must occur through a Hopf-like bifurcation, where one or more conjugate pairs of eigenvalues cross the imaginary axis. We then conduct a systematic numerical Evans function study, which shows that monotone and mildly oscillatory profiles in an adiabatic gas are spectrally stable for moderate values of shock and capillarity strengths. In particular, we show that the transition from monotone to nonmonotone profiles does not appear to trigger any instabilities.     

On wave propagation in a random conducting magneto-non-simple thermo-viscoelastic medium

The paper examines the problem of wave propagation in a random conducting magneto-non-simple thermo-viscoelastic medium. The medium has been assumed to be weakly conducting and weakly thermal. The thermomechanical coupling parameter and the conductivity are random functions, proportional to ε, with non-zero mean values, ε measuring the smallness of the scale of random fluctuation of inhomogeneities of the medium. The smooth perturbation technique enunciated by Keller (1964) has been employed to analyze the appropriate dispersion equation in non-simple thermoelastic medium. The longitudinal and transverse waves were discussed by using a particular form of thermomechanical coupling parameter representing the corresponding auto-correlation function. The effect of magnetic conductivity has been investigated. The phenomena of attenuation of waves and change of phase speed were discussed numerically in details.