Jump relations for magnetohydrodynamic shock waves in non-ideal gas flow

The generalized jump relations across the magnetohydrodynamic (MHD) shock front in non-ideal gas are derived considering the equation of state for non-ideal gas as given by Landau and Lifshitz. The jump relations for pressure, density, and particle velocity have been derived, respectively in terms of a compression ratio. Further, the simplified forms of the MHD shock jump relations have been obtained in terms of non-idealness parameter, simultaneously for the two cases viz., (i) when the shock is weak and, (ii) when it is strong. Finally, the cases of strong and weak shocks are explored under two distinct conditions viz., (i) when the applied magnetic field is strong and, (ii) when the field is weak. The aim of this paper is to contribute to the understanding of how shock waves behave in magnetized environment of non-ideal gases.     

Analytical solution of magnetogasdynamic cylindrical shock waves in self-gravitating and rotating gas,II

Using the C.C.W. method, propagation of diverging cylindrical shock wave in a self-gravitating and rotating gas under the influence of a constant axial magnetic field has been studied for two cases of weak and strong shocks. Medium ahead of the shock is supposed to be homogeneous. Analytical relations for shock velocity and shock strength along with the expressions for the pressure, density, and particle velocity just behind the shock wave have been also obtained for both cases.     

Shock jump relations for a dusty gas atmosphere

This paper presents simplified forms of jump relations for one dimensional shock waves propagating in a dusty gas. The dusty gas is assumed to be a mixture of a perfect gas and spherically small solid particles, in which solid particles are continuously distributed. The simplified jump relations for the pressure, the temperature, the density, the velocity of the mixture and the speed of sound have been derived in terms of the upstream Mach number. The expressions for the adiabatic compressibility of the mixture and the change-in-entropy across the shock front have also been derived in terms of the upstream Mach number. Further, the handy forms of shock jump relations have been obtained in terms of the initial volume fraction of small solid particles and the ratio of specific heats of the mixture, simultaneously for the two cases viz., (i) when the shock is weak and, (ii) when it is strong. The simplified shock jump relations reduce to the Rankine-Hugoniot conditions for shock waves in an ideal gas when the mass fraction (concentration) of solid particles in the mixture becomes zero. Finally, the effects due to the mass fraction of solid particles in the mixture, and the ratio of the density of solid particles to the initial density of the gas are studied on the pressure, the temperature, the density, the velocity of the mixture, the speed of sound, the adiabatic compressibility of the mixture and the change-in-entropy across the shock front. The results provided a clear picture of whether and how the presence of dust particles affects the flow field behind the shock front. The aim of this paper is to contribute to the understanding of how the shock waves behave in the gas-solid particle two-phase flows.     

An isothermal shock wave in a perfectly-conducting and non-homogeneous stellar interior

In the present paper, we have obtained some exact analytic self-similar solutions for a zero-temperature gradient behind a magnetogasdynamic shock wave produced by stellar explosions. The initial density of the medium is taken to vary as some power of the distance from the point of explosion. The solutions are obtained for the cases when the energy of the shocked gas is constant, the energy is varying, and the shock velocity is constant. General solutions are also obtained. We have also analytically obtained the position of the singular surface behind the shock wave.     

Effect of overtaking disturbances on the propagation of spherical shock wave through self-gravitating gas

Effect of overtaking disturbances on the propagation of a spherical shock wave in self gravitating gas has been studied by the technique developed by the first author [Mod. Meas. Cont. B,46(4), 1 (1992)]. The analytical expressions for modified shock velocity and shock strength have been obtained for an initial density distribution0 =r ^–w, where is the density at the axis of symmetry andw is a constant; simultaneously, for the two cases viz.; (i) when the shock is strong and ii) when it is weak. The results accomplished here have been compared with those for freely propagation of shock.It is observed that the conclusions arrived at here agree with experimental observations. Finally, the modified expressions for the pressure, the density and the particle velocity immediately behind the shock have also been derived from, for both cases.     

Hydromagnetic cylindrical shock in self-gravitating gas

Chisnell-Chester-Whitham method has been used to study the propagation of diverging hydromagnetic cylindrical shock through an infinitely electrically conducting self-gravitating gas having an initial density distribution ₀= r^-w where is the density at the axis of symmetry andw is a constant, simultaneously for the two cases, viz.: (i) when the shock is weak and (ii) when it is strong. The magnetic field is taken to be axial and initially of constant strength. Analytical relations for shock velocity and shock strength have been obtained. the expressions for the pressure, the density and the particle velocity immediately behind the shock have also been derived.     

A model of flare-produced magneto-radiative shock with increasing energy

An exact solution for a spherically-symmetric model of a magneto-radiative shock wave in the solar wind caused by the explosive energy release of a solar flare has been, obtained in the case when energy released is an increasing function of the time. It has been shown that due to increasing energy, density, pressure, radiation flux, magnetic field and shock velocity change considerably.     

Spherical shock wave propagation in self-gravitating stars

The problem of shock wave propagation in a heat-conducting and self-gravitating medium has been studied. The shock is strong enough so that the ambient gas pressure can be neglected. The variation of velocity, density, temperature, and mass distributions behind the shock have been obtained from a numerical solution of similarity equations involved.     

The emergence of a spherical magnetogasdynamic shock wave at the surface of a star

The propagation of a magnetogasdynamic shock wave originating in a stellar interior, is ocnsidered when it approaches the surfaces of the star. The flow behind the shock wave is assumed isothermal rather than adiabatic to stimulate the conditions of large radiative transfer near the stellar surface. The product solution of McVittie has been used to obtain exact solution of the problem. It has been obtained that velocity, density, pressure and magnetic field increases as we move from shock surface towards the nucleus of the star.     

Shock waves in a self-gravitating gas

The CCW method (see Chester, 1854; Chisnell, 1955; Whitham, 1958) has been used to investigate the propagation of diverging shock waves through an ideal gas under its own gravitation having an initial density distribution ₀ = exp(–r , where is the density at the plane/axis/origin, respectively, for plane, cylindrical, and spherical symmetry of the shock and, is non-dimensional constant, for the two situations: viz., (i) when the shock is weak and (ii) when it is strong, simultaneously. Analytical relations for shock velocity and shock strength have been obtained. Expressions for the pressure, the density and the particle velocity immediately behind the shock have been derived. Their numberical estimates for plane and cylindrical symmetry of the shock, have been computed.     

Propagation of a strong cylindrical shock wave in a rotational axisymmetric dusty gas with exponentially varying density

Non-similarity solutions are obtained for one-dimensional isothermal and adiabatic flow behind strong cylindrical shock wave propagation in a rotational axisymmetric dusty gas,which has a variable azimuthal and axial fluid velocity.The dusty gas is assumed to be a mixture of small solid particles and perfect gas.The equilibrium flow conditions are assumed to be maintained,and the density of the mixture is assumed to be varying and obeying an exponential law.The fluid velocities in the ambient medium are assumed to obey exponential laws.The shock wave moves with variable velocity.The effects of variation of the mass concentration of solid particles in the mixture,and the ratio of the density of solid particles to the initial density of the gas on the flow variables in the region behind the shock are investigated at given times.Also,a comparison between the solutions in the cases of isothermal and adiabatic flows is made.     

Plane and cylindrical hydromagnetic shock waves

The Chisnell-Chester-Whitham method has been used to investigate the propagation of diverging plane and cylindrical shock waves through an ideal gas in presence of a magnetic field having only constant axial and variable azimuthal components, simultaneously for both weak and strong cases. Assuming an initial density distribution ₀=r ^–w , where is the density at the plane/axis of symmetry andw is a constant, the analytical expressions for shock velocity and shock strength have been obtained. The expressions for the pressure, the density, and the particle velocity immediately behind the shock have also been derived for both cases.     

Structure of weak shock wave discontinuities in magnetohydrodynamics

In this paper, we study the propagation of a weak shock wave in a medium of initially constant fluid velocity, magnetic field and thermodynamic parameters. The structure of discontinuities for such a shock in real cases will be analyzed. By examining the change in variables inside the relaxation transition region, the length of the latter, i.e. of the disturbed region will be obtained. In order to derive the physical model explaining the finite shock length, several assumptions have been made: the medium has been treated as a very large layer of non-negligible viscosity and thermal conductivity. Starting from basic MHD relations, the invariants on the shock fronts, taking into consideration the process inside the disturbed region, have been calculated. Modified Rankine-Hugoniot equation discussing the process inside the relaxation region has been derived therefrom. Finally, the dependence of pressure upon distance has been examined under the assumptions: the fluid is considered as polytropic. Hence, by approximate integration of an obtained transcendental function, we get the length of relaxation region and discuss the result obtained.     

Anisotropic propagation of magnetogasdynamic sonic waves in conducting and radiating atmosphere

The propagation of sonic discontinuity in conducting and radiating atmosphere has been discussed under the influence of magnetic field. The velocity of sonic wave and its termination into shock wave has been obtained. We have also obtained the critical time at which sonic wave terminates into shock wave. There is significant effect of magnetic field on sonic velocity and its termination into shock wave.     

Jump relations across a shock in non-ideal gas flow

Generalized forms of jump relations are obtained for one dimensional shock waves propagating in a non-ideal gas which reduce to Rankine-Hugoniot conditions for shocks in idea gas when non-idealness parameter becomes zero. The equation of state for non-ideal gas is considered as given by Landau and Lifshitz. The jump relations for pressure, density, temperature, particle velocity, and change in entropy across the shock are derived in terms of upstream Mach number. Finally, the useful forms of the shock jump relations for weak and strong shocks, respectively, are obtained in terms of the non-idealness parameter. It is observed that the shock waves may arise in flow of real fluids where upstream Mach number is less than unity.     

Effect of overtaking disturbances on a strong cylindrical hydromagnetic shock wave in a self-gravitating gas

The effect of overtaking disturbances upon the free propagation of strong cylindrical hydromagnetic shock through a self-gravitating gas has been studied by an approximating technique developed by Yadav. Assuming an initial density distribution law as ₀=r^–w, where is the density at the axis of symmetry and is a constant, the analytical relation for shock velocity and shock strength modified by overtaking waves has been obtained under two conditions: viz., (i) when the applied axial magnetic field is strong and (ii) when the field is weak. The results obtained here are compared with those for a freely propagating shock. The conclusions arrived at agreed with experimental results.It is shown that the applications of the CCW method and the neglect of overtaking disturbances are equivalent.     

Propagation of spherical shock waves in non-homogeneous medium with idealized magnetic field

By use of the approximate method of Whitham (1958), the propagation of magnetogasdynamic spherical shock waves is considered for adiabatic and isothermal flows in a decreasing density medium. The effect of initial magnetic fields on the shock velocity is discussed; and a comparison made between adiabatic and isothermal cases.     

An exact similarity solution for a spherical shock wave in a self-gravitating system

In this paper we have obtained a similarity solution for a spherical magneto-gas dynamic shock wave in a self-gravitating system. It is observed that the total energy of the shock wave is not a constant, but it decreases with time. A remarkable change in radiation flux is also being observed here because of the presence of the magnetic field while there is no change in density, velocity and pressure.     

Self-similar flow behind a spherical shock with varying strength in an inhomogeneous self-gravitating medium

The self-similar model of a shock wave, produced on account of an instantaneous release of energy in an inhomogeneous self-gravitating gaseous mass, has been discussed with the help of equations of motion and equilibrium conditions. The disturbances are headed by a shock of variable strength. The variation of velocity density, pressure, and mass have been discussed for the different values of strength of the shock.     

Similarity solution for the propagation of shock waves with varying energy in non-uniform medium

A model of self-similar propagation of shock waves driven by a flare energy release in a non-uniform atmosphere has been considered. The total energy content of the model is assumed to be increased with time within the inner expanding surface and shock front. Finally the variation of velocity, pressure, density, and energy of the model have been discussed. The gas is assumed to be grey and opaque.