Mixture Of Small Solid Particles Research Articles

A self-similar solution for unsteady isothermal and adiabatic flows behind the shock wave in a dusty non-ideal rotating medium under the action of magnetic field

In this paper, the similarity solutions using Lie group analysis for shock wave propagation in a rotational axisymmetric mixture of small solid particles of micro-size and a non-ideal gas with azimuthal magnetic field are obtained. In the mixture, the solid particles are distributed uniformly. For isothermal and adiabatic flows, the Lie group analysis is used to discuss all possible cases of similarity solutions. The local Lie group generator’s arbitrary constants generate a range of solutions with power and exponential law shock paths. A detailed discussion of the similarity solution for isothermal and adiabatic flows with a power-law shock path is presented. The constant density for the ambient medium has been assumed. The effect of changing the Alfven Mach number, non-idealness parameter, and other parameters on shock wave strength and flow variables is explored. These parameters are observed to have a decaying impact on shock waves. The effect of varying the Alfven Mach number, non-idealness parameter and other parameters on shock wave strength and flow variables is investigated. The parameter of non-idealness and Alfven Mach number are noted to have a decaying effect on shock waves.

Approximate analytical solution for the propagation of shock wave in a mixture of small solid particles and non-ideal gas: isothermal flow

Abstract This paper presents the development of mathematical model to obtain the approximate analytical solutions for isothermal flows behind the strong shock (blast) wave in a van der Waals gas and small solid particles mixture. The small solid particles are continuously distributed in the mixture and the equilibrium conditions for flow are maintained. To derive the analytical solutions, the physical variables such as density, pressure, and velocity are expanded using perturbation method in power series. The solutions are derived in analytical form for first approximation, and for second order approximation the set of differential equations are also obtained. The effects of an increase in the problem parameters value on the physical variables are investigated for first order approximation. A comparison is also, made between the solution of cylindrical shock and spherical shock. It is found that the fluid density and fluid pressure become zero near the point or axis of symmetry in spherical or cylindrical symmetry, respectively, and therefore a vacuum is created near the point or axis of symmetry which is in tremendous conformity with the physical condition in laboratory to generate the shock wave.

Shock Wave Driven Out by a Piston in a Mixture of a Non-ideal Gas and Small Solid Particles Under the Influence of Azimuthal or Axial Magnetic Field

One-dimensional self-similar unsteady flows behind a spherical or cylindrical shock wave driven out by a piston moving with time according to a power law in a dusty gas is investigated. The medium is assumed to be the mixture of small solid particles of micro-size and a non-ideal gas. The solid particles are uniformly distributed in the mixture, and the shock wave is assumed to be driven by the inner surface (piston). It is assumed that the equilibrium flow-conditions are maintained and the moving piston continuously supplies the variable energy input. Similarity solutions exist only when the surrounding medium is of constant density. Solutions are obtained, in both cases, when the flow between the shock and the piston is isothermal or adiabatic. The shock waves in non-ideal dusty gas can be important for the description of shocks in supernova explosions, in the study of a flare produced shock in the solar wind, the central part of starburst galaxies, nuclear explosion, rupture of the pressurized vessel, in the analysis of data from exploding wire experiments, and cylindrically symmetric hypersonic flow problems associated with meteors or re-entry vehicles, etc. The findings of the present work provided a clear picture of whether and how the non-idealness of the gas and the presence of the magnetic field affect the propagation of shock and the flow behind it. It is interesting to note that in the presence of azimuthal magnetic field, the pressure and density vanish at the piston and hence a vacuum is formed at the center of symmetry for both the isothermal and adiabatic flows, which is in excellent agreement with the laboratory condition to produce the shock wave.

Spherical and cylindrical shocks in a non-ideal dusty gas with magnetic field under the action of heat conduction and radiation heat flux

In this article, the propagation of spherical or cylindrical shock waves in a mixture of small solid particles of microsize and a non-ideal gas with conductive as well as radiative heat fluxes are studied under the influence of an azimuthal or axial magnetic field. The solid particles are uniformly distributed in the mixture, and the shock wave is assumed to be driven by a piston. It is assumed that the equilibrium flow conditions are maintained and the moving piston continuously supplies the variable energy input. The density of the undisturbed medium is assumed to be constant in order to obtain the self-similar solutions. Heat conduction is expressed in terms of Fourier’s law, and the radiation is considered to be of diffusion type for an optically thick gray gas model. The thermal conductivity and the absorption coefficient are assumed to vary with temperature and density. Numerical calculations have been performed to obtain the flow profiles of variables. The effects of different values of the non-idealness parameter, the strength of the magnetic field, the mass concentration, the ratio of the density of solid particles to the initial density of the gas, the piston velocity index, and the adiabatic index are shown in detail. It is interesting to note that in the presence of an azimuthal magnetic field, the pressure and density vanish at the piston, and hence, a vacuum is formed at the center of symmetry, which is in excellent agreement with the laboratory condition to produce the shock wave. The author of the article agrees to the retraction of the article effective AUGUST 20, 2021.

Propagation of strong cylindrical shock wave in a self-gravitating rotational axisymmetric mixture of small solid particles and perfect gas with density varying exponentially

The non-similarity solution for the propagation of a strong cylindrical shock wave in a self-gravitating and rotational axisymmetric dusty gas, having variable azimuthal and axial fluid velocities is obtained. The dusty gas is assumed to be a mixture of small solid particles of micro size and perfect gas. The equilibrium flow conditions are assumed to be maintained. The density of the mixture and the fluid velocities in the ambient medium are assumed to be varying and obeying an exponential law. The shock wave moves with variable velocity and the total energy of the wave is non-constant. The effects of variation of the mass concentration of solid particles in the mixture, the ratio of the density of solid particles to the initial density of the gas, and the gravitational parameter on the flow variables in the region behind the shock are investigated at a given time in both the rotating and non-rotating cases. Our analysis reveals that an increase in gravitational parameter or in the ratio of the density of solid particles to the initial density of the gas surprisingly the shock strength increases and remarkable differences are found in the distribution of flow variables in both the rotating and non-rotating cases. An increase in time also, increases the shock strength in the case of isothermal flow, but to decrease the shock strength in the case of adiabatic flow in general. Further, it is investigated that the consideration of isothermal flow increases the shock strength. A comparison is made between the solutions in isothermal and adiabatic flow cases with or without gravitational effects in both the rotating and non-rotating medium. The obtained solutions are applicable for arbitrary values of time.

Shock wave driven out by a piston in a mixture of a non-ideal gas and small solid particles under the influence of the gravitation field with monochromatic radiation

Similarity solutions for a spherical shock wave in a mixture of small solid particles of micro size and a non-ideal gas are discussed under the influence of the gravitational field with monochromatic radiation. The solid particles are uniformly distributed in the mixture, and the shock wave is assumed to be driven by a piston. It is assumed that the equilibrium flow-conditions are maintained and the moving piston continuously supplies the variable energy input. Due to the central mass (m¯) at the origin (Roche model), the medium is considered to be under the influence of the gravitational field. In comparison to the attraction of the central mass at the origin, the gravitational effect of the mixture itself is neglected. The density of the undisturbed medium is assumed to be constant in order to obtain the self-similar solutions. The effect of the parameter of non-idealness of the gas b¯, the mass concentration of solid particles in the mixture μp, the ratio of the density of solid particles to the initial density of the gas Ga and the gravitational parameter G0 are obtained. It is shown that due to an increase in the gravitational parameter the compressibility of the medium at any point in the flow field behind the shock front decrease and the flow variables velocity, pressure, radiation flux and shock strength are increased. Also, an increase in the ratio of the density of solid particles to the initial density of the gas Ga and the gravitational parameter G0 has the same effect on the shock strength and the reverse effect on the compressibility. The non-idealness of the gas causes a decrease in the shock strength and widens the disturbed region between the piston and the shock.

Flow behind an exponential cylindrical shock in a rotational axisymmetric mixture of small solid particles of micro size and non-ideal gas with conductive and radiative heat fluxes

Flow behind an exponential cylindrical shock in a rotational axisymmetric mixture of small solid particles of micro size and non-ideal gas with conductive and radiative heat fluxes

Flow behind an exponential shock in a rotational axisymmetric mixture of non-ideal gas and small solid particles with heat conduction and radiation heat flux

The propagation of exponential cylindrical shock wave in a rotational axisymmetric dusty gas with heat conduction and radiation heat flux, which has variable azimuthal and axial fluid velocities, is studied. The shock wave is driven out by a piston moving with time according to exponential law. Similarity solutions exist only when the surrounding medium is of constant density and constant angular velocity. The azimuthal and axial components of the fluid velocity in the ambient medium are assumed to be varying and obeying exponential laws. The dusty gas is assumed to be a mixture of small solid particles and non-ideal (or perfect) gas, in which solid particles are continuously distributed. In our model, it is assumed that the small solid particles are pseudo-fluid and the equilibrium flow-conditions are maintained in the flow-field. The heat conduction is expressed 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 αR are assumed to vary with density and temperature. The effects of the variation of the heat transfer parameters, mass concentration of solid particles in the mixture Kp, the ratio of the density of solid particles to the initial density of the gas Ga and the parameter of the non-idealness of the gas b¯ are worked out in detail. It is found that an increase in the ratio of the density of solid particles to the initial density of the gas or the conductive heat transfer parameter Γc or the radiative transfer parameter ΓR increases the compressibility of the mixture in the flow field behind the shock, and hence there is an increase in the shock strength. Also, it is shown that an increase in the parameter of non-idealness of the gas b¯ has decaying effect on the shock wave.

Self-similar Solution of a Cylindrical Shock Wave under the Action of Monochromatic Radiation in a Rotational Axisymmetric Dusty Gas

A self-similar flow behind a cylindrical shock wave is studied under the action of monochromatic radiation in a rotational axisymmetric dusty gas. The dusty gas is taken to be a mixture of small solid particles and perfect gas, and solid particles are continuously distributed in the mixture. The similarity solutions are obtained and the effects of the variation of the radiation parameter, the ratio of the density of solid particles to the initial density of the gas, the mass concentration of solid particles in the mixture and the index for the time dependent energy law are investigated. It is observed that an increase in the radiation parameter has decaying effect on the shock waves; whereas the shock strength increases with an increase in the ratio of the density of solid particles to the initial density of the gas or the index for the time dependent energy law. Also, it is found that an increase in the radiation parameter has effect to decrease the flow variables except the density and the azimuthal component of fluid velocity. A comparison is also made between rotating and non-rotating cases.

Propagation of a strong spherical shock wave in a gravitating or non-gravitating dusty gas with exponentially varying density

The propagation of a strong spherical shock wave in a dusty gas with or without self-gravitational effects is investigated in the case of isothermal and adiabatic flows. 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. Non-similarity solutions are obtained and the effects of variations 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, and the presence of self-gravitational field on the flow variables are investigated at given times. Our analysis reveals that after inclusion of gravitational field effects surprisingly the shock strength increases and remarkable differences are found in the distribution of flow variables. An increase in time also, increases the shock strength. Further, it is investigated that the consideration of isothermal flow increases the shock strength, and removes the singularity in the density distribution. Also, the presence of gravitational field increases the compressibility of the medium, due to which it is compressed and therefore the distance between the inner contact surface and the shock surface is reduced. The shock waves in self-gravitating dusty gas can be important for description of shocks in supernova explosions, in the study of central part of star burst galaxies, star formation and shocks in stellar explosion, nuclear explosion, in industry, rupture of a pressurized vessel and explosion in the ionosphere. Other potential applications of this study include analysis of data from exploding wire experiments and cylindrically symmetric hypersonic flow problems associated with meteors or re-entry of vehicles etc. A comparison is made between the solutions in the cases of the gravitating and the non-gravitating media. The obtained solutions are applicable for arbitrary values of time.

Self-similar solutions for unsteady flow behind an exponential shock in an axisymmetric rotating dusty gas

Similarity solutions are obtained for one-dimensional unsteady isothermal and adiabatic flows behind a strong exponential cylindrical shock wave propagating in a rotational axisymmetric dusty gas, which has variable azimuthal and axial fluid velocities. The shock wave is driven by a piston moving with time according to an exponential law. Similarity solutions exist only when the surrounding medium is of constant density. The azimuthal and axial components of the fluid velocity in the ambient medium are assumed to obey exponential laws. The dusty gas is assumed to be a mixture of small solid particles and a perfect gas. To obtain some essential features of the shock propagation, small solid particles are considered as a pseudo-fluid; it is assumed that the equilibrium flow conditions are maintained in the flow field, and that the viscous stresses and heat conduction in the mixture are negligible. Solutions are obtained for the cases when the flow between the shock and the piston is either isothermal or adiabatic, by taking into account the components of the vorticity vector. It is found that the assumption of zero temperature gradient results in a profound change in the density distribution as compared to that for the adiabatic case. The effects of the variation of the mass concentration of solid particles in the mixture \(K_p\), and the ratio of the density of solid particles to the initial density of the gas \(G_a\) are investigated. A comparison between the solutions for the isothermal and adiabatic cases is also made.

Self-similar flow of a mixture of a non-ideal gas and small solid particles behind a shock wave in presence of heat conduction, radiation heat flux and a gravitational field

Similarity solutions are obtained for one-dimensional unsteady flow of a dusty gas behind a spherical shock wave with heat conduction and radiation heat flux under a gravitational field of heavy nucleus at the centre (Roche model). The dusty gas is assumed to be a mixture of small solid particles and a non-ideal gas. The equilibrium flow conditions are assumed to be maintained, and the heat conduction is expressed 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 αR are assumed to vary with temperature and density. In order to obtain similarity solutions the density of the undisturbed medium is assumed to be constant. The effects of an increase in the value of the parameter of non-idealness of the gas in the mixture \(\bar{b}\), the mass concentration of the solid particles in the mixture Kp, the ratio of the density of the solid particles to the initial density of the gas G1 and the variation of the heat transfer parameters ΓR and Γc are obtained.

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.

A self- similar solution of a shock propagation in a mixture of a non-ideal gas and small solid particles

Similarity solutions are obtained for unsteady, one-dimensional self-similar flow behind a strong shock wave, driven by a moving piston, in a dusty gas. The dusty gas is assumed to consist of a mixture of small solid particles and a non-ideal gas, 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. Solutions are obtained under both the isothermal and adiabatic conditions of the flow-field. The spherical case is worked out in detail to investigate to what extent the flow-field behind the shock is influenced by the non-idealness of the gas in the mixture as well as by the mass concentration of the solid particles, by the ratio of density of the solid particles to the initial density of the mixture and by the energy input due to moving piston. A comparison is also made between isothermal and adiabatic cases.

Propagation of shock waves in a dusty gas with heat conduction, radiation heat flux and exponentially varying density

The propagation of shock waves in a dusty gas with heat conduction and radiation heat flux, in which density varies exponentially, is investigated. The dusty gas is assumed to be a mixture of small solid particles and a perfect gas. The equilibrium flow conditions are assumed to be maintained, and the heat conduction is expressed 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 αR are assumed to vary with temperature and density. The shock wave moves with variable velocity and the total energy of the wave is non-constant. Non-similar solutions are obtained and the effects of variation of the heat transfer parameters and time 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 in the region behind the shock are also investigated at given times.

Similarity solutions for unsteady flow behind an exponential shock in a dusty gas

One-dimensional self-similar unsteady flow behind a strong exponential shock driven out by a piston moving with time according to an exponential law is investigated. The medium is assumed to be a mixture of small solid particles and a perfect gas. In order to get some essential features of the shock propagation, small solid particles are considered as pseudo-fluid and it is assumed that the equilibrium flow condition is maintained in the flow-field, and that the viscous-stress and heat conduction of the mixture are negligible. Solutions are obtained, in both cases, when the flow between the shock and the piston is isothermal or adiabatic. It is found that the assumption of zero temperature gradient brings a profound change in the density distribution as compared to that of the adiabatic case. Effects of a change in the mass concentration of the solid particles in the mixture Kp and the ratio of the density of solid particles to that of initial density of the gas G1 are also obtained.

Propagation of Strong Spherical Shock Waves in a Dusty Gas

In this paper, we have examined one-dimensional unsteady self-similar adiabatic flow of a dusty gas behind a spherical shock wave with time dependent energy input. The dusty gas is assumed to be a mixture of small solid particles and perfect gas. The solid particles are continuously distributed in the perfect gas. In order to get some essential features of shock propagation, the solid particles are considered as a pseudo-fluid and it is assumed that the equilibrium flow condition is maintained in the flow-field, and viscous stress and heat conduction of the mixture are negligible. The presence of small solid particles affects the medium in two ways. On the one hand, the volume fraction of solid particles lowers the compressibility of the mixture; and on the other hand, the particle load increases the inertia of the mixture. It is found that if the density of the solid particles and that of the perfect gas are equal, by an increase in the mass fraction of solid particles in the mixture, the strength of the shock is decreases and its distance from the inner expanding surface is increased. Also, by an increase in the ratio of density of solid particles and that of the perfect gas in the mixture, the strength of the shock is increased and its distance from the inner surface is decreased. Further, an increase in the rate of energy input increases the shock velocity and decreases the distance between the shock front and the inner surface.

A method of control of transfer processes

The effective medium theory for the coupled phenomena of heat conduction and mass diffusion of solute in heterogeneous media has been derived. The theory has been applied to a barrier with adaptive conductive and diffusive characteristics to be used for control of heat flow and mass flow of species. The barrier consists of a mixture of small solid particles of ellipsoidal shape which are randomly distributed in a carrier fluid. Volume fraction of the particles is small and variation in conductive–diffusive properties of the barrier is achieved by changing orientation of the particles. This analysis gives an estimate to what a degree the solute flow can be controlled by both bulk concentration and temperature gradient applied to the walls of the barrier. Influence of different parameters like ratios of conductive and diffusive properties of particles and the carrier fluid, solubility ratio of the diffusing species in the fluid and in the particle, class of particle shapes, particle aspect ratio and volume fraction as well as thickness of the barrier on the species mass flow have been studied in detail.