Rotational Axisymmetric Non-ideal Gas Research Articles

Approximate analytical solution using power series method for the propagation of blast waves in a rotational axisymmetric non-ideal gas

In this paper, the propagation of the blast (shock) waves in non-ideal gas atmosphere in rotational medium is studied using a power series method in cylindrical geometry. The flow variables are assumed to be varying according to the power law in the undisturbed medium with distance from the axis of symmetry. To obtain the similarity solution, the initial density is considered as constant in the undisturbed medium. Approximate analytical solutions are obtained using Sakurai’s method by extending the power series of the flow variables in power of a0U2, where U and a0 are the shock velocity and speed of sound, respectively, in undisturbed fluid. The strong shock wave is considered for the ratio a0U2 which is considered to be a small quantity. With the aid of that method, the closed-form solutions for the zeroth-order approximation is given as well as first-order approximate solutions are discussed. Also, with the help of graphs behind the blast wave for the zeroth-order, a comparison between the numerical solution and the approximate analytical solution are shown. The distributions of flow variables such as density, radial velocity, pressure and azimuthal velocity are analysed. The results for the rotationally axisymmetric non-ideal gas environment are compared to those for the ideal gas atmosphere. The velocity-distance and time-distance curves are also shown to analyse the decaying characteristic of a blast wave.

A self-similar solution for unsteady adiabatic and isothermal flows behind the shock wave in a non-ideal gas using Lie group analysis method with azimuthal or axial magnetic field in rotating medium

The similarity solutions using Lie group analysis for shock wave propagation in a rotational axisymmetric non-ideal gas with azimuthal or axial magnetic field in the case of isothermal and adiabatic flows are obtained. All possible cases of similarity solutions are discussed using the Lie group analysis for the isothermal and adiabatic flows. The arbitrary constants involved in the generators of local Lie groups bring various possible cases of solutions with exponential law and power law shock paths. Similarity solution for isothermal and adiabatic flows with power law shock path is discussed in detail. The density of ambient medium is taken to be constant. The axial and azimuthal fluid velocities and magnetic field in the ambient medium are assumed to be varying according to power law. The effect on shock wave strength and that on the flow variables due to variation of the Alfven Mach number, adiabatic index of the gas, non-idealness parameter, rotational parameter and initial magnetic field variation exponent are investigated. It is found that these parameters have decaying effects on shock wave. The obtained results in the case of isothermal and adiabatic flows are also compared with each other.

Analytical solution for unsteady flow behind ionizing shock wave in a rotational axisymmetric non-ideal gas with azimuthal or axial magnetic field

Abstract The approximate analytical solution for the propagation of gas ionizing cylindrical blast (shock) wave in a rotational axisymmetric non-ideal gas with azimuthal or axial magnetic field is investigated. The axial and azimuthal components of fluid velocity are taken into consideration and these flow variables, magnetic field in the ambient medium are assumed to be varying according to the power laws with distance from the axis of symmetry. The shock is supposed to be strong one for the ratio C 0 V s 2 ${\left(\frac{{C}_{0}}{{V}_{s}}\right)}^{2}$ to be a negligible small quantity, where C 0 is the sound velocity in undisturbed fluid and V S is the shock velocity. In the undisturbed medium the density is assumed to be constant to obtain the similarity solution. The flow variables in power series of C 0 V s 2 ${\left(\frac{{C}_{0}}{{V}_{s}}\right)}^{2}$ are expanded to obtain the approximate analytical solutions. The first order and second order approximations to the solutions are discussed with the help of power series expansion. For the first order approximation the analytical solutions are derived. In the flow-field region behind the blast wave the distribution of the flow variables in the case of first order approximation is shown in graphs. It is observed that in the flow field region the quantity J 0 increases with an increase in the value of gas non-idealness parameter or Alfven-Mach number or rotational parameter. Hence, the non-idealness of the gas and the presence of rotation or magnetic field have decaying effect on shock wave.

Magnetogasdynamic exponential shock wave in a self-gravitating, rotational axisymmetric non-ideal gas under the influence of heat-conduction and radiation heat-flux

Similarity solutions are obtained for one-dimensional cylindrical shock wave in a self-gravitating, rotational axisymmetric non-ideal gas with azimuthal or axial magnetic field in the presence of conductive and radiative heat fluxes. The total energy of the wave is non-constant. It is obtained that the increase in the Cowling number, in the parameters of radiative as well as conductive heat transfer and the parameter of the non-idealness of the gas have a decaying effect on the shock wave however increase in the value of gravitational parameter has reverse effect on the shock strength. It is manifested that the presence of azimuthal magnetic field removes the singularities which arise in some cases of the presence of axial magnetic field. Also, it is observed that the effect of the parameter of non-idealness of the gas is diminished by increasing the value of the gravitational parameter.

Approximate analytical solution for ionizing cylindrical shock wave in rotational axisymmetric non-ideal gas: isothermal flow

The propagation of an ionizing cylindrical shock wave in rotational axisymmetric non-ideal gas under isothermal flow condition with an azimuthal magnetic field is investigated. The electrical conductivity is assumed to be negligible in the medium ahead of the shock wave, which after the passage of the shock wave becomes infinitely large. The magnetic pressure, azimuthal fluid velocity, and axial fluid velocity are assumed to be varying according to the power law with distance from the axis of symmetry in the undisturbed medium. The zeroth and first-order approximations are discussed by the aid of the power series method. Solutions for the zeroth-order approximation are constructed in analytical form. Distributions of hydrodynamical quantities are discussed. The effect of flow parameters, namely, shock wave Cowling number c∗, adiabatic exponent γ, rotational parameter L, and gas non-idealness parameter [Formula: see text] are studied on the flow variables. Due to the consideration of a rotating medium or due to the presence of magnetic field, the total energy of the disturbance increases, while with an increase in adiabatic exponent γ the total energy of the disturbance decreases. Density and pressure vanish near the axis of symmetry, thus forming a vacuum there.

Magnetogasdynamic shock waves in a rotating axisymmetric non-ideal gas with increasing energy, conductive, and radiative heat fluxes

Self-similar solutions are obtained for one-dimensional unsteady adiabatic flow behind a magnetogasdynamics cylindrical shock wave propagating in a rotational axisymmetric non-ideal gas with increasing energy and conductive and radiative heat fluxes in the presence of an azimuthal magnetic field. The azimuthal and axial components of the fluid velocity and the azimuthal magnetic field in the ambient medium are assumed to be varying and obeying power laws. In order to find the similarity solution, the angular velocity of the ambient medium is taken to be decreasing as the distance from the axis increases. Fourier’s law is used to express the heat conduction and the radiation is assumed to be of diffusion type for an optically thick gray gas model. The absorption coefficient and thermal conductivity are assumed to be dependent on temperature and density. The effects of the non-idealness of the gas and the magnetic field on the shock propagation and on the flow variables behind the shock front are investigated. A comparison between rotating and non-rotating medium is also, made. It is shown that the non-idealness of the gas and the magnetic field have a decaying effect on the shock wave, whereas the shock strength increases due to the consideration of rotating medium.

Self-similar solution for the flow behind an exponential shock wave in a rotational axisymmetric non-ideal gas with magnetic field

The propagation of a cylindrical shock wave driven out by a piston moving with time according to an exponential law, in a rotational axisymmetric non-ideal gas with azimuthal magnetic field is investigated. It is assumed that the ambient medium possess radial, azimuthal and axial components of fluid velocity. The magnetic field and the component of fluid velocity are assumed to be varying according to an exponential law. The solutions are obtained for both the cases of adiabatic and isothermal flows between the piston and the shock by considering the components of vorticity vector. The effects of variation of ambient magnetic field, non-idealness of the gas and adiabatic exponent are worked out in detail. It is shown that the increase in the non-idealness of the gas or the adiabatic exponent of the gas or presence of magnetic field have decaying effect on the shock wave. The zero temperature gradient assumption decreases the shock strength and brings a profound change in the density, the magnetic field and the non-dimensional axial components of vorticity vector distributions as compare to those of the adiabatic case. Also, a comparison between the solutions obtained in the case of isothermal and adiabatic flows is done.

An exact solution for the propagation of cylindrical shock waves in a rotational axisymmetric non-ideal gas with axial magnetic field and radiative heat flux

An exact solution for the propagation of cylindrical shock waves in a rotational axisymmetric non-ideal gas with axial magnetic field and radiative heat flux

Cylindrical shock wave generated by a moving piston in a rotational axisymmetric non-ideal gas with conductive and radiative heat-fluxes in the presence of azimuthal magnetic field

The propagation of a cylindrical shock wave in a rotational axisymmetric non-ideal gas with heat conduction and radiation heat-flux, in the presence of a spatially decreasing azimuthal magnetic field, driven out by a moving piston is investigated. The fluid velocities and the azimuthal magnetic field in the ambient medium are assume to be varying and obeying power laws. The gas is assumed to be electrically conducting and obey a simplified van der Waals equation of state. The shock wave moves with variable velocity and the total energy of the wave is non-constant. Similarity solutions are obtained for the flow-field behind the shock and the effects of variation of Alfven Mach number, the conductive and radiative heat transfer parameters, the parameter of the non-idealness are worked out in detail. An increase in the value of Alfven Mach number or in the non-idealness parameter of the gas, decreases the compressibility of the gas and hence there is a decrease in the shock strength; whereas the reverse effects on compressibility and shock strength are obtained with an increase in conductive or radiative heat transfer parameter. It is shown that due to the consideration of rotating medium the compressibility of the gas and the shock strength increase. Further, it is investigated that with an increase in the parameters of conductive and radiative heat transfer the tendency of formation of maxima in the distribution of pressure and minimum in magnetic field and non-dimensional axial component of vorticity vector decreases. It is interesting to note that the pressure and density vanish at the inner surface (piston) and hence a vacuum is formed at the axis of symmetry, which is in excellent agreement with the laboratory conditions to produce a shock wave. A comparison between the cases of rotating and non-rotating medium is also made.

Flow Behind an Exponential Shock Wave in a Rotational Axisymmetric Non-ideal Gas with Conduction and Radiation Heat Flux

A self-similar model for one-dimensional unsteady adiabatic flows behind a cylindrical shock wave driven out by a piston moving with time according to an exponential law in non-ideal gas in the presence of conductive and radiative heat fluxes is discussed in a rotating atmosphere. The ambient medium is assumed to have axial, azimuthal and radial component of fluid velocities. The axial and azimuthal component of fluid velocity in the ambient medium is assumed to be varying according to exponential laws. The initial density and angular velocity of the ambient medium are taken to be constant for the existence of the similarity solutions. It is shown that increase in the parameter of non-idealness of the gas and the conductive or radiative heat transfer parameters have decaying effect on the shock wave. It is obtained that the parameter of non-idealness of the gas and heat transfer parameters have same effects on shock strength, density, pressure, axial and azimuthal component of fluid velocity and vorticity vector.

Unsteady Adiabatic Flow Behind a Cylindrical Shock in a Rotational Axisymmetric Non-Ideal Gas Under the Action of Monochromatic Radiation

A self-similar model for the unsteady, one-dimensional cylindrical shock waves in non-ideal gas is discussed in a rotating atmosphere under the action of monochromatic radiation. The ambient medium is assumed to possess radial, axial and azimuthal component of fluid velocities. The fluid velocities within the ambient medium are assumed to be varying and obeying a power law. Similarity solutions exist only when the surrounding medium is of constant density. The gas is assumed to be non-ideal and the angular velocity of the ambient medium is assumed to be decreasing as the distance from the axis increases. It is investigated that increase in the non-idealness of the gas and the adiabatic exponent of the gas decay the shock wave. It is observed that the non-idealness of the gas and adiabatic exponent of the gas have same effects on the flow variables except radial component of fluid velocity. Also, the non-idealness of the gas decreases the radiation flux and increases pressure and density, whereas the shock-Mach number have opposite behaviour on them.

Similarity solutions for unsteady flow behind an exponential shock in an axisymmetric rotating non-ideal gas

One-dimensional self-similar unsteady isothermal and adiabatic flows behind a strong exponential shock wave driven out by a cylindrical piston moving with time according to an exponential law in a rotational axisymmetric non-ideal gas is investigated. The medium is assumed to be non-ideal gas rotating about the axis of symmetry. The fluid velocities in the ambient medium are assumed to be varying with time according to an exponential law. Similarity solutions exist only when the surrounding medium is of constant density. Solutions are obtained, in both the cases, when the flow between the shock and the piston is isothermal or adiabatic by taking into account components of vorticity vector. It is found that the assumption of zero temperature gradient brings a profound change in the density and compressibility distributions as compared to that of the adiabatic case. The effect of an increase in the value of the parameter of the non-idealness of the gas is investigated. Also, a comparison between the solutions in the cases of isothermal and adiabatic flows is made. Further, it is shown that the consideration of zero temperature gradient and the effect of variation of the parameter of non-idealness of the gas decrease the shock strength and widens the disturbed region between the shock and piston. The shock waves in non-ideal gas can be important for description of shocks in supernova explosions, in the study of a flare produced shock in solar wind, central part of star burst galaxies, nuclear explosion, rupture of pressurized vessel, in the analysis of data from exploding wire experiments, and cylindrically symmetric hypersonic flow problems associated with meteors or reentry vehicles, etc. The findings of the present work provided a clear picture of whether and how the non-idealness of the gas and consideration of zero temperature gradient affect the propagation of shock and the flow behind it.

Magnetohydrodynamic Cylindrical Shock in a Rotational Axisymmetric Non-Ideal Gas Under the Action of Monochromatic Radiation

A self-similar MHD cylindrical shock wave is investigated in a non-uniform rotating non-ideal gas under the action of monochro- matic radiation. The fluid velocities and the initial magnetic field of the ambient medium are assumed to be varying and obeying power laws. The gas is assumed to be non-ideal having infinite electrical conductivity and as the distance from the axis in- creases the angular velocity of the ambient medium decreases. Also, when the surrounding medium is of constant density then only similarity solutions exist. It is investigated that increase in the non-idealness of the gas, the adiabatic exponent of the gas and the strength of ambient magnetic field have decaying effect on the shock wave. Also, it is observed that an increase in the non-idealness of the gas and adiabatic exponent of the gas has same effect on the flow variables except the radial component of fluid velocity, the pressure and the magnetic field.

Self-similar Cylindrical Ionizing Shock Waves in a Rotational Axisymmetric Non-ideal Gas with Radiation Heat Flux

Similarity solutions are obtained for one-dimensional adiabatic flow behind a gas-ionizing cylindrical shock wave propagating in a rotational axisymetric non-ideal gas with radiation heat flux, in presence of an azimuthal magnetic field. The electrical conductivity in the medium ahead of the shock is assumed to be negligible, which becomes infinitely large after passage of the shock . The ambient medium is considered to have variable axial velocity component in addition to the variable azimuthal velocity. In order to obtain the similarity solutions, the initial density of the medium is assumed to be constant and the initial angular velocity to be obeying a power law and to be decreasing as the distance from the axis increases . Effects of an increase in the value of the parameter of non-idealness of the gas, in the value of radiation parameter and in the value of the index for variation of azimuthal velocity of the ambient medium ( or in the value of the index for variation of the ambient magnetic field ) on the shock propagation are investigated. It is observed that the non-idealness of the gas or radiation heat-flux has decaying effect on the shock wave.

Propagation of a cylindrical shock wave in a rotational axisymmetric isothermal flow of a non-ideal gas in magnetogasdynamics

Self-similar solutions are obtained for unsteady, one-dimensional isothermal flow behind a shock wave in a rotational axisymmetric non-ideal gas in the presence of an azimuthal magnetic field. The shock wave is driven out by a piston moving with time according to power law. The fluid velocities and the azimuthal magnetic field in the ambient medium are assumed to be varying and obeying a power law. The density of the ambient medium is assumed to be constant. The gas is assumed to be non-ideal having infinite electrical conductivity and the angular velocity of the ambient medium is assumed to be decreasing as the distance from the axis increases. It is expected that such an angular velocity may occur in the atmospheres of rotating planets and stars. The effects of the non-idealness of the gas and the Alfven-Mach number on the flow-field are obtained. It is shown that the presence of azimuthal magnetic field and the rotation of the medium has decaying effect on the shock wave. Also, a comparison is made between rotating and non-rotating cases.