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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. [ABSTRACT FROM AUTHOR]

The approximate analytical solutions are obtained for adiabatic and isothermal flows behind a cylindrical shock wave in a dusty gas. A mixture of perfect gas and micro size small inert solid particles is taken as the dusty gas. The inert solid particles are distributed continuously in the mixture. It is considered that the equilibrium flow conditions are maintained. The flow variables are expanded in power series to obtain the solution of the problem. The analytical solutions are obtained for the first order approximation in both the adiabatic and isothermal cases. Also, the system of ordinary differential equations for second order approximations to the solution is obtained. The influence of an increase in the ratio of the density of the inert solid particles to the initial density of the perfect gas, the rotational parameter and the mass concentration of inert solid particles in the mixture are discussed on the flow variables for first approximation. Our first approximation to the solution corresponds to the Taylor's solution for the creation of a blast wave by a strong explosion. A comparison is also made between the solutions for isothermal and adiabatic flows. It is investigated that the density and pressure near the line of symmetry in the case of isothermal flow become zero and hence a vacuum is formed at the axis of symmetry when the flow is isothermal. Also, it is found that an increase in the value of rotational parameter or the mass concentration of solid particles in the mixture has a decaying effect on shock wave. The present work may be used to verify the correctness of the solution obtained by self-similarity and numerical methods. [ABSTRACT FROM AUTHOR]

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 C0is the sound velocity in undisturbed fluid and VS 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 J0 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. [ABSTRACT FROM AUTHOR]

The present paper investigates the motion of the variable infinitesimal body in circular restricted four variable bodies problem. We have constructed the equations of motion of the infinitesimal variable mass under the effect of source of radiation pressure due to which albedo effects are produced by another two primaries and one primary is considered as an oblate body which is placed at the triangular equilibrium point of the classical restricted three-body problem and also the variation of Jacobi Integral constant has been determined. We have studied numerically the equilibrium points, Poincaré surface of sections and basins of attraction in five cases (i. Third primary is placed at one of the triangular equilibrium points of the classical restricted three-body problem, ii. Variation of masses, iii. Solar radiation pressure, iv. Albedo effect, v. Oblateness effect.) by using Mathematica software. Finally, we have examined the stability of the equilibrium points and found that all the equilibrium points are unstable.