Analytical solution for unsteady adiabatic and isothermal flows behind the shock wave in a rotational axisymmetric mixture of perfect gas and small solid particles

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.