Approximate analytical solution for the propagation of shock waves in self-gravitating perfect gas via power series method: isothermal flow

Abstract

For the propagation of a shock (blast) in a self-gravitating perfect gas in case of spherical and cylindrical symmetry, an approximate analytical solution is investigated. The shock wave is considered to be a strong one, with the ratio $\left( {\frac{C}{{V_{S} }}} \right)^{2}$ to be a small quantity, where $c$ is the sound speed in an undisturbed medium and $V_{S}$ is the shock wave velocity. The initial density in the undisturbed medium is taken to be varying according to a power law. To obtain the approximate closed-form similarity solution, the flow variables are expanded in a power series of $\left( {\frac{C}{{V_{S} }}} \right)^{2}$. The first- and second-order approximations are discussed with the help of power series expansion. The analytical solutions are constructed for the first-order approximation. The distribution of the flow variables for first-order approximation in the flow field region behind the shock wave is shown in graphs for both the cylindrical and spherical geometries. The effect of flow parameters, namely, ambient density variation index $\alpha$, adiabatic exponent $\gamma$and gravitational parameter $G_{0}$, are studied on the flow variables and on the total energy of disturbance in the case of the first approximation to the solutions. It is shown that the total energy of the disturbance in the flow field region behind the shock wave decreases with an increase in initial density variation index or adiabatic exponent, i.e. shock strength increases with increase in the value of adiabatic exponent or initial density variation index. A comparison is also made between the solutions obtained for non-gravitating and self-gravitating gases.