Kinematics of one-dimensional spherical shock waves in interstellar van der Waals gas clouds

In this work, a system of non-linear partial differential equations, which describes one-dimensional motion of an inviscid, self-gravitating, and spherically symmetric van der Waals gas cloud, is considered. By using the method based on the kinematics of shock waves, the evolution equation for spherical shock wave in an interstellar van der Waals gas cloud is derived. By applying the truncation approximation procedure, an infinite system of transport equations, which governs the shock propagation, is derived to study the kinematics of shock waves for the one-dimensional motion. The first, second, and third order transport equations, which describe the shock strength and the induced discontinuity behind it, are used to analyze the decay and growth behavior of the shock waves in a non-ideal gas. The results are obtained for the exponent obtained from the first, second, and third order approximations and compared with the results obtained by Whitham's characteristic rule (Chester-Chisnell-Whitham approximation). In addition, the effects of the parameters of non-idealness and cooling-heating function on the evolutionary behavior of shocks are discussed and shown graphically.