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This manuscript aims to deal with the one-dimensional unsteady flow of non-ideal gas with monochromatic radiation under the influence of a magnetic field, where the invariance method of the Lie group of transformations is applied in a cylindrically symmetric motion. The density is assumed to be uniform in the undisturbed medium. The radiation flux is considered to move through the non-ideal gas. The self-similar solutions of the considered model are obtained for the power-law and the exponential-law shock paths. The effects of the adiabatic exponent, radiation parameter, ambient magnetic field, and the parameter of non-idealness on the flow variables are shown graphically. Numerical calculations are performed to obtain the self-similar solutions.

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.