In the present paper, we use a Lie group of transformations to obtain a class of similarity solutions to a problem of cylindrically symmetric strong shock waves propagating through one-dimensional, unsteady, and isothermal flow of a self-gravitating ideal gas under the influence of azimuthal magnetic field. The density of the ambient medium is assumed to be non-uniform ahead of the shock. The generators of the Lie group of transformations involve arbitrary constants which yield four different cases of possible solutions. Out of all possibilities, only two cases hold similarity solutions. One is with a power law shock path, and the other one is with an exponential law shock path. We present a detailed investigation for the case of power law shock path. Numerical computations have been performed to find out the flow patterns in the flow-field behind the shock. Also, we have analyzed the effects of variation in adiabatic index, ambient density exponent, gravitational parameter, and Alfven-Mach number on the flow variables behind the shock. All computations have been done using the software package MATLAB.

在本文中，我们使用李群变换来获得一类相似解，以解决圆柱对称强冲击波在重力作用下通过自重理想气体的一维，非定常和等温流传播的问题。方位角磁场。假定在冲击之前环境介质的密度不均匀。Lie变换组的生成器包含任意常数，这些常数会产生四种不同情况的可能解。在所有可能性中，只有两种情况具有相似性解决方案。一种是幂律冲击路径，另一种是幂律冲击路径。我们对电源法冲击路径的情况进行详细调查。已经进行了数值计算以找出冲击后的流场中的流型。另外，我们分析了绝热指数，环境密度指数，重力参数和Alfven-Mach数变化对冲击后流动变量的影响。所有计算已使用软件包MATLAB进行。