Symmetry analysis and exact solutions of magnetogasdynamic equations

Abstract

Using the invariance group properties of the governing system of partial differential equations (PDEs), admitting Lie group of point transformations with commuting infinitesimal generators, we obtain exact solutions to the system of PDEs describing one-dimensional unsteady planar and cylindrically symmetric motions in magnetogasdynamics involving shock waves. Some appropriate canonical variables are characterised that transform the equations at hand to an equivalent autonomous form, the constant solutions of which correspond to non-constant solutions of the original system. The governing system of PDEs includes as a special case the Euler's equations of non-isentropic gasdynamics. It is interesting to remark that in the absence of magnetic field, one of the exact solutions obtained here is precisely the blast wave solution obtained earlier using a different method of approach. A particular solution to the governing system, which exhibits space-time dependence, is used to study the wave pattern that finally develops when a magnetoacoustic wave impacts with a shock. The influence of magnetic field strength on the evolutionary behaviour of incident and reflected waves and the jump in shock acceleration, after collision, are studied.