Symmetries of Discontinuous Flows and the Dual Rankine-Hugoniot Conditions in Fluid Dynamics

摘要

It has recently been shown that the maximal kinematical invariance group ofpolytropic fluids, for smooth subsonic flows, is the semidirect product ofSL(2,R) and the static Galilei group G. This result purports to offer atheoretical explanation for an intriguing similarity, that was recentlyobserved, between a supernova explosion and a plasma implosion. In this paperwe extend this result to discuss the symmetries of discontinuous flows, whichfurther validates the explanation by taking into account shock waves, which arethe driving force behind both the explosion and implosion. This is accomplishedby constructing a new set of Rankine-Hugoniot conditions, which follow fromNoether's conservation laws. The new set is dual to the standardRankine-Hugoniot conditions and is related to them through the SL(2,R)transformations. The entropy condition, that the shock needs to satisfy forphysical reasons, is also seen to remain invariant under the transformations.