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

TL;DR

This paper extends the symmetry analysis of polytropic fluid flows to include discontinuities like shock waves, introducing dual Rankine-Hugoniot conditions linked by SL(2,R) transformations, enhancing understanding of explosion and implosion phenomena.

Contribution

It develops a dual set of Rankine-Hugoniot conditions for discontinuous flows, connecting them through SL(2,R) symmetry, and validates the invariance of physical entropy conditions.

Findings

The dual Rankine-Hugoniot conditions are related via SL(2,R) transformations.

Shock entropy conditions remain invariant under the new symmetry.

Symmetry considerations apply to both smooth and discontinuous fluid flows.

Abstract

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