Rankine--Hugoniot conditions for fluids whose energy depends on space and time derivatives of density

TL;DR

This paper derives Rankine-Hugoniot conditions for fluids with energy depending on space and time derivatives of density, using Hamilton's principle, and discusses shock wave discontinuities in dispersive systems.

Contribution

It introduces a novel derivation of shock conditions for dispersive fluids where energy depends on derivatives, expanding classical shock theory.

Findings

Derived governing equations using Hamilton's principle.

Established additional Rankine-Hugoniot relations for dispersive systems.

Discussed the well-posedness of shock discontinuities in such fluids.

Abstract

By using the Hamilton principle of stationary action, we derive the governing equations and Rankine-Hugoniot conditions for continuous media where the specific energy depends on the space and time density derivatives. The governing system of equations is a time reversible dispersive system of conservation laws for the mass, momentum and energy. We obtain additional relations to the Rankine-Hugoniot conditions coming from the conservation laws and discuss the well-founded of shock wave discontinuities for dispersive systems.