Hamilton's Principle and Rankine-Hugoniot Conditions for General Motions of Mixtures

TL;DR

This paper extends the theoretical framework for two-fluid mixtures by deriving governing equations and jump conditions using Hamilton's principle, accounting for component entropies and revealing differences from classical conservation laws.

Contribution

It introduces a novel approach to derive Rankine-Hugoniot conditions for general two-fluid mixtures considering entropies, beyond traditional conservation law systems.

Findings

Governing equations for each component are established.

Rankine-Hugoniot conditions are derived using Hamilton's principle.

Jump conditions differ from classical conservation laws in the two-fluid case.

Abstract

In previous papers, we have presented hyperbolic governing equations and jump conditions for barotropic fluid mixtures. Now we extend our results to the most general case of two-fluid conservative mixtures taking into account the entropies of components. We obtain governing equations for each component of the medium. This is not a system of conservation laws. Nevertheless, using Hamilton's principle we are able to obtain a complete set of Rankine-Hugoniot conditions. In particular, for the gas dynamics they coincide with classical jump conditions of conservation of momentum and energy. For the two-fluid case, the jump relations do not involve the conservation of the total momentum and the total energy.