On the admissibility of hydraulic jumps in two-layer fluids

Summary

The present paper deals with the propagation of nearly resonant gravity waves in two-layer fluids assuming that both fluids are incompressible and inviscid. Also, in accordance with hydraulic theory, the pressure distribution inside both layers is taken to be hydrostatic. As a consequence, equations which govern the evolution of weakly nonlinear surface layer and internal layer waves are found to agree in form with transport equations known from investigations of acoustic waves in dense gases. In general, these solutions of steady as well as unsteady flows contain regions of multivaluedness, which have to be eliminated by the insertion of discontinuities in the flow quantities, but now representing hydraulic jumps or bores rather than shocks. These discontinuities are not uniquely defined by the initial data and boundary conditions for the problem under consideration and, therefore, a central issue is the identification of physically acceptable, i.e. admissible weak solutions. In contrast to gas dynamics, however, the balance of mass and total momentum for a two-layer flow are not sufficient to derive the jump relationships. To resolve this difficulty, a number of additional model equations have been proposed in the literature. Here we investigate their consequences and, in particular, their compatibility with the second law in the small amplitude limit. Special emphasis is placed on the existence of inviscid bores, which have no counterpart in gas dynamics.