Large-amplitude internal fronts in two-fluid systems

TL;DR

This paper demonstrates the existence of large-amplitude internal hydrodynamic bores in two-fluid systems, revealing complex behaviors such as overturning and contact with boundaries, using a novel global continuation theory for elliptic equations.

Contribution

It introduces a new abstract machinery for global continuation of monotone front solutions in elliptic equations, applicable to fully nonlinear and quasilinear problems with transmission boundary conditions.

Findings

Existence of continuous solution families bifurcating from flat interfaces.

Internal interfaces can overturn, contact upper walls, or develop double stagnation points.

Robust new method for analyzing nonlinear elliptic problems in fluid dynamics.

Abstract

In this announcement, we report results on the existence of families of large-amplitude internal hydrodynamic bores. These are traveling front solutions of the full two-phase incompressible Euler equation in two dimensions. The fluids are bounded above and below by flat horizontal walls and acted upon by gravity. We obtain continuous curves of solutions to this system that bifurcate from the trivial solution where the interface is flat. Following these families to the their extreme, the internal interface either overturns, comes into contact with the upper wall, or develops a highly degenerate "double stagnation" point. Our construction is made possible by a new abstract machinery for global continuation of monotone front-type solutions to elliptic equations posed on infinite cylinders. This theory is quite robust and, in particular, can treat fully nonlinear equations as well as quasilinear problems with transmission boundary conditions.