Traveling wave solutions to the multilayer free boundary incompressible Navier-Stokes equations

TL;DR

This paper proves the existence of traveling wave solutions in multilayer incompressible Navier-Stokes fluids with free boundaries, considering gravity, surface tension, and external forces, extending previous one-layer results to multiple layers.

Contribution

It is the first to construct traveling wave solutions for the multilayer incompressible Navier-Stokes equations with free boundaries.

Findings

Existence of traveling wave solutions for small forces and stresses.

Extension of one-layer results to multilayer fluid configurations.

Applicability to gravity, surface tension, and external forces.

Abstract

For a natural number $m \ge 2$, we study $m$ layers of finite depth, horizontally infinite, viscous, and incompressible fluid bounded below by a flat rigid bottom. Adjacent layers meet at free interface regions, and the top layer is bounded above by a free boundary as well. A uniform gravitational field, normal to the rigid bottom, acts on the fluid. We assume that the fluid mass densities are strictly decreasing from bottom to top and consider the cases with and without surface tension acting on the free surfaces. In addition to these gravity-capillary effects, we allow a force to act on the bulk and external stress tensors to act on the free interface regions. Both of these additional forces are posited to be in traveling wave form: time-independent when viewed in a coordinate system moving at a constant, nontrivial velocity parallel to the lower rigid boundary. Without surface tension in the case of two dimensional fluids and with all positive surface tensions in the higher dimensional cases, we prove that for each sufficiently small force and stress tuple there exists a traveling wave solution. The existence of traveling wave solutions to the one layer configuration ($m=1$) was recently established and, to the best of our knowledge, this paper is the first construction of traveling wave solutions to the incompressible Navier-Stokes equations in the $m$-layer arrangement.