Junichi Koganemaru | Searcharxiv

arXiv2023

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

In this paper we study traveling wave solutions to the free boundary incompressible Navier-Stokes system with generalized Navier-slip conditions. The fluid is assumed to occupy a horizontally infinite strip-like domain that is bounded below by a flat rigid surface and above by a moving surface. We assume that the fluid is acted upon by a bulk force and a surface stress that are stationary in a coordinate system moving parallel to the fluid bottom, and a uniform gravitational force that is perpendicular to the flat rigid surface. We construct our solutions via an implicit function argument, and show that as the slip parameter shrinks to zero, the Navier-slip solutions converge to solutions to the no-slip problem obtained previously.

math.AP↗

arXiv2022

Traveling wave solutions to the inclined or periodic free boundary incompressible Navier-Stokes equations

This paper concerns the construction of traveling wave solutions to the free boundary incompressible Navier-Stokes system. We study a single layer of viscous fluid in a strip-like domain that is bounded below by a flat rigid surface and above by a moving surface. The fluid is acted upon by a bulk force and a surface stress that are stationary in a coordinate system moving parallel to the fluid bottom. We also assume that the fluid is subject to a uniform gravitational force that can be resolved into a sum of a vertical component and a component lying in the direction of the traveling wave velocity. This configuration arises, for instance, in the modeling of fluid flow down an inclined plane. We also study the effect of periodicity by allowing the fluid cross section to be periodic in various directions. The horizontal component of the gravitational field gives rise to stationary solutions that are pure shear flows, and we construct our solutions as perturbations of these by means of an implicit function argument. An essential component of our analysis is the development of some new functional analytic properties of a scale of anisotropic Sobolev spaces, including that these spaces are an algebra in the supercritical regime, which may be of independent interest.

math.AP↗