Decay of solutions to the linearized free surface Navier-Stokes equations with fractional boundary operators

TL;DR

This paper investigates how the decay rate to equilibrium in a viscous fluid with a free surface is affected by replacing surface tension with a fractional boundary operator, revealing a transition in decay behavior.

Contribution

It introduces a fractional boundary operator to interpolate between no surface tension and full surface tension, analyzing its impact on decay rates in linearized free surface Navier-Stokes equations.

Findings

Identifies a critical order of the fractional operator where decay rate transitions

Shows decay behavior depends on the fractional operator's order and cross-section

Provides a continuum of decay rates between no surface tension and classical surface tension cases

Abstract

In this paper we consider a slab of viscous incompressible fluid bounded above by a free boundary, bounded below by a flat rigid interface, and acted on by gravity. The unique equilibrium is a flat slab of quiescent fluid. It is well-known that equilibria are asymptotically stable but that the rate of decay to equilibrium depends heavily on whether or not surface tension forces are accounted for at the free interface. The aim of the paper is to better understand the decay rate by studying a generalization of the linearized dynamics in which the surface tension operator is replaced by a more general fractional-order differential operator, which allows us to continuously interpolate between the case without surface tension and the case with surface tension. We study the decay of the linearized problem in terms of the choice of the generalized operator and in terms of the horizontal cross-section. In the case of a periodic cross-section we identify a critical order of the differential operator at which the decay rate transitions from almost exponential to exponential.