Decay Rates for Viscous Surface Waves of Isotropic Micropolar Fluids With or Without Surface Tension

TL;DR

This paper analyzes the decay rates of viscous micropolar surface waves under gravity, showing surface tension accelerates decay and establishing global well-posedness for both cases.

Contribution

It provides the first comprehensive study of decay rates for viscous micropolar fluids with and without surface tension in a three-dimensional setting.

Findings

Surface tension enhances decay rate of solutions.

Solutions decay exponentially with surface tension.

Solutions decay almost exponentially without surface tension.

Abstract

In this paper, we consider a layer of viscous incompressible isotropic micropolar fluid in a uniform gravitational field of finite depth, lying above a flat rigid bottom and below the atmosphere in a three-dimensional horizontally periodic setting. The fluid dynamics are governed by gravity-driven incompressible micropolar equations. We investigate the global well-posedness for both the cases with and without surface tension. On one hand, in the case with surface tension (i.e. {\sigma} > 0), we show that the global solution decays to the equilibrium exponentially. On the other hand, in the case without surface tension (i.e. {\sigma} = 0), the solution decays to the equilibrium at an almost exponential rate. Comparing the two different cases for {\sigma} > 0 and {\sigma} = 0 reveals that the surface tension can enhance the decay rate.