Derivation of a viscous Boussinesq system for surface water waves

TL;DR

This paper derives a new viscous Boussinesq system for surface water waves directly from Navier-Stokes equations without common simplifying assumptions, including the effects of bottom shear stress and decay rates, and extends it to 3-D.

Contribution

It introduces a novel derivation of a viscous Boussinesq system that does not rely on irrotationality or Zakharov-Craig-Sulem formulation, including new physical effects and 3-D extension.

Findings

Derived the viscous Korteweg-de Vries equation from the new system

Compared the derived equations to existing models in literature

Identified decay rates and bottom shear stress effects

Abstract

In this article, we derive a viscous Boussinesq system for surface water waves from Navier-Stokes equations. We use neither the irrotationality assumption, nor the Zakharov-Craig-Sulem formulation. During the derivation, we find the bottom shear stress, and also the decay rate for shallow water. In order to justify our derivation, we derive the viscous Korteweg-de Vries equation from our viscous Boussinesq system and compare it to the ones found in the bibliography. We also extend the system to the 3-D case.