Reconciling different formulations of viscous water waves and their mass conservation

TL;DR

This paper compares three models of viscous water waves with vorticity, analyzing their differences in boundary conditions, rotational effects, and mass conservation to clarify their assumptions and relations.

Contribution

It provides a detailed comparison of three common water wave models incorporating viscosity and vorticity, highlighting their assumptions and differences.

Findings

Links rotational pressure to vorticity in the boundary layer

Identifies where nonlinear vorticity terms appear in models

Analyzes mass conservation across different formulations

Abstract

The viscosity of water induces a vorticity near the free surface boundary. The resulting rotational component of the fluid velocity vector greatly complicates the water wave system. Several approaches to close this system have been proposed. Our analysis compares three common sets of model equations. The first set has a rotational kinematic boundary condition at the surface. In the second set, a gauge choice for the velocity vector is made that cancels the rotational contribution in the kinematic boundary condition, at the cost of rotational velocity in the bulk and a rotational pressure. The third set circumvents the problem by introducing two domains: the irrotational bulk and the vortical boundary layer. This comparison puts forward the link between rotational pressure on the surface and vorticity in the boundary layer, addresses the existence of nonlinear vorticity terms, and shows where approximations have been used in the models. Furthermore, we examine the conservation of mass for the three systems, and how this can be compared to the irrotational case.