Reconciling different formulations of viscous water waves and their mass conservation

Abstract

The viscosity of water induces a vorticity near the free surface boundary, giving a rotational component to the fluid velocity. Our analysis reveals the correspondence between three common sets of model equations that deal with this vorticity in different ways. 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 explicitly splitting the problem into two domains: the irrotational bulk and the vortical boundary layer. The correspondence between the three systems in terms of mass conservation disproves the speculation that mass is not conserved in a rotational model, and exemplifies the correspondence between rotational pressure on the surface and vorticity in the boundary layer.