Water waves over a muddy bed: a two-layer Stokes' boundary layer model

This study aims to develop an asymptotic theory for the flow kinematics of a thin layer of viscous mud under water surface waves. The mud depth, the mud Stokes' boundary layer thickness, and the wave amplitude are assumed to be comparable with one another, and much smaller than the wavelength. By virtue of this sharp contrast in length scales, boundary layer equations are used to describe motion of both the mud and the immediately overlying water. Analytical solutions are sought and explicit expressions are obtained, under progressive waves, for the fluid velocity fields, interface wave characteristics, and wave-damping rate at the first-order, and the steady mean discharge rate, and mass-transport velocity of mud at the second-order. Effects of the mud layer thickness, density, and viscosity ratios on these kinematic quantities are examined in detail.