The propagation of long wavelength disturbances on the surface of a fluid layer of finite depth is considered. Attention is focused on the effect of stress applied at the surface. Constant surface tension leads to a normal stress at the surface, but the presence of a surfactant or the application of an electric field can give rise to tangential stresses. In the large Reynolds number limit, the evolution equation for the surface elevation contains contributions from both boundary layers in the flow; one is adjacent to the free surface while the other lies at the base of the fluid layer.Aweakly non-linear analysis is performed leading to an evolution equation similar to the classic Korteweg-de Vries equation, but modified by additional terms due to the viscosity and to the tangential and normal stress at the surface. It is demonstrated that careful treatment of the boundary layer at the free surface is necessary when the tangential stress at the surface is non-zero. Particular cases of flows with tangential surface stress due the presence of a surfactant or due to an electric field are discussed, and a pseudo-spectral scheme is used in order to obtain some typical numerical results.
|Number of pages
|Quarterly Journal of Mechanics and Applied Mathematics
|Published - 1 Aug 2013