Calculation of the wave motion on the uniformly charged interface between two immiscible viscous liquids on the basis of the boundary layer theory

A modified theory of the boundary layer associated with a periodic capillary-gravity wave on the uniformly charged interface between two immiscible viscous incompressible liquids is proposed. A model problem simpler than the exact problem is proposed for describing the boundary layer flows in the upper and lower liquids. The structure of the solution of this problem reflects the principal features of the exact asymptotic solution, namely, the rapidity of the decrease in the vortex part of the flow as a function of the depth and the insignificance of certain components. Estimates of the boundary layer thicknesses for which the difference between the exact asymptotic solution and the solution of the simplified model problem (formulated within the framework of the theory proposed) can be specified with a predetermined accuracy are obtained.