Analytical solutions for the interfacial viscous capillary-gravity waves due to an oscillating Stokeslet

Abstract

The fundamental solutions of the Stokes/Oseen equations due to a point force in an unbounded viscous fluid are referred to as the Stokeslet/Oseenlet, for which a systematic derivation are analytically presented here in terms of a uniform expression. By means of integral transforms, the closed-form solutions are explicitly deduced in a formula which involves the Hamiltonian, Hessian, and Laplacian operators, and elementary functions. Secondly, interfacial viscous capillary-gravity waves between two semi-infinite fluids due to oscillating singularities, including a simple source in the upper inviscid fluid and a Stokeslet in the low viscous fluid, were analytically studied by the Laplace-Fourier integral transform and asymptotic analysis. The dynamics responses consist of the transient and steady-state components, which are dealt with by the method of stationary phase and the Cauchy residue theorem, respectively. The transient response is made up of one short capillarity-dominated and one long gravity-dominated wave with the former riding on the latter. The steady-state wave has the same frequency as that of oscillating singularities. Asymptotic solutions for the wave profiles and the exact solution for the wave number are analytically derived, which show the combined effects of fluid viscosity, surface capillarity and an upper layer fluid.