Generation of free-surface gravity waves by an unsteady Stokeslet

The interaction of unsteady Stokeslets with the free surface of an initially quiescent incompressible fluid of infinite depth is investigated analytically for two- and three-dimensional cases. The disturbed flows are generated by an unsteady singular force moving perpendicularly downwards away from the surface. The analysis is based on the assumption that the motion satisfies the linearized unsteady Navier–Stokes equations with linear kinematic and dynamic boundary conditions. Firstly, the asymptotic representation for the transient free-surface waves due to an instantaneous Stokeslet is derived for a large time with a fixed distance-to-time ratio. As is well known, the corresponding inviscid waves predicted by the potential theory do not decay to zero as the time goes to infinity. In the present study, the transient waves predicted by the viscous theory eventually vanish due to the presence of viscosity, which is consistent with reality from the physical point of view. Secondly, the asymptotic solutions are obtained for the unsteady free-surface waves due to a harmonically oscillating Stokeslet. It is found that the unsteady waves can be decomposed into steady-state and transient responses. The steady state can be attained as time approaches infinity. It is shown that the viscosity of the fluid plays an important role in the evolution of the singularity-induced waves.