Unsteady waves due to an impulsive Oseenlet beneath the capillary surface of a viscous fluid

Abstract

The two-dimensional free-surface waves due to a point force steadily moving beneath the capillary surface of an incompressible viscous fluid of infinite depth were analytically investigated. The unsteady Oseen equations were taken as the governing equations for the viscous flows. The kinematic and dynamic conditions including the combined effects of surface tension and viscosity were linearized for small-amplitude waves on the free-surface. The point force is modeled as an impulsive Oseenlet. The complex dispersion relation for the capillary-gravity waves shows that the wave patterns are characterized by the Weber number and the Reynolds number. The asymptotic expansions for the wave profiles were explicitly derived by means of Lighthill's theorem for the Fourier transform of a function with a finite number of singularities. Furthermore, it is found that the unsteady wave system consists of four families, that is, the steady-state gravity wave, the steady-state capillary wave, the transient gravity wave, and the transient capillary wave. The effect of viscosity on the capillary-gravity was analytically expressed.