Withdrawal of a fluid of finite depth through a line sink with a cusp in the free surface

Abstract

The steady withdrawal of a fluid of finite depth into a line sink is considered. The problem is solved numerically by a boundary integral equation method. It is shown that the flow depends on the Froude number F=m( gH 3) −1/2 and the nondimensional sink depth β=H S/ H, where m is the sink strength, g the acceleration of gravity, H is the total depth and H S is the depth of the sink. For given values of β and F there is a one-parameter family of solutions with a cusp on the free surface above the sink. It is found that in general there is a train of steady waves on the free surface. For particular values of the parameters the amplitude of the waves vanishes and the solutions reduce to those computed by Vanden-Broeck and Keller. These findings confirm and generalize the calculations of Vanden-Broeck where the free surface was covered by a lid everywhere but close to the sink.