Two-dimensional Stokes flow due to a pair of vortices below the free surface

Abstract

A two-dimensional Stokes flow due to a pair of counter-rotating vortices of equal strength below the free surface is analyzed, and the streamline pattern and free-surface deformation are discussed. Two vortices are placed at a fixed depth and an arbitrary distance between each other. In the analysis, Stokes’ approximation is used and surface tension effects are included, but gravity is neglected. The solution is obtained by using conformal mapping and complex function theory. From the solution, typical flow patterns are seen, depending on the capillary number, Ca, and the distance between the two vortices, and some interesting results are obtained. For separation distances below a critical distance, a cusp occurs at the center of the free surface as Ca→∞, following the results of Jeong and Moffatt [“Free-surface cusps associated with flow at low Reynolds number,” J. Fluid Mech. 241, 1 (1992)] for no separation (distance of zero). However, above the critical distance, the cusp disappears and a smooth, trough-shaped interface is formed. At even greater separation distances, a pair of viscous eddies exists near the free surface beyond some critical values of Ca. As the capillary number vanishes, the solution is reduced to that of a linearized potential flow.