Free surface deformation due to a source and a sink of equal strength in Stokes flow

Two-dimensional Stokes flow due to a source and a sink of equal strength below the free surface is analyzed and free surface shape and cusp formation are discussed. The source-sink pair below the free surface are aligned vertical to the free surface. In the analysis, the 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 free surface shapes are shown and formation of a cusp on the free surface is discussed. As the capillary number increases, the converging free surface shape becomes singular and tends to form a cusp for sufficiently large capillary number. Typically, streamline patterns for some capillary numbers are also shown. As the capillary number vanishes, the solution is reduced to the linearized potential flow solution.