Streamline Patterns and Eddies in Slipping Stokes Flow

Streamline topologies are analyzed in the vicinity of boundaries in the limit of Stokes flow with Navier slip boundary conditions for some simple flows involving two- and three-dimensional configurations. It is found that the streamline pattern transformations, and consequently the flow fields are sensitive to the non-dimensional slip parameter $\lambda$. For two-dimensional flows, the separated/attached eddies - that are known to exist in the no-slip case at the contour - get destroyed or pushed away from the boundary as the slip is varied. Analysis of flow generated by a point force (stokeslet) inside a spherical container reveals that when the stokeslet is positioned at the center of the container, the eddy pattern - that is noted in the no-slip case - undergoes a series of transformations due to slip variations and eventually disappears. Furthermore, the parameter $\lambda$ dictates the locations of the stagnation point and the point of zero vorticiy in the flow domain. Our analytical solution indicates that the {\it co-existence of a stagnation point ($r_{stag}$) and a point of zero vorticity ($r_{\Omega=0}$) in the flow region is necessary for the occurrence of closed eddies}. The results may be of some interest in small scale hydrodynamics in which Stokes flow occurs.