The Stokes flow past the unusual geometry of two fused (overlapping) spheres, somewhat surprisingly, has an exact analytic solution, provided that the spheres intersect orthogonally. The problem of flow past a partially encapsulated droplet is considered, to illustrate the basic idea. The simple nature of the analytical solution is of some interest in biophysical models of molecular assemblies involving a combination of stick and slip surfaces. ©1997 American Institute of Physics.