Using toroidal coordinates, an exact solution is derived for the velocity field induced in two immiscible semi-infinite fluids possessing a plane interface, by the slow rotation of an axially symmetric body partly immersed in each fluid. The surface of the body is assumed to be formed from two intersecting spheres, or a sphere and a circular disc, with the circle of intersection of the composite surfaces lying in th interface.
It is shown that when the rotating body possesses reflection symmetry about the plane of the interface of the fluids, the velocity field in either fluid is independent of the viscosities of the fluids. The torque exerted on the body is then proportional to the sum of the viscosities. Analytic closed-form expressions are derived for the torque when the body is either a sphere, a circular disc, or a tangent-sphere dumbbell, and for a hemisphere rotating in an infinite homogeneous fluid. Closed-form results are also given for an immersed sphere, tangent to a free surface. For other geometrical configurations, numerical values of the torque are provided for a variety of body shapes and two-fluid systems of various viscosity ratios.