The slow rotation of a sphere submerged in a fluid with a surfactant surface layer

The effect of a layer of an adsorbed surfactant monomolecular film of fluid which covers the surface of a large volume of a different substrate fluid is considered with respect to the fluid motion caused by the slow rotation of a submerged sphere. For a semi-infinite substrate, the boundary value problem posed with the surfactant boundary condition of Scriven and Goodrich is solved exactly for any depth of the submerged sphere. Comprehensive numerical calculations are given for the torque and surface velocity for various values of the parameters defining the depth of the sphere and the surface shear viscosity. Asymptotic expressions for the solution are given for the cases of a deeply submerged sphere or when the substrate has a finite depth. The relevance of the work to providing an experimental technique for measuring surface shear viscosity is also considered.