The effect of interfacial slip on the motion and deformation of a droplet in an unbounded arbitrary Stokes flow

The motion and deformation of a droplet suspended in an unbounded fluid with an arbitrary, but Stokesian, imposed flow is investigated when there is a slip at the interface between the two liquids. The boundary condition at the interface is accounted by means of a simple Navier slip condition. Expressions are derived considering the effect of slip on the velocity and the shape deformation of the droplet for any arbitrary imposed flow field, and results are presented for the specific cases of shear flow and Poiseuille flow with the results of Hetsroni and Haber (J. Fluid Mech., 1970, vol. 41(04), pp. 689-705); and Ramachandran and Leal (J. Rheol., 2012, vol. 56(6), pp. 1555-1587) as the limiting cases of our general expressions. The modification to Fax\'en's law is also presented in the above perspective.