Translation and rotation of slightly deformed colloidal spheres experiencing slip

Abstract

The steady translation and rotation of a rigid, slightly deformed colloidal sphere in arbitrary directions in a viscous fluid are analyzed in the limit of small Reynolds number. The fluid is allowed to slip frictionally at the surface of the particle, and the Stokes equations are solved asymptotically using a method of perturbed expansions. To the second order in the small parameter characterizing the deformation of the particle from the spherical shape, the resistance problem is formulated for the general case and explicit expressions for the hydrodynamic drag force and torque exerted on the particle are obtained for the special cases of prolate and oblate spheroids. The agreement between our asymptotic results for a slip-surface spheroid and the relevant exact solutions in the literature is very good, even if the particle deformation from the spherical shape is not very small. As expected, the second-order expansions for the translational and rotational resistances in powers of the small deformation parameter make better consistency with the available exact results than the first-order expansions do. Depending on the value of the slip parameter, the hydrodynamic drag force and torque acting on a moving spheroid normalized by the corresponding values for a spherical particle with equal equatorial radius are not necessarily monotonic functions of the aspect ratio of the spheroid. Noticeable behavior of the drag force and torque is grasped in the second-order expansions; e.g., the torque exerted on a perfect-slip rotating spheroid is not necessarily zero. For a moving spheroid with a fixed aspect ratio, its normalized hydrodynamic drag force and torque decrease monotonically with an increase in the slip capability of the particle.