Slow Rotation of a Soft Colloidal Sphere Normal to Two Plane Walls

Abstract

The creeping flow of a viscous fluid around a soft colloidal sphere rotating about a diameter normal to two planar walls at an arbitrary position between them is theoretically investigated in the steady limit of small Reynolds numbers. The fluid velocity outside the particle consists of the general solutions of the Stokes equation in circular cylindrical and spherical coordinates, while the fluid velocity inside the porous surface layer of the particle is expressed by the general solution of the Brinkman equation in spherical coordinates. The boundary conditions are implemented first on the planar walls by means of the Hankel transforms and then at the particle and hard-core surfaces by a collocation technique. The torque exerted on the particle by the fluid is calculated as a function of the ratio of the core-to-particle radii, ratio of the particle radius to the flow penetration length of the porous layer, and relative particle-to-wall spacings over the entire range. The wall effect on the rotating soft particle can be significant. The hydrodynamic torque exerted on the confined soft sphere increases as the relative particle-to-wall spacings decrease and stays finite even when the soft sphere contacts the plane walls. It is smaller than the torque on a hard sphere (or soft one with a reduced thickness or penetration length of the porous layer), holding the other parameters constant. For a given relative wall-to-wall spacing, this torque is minimal when the particle is situated midway between the walls and rises as it locates closer to either wall.