Slow axisymmetric rotation of a soft sphere in a circular cylinder

Abstract

A semi-analytical study of the creeping flow of an incompressible Newtonian fluid around a soft spherical particle (a hard core covered by a permeable porous layer) rotating about a diameter lying on the axis of a long circular cylinder is presented. To solve the Stokes and Brinkman equations, a solution comprising of the general solutions in spherical and cylindrical coordinates is utilized. The boundary conditions are applied first at the confining cylinder wall through the Fourier transform and then at the particle and hard-core surfaces via a collocation method. Accurate results of the hydrodynamic torque acting on the particle can be obtained for various values of the particle-to-cylinder radius ratio, core-to-particle radius ratio, and ratio of the particle radius to the porous layer’s flow penetration length. The boundary effect of the cylinder on the rotation of the soft particle is significant. The hydrodynamic torque exerted on the confined soft sphere increases with an increase in the particle-to-cylinder radius ratio, and in general remains finite even as the soft particle fills the cylinder. This torque is less than that on an equal-sized hard sphere (or soft sphere having a smaller thickness or lower permeability of its porous layer).