Start-Up Rotation of a Porous Colloidal Sphere in a Cavity

Abstract

The starting rotation of a porous sphere induced by the sudden application of a continuous torque about its diameter at the center of a spherical cavity filled with an incompressible Newtonian fluid at low Reynolds numbers is analyzed. The unsteady Stokes and Brinkman equations governing the fluid velocities outside and inside the porous particle, respectively, are solved via the Laplace transform, and an explicit formula of its dynamic angular velocity as a function of the pertinent parameters is obtained. The behavior of the start-up rotation of an isolated porous particle and the cavity wall effect on the particle rotation are interesting. The angular velocity of the particle grows incessantly over time from an initial zero to its final value, while the angular acceleration declines with time continuously. In general, the transient angular velocity is an increasing function of the porosity of the particle. A porous sphere with higher fluid permeability rotates at higher angular velocity and acceleration relative to the reference particle at any elapsed time but lags behind the reference particle in the percentage growth of angular velocity towards the respective terminal values. The transient angular velocity decreases with increasing particle-to-cavity radius ratio, but it is not a sensitive function of the radius ratio when the resistance to fluid flow inside the porous particle or the radius ratio itself is small.