Transient slow motions of a composite spherical particle

The transient translation and rotation of a composite spherical particle, consisting of an impermeable hard sphere core and a permeable porous surface layer, in a viscous fluid at low Reynolds number induced by suddenly applied continuous force and torque are analytically studied. By solving the unsteady Stokes and Brinkman equations that respectively govern the fluid flows outside and inside the porous surface layer through the Laplace transform, closed-form formulas of the starting linear and angular velocities of the composite sphere as functions of the relevant parameters are obtained. These velocities increase over time from initial values of zero to their final values, while the linear and angular accelerations of the composite sphere decrease over time, eventually approaching zero. At any elapsed time, these velocities and accelerations increase monotonically and significantly with increasing relative spatial volume and fluid permeability of the porous surface layer of the composite sphere. The transient linear and angular velocities of the composite sphere are generally increasing functions of the porosity of the surface layer, but may decrease slightly with increasing porosity when the particle-to-fluid density ratio is small. The linear and angular accelerations increase with the increase of surface layer porosity in the early stage, no longer change monotonically with its increase in the middle stage, and decrease with its increase in the later stage. The transient linear and angular velocities of the composite sphere decrease with the increase of its relative density, but the linear and angular accelerations only decrease with the increase of the relative density in the early stage and increase with the increase of the relative density in the later stage.