Transient motion of a rotating slippery spherical particle within a spherical cavity filled with a porous medium

A semi-analytical investigation is presented to examine the transient slow rotation of a spherical particle situated concentrically within a spherical cavity filled with a Brinkman porous medium. The surface of the particle and the internal surface of the cavity both feature slip surfaces. Rotation is induced by the sudden application of a constant torque along a common diameter. In the porous medium, idealized hydrodynamic frictional segments are assumed to be evenly distributed. The spherical particle and the cavity rotate at distinct angular velocities around an axis aligned with their shared diameter. The Laplace transform is employed to address the unsteady Brinkman equations that govern fluid velocity. Through this process, we derive an analytical expression for the transformed transient angular velocity of the particle in terms of relevant dimensionless parameters. Numerical findings suggest that, when the permeability parameter is increased for finite values of the particle-to-cavity radius ratio, the timescale for the evolution of angular velocity in a Brinkman medium exceeds that of viscous fluid flow. Additionally, as the permeability parameter increases, angular acceleration consistently decreases. The time-dependent angular velocity of the particle within the cavity undergoes a continuous and rapid increase from zero to its eventual steady-state value, while angular acceleration decreases over time. Moreover, the normalized transient angular velocity consistently declines as the particle-to-medium density ratio increases. It has been observed that a high-density particle takes longer to reach its equilibrium angular velocity compared to a low-density particle. In general, the dimensionless time-dependent angular velocity decreases as the following parameters increase: particle-to-cavity radius ratio, Navier slip coefficients at the particle surface and cavity wall, and relative angular velocity. In the limit as the permeability parameter approaches zero, our results align perfectly with existing literature. This study provides insights relevant to biological and filtration systems where porous media and rotational motion influence particle transport. Applications include understanding the motion of proteins or cellular components in confined biological fluids and optimizing separation efficiency in centrifugal and dynamic filtration technologies.