Two-particle Motion in a Brinkman Micropolar Medium

Abstract An effective-medium framework based on the Brinkman micropolar equations is employed to analyze the axisymmetric, quasi-steady motion of two spherical colloidal particles within a hydrogel medium. The hydrogel is idealized as a homogeneous, isotropic porous matrix saturated with a microstructured fluid that exhibits micropolar behavior. The analysis assumes a low Reynolds number regime, allowing for a linearized description of the flow field and accounting for differences in particle size and translational velocities along the line connecting their centers. A general solution is constructed by superimposing fundamental flow solutions expressed in two spherical coordinate systems, each centered at a particle. A collocation method is employed to enforce the no-slip and no-spin boundary conditions on the surfaces of the particles. Numerical results for the normalized drag force on each particle are computed with rapid convergence across a range of values for the size ratio, separation distance, velocity ratio, and inverse permeability parameter. As the distance between particle centers increases, the normalized drag force on each particle approaches the single-particle limit, indicating that the particles effectively translate independently. The accuracy of the numerical approach is validated by comparison with known solutions available in literature. This analysis has potential applications in the design of targeted drug delivery systems, where understanding particle transport through hydrogels is critical.