The creeping movement of a soft colloidal particle normal to a planar interface

Abstract

A methodological blend of analytical and numerical strategies employing collocation techniques is presented to investigate the task of describing the Stokes flow generated by a soft particle (composite sphere) moving perpendicularly to a planar interface of infinite extent, separating two semi-infinite, immiscible viscous fluid domains. The particle consists of a solid core enclosed by a porous membrane allowing fluid passage. The movement of the soft nanoparticle has been examined through a continuum mathematical model. This model incorporates the Stokes and Brinkman equations, accounting for the hydrodynamic fields both outside and within the porous membrane layer, respectively. The motion is investigated under conditions characterized by low Reynolds and capillary numbers, where the interface experiences negligible deformation. The solution combines cylindrical and spherical fundamental solutions via superposition. Initially, the boundary conditions at the fluid–fluid interface are satisfied utilizing Fourier–Bessel transforms, subsequently addressing the conditions at the soft particle's surface through a collocation method. The normalized drag force exerted on the particle is accurately calculated, exhibiting robust convergence across various geometric and physical parameters. These findings are effectively visualized via graphs and tables. We juxtapose our drag force coefficient results with established literature data, particularly focusing on the extreme cases. The findings highlight the substantial impact of the interface on the drag force coefficient. Across the full range of viscosity ratios, the normalized drag force decreases as the relative thickness of the porous layer increases. These results enhance the understanding of practical systems and industrial processes such as sedimentation, flotation, electrophoresis, and agglomeration.