Hydrophobic core influences and stress jump conditions in creeping flow over two soft spherical particles

The hydrodynamic interactions of two soft composite spheres in a creeping flow regime are investigated using a semi-analytical framework that combines separation of variables with a boundary collocation scheme. The formulation incorporates non-ideal boundary conditions, including interfacial slip and stress jump, together with complex internal architectures comprising a solid core enveloped by a permeable Brinkman shell. Our analysis quantifies the role of these features and reveals a distinct hierarchy of effects: hydrodynamic confinement due to inter-particle proximity governs the overall drag, while internal structure and interfacial properties modulate its magnitude. High shell permeability can paradoxically increase drag through internal viscous dissipation, whereas an impermeable high-slip core provides a “hydrodynamic cushion” that mitigates resistance. Importantly, these influences diminish in the impermeable limit, where particle behavior is dictated solely by external geometry. Our results highlight that, counterintuitively, a highly permeable shell may amplify drag through enhanced internal dissipation, whereas reduced permeability suppresses this mechanism. Moreover, the overall drag is overwhelmingly dictated by external hydrodynamic confinement, while interfacial slip and stress jump conditions act only as secondary modulators. Limiting-case comparisons and spectral-convergence tests confirm the reliability of the semi-analytical scheme. This work establishes a comprehensive physical framework for predicting particulate transport, clarifying how fluid–structure interactions across multiple scales govern the behavior of soft matter systems.