Oscillatory interactions of two spheres in an unbounded couple stress fluid.

This study investigates the rectilinear oscillations of two coaxially aligned spherical particles in an unbounded couple stress fluid at low Reynolds numbers, addressing a fundamental problem in microfluidics and biomechanics where microstructure effects dominate. The importance lies in applications such as drug delivery and material processing, where understanding particle-fluid interactions is critical. The unsteady Stokes equations were solved using a superposition of fundamental solutions in spherical coordinates, centered on each particle, with no-slip boundary conditions enforced via a collocation method. Key results include the quantification of in-phase and out-of-phase drag force coefficients, revealing that increasing the couple stress parameter ([Formula: see text]) enhances drag forces by up to 50% for [Formula: see text] compared to Newtonian cases ([Formula: see text]). Numerical simulations demonstrated robust convergence across dimensionless parameters (e.g., separation distance δ, frequency α), with tabulated data showing agreement within % of established solutions for steady-state and single-sphere oscillations. Novelty arises from extending prior work on viscous fluids to couple stress fluids, uncovering how microstructural effects amplify drag and alter oscillation dynamics. For instance, at [Formula: see text], drag forces increased by 30% for closely spaced spheres ([Formula: see text]), highlighting the interplay between frequency and microstructure. This work advances predictive models for complex fluids and provides design insights for microfluidic systems.