Interception of two spherical particles with arbitrary radii in simple shear flow

Abstract

The interception of two force-free and torque-free spherical particles with arbitrary radii freely suspended in simple shear flow is investigated in the limit of vanishing Reynolds number. At any instant, the flow is computed in a frame of reference with origin at the center of one particle using a cylindrical polar coordinate system whose axis of revolution passes through the center of the second particle. The problem is formulated as a decoupled system of integral equations for the zeroth, first, and second Fourier coefficients of the boundary traction with respect to the meridional angle. The derived integral equations are solved with high accuracy using a boundary element method that features adaptive discretization and automatic time-step adjustment according to the inter-particle gap. The results illustrate particle trajectories and describe the particle rotation and evolution of the stress tensor during the interception. The particle interaction is found to always cause a positive shift in the rotational phase angle due to the rolling motion at close contact. As the gap between two particles tends to zero, the shear stress diverges even though the net force and torque exerted on each particle remain zero, independent of the particle relative radius. A frictional force for rough surfaces and small gaps eliminates the slip velocity and promotes the rolling motion.