Hydrodynamic interactions of spherical particles in quadratic Stokes flows

Abstract

The quasi-steady hydrodynamic Stokes drag force and torque exerted on each of N non-identical particles immersed in a general quadratic undisturbed flow field at infinity is analytically investigated. Explicit results are given for the case of two spherical particles. In particular, expressions are obtained for these forces and torques in terms of: (1) the linear and angular velocities of each of the particles together with the local shear rate of the quadratic flow, all evaluated at the center of volume of the two-sphere system; and (2) the position-independent triadic shear-rate gradient characterizing the undisturbed quadratic flow field. These tensorial (matrix) formulas, obtained by the method of reflections for the intrinsic Stokes resistance coefficients for each of the spheres, are expressed nondimensionally in terms of the respective sphere radii, the center-to-center distance 2 H between spheres, a unit vector e lying along the line-of-centers, and a characteristic length-scale R 0 of the undisturbed quadratic flow field — typically the radius of a tube bounding an axisymmetric Poiseuille flow in which the two spheres are suspended. The reflection scheme is accurate within an error of O r/H α r/R 0 β , wherein α+ β≤6 and in which r is a characteristic sphere radius. The respective translational velocities of the two spheres are derived for the particular case where each is freely suspended in an unbounded Poiseuille flow (i.e., wall effects are neglected). It is shown in this case that a net radial motion of the pair of neutrally buoyant spheres ensues across the undisturbed streamlines as a consequence of the quadratic nature of the flow field (coupled with the interparticle hydrodynamic interaction). The direction of net migration calculated for the two spheres is from low to high shear rates. This contrasts with shear-induced migration phenomena observed in concentrated suspensions subjected to inhomogeneous shearing flows, where the direction of particle migration is from high to low shear rates. The two opposing effects acting in concert may help explain how a steady-state radial particle concentration distribution can be achieved for suspensions of non-Brownian particles.