(168w) On the Minimum Separation Attained During the Motion of a Sphere Past a Fixed Spherical or Cylindrical Obstacle

We present an analytical expression for the minimum surface-to-surface separation attained during the motion of a spherical particle past a fixed sphere or cylinder in the limit of zero Reynolds number, as a function of the impact parameter - defined as the initial perpendicular distance between the line of motion and the coordinate axis parallel to the line of motion. We take advantage of the planar character of the particle trajectories to implement an approach analogous to the one introduced by Batchelor and Green in the study of two-particle trajectories in a linear shear flow. We investigate each of the above cases - that of a fixed cylinder and that of a fixed sphere - for a uniform ambient flow and for a constant external force acting on the moving sphere in quiescent ambient fluid. Our treatment is valid for arbitrary values of ratio of radii of the sphere and the obstacle, and we have compared it with available results from literature for the special case in which both radii are equal. We also obtain an important result - similar to Batchelor and Green - that the probability distribution related with finding a spherical particle at a given center-to-center separation is function of center-to-center separation only; it does not exhibit any angular dependence. We observe that the sphere comes closer to the fixed body when acted upon by an external force than left force-free in uniform flow. We have successfully compared our analytical expression for the case of two spheres with results obtained from Stokesian Dynamics direct numerical simulations. The current work includes such a comparison for the case of a cylinder and a sphere. This analytical expression is expected to strengthen our understanding of directional locking and deterministic lateral displacement of suspended particles through microfluidic obstacle arrays.