Far-field disturbance flow induced by a small non-neutrally buoyant sphere in a linear shear flow

The disturbance flow due to the motion of a small sphere parallel to the streamlines of an unbounded linear shear flow is evaluated at small Reynolds number using the method of matched expansions. Decaying laws are obtained for all velocity components in a far inviscid region and viscous wakes. The z component (in the direction of the shear-rate gradient) of the disturbance velocity is cylindrically symmetric in the inviscid region. It decays with the distance r from the sphere like r(-5/3), while the y component (in the direction of vorticity) decays like r(-4/3). The widths of two viscous wakes, located upstream and downstream of the sphere, grow with the longitudinal coordinate x as y(w) similar to z(w) similar to vertical bar x vertical bar(1/3). The maximum x and z components of the velocity are located in the wake cores; they scale like vertical bar x vertical bar(-2/3) and vertical bar x vertical bar(-1) respectively. For two particles interacting through their outer regions, the migration velocity of each particle is the sum of the velocity of an isolated particle and of a disturbance velocity induced by the other one. Particles placed in the normal or transversal directions repel each other. When each particle is located in a wake of the other one, they may either attract or repel each other.