Three-sphere low-Reynolds-number swimmer near a wall

We study the influence of a wall on the dynamics of a low-Reynolds-number three-sphere swimmer. A far swimmer whose arm makes an angle theta with the horizon experiences the wall presence as an angle-dependent quadrupole force proportional to (a/L)(2)(L/z)(2)cos theta, where a, L, and z are the radius of spheres, the arm length, and the swimmer distance to the wall, respectively. The wall-induced translational velocity of swimmer is perpendicular to the arms. A far swimmer prefers to orient its arms parallel to the plate. This state is stable. Remarkably, the parallel state is unstable when the swimmer is close to the wall. In this regime, the velocity of swimmer decreases as (z/L)(2). Numerical solution of the equations of motion for arbitrary initial z/L and theta reveals four different phases of locomotion.