Resistance and stability of a line of particles moving near a wall

Abstract

Corrections to Stokes’ law are determined to first order in a/b and a/h for a sphere of radius a in a one-dimensional array of identical spheres having centre-to-centre-spacing b and translating a distance h from a no-slip wall. When h/b is small the drag is greater than that given by Stokes’ law; as h/b increases, the drag generally decreases and becomes less than that given by Stokes’ law. Stability of the array is examined. Motion along the line of centres is found to be stable, but the other two motions are unstable. The wall is a stabilizing influence when motion is toward the wall and a destabilizing influence when motion is away from the wall. For motion parallel to the wall, the presence of the wall shifts the region of maximum instability to smaller wavelengths. Crowley's results, which neglect any influence of the wall, are approached for h/b greater than about 5.