A Slow motion of viscous liquid caused by a slowly moving solid sphere

1. A slow steady motion of incompressible viscous liquid, bounded by an infinite rigid plane, which is generated when a rigid sphere of radius a moves steadily without rotation in a direction parallel to, and at a distance d from, the plane is considered. Use is made of bispherical coordinates, which were employed some years ago by G. B. Jeffery [1] and Stimson and Jeffery [2] in solving the axi-symmetrical problems in which the sphere is fixed and rotates about a diameter perpendicular to the plane, or when two spheres move without rotation along their line of centres in infinite liquid. The coordinate system has been used recently by Dean and O'Neill [3] in solving the problem in which the sphere is fixed and rotates about a diameter parallel to the plane. Since the equations governing the motion of the liquid are linear, the solution of the problem in which the sphere has a uniform velocity (U, V, W) of translation and (O1, Q2, Q3) of rotation, referred to a system of Cartesian coordinates in which the plane is given by z = 0 and the coordinates of the centre of the sphere by (0, 0, d), may be obtained by combining the solutions of the problems in which only one of U, V, W, D,v Q2> ^3 * s non-zero. For those problems not discussed here, the solutions at any instant may be obtamed by the method of [1] for the case when Q3 ^ 0, by a method similar to that of [2] for the case when W ^ 0, and by the method of [3] for the cases when Qx or Q2 is non-zero.