Slow motion of an arbitrary axisymmetric body along its axis of revolution and normal to a plane surface

Abstract

This paper presents a combined analytical-numerical study for the Stokes flow caused by an arbitrary body of revolution translating axisymmetrically in viscous fluid toward an infinite plane, which can be either a solid wall or a free surface. A singularity method based on the principle of distribution of a set of Sampson spherical singularities along the axis of revolution within a prolate body or on the fundamental plane within an oblate body is used to find the general solution for the fluid velocity field which satisfies the boundary condition at the infinite plane. The no-slip condition on the surface of the translating body is then satisfied by applying a boundary collocation technique to this general solution to dctermine the unknown coefficients. The hydrodynamic drag exerted on the body is evaluated with good convergence behavior for various cases of the body shape and the separation between the plane and the body. For file motion of a sphere normal to a solid plane or a planar free surface, our drag results agree very well with the exact solutions obtained by utilizing spherical bipolar coordinates. For the translation of a spheroid, prolate or oblate, along its axis of symmetry and perpendicular to a plane wall, the agreement betwecn our results and the numerical solutions obtained using the boundary integral method is also quite good. In addition to the solutions for a spheroidal body, the drag results for the axially symmetric motions of a Cassini oval towards a solid plane and a planar free surface are also presented.