This paper proposes a combined multipole-series representation and integral-equation method for solving the low-Reynolds-number hydrodynamic interaction of a finite sphere at the entrance of a circular orifice. This method combines the flexibility of the intergral-equation method in treating complicated geometries and the accuracy and computational efficiency of the multipole-series-representation technique. For the axisymmetric case, the hydrodynamic force has been solved for the difficult case where the sphere intersects the plane of the orifice opening, which could not be treated by previous methods. For the three-dimensional case, the first numerical solutions have been obtained for the spatial variation of the twelve force and torque correction factors describing the translation or rotation of the sphere in a quiescent fluid at a pore entrance or the Sampson flow past a fixed sphere. Restricted by excessive computation time, accurate three-dimensional solutions are presented only for a sphere which is one-half the orifice diameter. However, based on an analysis of the behaviour of the force and torque correction factors for this case, approximate interpolation formulas utilizing the results on or near the orifice axis and in the far field are proposed for other diameter ratios, thus greatly extending the usefulness of the present solution.
