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Analytical solutions are presented for axisymmetric creeping flows around a spherical particle near a wall, when the unperturbed flow velocity varies as polynomial with the coordinates. The perturbed fluid velocity and pressure are calculated directly with the method of bipolar coordinates. They are obtained with a 10−16 precision, even for small gaps of the order of 10−6 sphere radius. For a constant unperturbed flow, the problem is equivalent to that of a sphere moving perpendicularly to a wall in a fluid at rest. An alternative indirect solution for the fluid velocity and pressure is obtained in this case from the solution of Brenner [Chem. Eng. Sci. 16, 242 (1961)] and Maude [Brit. J. Appl. Phys. 12, 293 (1961)] for the stream function. For a small gap between the sphere and the wall, the values of the pressure are compatible with the ones from the lubrication approximation but are systematically larger; this may be important for applications. Calculations are also performed when the unperturbed flow velocity is a polynomial of degree 2 and 3 in the coordinates, viz., for different types of stagnation point flows. Various flow structures are obtained, depending on the particle to wall distance. When the sphere approaches the wall, there is an increasing number of nested toroidal eddies, providing a link between the case of a single toroidal eddy when the sphere is far from the wall and the infinite set of Moffatt eddies in the gap between bodies in contact. The flow structure is analogous to that for two equal spheres in a uniform flow field, cf. Davis, O’Neill, Dorrepaal, and Ranger [J. Fluid Mech. 77, 625 (1976)]. Precise results for the force on the sphere are provided.

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