First-order wall curvature effects upon the Stokes resistance of a spherical particle moving in close proximity to a solid wall

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Abstract
A method for calculating the effect of the curvature of a solid wall bounding a viscous fluid upon the quasi-static Stokes force F and torque T experienced by a spherical particle performing arbitrarily directed translational and rotational motions in close proximity to the wall is given. The results presented herein are valid for values of the ratio κ = a/d (a = sphere radius, d = shortest perpendicular distance from the sphere centre to the wall) over the entire range 0 [les ] κ [les ] 1, provided that β = a/R0 [Lt ] 1 and, simultaneously, d/R0 ≡ β/κ [Lt ] 1 (R0 = characteristic radius of curvature of the wall). Unlike existing wall-effect theories, our results are valud for κ = O(1). It is shown that to the first-order in β (and, concomitantly, in d/R0, wall curvature effects upon F and T depend linearly upon two scalar principal curvature coefficients of the wall at the foot of the shortest normal to the wall from the sphere centre. This single-particle analysis is used to resolve a ‘paradox’ relating to macroscopic slip boundary conditions prevailing at a wall bounding a dilute ferrofluid suspension undergoing rotation relative to a magnetic field.