Creeping Flow around a Sphere in a Shear Flow Close to a Wall

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Abstract
The creeping flow around a sphere close to a wall is calculated by revisiting the bipolar coordinates solution technique. The coefficients in the expansions of the spherical harmonics are easily calculated from two recurrence relationships, following the idea of O'Neill and Bhatt (1991). That is, there is no need to solve successively larger truncated linear systems to obtain the coefficients, as was done in earlier papers for a sphere translating along a wall by (O'Neill 1964, 1967) or rotating by (Dean and O'Neill 1963) and for a shear flow around a fixed sphere by (Tözeren and Skalak 1977). Consequently, results can be obtained even for small gaps between the sphere and the wall without excessive computer resources. Thus, the present technique provides a reference to obtain all the friction coefficients and flow fields with excellent precision. Results for the friction coefficients are given with up to 17 significant digits for later reference. Earlier results for the drag force and torque coefficients by (Goldman, Cox and Brenner 1967) are recovered and in some cases corrected. The exact solutions for sphere translation and rotation are further exploited to provide approximate formulae with a precision of 10−9, based on expansions for large and small gaps. For the shear flow problem, there is no singularity near contact and a polynomial interpolation gives a precision of 10−11. The exact solutions for a shear flow around a fixed sphere and the flows due to a translating sphere and a rotating sphere in a fluid at rest are then exploited to provide approximation formulae with a precision of 10−11 for the translation and rotation velocities of a neutrally buoyant sphere freely moving along a wall in a shear flow.

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