The Analytic Structure of Oseen Flow Past a Sphere as a Function of Reynolds Number

The feasibility of Van Dyke’s [28] proposal that one go from zero to infinite Reynolds number R by computer extension of a Stokes series in powers of R depends on the analytic structure of the flow as a function of R. This structure is investigated in detail for Oseen flow past a sphere. We find that there is a doubly infinite array of simple pole singularities in the left half complex R-plane, each of which lies close to one of the n zeros of the modified Bessel functions Kn + (1/2) (R), for some positive integer n. Consequently, R = ∞ is an accumulation point of poles and hence a nonisolated singularity. We have extended the Stokes series for the drag coefficient CD (R) to 66 terms in double precision, and have sought the asymptotic behavior of CD (R) as R→ ∞. Neither Padé approximants nor an Euler transformation formed from our long series give good convergence. We suggest that the asymptotic behavior of CD (R) is primarily as an expansion in powers of R- 2/3 though with additional small correction terms. Short expansions of this type can be fitted to computed values of CD (R) and give the best extrapolations from finite R to R = ∞, with errors of the order of $.05%$