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

Abstract

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 $K_{n + (1/2)} (R)$, for some positive integer n. Consequently, $R = \infty $ is an accumulation point of poles and hence a nonisolated singularity. We have extended the Stokes series for the drag coefficient $C_D (R)$ to 66 terms in double precision, and have sought the asymptotic behavior of $C_D (R)$ as $R\to \infty$. Neither Padé approximants nor an Euler transformation formed from our long series give good convergence. We suggest that the asymptotic behavior of $C_D (R)$ 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 $C_D (R)$ and give the best extrapolations from finite R to $R = \infty $, with errors of the order of $.05\%$