A Hybrid Asymptotic-Numerical Method for Calculating Low Reynolds Number Flows Past Symmetric Cylindrical Bodies

The classical problem of slow, steady, two-dimensional ow of a viscous incompressible uid around an innnitely long straight cylinder is considered. The cylinder cross-section is symmetric about the direction of the oncoming stream, but otherwise is arbitrary. For low Reynolds number, the well-known singular perturbation analysis for this problem shows that the asymptotic expansions of the drag coeecient and of the ow eld start with innnite logarithmic series. We show that the entire innnite logarithmic expansions of the ow eld and of the drag coeecient are contained in the solution to a certain related problem that does not involve the cross-sectional shape of the cylinder. The solution to this related problem is computed numerically using a straightforward nite-diierence scheme. The drag coeecient for a cylinder of a speciic cross-sectional shape, which is asymptotically correct to within all logarithmic terms, is given in terms of a single shape-dependent constant that is determined by the solution to a canonical Stokes ow problem. The resulting hybrid asymptotic-numerical method is illustrated for cylinders of various cross-sectional shapes. For a circular cylinder, our results for the drag coeecient are compared with experimental results, with the explicit three-term asymptotic theory of Kaplun, and with numerical results computed from the full problem. A similar hybrid approach is used to sum innnite logarithmic expansions for a generalized version of Lagerstrom's ordinary diierential equation model of slow viscous ow.