A Hybrid Asymptotic-Numerical Method for Low Reynolds Number Flows Past a Cylindrical Body

The classical problem of slow, steady, two-dimensional flow of a viscous incompressible fluid around an infinitely 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 coefficient and of the flow field start with infinite logarithmic series. We show that the entire infinite logarithmic expansions of the flow field and of the drag coefficient 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 finite-difference scheme. The drag coefficient for a cylinder of a specific 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 flow 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 coefficient 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 infinite logarithmic expansions for a generalized version of Lagerstrom's ordinary differential equation model of slow viscous flow.