Optimum profiles in two-dimensional Stokes flow

Abstract

We consider the problem of designing the section of a cylinder to minimize the drag per unit length it experiences when placed perpendicular to a uniform stream at low Reynolds number; we suppose the area of the cross-section to be given, and the flow to be two-dimensional. The relevant properties of a cylinder of general cross-section in a particular orientation can conveniently be expressed in terms of its equivalent radius; when the drag and flow at infinity are parallel, this equivalent radius is the radius of the circular cylinder giving rise to the same drag per unit length. We obtain a variational formula for this equivalent radius when the surface of the cylinder is perturbed; this shows that the optimum profile we seek must be such that the flow past it has a vorticity of constant magnitude at its surface, and this fact enables the optimum to be determined analytically. The efficacy of a particular section may be measured by its effective radius, this being the equivalent radius when the length scale is chosen to give the section an area π ; thus a circular cylinder has an effective radius of 1. The minimum possible effective radius, achieved by the optimum profile, is 0.88876. To illustrate some of the arguments we exploit in a more familiar setting, we also obtain a variational formula for the drag on a three-dimensional body in Stokes flow when its surface is perturbed.