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Optimum Profiles in Two-dimensional Stokes Flow

S. Richardson
Journal / Anthology

Proc. R. Soc. Lond. A
Year: 1995
Volume: 450
Page range: 603-622

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 or 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 it surface, and this fact enables the optimum to be determined analytically. The efficacy of the particular section may be measured by its effective radius, this being the equivalent radius when the length scale is chosen to give the section and area p; thus a circular cylinder has an effective radius of 1. The minimum possible effective radius, achieved by the optimum profile, is 0.88876.

*Science > Physics > Fluid Mechanics