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Starting from a complex variable formulation of the exact Navier-Stokes equations for the steady two-dimensional motion of an incompressible viscous fluid, a Burgers type linearization is introduced at small Reynolds number which results in a conformally invariant Oseen equation. The independent variables are the velocity potential and stream function for the corresponding irrotational flow. There are certain analytical advantages in this approach for the determination of uniformly valid approximations to flow at small Reynolds numbers and the method is illustrated by various examples. Introduction and the Oseen approximation. For two-dimensional flow the fluid velocity can be prescribed in terms of a stream function by q = cur^-^fc), (A) where k is perpendicular to the plane of motion and the Oseen linearization of the equations of motion can be written in terms of ¥ as d2 dx2 + dy2' R is the Reynolds number based on the free stream speed at infinity, the length scale for the boundary geometry under consideration, and the kinematic viscosity. The Oseen approximation is suitable for fluid domains of infinite extent in which the flow converges to a uniform stream at infinity, that is ~ y as x2 + y2 —* oo. The essence of the approximation is to account for the forced convection produced by the uniform stream, and the convection term i?(q V)q which appears in the NavierStokes equation is linearized by replacement with q. Not only does the Oseen approximation resolve the Stokes paradox, but to leading order the solutions provide uniformly valid composite expressions for the stream function at small Reynolds numbers R. For example, the boundary value problem for flow past a circular cylinder may be expressed in terms of polar coordinates as n, na\,„ n _2 d2 1 d 1 d2 d a d sin 6 8 V2 + R— »P = 0, V2 =-T + —+ ^—y, — = cos d — ~ V2 (v2 + /?^ ^ = 0, V2 = ^y + ^. (B) dx) ' dr2 r dr r2d62' dx dr r 86' (C) Received May 4, 1989. 189 ©1990 Brown University

A conformally invariant Oseen equation for flow at small Reynolds numbers