Low Reynolds number

Introduction
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The difficulty in establishing a sensible global solution to the problem of low $R_e$ (Reynolds number) viscous flow around simple objects, where the flow is uniform at infinity, has fascinated applied mathematicians for just over 150 years. Stokes (1851) established that there was no solution to the two-dimensional, steady, incompressible, equations for asymptotically uniform flow around a cylinder in the biharmonic limit. This situation is now routinely described in modern tutorial discussions. See, for example, Chapter 7 of Acheson (1990) for an exercise on Stokes’ paradox and a discussion of the corresponding situation for the sphere. The biharmonic equation can be solved in a neighbourhood of the cylinder (i.e. a circle) by any stream function of the form $$\psi = C\sin\theta(r \log r - \frac{1}{2}r + \frac{1}{2r})$$ but there is no choice of $C$ for which $\psi \sim r \sin\theta$ for large $r$. The analysis of this problem has lead to several classic papers (Oseen, 1910; Lamb, 1911) and its understanding through the use of matched asymptotic expansions (MAE) is one of the triumphs of perturbation theory. The reader is referred to Van Dyke (1964) for his classic survey of the work of Kaplun (1957), 