= Stokes' paradox
{c}
The unbounded two-dimensional <Stokes flow> past a no-slip circular cylinder has no solution approaching a nonzero uniform stream. With radius and speed scaled to one, the forced angular mode has <stream function> $\psi=f(r)\sin\theta$. The <biharmonic equation> gives $f=Ar^3+Br\log r+Cr+D/r$. Uniform flow at infinity forces $A=B=0$, $C=1$; impermeability forces $D=-1$, but then $f'(1)=2$ violates no slip. The failure belongs to a nonuniform small-<Reynolds number> limit. In a distant outer region inertia must be retained through an <Oseen approximation> or the full <Navier-Stokes equations>, matched to an inner Stokes region. A finite outer wall changes the problem and removes this particular infinity obstruction.
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