Stokes' paradox

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 $f=A{r}^3+Br\log r+Cr+D/r$. Uniform flow at infinity forces $A=B=0$, $C=1$; impermeability forces $D=-1$, but then $f^{\prime }(1)=2$ 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.