Stokes' paradox

In the science of fluid flow, Stokes' paradox is the phenomenon that there can be no creeping flow of a fluid around a disk in two dimensions; or, equivalently, the fact there is no non-trivial, steady state solution for the Stokes equations around an infinitely long cylinder. This is opposed to the 3-dimensional case, where Stokes' method provides a solution to the problem of flow around a sphere.

The velocity vector ${\displaystyle u}$ of the fluid may be written in terms of the stream function ${\displaystyle \psi }$ as:

As the stream function in a Stokes flow problem, ${\displaystyle \psi }$ satisfies the biharmonic equation. Since the plane may be regarded to as the complex plane, the problem may be dealt with using methods of complex analysis. In this approach, ${\displaystyle \psi }$ is either the real or imaginary part of:

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