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.

## Derivation

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

$\mathbf {u} =\left({\frac {\partial \psi }{\partial y}},-{\frac {\partial \psi }{\partial x}}\right).$ The stream function in a Stokes flow problem, $\psi$ satisfies the biharmonic equation. By regarding the $(x,y)$ -plane as the complex plane, the problem may be dealt with using methods of complex analysis. In this approach, $\psi$ is either the real or imaginary part of

${\bar {z}}f(z)+g(z)$ .

Here $z=x+iy$ , where $i$ is the imaginary unit, ${\bar {z}}=x-iy$ , and $f(z),g(z)$ are holomorphic functions outside of the disk. We will take the real part without loss of generality. Now the function $u$ , defined by $u=u_{x}+iu_{y}$ is introduced. $u$ can be written as $u=-2i{\frac {\partial \psi }{\partial {\bar {z}}}}$ , or ${\frac {1}{2}}iu={\frac {\partial \psi }{\partial {\bar {z}}}}$ (using the Wirtinger derivatives). This is calculated to be equal to

${\frac {1}{2}}iu=f(z)+z{\bar {f\prime }}(z)+{\bar {g\prime }}(z).$ Without loss of generality, the disk may be assumed to be the unit disk, consisting of all complex numbers z of absolute value smaller or equal to 1.

The boundary conditions are:

$\lim _{z\to \infty }u=1,$ $u=0,$ whenever $|z|=1$ , and by representing the functions $f,g$ as Laurent series:

$f(z)=\sum _{n=-\infty }^{\infty }f_{n}z^{n},\quad g(z)=\sum _{n=-\infty }^{\infty }g_{n}z^{n},$ the first condition implies $f_{n}=0,g_{n}=0$ for all $n\geq 2$ .

Using the polar form of $z$ results in $z^{n}=r^{n}e^{in\theta },{\bar {z}}^{n}=r^{n}e^{-in\theta }$ . After deriving the series form of u, substituting this into it along with $r=1$ , and changing some indices, the second boundary condition translates to

$\sum _{n=-\infty }^{\infty }e^{in\theta }\left(f_{n}+(2-n){\bar {f}}_{2-n}+(1-n){\bar {g}}_{1-n}\right)=0.$ Since the complex trigonometric functions $e^{in\theta }$ compose a linearly independent set, it follows that all coefficients in the series are zero. Examining these conditions for every $n$ after taking into account the condition at infinity shows that $f$ and $g$ are necessarily of the form

$f(z)=az+b,\quad g(z)=-bz+c,$ where $a$ is an imaginary number (opposite to its own complex conjugate), and $b$ and $c$ are complex numbers. Substituting this into $u$ gives the result that $u=0$ globally, compelling both $u_{x}$ and $u_{y}$ to be zero. Therefore, there can be no motion – the only solution is that the cylinder is at rest relative to all points of the fluid.

## Resolution

The paradox is caused by the limited validity of Stokes' approximation, as explained in Oseen's criticism: the validity of Stokes' equations relies on Reynolds number being small, and this condition cannot hold for arbitrarily large distances $r$ .

A correct solution for a cylinder was derived using Oseen's equations, and the same equations lead to an improved approximation of the drag force on a sphere.