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.[1][2]

## Derivation

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

$\mathbf{u} = \begin{pmatrix} {\partial \psi \over \partial y} & - {\partial \psi \over \partial x} \end{pmatrix}$

As the stream function in a Stokes flow problem, $\psi$ satisfies the biharmonic equation.[3] 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, $\psi$ is either the real or imaginary part of:

$\bar{z} f(z)+g(z)$[4]

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:

$\lim_{z \to \infty} u=1$

and

$u = 0$

whenever $|z| =1$, [5][6] and by representing the functions $f,g$ as Laurent series:[7]

$f(z)=\sum_{n= -\infty}^\infty f_n z^n,g(z)=\sum_{n= -\infty}^\infty g_n z^n$

the first condition implies $f_n=0,g_n=0$ for all $n\geq2$.

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 and 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:

$f(z)=az+b,g(z)=-bz+c$

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 $r$.[8][2]

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.[9][10]