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

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

As the stream function in a Stokes flow problem, ${\displaystyle \psi }$ 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, ${\displaystyle \psi }$ is either the real or imaginary part of:

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

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

${\displaystyle {\frac {1}{2}}iu=f(z)+z{\bar {f\prime }}(z)+{\bar {g\prime }}(z)}$

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:

${\displaystyle \lim _{z\to \infty }u=1}$

and

${\displaystyle u=0}$

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

${\displaystyle f(z)=\sum _{n=-\infty }^{\infty }f_{n}z^{n},g(z)=\sum _{n=-\infty }^{\infty }g_{n}z^{n}}$

the first condition implies ${\displaystyle f_{n}=0,g_{n}=0}$ for all ${\displaystyle n\geq 2}$.

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

${\displaystyle \sum _{n=-\infty }^{\infty }e^{in\theta }\left(f_{n}+(2-n){\bar {f}}_{2-n}+(1-n){\bar {g}}_{1-n}\right)=0}$.

Since the complex trigonometric functions ${\displaystyle e^{in\theta }}$ compose a linearly independent set, it follows that all coefficients in the series are zero. Examining these conditions for every ${\displaystyle n}$ after taking into account the condition at infinity shows that ${\displaystyle f}$ and ${\displaystyle g}$ are necessarily of the form:

${\displaystyle f(z)=az+b,g(z)=-bz+c}$

where ${\displaystyle a}$ is an imaginary number (opposite to its own complex conjugate) and ${\displaystyle b}$ and ${\displaystyle c}$ are complex numbers. Substituting this into ${\displaystyle u}$ gives the result that ${\displaystyle u=0}$ globally, compelling both ${\displaystyle u_{x}}$ and ${\displaystyle u_{y}}$ 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 ${\displaystyle r}$.[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]