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. Derivation. The velocity vector formula_1 of the fluid may be written in terms of the stream function formula_2 as The stream function in a Stokes flow problem, formula_2  satisfies the biharmonic equation. By regarding the formula_5-plane as the complex plane, the problem may be dealt with using methods of complex analysis. In this approach, formula_2 is either the real or imaginary part of Here formula_8, where formula_9 is the imaginary unit, formula_10, and formula_11 are holomorphic functions outside of the disk. We will take the real part without loss of generality. Now the function formula_12, defined by formula_13 is introduced. formula_12 can be written as formula_15, or formula_16 (using the Wirtinger derivatives). This is calculated to be equal to 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: whenever formula_20, and by representing the functions formula_21 as Laurent series: the first condition implies formula_23 for all formula_24. Using the polar form of formula_25 results in formula_26. After deriving the series form of "u", substituting this into it along with formula_27, and changing some indices, the second boundary condition translates to Since the complex trigonometric functions formula_29 compose a linearly independent set, it follows that all coefficients in the series are zero. Examining these conditions  for every formula_30 after taking into account the condition at infinity shows that formula_31 and formula_32 are necessarily of the form where formula_34 is an imaginary number (opposite to its own complex conjugate), and formula_35 and formula_36 are complex numbers. Substituting this into formula_12 gives the result that formula_38 globally, compelling both formula_39 and formula_40 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 formula_41. 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.