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. Stokes' paradox was resolved by Carl Wilhelm Oseen in 1910, by introducing the Oseen equations which improve upon the Stokes equations – by adding convective acceleration.
Derivation The velocity vector u {\displaystyle \mathbf {u} } of the fluid may be written in terms of the stream function ψ {\displaystyle \psi } as
u = ( ∂ ψ ∂ y , − ∂ ψ ∂ x ) . {\displaystyle \mathbf {u} =\left({\frac {\partial \psi }{\partial y}},-{\frac {\partial \psi }{\partial x}}\right).}
The stream function in a Stokes flow problem, ψ {\displaystyle \psi } satisfies the biharmonic equation. By regarding the ( x , y ) {\displaystyle (x,y)} -plane as the complex plane, the problem may be dealt with using methods of complex analysis. In this approach, ψ {\displaystyle \psi } is either the real or imaginary part of
z ¯ f ( z ) + g ( z ) {\displaystyle {\bar {z}}f(z)+g(z)} . Here z = x + i y {\displaystyle z=x+iy} , where i {\displaystyle i} is the imaginary unit, z ¯ = x − i y {\displaystyle {\bar {z}}=x-iy} , and f ( z ) , g ( z ) {\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 u {\displaystyle u} , defined by u = u x + i u y {\displaystyle u=u_{x}+iu_{y}} is introduced. u {\displaystyle u} can be written as u = − 2 i ∂ ψ ∂ z ¯ {\displaystyle u=-2i{\frac {\partial \psi }{\partial {\bar {z}}}}} , or 1 2 i u = ∂ ψ ∂ z ¯ {\displaystyle {\frac {1}{2}}iu={\frac {\partial \psi }{\partial {\bar {z}}}}} (using the Wirtinger derivatives). This is calculated to be equal to
1 2 i u = f ( z ) + z f ′ ¯ ( z ) + g ′ ¯ ( z ) . {\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:
lim z → ∞ u = 1 , {\displaystyle \lim _{z\to \infty }u=1,}
u = 0 , {\displaystyle u=0,}
whenever | z | = 1 {\displaystyle |z|=1} , and by representing the functions f , g {\displaystyle f,g} as Laurent series:
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