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]

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

${\displaystyle \mathbf {u} =\left({\frac {\partial \psi }{\partial y}},-{\frac {\partial \psi }{\partial x}}\right).}$

The stream function in a Stokes flow problem, ${\displaystyle \psi }$ satisfies the biharmonic equation.[3] By regarding the ${\displaystyle (x,y)}$-plane 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:

${\displaystyle \lim _{z\to \infty }u=1,}$
${\displaystyle u=0,}$

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

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

the first condition implies ${\displaystyle f_{n}=0,g_{n}=0}$ 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, 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

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

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 ${\displaystyle r}$.[7][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.[8][9]

## Unsteady-state flow around a circular cylinder

On the contrary to Stokes' paradox, there exists the unsteady-state solution of the same problem which models a fluid flow moving around a circular cylinder with Reynolds number being small. This solution can be given by explicit formula in terms of vorticity of the flow's vector field.

### Formula of the Stokes Flow around a circular cylinder

The vorticity of Stokes' flow is given by the following relation:[10]

${\displaystyle w_{k}(t,r)=W_{|k|,|k|-1}^{-1}\left[e^{-\lambda ^{2}t}W_{|k|,|k|-1}[w_{k}(0,\cdot )](\lambda )\right](t,r).}$

Here ${\displaystyle w_{k}(t,r)}$ - are the Fourier coefficients of the vorticity's expansion by polar angle which are defined on ${\displaystyle (r_{0},\infty )}$, ${\displaystyle r_{0}}$ - radius of the cylinder, ${\displaystyle W_{|k|,|k|-1}}$, ${\displaystyle W_{|k|,|k|-1}^{-1}}$ are the direct and inverse special Weber's transforms,[11] and initial function for vorticity ${\displaystyle w_{k}(0,r)}$ satisfies no-slip boundary condition.

Special Weber's transform has a non-trivial kernel, but from the no-slip condition follows orthogonality of the vorticity flow to the kernel.[10]

### Derivation

#### Special Weber's transform

Special Weber's transform[11] is an important tool in solving problems of the hydrodynamics. It is defined for ${\displaystyle k\in \mathbb {R} }$ as

${\displaystyle W_{k,k-1}[f](\lambda )=\int _{r_{0}}^{\infty }{\frac {J_{k}(\lambda s)Y_{k-1}(\lambda r_{0})-Y_{k}(\lambda s)J_{k-1}(\lambda r_{0})}{\sqrt {J_{k-1}^{2}(\lambda r_{0})+Y_{k-1}^{2}(\lambda r_{0})}}}f(s)sds,}$
where ${\displaystyle J_{k}}$, ${\displaystyle Y_{k}}$ are the Bessel functions of the first and second kind[12] respectively. For ${\displaystyle k>1}$ it has a non-trivial kernel[13][10] which consists of the functions ${\displaystyle C/r^{k}\in \ker(W_{k,k-1})}$.

The inverse transform is given by the formula

${\displaystyle W_{k,k-1}^{-1}[{\hat {f}}](r)=\int _{0}^{\infty }{\frac {J_{k}(\lambda r)Y_{k-1}(\lambda r_{0})-Y_{k}(\lambda s)J_{k-1}(\lambda r_{0})}{\sqrt {J_{k-1}^{2}(\lambda r_{0})+Y_{k-1}^{2}(\lambda r_{0})}}}{\hat {f}}(\lambda )\lambda d\lambda .}$

Due to non-triviality of the kernel, the inversion identity

${\displaystyle f(r)=W_{k,k-1}^{-1}\left[W_{k,k-1}[f]\right](r)}$
is valid if ${\displaystyle k\leq 1}$. Also it is valid in the case of ${\displaystyle k>1}$ but only for functions, which are orthogonal to the kernel of ${\displaystyle W_{k,k-1}}$ in ${\displaystyle L_{2}(r_{0},\infty )}$ with infinitesimal element ${\displaystyle rdr}$:
${\displaystyle \int _{r_{0}}^{\infty }{\frac {1}{r^{k}}}f(r)rdr=0,~k>1.}$

#### No-slip condition and Biot–Savart law

In exterior of the disc of radius ${\displaystyle r_{0}}$ ${\displaystyle B_{r_{0}}=\{\mathbf {x} \in \mathbb {R} ^{2},~\vert \mathbf {x} \vert >r_{0}\}}$ the Biot-Savart law

${\displaystyle \mathbf {v} (\mathbf {x} )={\frac {1}{2\pi }}\int _{B_{r_{0}}}{\frac {(\mathbf {x} -\mathbf {y} )^{\perp }}{\vert \mathbf {x} -\mathbf {y} \vert ^{2}}}w(\mathbf {y} )\operatorname {d\mathbf {y} } +\mathbf {v} _{\infty },}$
restores the velocity field ${\displaystyle \mathbf {v} (\mathbf {x} )}$ which is induced by the vorticity ${\displaystyle w(\mathbf {x} )}$ with zero-circularity and given constant velocity ${\displaystyle \mathbf {v} _{\infty }}$ at infinity.

No-slip condition for ${\displaystyle \mathbf {x} \in S_{r_{0}}=\{\mathbf {x} \in \mathbb {R} ^{2},~\vert \mathbf {x} \vert =r_{0}\}}$

${\displaystyle {\frac {1}{2\pi }}\int _{B_{r_{0}}}{\frac {(\mathbf {x} -\mathbf {y} )^{\perp }}{\vert \mathbf {x} -\mathbf {y} \vert ^{2}}}w(\mathbf {y} )\operatorname {d\mathbf {y} } +\mathbf {v} _{\infty }=0}$
leads to the relations for ${\displaystyle k\in \mathbf {Z} }$:
${\displaystyle \int _{r_{0}}^{\infty }r^{-\vert k\vert +1}w_{k}(r)dr=d_{k},}$
where ${\displaystyle d_{k}=\delta _{\vert k\vert ,1}(v_{\infty ,y}+ikv_{\infty ,x}),}$ ${\displaystyle \delta _{\vert k\vert ,1}}$ is the Kronecker delta, ${\displaystyle v_{\infty ,x}}$, ${\displaystyle v_{\infty ,y}}$ are the cartesian coordinates of ${\displaystyle \mathbf {v} _{\infty }}$.

In particular, from the no-slip condition follows orthogonality the vorticity to the kernel of the Weber's transform ${\displaystyle W_{k,k-1}}$:

${\displaystyle \int _{r_{0}}^{\infty }r^{-\vert k\vert +1}w_{k}(r)dr=0~for~|k|>1.}$

#### Vorticity flow and its boundary condition

Vorticity ${\displaystyle w(t,\mathbf {x} )}$ for Stokes flow satisfies to the vorticity equation

${\displaystyle {\frac {\partial w(t,\mathbf {x} )}{\partial t}}-\Delta w=0,}$
or in terms of the Fourier coefficients in the expansion by polar angle
${\displaystyle {\frac {\partial w_{k}(t,r)}{\partial t}}-\Delta w_{k}=0,}$
where
${\displaystyle \Delta _{k}w_{k}(t,r)={\frac {1}{r}}{\frac {\partial }{\partial r}}\left(r{\frac {\partial }{\partial r}}w_{k}(t,r)\right)-{\frac {k^{2}}{r^{2}}}w_{k}(t,r).}$

From no-slip condition follows

${\displaystyle {\frac {d}{dt}}\int _{r_{0}}^{\infty }r^{-\vert k\vert +1}w_{k}(t,r)dr=0.}$

Finally, integrating by parts, we obtain the Robin boundary condition for the vorticity:

${\displaystyle \int _{r_{0}}^{\infty }s^{-|k|+1}\Delta _{k}w_{k}(t,r)dr=-r_{0}^{-|k|}\left(r_{0}{\frac {\partial w_{k}(t,r)}{\partial r}}{\Big |}_{r=r_{0}}+|k|w_{k}(t,r_{0})\right)=0.}$
Then the solution of the boundary-value problem can be expressed via Weber's integral above.

### Remark

Formula for vorticity can give another explanation of the Stokes' Paradox. The functions ${\displaystyle {\frac {C}{r^{k}}}\in ker(\Delta _{k}),~k>1}$ belong to the kernel of ${\displaystyle \Delta _{k}}$ and generate the stationary solutions of the vorticity equation with Robin-type boundary condition. From the arguments above any Stokes' vorticity flow with no-slip boundary condition must be orthogonal to the obtained stationary solutions. That is only possible for ${\displaystyle w\equiv 0}$.

## References

1. ^ a b Lamb, Horace (1945). Hydrodynamics (Sixth ed.). New York: Dover Publications. pp. 602–604.
2. ^ a b Van Dyke, Milton (1975). Perturbation Methods in Fluid Mechanics. Parabolic Press.
3. ^ Lamb, Horace (1945). Hydrodynamics (Sixth ed.). New York: Dover Publications. pp. 602.
4. ^ Weisstein, Eric W. (2002). CRC Concise Encyclopedia of Mathematics. CRC Press. ISBN 1584883472.
5. ^ Lamb, Horace (1945). Hydrodynamics (Sixth ed.). New York: Dover Publications. pp. 615.
6. ^ Sarason, Donald (1994). Notes on Complex Function Theory. Berkeley, California.{{cite book}}: CS1 maint: location missing publisher (link)
7. ^ Lamb, Horace (1945). Hydrodynamics (Sixth ed.). New York: Dover Publications. pp. 608–609.
8. ^ Lamb, Horace (1945). Hydrodynamics (Sixth ed.). New York: Dover Publications. pp. 609–616.
9. ^ Goldstein, Sydney (1965). Modern Developments in Fluid Dynamics. Dover Publications.
10. ^ a b c Gorshkov, A.V. (2019). "Associated Weber–Orr Transform, Biot–Savart Law and Explicit Form of the Solution of 2D Stokes System in Exterior of the Disc". J. Math. Fluid Mech. 21 (41): 41. arXiv:1904.12495. Bibcode:2019JMFM...21...41G. doi:10.1007/s00021-019-0445-2. S2CID 199113540.
11. ^ a b Titchmarsh, E.C. (1946). Eigenfunction Expansions Associated With Second-Order Differential Equations, Part I. Clarendon Press, Oxford.
12. ^ Watson, G.N. (1995). A Treatise on the Theory of Bessel Functions. Cambridge University Press.
13. ^ Griffith, J.L. (1956). "A note on a generalisation of Weber's transform". J. Proc. Roy. Soc. 90. New South Wales: 157–162.