Stokes' paradox

The velocity vector $\backslash mathbf\{u\}$ of the fluid may be written in terms of the stream function $\backslash psi$ as

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

The stream function in a Stokes flow problem, $\backslash psi$ satisfies the biharmonic equation.{{cite book|last=Lamb|first=Horace|title=Hydrodynamics|url=https://archive.org/details/hydrodynamics00lamb|year=1945|publisher=Dover Publications|location=New York|pages=[https://archive.org/details/hydrodynamics00lamb/page/602 602]|edition=Sixth}} By regarding the $(x,y)$-plane as the complex plane, the problem may be dealt with using methods of complex analysis. In this approach, $\backslash psi$ is either the real or imaginary part of

: $\backslash bar\{z\}\; f(z)\; +\; g(z)$.{{cite book|last=Weisstein|first=Eric W.|title=CRC Concise Encyclopedia of Mathematics|year=2002|publisher=CRC Press|isbn=1584883472}}

Here $z\; =\; x\; +\; iy$, where $i$ is the imaginary unit, $\backslash bar\{z\}\; =\; x\; -\; iy$, 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 $u\; =\; -2i\; \backslash frac\{\backslash partial\; \backslash psi\}\{\backslash partial\; \backslash bar\{z\}\}$, or $\backslash frac\{1\}\{2\}\; iu\; =\; \backslash frac\{\backslash partial\; \backslash psi\}\{\backslash partial\; \backslash bar\{z\}\}$ (using the Wirtinger derivatives).

This is calculated to be equal to

: $\backslash frac\{1\}\{2\}\; iu\; =\; f(z)\; +\; z\; \backslash bar\{f\; \backslash prime\}(z)\; +\; \backslash bar\{g\; \backslash prime\}(z).$

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:

: $\backslash lim\_\{z\; \backslash to\; \backslash infty\}\; u\; =\; 1,$

: $u\; =\; 0,$

whenever $|z|\; =\; 1$,{{cite book|last=Lamb|first=Horace|title=Hydrodynamics|url=https://archive.org/details/hydrodynamics00lamb|year=1945|publisher=Dover Publications|location=New York|pages=[https://archive.org/details/hydrodynamics00lamb/page/615 615]|edition=Sixth}}

and by representing the functions $f,\; g$ as Laurent series:{{cite book|last=Sarason|first=Donald|title=Notes on Complex Function Theory|year=1994|location=Berkeley, California}}

: $f(z)\; =\; \backslash sum\_\{n\; =\; -\backslash infty\}^\backslash infty\; f\_n\; z^n,\; \backslash quad\; g(z)\; =\; \backslash sum\_\{n\; =\; -\backslash infty\}^\backslash infty\; g\_n\; z^n,$

the first condition implies $f\_n\; =\; 0,\; g\_n\; =\; 0$ for all $n\; \backslash geq\; 2$.

Using the polar form of $z$ results in $z^n\; =\; r^n\; e^\{in\backslash theta\},\; \backslash bar\{z\}^n\; =\; r^n\; e^\{-in\; \backslash theta\}$.

After deriving the series form of u, substituting this into it along with $r\; =\; 1$, and changing some indices, the second boundary condition translates to

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

Since the complex trigonometric functions $e^\{in\; \backslash theta\}$ 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

: $f(z)\; =\; az\; +\; b,\; \backslash quad\; g(z)\; =\; -bz\; +\; c,$

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.

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 $r$.{{cite book|last=Lamb|first=Horace|title=Hydrodynamics|url=https://archive.org/details/hydrodynamics00lamb|year=1945|publisher=Dover Publications|location=New York|pages=[https://archive.org/details/hydrodynamics00lamb/page/608 608–609]|edition=Sixth}}

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.{{cite book|last=Lamb|first=Horace|title=Hydrodynamics|url=https://archive.org/details/hydrodynamics00lamb|year=1945|publisher=Dover Publications|location=New York|pages=[https://archive.org/details/hydrodynamics00lamb/page/609 609–616]|edition=Sixth}}{{cite book|last=Goldstein|first=Sydney|title=Modern Developments in Fluid Dynamics|date=1965|publisher=Dover Publications}}

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.

The vorticity of Stokes' flow is given by the following relation:{{cite journal |last1=Gorshkov |first1=A.V. |title=Associated Weber–Orr Transform, Biot–Savart Law and Explicit Form of the Solution of 2D Stokes System in Exterior of the Disc |journal=J. Math. Fluid Mech. |date=2019 |volume=21 |issue=41 |page=41 |doi=10.1007/s00021-019-0445-2 |arxiv=1904.12495 |bibcode=2019JMFM...21...41G |s2cid=199113540 }}

$$w\_k(t,r)\; =\; W^\{-1\}\_\{|k|,|k|-1\}\; \backslash left\; [\; e^\{-\backslash lambda^2\; t\}\; W\_\{|k|,|k|-1\}\; [w\_k(0,\backslash cdot)](\backslash lambda)\; \backslash right\; ](t,r).$$

Here $w\_k(t,r)$ - are the Fourier coefficients of the vorticity's expansion by polar angle which are defined on $(r\_0,\backslash infty)$, $r\_0$ - radius of the cylinder, $W\_\{|k|,|k|-1\}$, $W^\{-1\}\_\{|k|,|k|-1\}$ are the direct and inverse special Weber's transforms,{{cite book |date=1946|last1=Titchmarsh |first1=E.C. |title=Eigenfunction Expansions Associated With Second-Order Differential Equations, Part I. |publisher=Clarendon Press, Oxford}} and initial function for vorticity $w\_k(0,r)$ 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.

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

$$W\_\{k,k-\; 1\}[f](\backslash lambda)\; =\; \backslash int\_\{r\_0\}^\backslash infty\; \backslash frac\{J\_\{k\}(\backslash lambda\; s)Y\_\{k-\; 1\}(\backslash lambda\; r\_0)\; -\; Y\_\{k\}(\backslash lambda\; s)J\_\{k-\; 1\}(\backslash lambda\; r\_0)\}$$

{\sqrt{J_{k- 1}^2(\lambda r_0) + Y_{k- 1}^2(\lambda r_0)}} f(s) s ds,

where $J\_k$, $Y\_k$ are the Bessel functions of the first and second kind{{cite book |last1=Watson |first1=G.N. |title=A Treatise on the Theory of Bessel Functions |date=1995 |publisher=Cambridge University Press}} respectively. For $k>1$ it has a non-trivial kernel{{cite journal |last1=Griffith |first1=J.L. |title=A note on a generalisation of Weber's transform |journal=J. Proc. Roy. Soc. |date=1956 |volume=90 |publisher=New South Wales |pages=157–162}} which consists of the functions $C/r^k\; \backslash in\; \backslash ker(W\_\{k,k-1\})$.

The inverse transform is given by the formula

$$W^\{-1\}\_\{k,k-\; 1\}[\backslash hat\; f](r)\; =\; \backslash int\_\{0\}^\backslash infty\; \backslash frac\; \{J\_\{k\}(\backslash lambda\; r)Y\_\{k-\; 1\}(\backslash lambda\; r\_0)\; -\; Y\_\{k\}(\backslash lambda\; s)J\_\{k-\; 1\}(\backslash lambda\; r\_0)\}\{\backslash sqrt\{J\_\{k-\; 1\}^2(\backslash lambda\; r\_0)\; +\; Y\_\{k-\; 1\}^2(\backslash lambda\; r\_0)\}\}\; \backslash hat\; f\; (\backslash lambda)\; \backslash lambda\; d\backslash lambda.$$

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

f(r) = W^{-1}_{k,k-1}\left [W_{k,k-1} [f] \right ](r)

is valid if $k\backslash leq\; 1$. Also it is valid in the case of $k>\; 1$ but only for functions, which are orthogonal to the kernel of $W\_\{k,k-1\}$ in $L\_2(r\_0,\backslash infty)$ with infinitesimal element $rdr$:

\int_{r_0}^\infty \frac 1{r^k} f(r) r dr = 0,~k>1.

In exterior of the disc of radius $r\_0$ $B\_\{r\_0\}=\backslash \{\backslash mathbf\{x\}\; \backslash in\; \backslash mathbb\{R\}^2,~\backslash vert\; \backslash mathbf\{x\}\; \backslash vert\; >\; r\_0\; \backslash \}$ the Biot-Savart law

$$\backslash mathbf\{v\}(\backslash mathbf\{x\})\; =\backslash frac\; 1\{2\backslash pi\}\; \backslash int\_\{B\_\{r\_0\}\}\; \backslash frac\{(\backslash mathbf\{x\}-\backslash mathbf\{y\})^\backslash perp\}\{\backslash vert\backslash mathbf\{x\}-\backslash mathbf\{y\}\backslash vert^2\}\; w(\backslash mathbf\{y\})\; \backslash operatorname\{d\backslash mathbf\{y\}\}\; +\; \backslash mathbf\{v\}\_\backslash infty,$$

restores the velocity field $\backslash mathbf\{v\}(\backslash mathbf\{x\})$ which is induced by the vorticity $w(\backslash mathbf\{x\})$ with zero-circularity and given constant velocity $\backslash mathbf\{v\}\_\backslash infty$ at infinity.

No-slip condition for $\backslash mathbf\{x\}\backslash in\; S\_\{r\_0\}=\backslash \{\backslash mathbf\{x\}\; \backslash in\; \backslash mathbb\{R\}^2,~\backslash vert\; \backslash mathbf\{x\}\; \backslash vert\; =\; r\_0\; \backslash \}$

$$\backslash frac\; 1\{2\backslash pi\}\; \backslash int\_\{B\_\{r\_0\}\}\; \backslash frac\{(\backslash mathbf\{x\}-\backslash mathbf\{y\})^\backslash perp\}\{\backslash vert\backslash mathbf\{x\}-\backslash mathbf\{y\}\backslash vert^2\}\; w(\backslash mathbf\{y\})\; \backslash operatorname\{d\backslash mathbf\{y\}\}\; +\; \backslash mathbf\{v\}\_\backslash infty\; =0$$

leads to the relations for $k\backslash in\; \backslash mathbf\{Z\}$:

$$\backslash int\_\{r\_0\}^\backslash infty\; r^\{-\backslash vert\; k\; \backslash vert+1\}w\_k(r)dr\; =\; d\_k,$$

where

$d\_k=\backslash delta\_\{\backslash vert\; k\; \backslash vert,1\}\; (v\_\{\backslash infty,y\}\; +\; i\; k\; v\_\{\backslash infty,x\}),$

$\backslash delta\_\{\backslash vert\; k\; \backslash vert,1\}$ is the Kronecker delta, $v\_\{\backslash infty,x\}$, $v\_\{\backslash infty,y\}$ are the cartesian coordinates of $\backslash mathbf\{v\}\_\backslash infty$.

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

$$\backslash int\_\{r\_0\}^\backslash infty\; r^\{-\backslash vert\; k\; \backslash vert+1\}w\_k(r)dr\; =\; 0\; ~for~\; |k|>1.$$

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

$$\backslash frac\{\backslash partial\; w(t,\backslash mathbf\{x\})\}\{\backslash partial\; t\}\; -\; \backslash Delta\; w\; =\; 0,$$ or in terms of the Fourier coefficients in the expansion by polar angle

$$\backslash frac\{\backslash partial\; w\_k(t,r)\}\{\backslash partial\; t\}\; -\; \backslash Delta\; w\_k\; =\; 0,$$

where

$$\backslash Delta\_k\; w\_k(t,r)\; =\; \backslash frac\; 1r\; \backslash frac\; \{\backslash partial\}\{\backslash partial\; r\}\backslash left(r\; \backslash frac\; \{\backslash partial\}\{\backslash partial\; r\}w\_k(t,r)\backslash right)\; -\; \backslash frac\{k^2\}\{r^2\}\; w\_k(t,r).$$

From no-slip condition follows $$\backslash frac\; d\{dt\}\; \backslash int\_\{r\_0\}^\backslash infty\; r^\{-\backslash vert\; k\; \backslash vert+1\}w\_k(t,r)dr\; =\; 0.$$

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

$$\backslash int\_\{r\_0\}^\backslash infty\; s^\{-|k|+1\}\; \backslash Delta\_k\; w\_k(t,r)dr$$

= - 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.

Formula for vorticity can give another explanation of the Stokes' Paradox. The functions $\backslash frac\; C\{r^k\}\; \backslash in\; ker(\backslash Delta\_k),~k>1$ belong to the kernel of $\backslash Delta\_k$ 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 $w\backslash equiv\; 0$.

• Oseen's approximation
• Stokes' law

Category:Fluid dynamics

Category:Equations of fluid dynamics