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| {{one source|date=October 2013}}
| | Hi, everybody! My name is Samuel. <br>It is a little about myself: I live in Great Britain, my city of Rhuallt. <br>It's called often Northern or cultural capital of . I've married 2 years ago.<br>I have two children - a son (Tommie) and the daughter (Jonathan). We all like Travel.<br><br>my webpage :: [http://support.groupsite.com/entries/32420770-How-To-Upload-A-Website-To-Hostgator Hostgator Coupon] |
| 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 flow|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.<ref>{{cite book|last=Lamb|first=Horace|title=Hydrodynamics|year=1945|publisher=Dover Publications|location=New York|pages=602–604|edition=Sixth}}</ref>
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| ==Derivation==
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| The velocity vector <math>u</math> of the [[fluid]] may be written in terms of the [[stream function]] <math>\psi</math> as:
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| : <math>\mathbf{u} = \begin{pmatrix} | |
| {\partial \psi \over \partial y} & - {\partial \psi \over \partial x}
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| \end{pmatrix} </math>
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| As the stream function in a Stokes flow problem, <math>\psi</math> satisfies the [[biharmonic equation]].<ref>{{cite book|last=Lamb|first=Horace|title=Hydrodynamics|year=1945|publisher=Dover Publications|location=New York|pages=602|edition=Sixth}}</ref> 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, <math>\psi</math> is either the [[Real part|real]] or [[imaginary part]] of:
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| : <math>\bar{z} f(z)+g(z)</math><ref>{{cite book|last=Weisstein|first=Eric W.|title=CRC Concise Encyclopedia of Mathematics|year=2002|publisher=CRC Press|isbn=1584883472}}</ref>
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| Here <math>z=x+iy</math>, where <math>i</math> is the [[Imaginary number|imaginary]] unit, <math>\bar{z} = x-iy</math> and <math>f(z),g(z)</math> are [[holomorphic functions]] outside of the disk. We will take the real part [[without loss of generality]].
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| Now the function <math>u</math>, defined by <math>u=u_x +iu_y</math> is introduced. <math>u</math> can be written as <math>u=-2i \frac{\partial \psi} {\partial \bar{z}}</math>, or <math>\frac{1}{2} iu = \frac{\partial \psi}{\partial \bar{z}}</math> (using the [[Wirtinger derivatives]]).
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| This is calculated to be equal to:
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| : <math>\frac{1}{2} iu=f(z)+z \bar{f \prime} (z)+ \bar{g \prime} (z)</math> | |
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| Without loss of generality, the disk may be assumed to be the [[unit disk]], consisting of all [[complex numbers]] z of [[complex number#Polar form|absolute value]] smaller or equal to 1.
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| The [[boundary conditions]] are:
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| : <math> \lim_{z \to \infty} u=1 </math> | |
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| and
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| : <math> u = 0 </math>
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| whenever <math> |z| =1 </math>, <ref>{{cite book|last=Lamb|first=Horace|title=Hydrodynamics|year=1945|publisher=Dover Publications|location=New York|pages=602–604|edition=Sixth}}</ref><ref>{{cite book|last=Lamb|first=Horace|title=Hydrodynamics|year=1945|publisher=Dover Publications|location=New York|pages=615|edition=Sixth}}</ref>
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| and by representing the functions <math>f,g</math> as [[Laurent series]]:<ref>{{cite book|last=Sarason|first=Donald|title=Notes on Complex Function Theory|year=1994|location=Berkeley, California}}</ref>
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| : <math>f(z)=\sum_{n= -\infty}^\infty f_n z^n,g(z)=\sum_{n= -\infty}^\infty g_n z^n</math>
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| the first condition implies <math>f_n=0,g_n=0</math> for all <math>n\geq2</math>.
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| Using the polar form of <math>z</math> results in <math>z^n=r^n e^{in\theta} ,\bar{z}^n=r^n e^{-in \theta}</math>.
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| After deriving the series form of u and substituting this into it along with <math>r=1</math>, and changing some indices, the second boundary condition translates to:
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| : <math>\sum_{n=- \infty}^ \infty e^{in \theta} \left ( f_n + (2-n) \bar{f}_{2-n} + (1-n) \bar{g}_{1-n} \right ) = 0</math>.
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| Since the complex trigonometric functions <math>e^{in \theta}</math> compose a [[linearly independent]] set, it follows that all coefficients in the series are zero.
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| Examining these conditions for every <math>n</math> after taking into account the condition at infinity shows that <math>f</math> and <math>g</math> are necessarily of the form:
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| : <math>f(z)=az+b,g(z)=-bz+c</math>
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| where <math>a</math> is an imaginary number (opposite to its own [[complex conjugate]]) and <math>b</math> and <math>c</math> are complex numbers. Substituting this into <math>u</math> gives the result that <math>u=0</math> globally, compelling both <math>u_x</math> and <math>u_y</math> 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.
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| ==Resolution==
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| The paradox is caused by the limited validity of Stokes' approximation, as explained in [[Carl Wilhelm Oseen|Oseen's]] criticism: the validity of Stokes' equations relies on [[Reynolds number]] being small, and this condition cannot hold for arbitrarily large distances <math>r</math>.<ref>{{cite book|last=Lamb|first=Horace|title=Hydrodynamics|year=1945|publisher=Dover Publications|location=New York|pages=608–609|edition=Sixth}}</ref>
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| A correct solution for a cylinder was derived using [[Oseen's equations]], and the same equations lead to an improved approximation of the [[Stokes' law|drag force on a sphere]].<ref>{{cite book|last=Lamb|first=Horace|title=Hydrodynamics|year=1945|publisher=Dover Publications|location=New York|pages=609–616|edition=Sixth}}</ref>
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| ==See also==
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| * [[Oseen's approximation]]
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| * [[Stokes' law]]
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| ==References==
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| <references />
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| [[Category:Fluid dynamics]]
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| [[Category:Equations of fluid dynamics]]
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Hi, everybody! My name is Samuel.
It is a little about myself: I live in Great Britain, my city of Rhuallt.
It's called often Northern or cultural capital of . I've married 2 years ago.
I have two children - a son (Tommie) and the daughter (Jonathan). We all like Travel.
my webpage :: Hostgator Coupon