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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Stokes' paradox ： ウィキペディア英語版 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 $u$ of the fluid may be written in terms of the stream function $\backslash psi$ as: : $\backslash mathbf\; =\; \backslash begin$ & - \end As the stream function in a Stokes flow problem, $\backslash psi$ satisfies the biharmonic equation. 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, $\backslash psi$ is either the real or imaginary part of: : $\backslash bar\; f(z)+g(z)$ Here $z=x+iy$, where $i$ is the imaginary unit, $\backslash bar\; =\; x-iy$ 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 $u=-2i\; \backslash frac\; iu\; =\; \backslash frac\; iu=f(z)+z\; \backslash bar\; (z)+\; \backslash bar\; (z)$ 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: : $\backslash lim\_\; u=1$ and : $u\; =\; 0$ whenever $|z|\; =1$, and by representing the functions $f,g$ as Laurent series: : $f(z)=\backslash sum\_^\backslash infty\; f\_n\; z^n,g(z)=\backslash sum\_^\backslash infty\; g\_n\; z^n$ the first condition implies $f\_n=0,g\_n=0$ for all $n\backslash geq2$. Using the polar form of $z$ results in $z^n=r^n\; e^\; ,\backslash bar^n=r^n\; e^$. After deriving the series form of u and substituting this into it along with $r=1$, and changing some indices, the second boundary condition translates to: : $\backslash sum\_^\; \backslash infty\; e^\; \backslash left\; (\; f\_n\; +\; (2-n)\; \backslash bar\_\; +\; (1-n)\; \backslash bar\_\; \backslash right\; )\; =\; 0$. Since the complex trigonometric functions $e^$ 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: : $f(z)=az+b,g(z)=-bz+c$ 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Stokes' paradox」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.