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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 Hagen–Poiseuille flow from the Navier–Stokes equations ： ウィキペディア英語版 Hagen–Poiseuille flow from the Navier–Stokes equations In fluid dynamics, the derivation of the Hagen–Poiseuille flow from the Navier–Stokes equations shows how this flow is an exact solution to the Navier–Stokes equations. ==Derivation== The laminar flow through a pipe of uniform (circular) cross-section is known as Hagen–Poiseuille flow. The equations governing the Hagen–Poiseuille flow can be derived directly from the Navier–Stokes momentum equations in 3D cylidrical coordinated by making the following set of assumptions: # The flow is steady ( $\backslash partial(...)/\backslash partial\; t\; =\; 0$ ). # The radial and swirl components of the fluid velocity are zero ( $u\_r\; =\; u\_\backslash theta\; =\; 0$ ). # The flow is axisymmetric ( $\backslash partial(...)/\backslash partial\; \backslash theta\; =\; 0$ ) and fully developed ($\backslash partial\; u\_z/\backslash partial\; z\; =\; 0$ ). Then the angular equation in the momentum equations and the continuity equation are identically satisfied. The first momentum equation reduces to $\backslash partial\; p/\backslash partial\; r\; =\; 0$, i.e., the pressure $p$ is a function of the axial coordinate $z$ only. The third momentum equation reduces to: :$\backslash frac\backslash frac\backslash left(r\; \backslash frac\backslash right)=\; \backslash frac\; \backslash frac$ where $\backslash mu$ is the dynamic viscosity of the fluid. :The solution is :$u\_z\; =\; \backslash frac\; \backslash fracr^2\; +\; c\_1\; \backslash ln\; r\; +\; c\_2$ Since $u\_z$ needs to be finite at $r\; =\; 0$, $c\_1\; =\; 0$. The no slip boundary condition at the pipe wall requires that $u\_z\; =\; 0$ at $r\; =\; R$ (radius of the pipe), which yields :$c\_2\; =\; -\backslash frac\; \backslash fracR^2.$ Thus we have finally the following parabolic velocity profile: :$u\_z\; =\; -\backslash frac\; \backslash frac\; (R^2\; -\; r^2).$ The maximum velocity occurs at the pipe centerline ($r=0$): :$\_=\backslash frac\; \backslash left(-\backslash frac\backslash right).$ The average velocity can be obtained by integrating over the pipe cross section: : $\_\backslash mathrm=\backslash frac\; \backslash int\_0^R\; u\_z\; \backslash cdot\; 2\backslash pi\; r\; dr\; =\; 0.5\; \_\backslash mathrm.$ The Hagen–Poiseuille equation relates the pressure drop $\backslash Delta\; p$ across a circular pipe of length $L$ to the average flow velocity in the pipe $\_\backslash mathrm$ and other parameters. Assuming that the pressure decreases linearly across the length of the pipe, we have $-\; \backslash frac\; =\; \backslash frac$ (constant). Substituting this and the expression for $\_\backslash mathrm$ into the expression for $\_\backslash mathrm$, and noting that the pipe diameter $D\; =\; 2R$, we get: : $\_\; =\; \backslash frac\; \backslash frac.$ Rearrangement of this gives the Hagen–Poiseuille equation: : $\backslash Delta\; p\; =\; \backslash frac\}.$ 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「Hagen–Poiseuille flow from the Navier–Stokes equations」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.