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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 Potential flow around a circular cylinder ： ウィキペディア英語版 Potential flow around a circular cylinder In mathematics, potential flow around a circular cylinder is a classical solution for the flow of an inviscid, incompressible fluid around a cylinder that is transverse to the flow. Far from the cylinder, the flow is unidirectional and uniform. The flow has no vorticity and thus the velocity field is irrotational and can be modeled as a potential flow. Unlike a real fluid, this solution indicates a net zero drag on the body, a result known as d'Alembert's paradox. "The flow of an incompressible fluid past a cylinder is one of the first mathematical models that a student of fluid dynamics encounters. This flow is an excellent vehicle for the study of concepts that will be encountered numerous times in mathematical physics, such as vector fields, coordinate transformations, and most important, the physical interpretation of mathematical results." 〔http://library.wolfram.com/infocenter/Articles/2731/〕 == Mathematical solution == A cylinder (or disk) of radius $R$ is placed in two-dimensional, incompressible, inviscid flow. The goal is to find the steady velocity vector $\backslash vec$ and pressure $p$ in a plane, subject to the condition that far from the cylinder the velocity vector is :$\backslash vec=U\backslash widehat+0\backslash widehat,$ where $U$ is a constant, and at the boundary of the cylinder :$\backslash vec\backslash cdot\backslash widehat=0,$ where $\backslash widehat$ is the vector normal to the cylinder surface. The upstream flow is uniform and has no vorticity. The flow is inviscid, incompressible and has constant mass density $\backslash rho$. The flow therefore remains without vorticity, or is said to be ''irrotational'', with $\backslash nabla\; \backslash times\; \backslash vec=0$ everywhere. Being irrotational, there must exist a velocity potential $\backslash phi$: :$\backslash vec=\backslash nabla\backslash phi.$ Being incompressible, $\backslash nabla\backslash cdot\backslash vec=0$, so $\backslash phi$ must satisfy Laplace's equation: :$\backslash nabla^2\backslash phi=0.$ The solution for $\backslash phi$ is obtained most easily in polar coordinates $r$ and $\backslash theta$, related to conventional Cartesian coordinates by $x=r\backslash cos\backslash theta$ and $y=r\backslash sin\backslash theta$. In polar coordinates, Laplace's equation is: :$\backslash frac\; +\; \backslash frac\backslash frac\; +\; \backslash frac\; \backslash frac=0.$ The solution that satisfies the boundary conditions is〔William S. Janna, ''Introduction to Fluid Mechanics'', PWS Publishing Company, Boston (1993)〕 :$\backslash phi(r,\backslash theta)=U\backslash left(r+\backslash frac\backslash right)\backslash cos\backslash theta.$ The velocity components in polar coordinates are obtained from the components of $\backslash nabla\backslash phi$ in polar coordinates: :$V\_r=\backslash frac\; =\; U\backslash left(1-\backslash frac\backslash right)\backslash cos\backslash theta$ and :$V\_\backslash theta=\backslash frac\backslash frac\; =\; -\; U\backslash left(1+\backslash frac\backslash right)\backslash sin\backslash theta.$ Being invisicid and irrotational, Bernoulli's equation allows the solution for pressure field to be obtained directly from the velocity field: :$p=\backslash frac\backslash rho\backslash left(U^2-V^2\backslash right)\; +\; p\_\backslash infty,$ where the constants $U$ and $p\_\backslash infty$ appear so that $p\backslash rightarrow\; p\_\backslash infty$ far from the cylinder, where $V=U$. Using :$V^2=V\_r^2+V\_\backslash theta^2,$ :$p=\backslash frac\backslash rho\; U^2\backslash left(2\backslash frac\backslash cos(2\backslash theta)-\backslash frac\backslash right)\; +\; p\_\backslash infty.$ In the figures, the colorized field referred to as "pressure" is a plot of :$2\; \backslash frac\; =2\backslash frac\backslash cos(2\backslash theta)-\backslash frac.$ On the surface of the cylinder, or $r=R$, pressure varies from a maximum of 1 (red color) at the stagnation points at $\backslash theta=0$ and $\backslash theta=\backslash pi$ to a minimum of -3 (purple) on the sides of the cylinder, at $\backslash theta=\backslash tfrac\backslash pi$ and $\backslash theta=\backslash tfrac\backslash pi.$ Likewise, $V$ varies from V=0 at the stagnation points to $V=2U$ on the sides, in the low pressure. 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「Potential flow around a circular cylinder」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.