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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 Kelvin's circulation theorem ： ウィキペディア英語版 Kelvin's circulation theorem In fluid mechanics, Kelvin's circulation theorem (named after William Thomson, 1st Baron Kelvin who published it in 1869〔Katz, Plotkin: ''Low-Speed Aerodynamics''〕) states ''In a barotropic ideal fluid with conservative body forces, the circulation around a closed curve (which encloses the same fluid elements) moving with the fluid remains constant with time''.〔Kundu, P and Cohen, I: ''Fluid Mechanics'', page 130. Academic Press 2002〕 Stated mathematically: :$\backslash frac\; =\; 0$ where $\backslash Gamma$ is the circulation around a material contour $C(t)$. Stated more simply this theorem says that if one observes a closed contour at one instant, and follows the contour over time (by following the motion of all of its fluid elements), the circulation over the two locations of this contour are equal. This theorem does not hold in cases with viscous stresses, nonconservative body forces (for example a coriolis force) or non-barotropic pressure-density relations. ==Mathematical proof== The circulation $\backslash Gamma$ around a closed material contour $C(t)$ is defined by: :$\backslash Gamma(t)\; =\; \backslash oint\_C\; \backslash boldsymbol\; \backslash cdot\; \backslash boldsymbol$ where ''u'' is the velocity vector, and ''ds'' is an element along the closed contour. The governing equation for an inviscid fluid with a conservative body force is :$\backslash frac\}\; =\; -\; \backslash frac\backslash boldsymbolp\; +\; \backslash boldsymbol\; \backslash Phi$ where D/D''t'' is the convective derivative, ''ρ'' is the fluid density, ''p'' is the pressure and ''Φ'' is the potential for the body force. These are the Euler equations with a body force. The condition of barotropicity implies that the density is a function only of the pressure, i.e. $\backslash rho=\backslash rho(p)$. Taking the convective derivative of circulation gives :$\backslash frac\; =\; \backslash oint\_C\; \backslash frac\}\; \backslash cdot\; \backslash boldsymbol\; +\; \backslash oint\_C\; \backslash boldsymbol\; \backslash cdot\; \backslash fracs\}\}.$ For the first term, we substitute from the governing equation, and then apply Stokes' theorem, thus: :$\backslash oint\_C\; \backslash frac\}\; \backslash cdot\; \backslash boldsymbol\; =\; \backslash int\_A\; \backslash boldsymbol\; \backslash times\; \backslash left(\; -\backslash frac\; \backslash boldsymbol\; p\; +\; \backslash boldsymbol\; \backslash Phi\; \backslash right)\; \backslash cdot\; \backslash boldsymbol\; \backslash ,\; \backslash mathrmS\; =\; \backslash int\_A\; \backslash frac\; \backslash left(\; \backslash boldsymbol\; \backslash rho\; \backslash times\; \backslash boldsymbol\; p\; \backslash right)\; \backslash cdot\; \backslash boldsymbol\; \backslash ,\; \backslash mathrmS\; =\; 0.$ The final equality arises since $\backslash boldsymbol\; \backslash rho\; \backslash times\; \backslash boldsymbol\; p=0$ owing to barotropicity. We have also made use of the fact that the curl of any gradient is necessarily 0, or $\backslash boldsymbol\; \backslash times\; \backslash boldsymbol\; f=0$ for any function $f$. For the second term, we note that evolution of the material line element is given by :$\backslash fracs\}\}\; =\; \backslash left(\; \backslash boldsymbols\; \backslash cdot\; \backslash boldsymbol\; \backslash right)\; \backslash boldsymbol.$ Hence :$\backslash oint\_C\; \backslash boldsymbol\; \backslash cdot\; \backslash fracs\}\}\; =\; \backslash oint\_C\; \backslash boldsymbol\; \backslash cdot\; \backslash left(\; \backslash boldsymbols\; \backslash cdot\; \backslash boldsymbol\; \backslash right)\; \backslash boldsymbol\; =\; \backslash frac\; \backslash oint\_C\; \backslash boldsymbol\; \backslash left(\; |\backslash boldsymbol|^2\; \backslash right)\; \backslash cdot\; \backslash boldsymbol\; =\; 0.$ The last equality is obtained by applying Stokes theorem. Since both terms are zero, we obtain the result :$\backslash frac\; =\; 0.$ The theorem also applies to a rotating frame, with a rotation vector $\backslash boldsymbol$, if the circulation is modified thus: :$\backslash Gamma(t)\; =\; \backslash oint\_C\; (\backslash boldsymbol\; +\; \backslash boldsymbol\; \backslash times\; \backslash boldsymbol)\; \backslash cdot\; \backslash boldsymbol$ Here $\backslash boldsymbol$ is the position of the area of fluid. From Stoke's theorem, this is: :$\backslash Gamma(t)\; =\; \backslash int\_A\; \backslash boldsymbol\; \backslash times\; (\backslash boldsymbol\; +\; \backslash boldsymbol\; \backslash times\; \backslash boldsymbol)\; \backslash cdot\; \backslash boldsymbol\; \backslash ,\; \backslash mathrmS\; =\; \backslash int\_A\; (\backslash boldsymbol\; \backslash times\; \backslash boldsymbol\; +\; 2\; \backslash boldsymbol)\; \backslash cdot\; \backslash boldsymbol\; \backslash ,\; \backslash mathrmS$ 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「Kelvin's circulation theorem」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.