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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 Cauchy momentum equation ： ウィキペディア英語版 Cauchy momentum equation The Cauchy momentum equation is a vector partial differential equation put forth by Cauchy that describes the non-relativistic momentum transport in any continuum. In convective (or Lagrangian) form it is written: :$\backslash frac\; =\; \backslash frac\; 1\; \backslash rho\; \backslash nabla\; \backslash cdot\; \backslash boldsymbol\; +\; \backslash mathbf$ where $\backslash rho$ is the density at the point considered in the continuum (for which the continuity equation holds), $\backslash boldsymbol$ is the stress tensor, and $\backslash mathbf$ contains all of the body forces per unit mass (often simply gravitational acceleration). $\backslash mathbf$ is the flow velocity vector field, which depends on time and space. Notably, it can be written, through an appropriate change of variables, also in conservation (or Eulerian) form: :$\backslash frac\; +\; \backslash nabla\; \backslash cdot\; \backslash bold\; F\; =\; \backslash bold\; s$ where $j$ is the momentum density at the point considered in the continuum (for which the continuity equation holds), $F$ is the flux associated to the momentum density, and $\backslash mathbf$ contains all of the body forces per unit volume. ==Derivation== Applying Newton's second law ($i^$ component) to a control volume in the continuum being modeled gives: :$m\; a\_i\; =\; F\_i\backslash ,$ and basing on the Reynolds transport theorem and on the material derivative notation: :$\backslash int\_\; \backslash rho\; \backslash frac\; \backslash ,\; dV\; =\; \backslash int\_\; \backslash nabla\_j\backslash sigma\_i^j\; \backslash ,\; dV\; +\; \backslash int\_\; \backslash rho\; g\_i\; \backslash ,\; dV$ :$\backslash int\_\; (\backslash rho\; \backslash frac\; -\; \backslash nabla\_j\backslash sigma\_i^j\; -\; \backslash rho\; g\_i\; )\backslash ,\; dV\; =\; 0$ :$\backslash rho\; \backslash frac-\; \backslash nabla\_j\backslash sigma\_i^j\; -\; \backslash rho\; g\_i\; =\; 0$ :$\backslash frac-\; \backslash frac\; -\; g\_i\; =\; 0$ where $\backslash Omega$ represents the control volume. Since this equation must hold for any control volume, it must be true that the integrand is zero, from this the Cauchy momentum equation follows. The main step (not done above) in deriving this equation is establishing that the derivative of the stress tensor is one of the forces that constitutes $F\_i$.〔 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「Cauchy momentum equation」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.