翻訳と辞書 Words near each other ・ Clausilia cruciata ・ Clausilia dubia ・ Clausilia pumila ・ Clausilia rugosa ・ Clausiliidae ・ Clausilioidea ・ Clausilioides ・ Clausilium ・ Clausinella fasciata ・ Clausirion ・ Clausirion bicolor ・ Clausirion comptum ・ Clausius (crater) ・ Clausius theorem ・ Clausius–Clapeyron relation ・ Clausius–Duhem inequality ・ Clausius–Mossotti relation ・ Clausnitz ・ Clausnitzer Glacier ・ Clauson ・ Clausospicula ・ Clauss ・ Clauss Cutlery Company ・ Clauss Ice Arena ・ Claussen ・ Claussen and Claussen ・ Claussen House ・ Claussen pickles ・ Claussen's Bakery ・ Claussenius, California Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Clausius–Duhem inequality ： ウィキペディア英語版 Clausius–Duhem inequality The Clausius–Duhem inequality〔.〕〔.〕 is a way of expressing the second law of thermodynamics that is used in continuum mechanics. This inequality is particularly useful in determining whether the constitutive relation of a material is thermodynamically allowable.〔.〕 This inequality is a statement concerning the irreversibility of natural processes, especially when energy dissipation is involved. It was named after the German physicist Rudolf Clausius and French physicist Pierre Duhem. == Clausius–Duhem inequality in terms of the specific entropy == The Clausius–Duhem inequality can be expressed in integral form as :$$ \cfrac\left(\int_\Omega \rho~\eta~\text\right) \ge \int_ \rho~\eta~(u_n - \mathbf\cdot\mathbf)~\text - \int_ \cfrac}~\text + \int_\Omega \cfrac~\text. In this equation $t\backslash ,$ is the time, $\backslash Omega\backslash ,$ represents a body and the integration is over the volume of the body, $\backslash partial\; \backslash Omega\backslash ,$ represents the surface of the body, $\backslash rho\backslash ,$ is the mass density of the body, $\backslash eta\backslash ,$ is the specific entropy (entropy per unit mass), $u\_n\backslash ,$ is the normal velocity of $\backslash partial\; \backslash Omega\backslash ,$, $\backslash mathbf$ is the velocity of particles inside $\backslash Omega\backslash ,$, $\backslash mathbf$ is the unit normal to the surface, $\backslash mathbf$ is the heat flux vector, $s\backslash ,$ is an energy source per unit mass, and $T\backslash ,$ is the absolute temperature. All the variables are functions of a material point at $\backslash mathbf$ at time $t\backslash ,$. In differential form the Clausius–Duhem inequality can be written as :$$ \rho~\dot \ge - \boldsymbol \cdot \left(\cfrac\right) + \cfrac where $\backslash dot$ is the time derivative of $\backslash eta\backslash ,$ and $\backslash boldsymbol\; \backslash cdot\; (\backslash mathbf)$ is the divergence of the vector $\backslash mathbf$. (\rho~\eta)~\text \ge -\int_ \rho~\eta~(\mathbf\cdot\mathbf)~\text - \int_ \cfrac}~\text + \int_\Omega \cfrac~\text. Using the divergence theorem, we get :$$ \int_\Omega \frac(\rho~\eta)~\text \ge -\int_\Omega \boldsymbol \cdot (\rho~\eta~\mathbf)~\text - \int_\Omega \boldsymbol \cdot \left(\cfrac\right)~\text + \int_\Omega \cfrac~\text. Since $\backslash Omega$ is arbitrary, we must have :$$ \frac(\rho~\eta) \ge -\boldsymbol \cdot (\rho~\eta~\mathbf) - \boldsymbol \cdot \left(\cfrac\right) + \cfrac. Expanding out :$$ \frac~\eta + \rho~\frac \ge -\boldsymbol (\rho_\eta)\cdot\mathbf - \rho~\eta~(\boldsymbol \cdot \mathbf) - \boldsymbol \cdot \left(\cfrac\right) + \cfrac or, :$$ \frac~\eta + \rho~\frac \ge -\eta~\boldsymbol \rho\cdot\mathbf - \rho~\boldsymbol \eta\cdot\mathbf - \rho~\eta~(\boldsymbol \cdot \mathbf) - \boldsymbol \cdot \left(\cfrac\right) + \cfrac or, :$$ \left(\frac + \boldsymbol \rho\cdot\mathbf + \rho~\boldsymbol \cdot \mathbf\right) ~\eta + \rho~\left(\frac + \boldsymbol \eta\cdot\mathbf\right) \ge -\boldsymbol \cdot \left(\cfrac\right) + \cfrac. Now, the material time derivatives of $\backslash rho$ and $\backslash eta$ are given by :$$ \dot = \frac + \boldsymbol \rho\cdot\mathbf ~;~~ \dot = \frac + \boldsymbol \eta\cdot\mathbf. Therefore, :$$ \left(\dot + \rho~\boldsymbol \cdot \mathbf\right)~\eta + \rho~\dot \ge -\boldsymbol \cdot \left(\cfrac\right) + \cfrac. From the conservation of mass $\backslash dot\; +\; \backslash rho~\backslash boldsymbol\; \backslash cdot\; \backslash mathbf\; =\; 0$. Hence, :$$ \cdot \left(\cfrac\right) + \cfrac. } |} 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「Clausius–Duhem inequality」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.