查看“︁雷諾傳輸定理”︁的源代码

{{微積分學}}
'''雷諾傳輸定理'''也稱為'''萊布尼茲-雷諾傳輸定理'''或'''雷諾输运定理'''，是以[[積分符號內取微分]]聞名的萊布尼茲積分的三維推廣。
 
雷諾傳輸定理得名自[[奧斯鮑恩·雷諾]]（1842–1912），用來調整積分量的微分，用來推導[[連續介質力學]]的基礎方程。

考慮在時變的區域<math>\Omega(t)</math>積分<math>\mathbf{f} = \mathbf{f}(\mathbf{x},t)</math>，其邊界為<math>\partial \Omega(t)</math>，考慮上式對時間的微分：
:<math>
\cfrac{\mathrm{d}}{\mathrm{d}t}\int_{\Omega(t)} \mathbf{f}~\text{dV} ~.
</math>

若要求上述積分的導數，會有兩個問題，<math>\mathbf{f}</math>的時間相依性，及因<math>\Omega</math>動態的邊界而增加或減少的空間，雷諾傳輸定理提供了必要的框架。

== 通用型式 ==
雷諾傳輸定理可表為以下形式<ref name=LGLp23>L. Gary Leal, 2007, p. 23.</ref><ref name=OR14>O. Reynolds, 1903, Vol. 3, p. 12–13</ref><ref name=MTp23>J.E. Marsden and A. Tromba, 5th ed. 2003</ref>是：
:<math>
  \cfrac{\mathrm{d}}{\mathrm{d}t}\int_{\Omega(t)} \mathbf{f}~\text{dV} = 
     \int_{\Omega(t)} \frac{\partial \mathbf{f}}{\partial t}~\text{dV} + \int_{\partial \Omega(t)} (\mathbf{v}_{b}\cdot\mathbf{n})\mathbf{f}~\text{dA} ~
</math>
其中<math>\mathbf{n}(\mathbf{x},t)</math>為向外的單位法向量，<math>\mathbf{x}</math>為區域中的一點，也是積分變數，<math>\text{dV}</math> 及<math>\text{dA}</math>是位於<math>\mathbf{x}</math>的體積元素及表面元素，<math>\mathbf{v}_{b}(\mathbf{x},t)</math>為面積元素的速度而非流速。函數<math>\mathbf{f}</math>可以是[[張量]]、[[向量]]或[[标量 (数学)|純量]]函數<ref>H. Yamaguchi, ''Engineering Fluid Mechanics, ''Springer c2008 p23</ref>。注意等式左邊的積分只是時間的函數，所以採用全微分符號。

==針對流體塊的形式==
在連續介質力學中，此定理常用在沒有物質進來或離開的[[流體塊]]或固體中。若<math>\Omega(t)</math>為一流體塊，則存在速度函數<math>\mathbf{v}=\mathbf{v}(\mathbf{x},t)</math>及邊界元素符合下式
:<math>\mathbf{v}^{b}\cdot\mathbf{n}=\mathbf{v}\cdot\mathbf{n}.</math>
上式在替代後，可以得到以下的定理<ref name=BLM>T. Belytschko, W. K. Liu, and B. Moran, 2000, '' Nonlinear Finite Elements for Continua and Structures'', John Wiley and Sons, Ltd., New York.</ref>
:<math>
  \cfrac{\mathrm{d}}{\mathrm{d}t}\left(\int_{\Omega(t)} \mathbf{f}~\text{dV}\right) = 
     \int_{\Omega(t)} \frac{\partial \mathbf{f}}{\partial t}~\text{dV} + \int_{\partial \Omega(t)} (\mathbf{v}\cdot\mathbf{n})\mathbf{f}~\text{dA} ~.
</math>

:{| class="toccolours collapsible collapsed" width="50%" style="text-align:left"
!針對一流體塊的證明
|-
|
令<math>\Omega_0</math>為區域<math>\Omega(t)</math>的參考組態，令其運動及形變梯度為
:<math>
   \mathbf{x} = \boldsymbol{\varphi}(\mathbf{X}, t)~; \qquad\implies\qquad 
   \boldsymbol{F}(\mathbf{X},t) = \boldsymbol{\nabla}_{\circ} \boldsymbol{\varphi} ~.
</math>
令<math>J(\mathbf{X},t) = \det[\boldsymbol{F}(\mathbf{X},t)]</math>.
則目前組態及參考組態的積分有以下的關係
:<math>
   \int_{\Omega(t)} \mathbf{f}(\mathbf{x},t)~\text{dV} = 
      \int_{\Omega_0} \mathbf{f}[\boldsymbol{\varphi}(\mathbf{X},t),t]~J(\mathbf{X},t)~\text{dV}_0 =
      \int_{\Omega_0} \hat{\mathbf{f}}(\mathbf{X},t)~J(\mathbf{X},t)~\text{dV}_0 ~.
</math>
That this derivation is for a material element is implicit in the time constancy of the reference configuration: it is constant in material coordinates. 針對體積積分的微分定義為
:<math>
  \cfrac{\mathrm{d}}{\mathrm{d}t}\left( \int_{\Omega(t)} \mathbf{f}(\mathbf{x},t)~\text{dV}\right) = 
    \lim_{\Delta t \rightarrow 0} \cfrac{1}{\Delta t}
     \left(\int_{\Omega(t + \Delta t)} \mathbf{f}(\mathbf{x},t+\Delta t)~\text{dV} - 
           \int_{\Omega(t)} \mathbf{f}(\mathbf{x},t)~\text{dV}\right) ~.
</math>
將上式轉換為對參考組態的積分，可得
:<math>
  \cfrac{\mathrm{d}}{\mathrm{d}t}\left( \int_{\Omega(t)} \mathbf{f}(\mathbf{x},t)~\text{dV}\right) = 
    \lim_{\Delta t \rightarrow 0} \cfrac{1}{\Delta t}
     \left(\int_{\Omega_0} \hat{\mathbf{f}}(\mathbf{X},t+\Delta t)~J(\mathbf{X},t+\Delta t)~\text{dV}_0 - 
           \int_{\Omega_0} \hat{\mathbf{f}}(\mathbf{X},t)~J(\mathbf{X},t)~\text{dV}_0\right) ~.
</math>
因為<math>\Omega_0</math>和時間無關，可得
:<math>
  \begin{align}
  \cfrac{\mathrm{d}}{\mathrm{d}t}\left( \int_{\Omega(t)} \mathbf{f}(\mathbf{x},t)~\text{dV}\right) & = 
    \int_{\Omega_0} \left[\lim_{\Delta t \rightarrow 0} \cfrac{ 
           \hat{\mathbf{f}}(\mathbf{X},t+\Delta t)~J(\mathbf{X},t+\Delta t) - 
           \hat{\mathbf{f}}(\mathbf{X},t)~J(\mathbf{X},t)}{\Delta t} \right]~\text{dV}_0 \\
    & = \int_{\Omega_0} \frac{\partial }{\partial t}[\hat{\mathbf{f}}(\mathbf{X},t)~J(\mathbf{X},t)]~\text{dV}_0 \\
    & = \int_{\Omega_0} \left(
          \frac{\partial }{\partial t}[\hat{\mathbf{f}}(\mathbf{X},t)]~J(\mathbf{X},t)+
          \hat{\mathbf{f}}(\mathbf{X},t)~\frac{\partial }{\partial t}[J(\mathbf{X},t)]\right) ~\text{dV}_0 
  \end{align}
</math>
現在，<math>\det\boldsymbol{F}</math>的時間導數為 
<ref name=Gurtin>[[Morton Gurtin|Gurtin M. E.]], 1981, '' An Introduction to Continuum Mechanics''. Academic Press, New York, p. 77.</ref>
:<math>
   \frac{\partial J(\mathbf{X},t)}{\partial t} = \frac{\partial }{\partial t}(\det\boldsymbol{F}) = (\det\boldsymbol{F})(\boldsymbol{\nabla} \cdot \mathbf{v}) 
      = J(\mathbf{X},t)~\boldsymbol{\nabla} \cdot \mathbf{v}(\boldsymbol{\varphi}(\mathbf{X},t),t) 
      = J(\mathbf{X},t)~\boldsymbol{\nabla} \cdot \mathbf{v}(\mathbf{x},t) ~.
</math>
因此
:<math>
  \begin{align}
  \cfrac{\mathrm{d}}{\mathrm{d}t}\left( \int_{\Omega(t)} \mathbf{f}(\mathbf{x},t)~\text{dV}\right) & = 
     \int_{\Omega_0} \left(
          \frac{\partial }{\partial t}[\hat{\mathbf{f}}(\mathbf{X},t)]~J(\mathbf{X},t)+
          \hat{\mathbf{f}}(\mathbf{X},t)~J(\mathbf{X},t)~\boldsymbol{\nabla} \cdot \mathbf{v}(\mathbf{x},t)\right) ~\text{dV}_0 \\
     & = 
     \int_{\Omega_0} 
          \left(\frac{\partial }{\partial t}[\hat{\mathbf{f}}(\mathbf{X},t)]+
          \hat{\mathbf{f}}(\mathbf{X},t)~\boldsymbol{\nabla} \cdot \mathbf{v}(\mathbf{x},t)\right)~J(\mathbf{X},t) ~\text{dV}_0  \\
     & = 
     \int_{\Omega(t)} 
          \left(\dot{\mathbf{f}}(\mathbf{x},t)+
          \mathbf{f}(\mathbf{x},t)~\boldsymbol{\nabla} \cdot \mathbf{v}(\mathbf{x},t)\right)~\text{dV} 
  \end{align}
</math>
其中<math>\dot{\mathbf{f}}</math>為<math>\mathbf{f}</math>的{{link-en|材料導數|Material derivative}}，現在材料導數為
:<math>
  \dot{\mathbf{f}}(\mathbf{x},t) = 
    \frac{\partial \mathbf{f}(\mathbf{x},t)}{\partial t} + [\boldsymbol{\nabla} \mathbf{f}(\mathbf{x},t)]\cdot\mathbf{v}(\mathbf{x},t) ~.
</math>
因此
:<math>
  \cfrac{\mathrm{d}}{\mathrm{d}t}\left( \int_{\Omega(t)} \mathbf{f}(\mathbf{x},t)~\text{dV}\right) = 
     \int_{\Omega(t)} 
       \left(
         \frac{\partial \mathbf{f}(\mathbf{x},t)}{\partial t} + [\boldsymbol{\nabla} \mathbf{f}(\mathbf{x},t)]\cdot\mathbf{v}(\mathbf{x},t) +
         \mathbf{f}(\mathbf{x},t)~\boldsymbol{\nabla} \cdot \mathbf{v}(\mathbf{x},t)\right)~\text{dV} 
</math>
或者
:<math>
  \cfrac{\mathrm{d}}{\mathrm{d}t}\left( \int_{\Omega(t)} \mathbf{f}~\text{dV}\right) = 
     \int_{\Omega(t)} 
       \left(
         \frac{\partial \mathbf{f}}{\partial t} + \boldsymbol{\nabla} \mathbf{f}\cdot\mathbf{v} +
         \mathbf{f}~\boldsymbol{\nabla} \cdot \mathbf{v}\right)~\text{dV} ~.
</math>
利用以下的恆等式
:<math>
   \boldsymbol{\nabla} \cdot (\mathbf{v}\otimes\mathbf{w}) = \mathbf{v}(\boldsymbol{\nabla} \cdot \mathbf{w}) + \boldsymbol{\nabla}\mathbf{v}\cdot\mathbf{w} 
</math>
可得
:<math>
  \cfrac{\mathrm{d}}{\mathrm{d}t}\left( \int_{\Omega(t)} \mathbf{f}~\text{dV}\right) = 
     \int_{\Omega(t)} 
       \left(\frac{\partial \mathbf{f}}{\partial t} + \boldsymbol{\nabla} \cdot (\mathbf{f}\otimes\mathbf{v})\right)~\text{dV} ~.
</math>
利用[[高斯散度定理]]及恆等式 
<math>(\mathbf{a}\otimes\mathbf{b})\cdot\mathbf{n} = (\mathbf{b}\cdot\mathbf{n})\mathbf{a}</math>，可得
:<math>
  {
  \cfrac{\mathrm{d}}{\mathrm{d}t}\left( \int_{\Omega(t)} \mathbf{f}~\text{dV}\right) = 
     \int_{\Omega(t)}\frac{\partial \mathbf{f}}{\partial t}~\text{dV} + 
     \int_{\partial \Omega(t)}(\mathbf{f}\otimes\mathbf{v})\cdot\mathbf{n}~\text{dA}
     = \int_{\Omega(t)}\frac{\partial \mathbf{f}}{\partial t}~\text{dV} + 
     \int_{\partial \Omega(t)}(\mathbf{v}\cdot\mathbf{n})\mathbf{f}~\text{dA} \qquad \square
  }
</math>
|}

== 錯誤的引用 ==
此定理常被錯誤的引用為只針對物质体积（material volume）的形式，若將只針對物质体积應用於物质体积以外的區域中，就會出現問題。

==特別形式==
若<math>\Omega</math>不隨時間改變，則<math>\mathbf{v}_b=0</math>，且恆等式化簡為以下的形式 
:<math>
  \cfrac{\mathrm{d}}{\mathrm{d}t}\int_{\Omega} f~\text{dV} = 
     \int_{\Omega} \frac{\partial f}{\partial t}~\text{dV} ~,
</math>
不過若用了不正確的雷諾傳輸定理，無法進行上述的簡化。

===在一維下的詮釋及簡化===
此定理是[[積分符號內取微分]]的高維延伸，有些情形下可以簡化為積分符號內取微分。假設<math>f</math>和<math>y</math>和<math>z</math>無關，且<math>\Omega(t)</math>為<math>y-z</math>平面的單位方塊，且有<math>a(t)</math>及<math>b(t)</math>的極限，雷諾傳輸定理會簡化為
:<math>
  \cfrac{\mathrm{d}}{\mathrm{d}t}\int_{a(t)}^{b(t)} f~\text{dx} = 
     \int_{a(t)}^{b(t)} \frac{\partial f}{\partial t}~\text{dx} + 
\frac{\partial b(t)}{\partial t} f(b(t),t)
-\frac{\partial a(t)}{\partial t} f(a(t),t) ~,
</math>
上述是由[[積分符號內取微分]]來的表示式，但x及t變數已經對調。

==相關條目==
*[[積分符號內取微分]]
*{{link-en|萊布尼茲積分律|Leibniz integral rule}}

==腳註==
{{reflist|2}}

==參考資料==
*L. G. Leal, 2007, ''Advanced transport phenomena: fluid mechanics and convective transport processes'', Cambridge University Press, p.&nbsp;912.
*O. Reynolds, 1903, ''Papers on Mechanical and Physical Subjects'', Vol. 3, The Sub-Mechanics of the Universe, Cambridge University Press, Cambridge.
*J. E. Marsden and A. Tromba, 2003, ''Vector Calculus'', 5th ed., W. H. Freeman .

==外部連結==
* Osborne Reynolds, Collected Papers on Mechanical and Physical Subjects, in three volumes, published circa 1903, now fully and freely 

available in digital format:[http://www.archive.org/details/papersonmechanic01reynrich Volume 1], [http://www.archive.org/details/papersonmechanic02reynrich Volume 2], [http://www.archive.org/details/papersonmechanic03reynrich Volume 3],
* https://web.archive.org/web/20080327180821/http://www.catea.org/grade/mecheng/mod6/mod6.html#slide1
* http://planetmath.org/reynoldstransporttheorem {{Wayback|url=http://planetmath.org/reynoldstransporttheorem |date=20150402132050 }}

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