雷諾傳輸定理

雷諾傳輸定理也稱為萊布尼茲-雷諾傳輸定理或雷諾輸運定理，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${\displaystyle \Omega (t)}$積分${\displaystyle \mathbf {f} =\mathbf {f} (\mathbf {x} ,t)}$，其邊界為${\displaystyle \partial \Omega (t)}$，考慮上式對時間的微分：

${\displaystyle {\cfrac {\mathrm {d} }{\mathrm {d} t}}\int _{\Omega (t)}\mathbf {f} ~{\text{dV}}~.}$

若要求上述積分的導數，會有兩個問題，${\displaystyle \mathbf {f} }$的時間相依性，及因${\displaystyle \Omega }$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^[1][2][3]是：

${\displaystyle {\cfrac {\mathrm {d} }{\mathrm {d} t}}\int _{\Omega (t)}\mathbf {f} ~{\text{dV}}=\int _{\Omega (t)}{\frac {\partial \mathbf {f} }{\partial t}}~{\text{dV}}+\int _{\partial \Omega (t)}(\mathbf {v} _{b}\cdot \mathbf {n} )\mathbf {f} ~{\text{dA}}~}$

其中${\displaystyle \mathbf {n} (\mathbf {x} ,t)}$為向外的單位法向量，${\displaystyle \mathbf {x} }$為區域中的一點，也是積分變數，${\displaystyle {\text{dV}}}$ 及${\displaystyle {\text{dA}}}$是位於${\displaystyle \mathbf {x} }$的體積元素及表面元素，${\displaystyle \mathbf {v} _{b}(\mathbf {x} ,t)}$為面積元素的速度而非流速。函數${\displaystyle \mathbf {f} }$可以是張量、向量或純量函數^[4]。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${\displaystyle \Omega (t)}$為一流體塊，則存在速度函數${\displaystyle \mathbf {v} =\mathbf {v} (\mathbf {x} ,t)}$及邊界元素符合下式

${\displaystyle \mathbf {v} ^{b}\cdot \mathbf {n} =\mathbf {v} \cdot \mathbf {n} .}$

上式在替代後，可以得到以下的定理^[5]

${\displaystyle {\cfrac {\mathrm {d} }{\mathrm {d} t}}\left(\int _{\Omega (t)}\mathbf {f} ~{\text{dV}}\right)=\int _{\Omega (t)}{\frac {\partial \mathbf {f} }{\partial t}}~{\text{dV}}+\int _{\partial \Omega (t)}(\mathbf {v} \cdot \mathbf {n} )\mathbf {f} ~{\text{dA}}~.}$

針對一流體塊的證明

令${\displaystyle \Omega _{0}}$為區域${\displaystyle \Omega (t)}$的參考組態，令其運動及形變梯度為 ${\displaystyle \mathbf {x} ={\boldsymbol {\varphi }}(\mathbf {X} ,t)~;\qquad \implies \qquad {\boldsymbol {F}}(\mathbf {X} ,t)={\boldsymbol {\nabla }}_{\circ }{\boldsymbol {\varphi }}~.}$ 令${\displaystyle J(\mathbf {X} ,t)=\det[{\boldsymbol {F}}(\mathbf {X} ,t)]}$. 則目前組態及參考組態的積分有以下的關係 ${\displaystyle \int _{\Omega (t)}\mathbf {f} (\mathbf {x} ,t)~{\text{dV}}=\int _{\Omega _{0}}\mathbf {f} [{\boldsymbol {\varphi }}(\mathbf {X} ,t),t]~J(\mathbf {X} ,t)~{\text{dV}}_{0}=\int _{\Omega _{0}}{\hat {\mathbf {f} }}(\mathbf {X} ,t)~J(\mathbf {X} ,t)~{\text{dV}}_{0}~.}$ 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 ${\displaystyle {\cfrac {\mathrm {d} }{\mathrm {d} t}}\left(\int _{\Omega (t)}\mathbf {f} (\mathbf {x} ,t)~{\text{dV}}\right)=\lim _{\Delta t\rightarrow 0}{\cfrac {1}{\Delta t}}\left(\int _{\Omega (t+\Delta t)}\mathbf {f} (\mathbf {x} ,t+\Delta t)~{\text{dV}}-\int _{\Omega (t)}\mathbf {f} (\mathbf {x} ,t)~{\text{dV}}\right)~.}$ 將上式轉換為對參考組態的積分，可得 ${\displaystyle {\cfrac {\mathrm {d} }{\mathrm {d} t}}\left(\int _{\Omega (t)}\mathbf {f} (\mathbf {x} ,t)~{\text{dV}}\right)=\lim _{\Delta t\rightarrow 0}{\cfrac {1}{\Delta t}}\left(\int _{\Omega _{0}}{\hat {\mathbf {f} }}(\mathbf {X} ,t+\Delta t)~J(\mathbf {X} ,t+\Delta t)~{\text{dV}}_{0}-\int _{\Omega _{0}}{\hat {\mathbf {f} }}(\mathbf {X} ,t)~J(\mathbf {X} ,t)~{\text{dV}}_{0}\right)~.}$ 因為${\displaystyle \Omega _{0}}$和時間無關，可得 ${\displaystyle {\begin{aligned}{\cfrac {\mathrm {d} }{\mathrm {d} t}}\left(\int _{\Omega (t)}\mathbf {f} (\mathbf {x} ,t)~{\text{dV}}\right)&=\int _{\Omega _{0}}\left[\lim _{\Delta t\rightarrow 0}{\cfrac {{\hat {\mathbf {f} }}(\mathbf {X} ,t+\Delta t)~J(\mathbf {X} ,t+\Delta t)-{\hat {\mathbf {f} }}(\mathbf {X} ,t)~J(\mathbf {X} ,t)}{\Delta t}}\right]~{\text{dV}}_{0}\\&=\int _{\Omega _{0}}{\frac {\partial }{\partial t}}[{\hat {\mathbf {f} }}(\mathbf {X} ,t)~J(\mathbf {X} ,t)]~{\text{dV}}_{0}\\&=\int _{\Omega _{0}}\left({\frac {\partial }{\partial t}}[{\hat {\mathbf {f} }}(\mathbf {X} ,t)]~J(\mathbf {X} ,t)+{\hat {\mathbf {f} }}(\mathbf {X} ,t)~{\frac {\partial }{\partial t}}[J(\mathbf {X} ,t)]\right)~{\text{dV}}_{0}\end{aligned}}}$ 現在，${\displaystyle \det {\boldsymbol {F}}}$的時間導數為 [6] ${\displaystyle {\frac {\partial J(\mathbf {X} ,t)}{\partial t}}={\frac {\partial }{\partial t}}(\det {\boldsymbol {F}})=(\det {\boldsymbol {F}})({\boldsymbol {\nabla }}\cdot \mathbf {v} )=J(\mathbf {X} ,t)~{\boldsymbol {\nabla }}\cdot \mathbf {v} ({\boldsymbol {\varphi }}(\mathbf {X} ,t),t)=J(\mathbf {X} ,t)~{\boldsymbol {\nabla }}\cdot \mathbf {v} (\mathbf {x} ,t)~.}$ 因此 ${\displaystyle {\begin{aligned}{\cfrac {\mathrm {d} }{\mathrm {d} t}}\left(\int _{\Omega (t)}\mathbf {f} (\mathbf {x} ,t)~{\text{dV}}\right)&=\int _{\Omega _{0}}\left({\frac {\partial }{\partial t}}[{\hat {\mathbf {f} }}(\mathbf {X} ,t)]~J(\mathbf {X} ,t)+{\hat {\mathbf {f} }}(\mathbf {X} ,t)~J(\mathbf {X} ,t)~{\boldsymbol {\nabla }}\cdot \mathbf {v} (\mathbf {x} ,t)\right)~{\text{dV}}_{0}\\&=\int _{\Omega _{0}}\left({\frac {\partial }{\partial t}}[{\hat {\mathbf {f} }}(\mathbf {X} ,t)]+{\hat {\mathbf {f} }}(\mathbf {X} ,t)~{\boldsymbol {\nabla }}\cdot \mathbf {v} (\mathbf {x} ,t)\right)~J(\mathbf {X} ,t)~{\text{dV}}_{0}\\&=\int _{\Omega (t)}\left({\dot {\mathbf {f} }}(\mathbf {x} ,t)+\mathbf {f} (\mathbf {x} ,t)~{\boldsymbol {\nabla }}\cdot \mathbf {v} (\mathbf {x} ,t)\right)~{\text{dV}}\end{aligned}}}$ 其中${\displaystyle {\dot {\mathbf {f} }}}$為${\displaystyle \mathbf {f} }$的材料導數（英語：Material derivative），現在材料導數為 ${\displaystyle {\dot {\mathbf {f} }}(\mathbf {x} ,t)={\frac {\partial \mathbf {f} (\mathbf {x} ,t)}{\partial t}}+[{\boldsymbol {\nabla }}\mathbf {f} (\mathbf {x} ,t)]\cdot \mathbf {v} (\mathbf {x} ,t)~.}$ 因此 ${\displaystyle {\cfrac {\mathrm {d} }{\mathrm {d} t}}\left(\int _{\Omega (t)}\mathbf {f} (\mathbf {x} ,t)~{\text{dV}}\right)=\int _{\Omega (t)}\left({\frac {\partial \mathbf {f} (\mathbf {x} ,t)}{\partial t}}+[{\boldsymbol {\nabla }}\mathbf {f} (\mathbf {x} ,t)]\cdot \mathbf {v} (\mathbf {x} ,t)+\mathbf {f} (\mathbf {x} ,t)~{\boldsymbol {\nabla }}\cdot \mathbf {v} (\mathbf {x} ,t)\right)~{\text{dV}}}$ 或者 ${\displaystyle {\cfrac {\mathrm {d} }{\mathrm {d} t}}\left(\int _{\Omega (t)}\mathbf {f} ~{\text{dV}}\right)=\int _{\Omega (t)}\left({\frac {\partial \mathbf {f} }{\partial t}}+{\boldsymbol {\nabla }}\mathbf {f} \cdot \mathbf {v} +\mathbf {f} ~{\boldsymbol {\nabla }}\cdot \mathbf {v} \right)~{\text{dV}}~.}$ 利用以下的恆等式 ${\displaystyle {\boldsymbol {\nabla }}\cdot (\mathbf {v} \otimes \mathbf {w} )=\mathbf {v} ({\boldsymbol {\nabla }}\cdot \mathbf {w} )+{\boldsymbol {\nabla }}\mathbf {v} \cdot \mathbf {w} }$ 可得 ${\displaystyle {\cfrac {\mathrm {d} }{\mathrm {d} t}}\left(\int _{\Omega (t)}\mathbf {f} ~{\text{dV}}\right)=\int _{\Omega (t)}\left({\frac {\partial \mathbf {f} }{\partial t}}+{\boldsymbol {\nabla }}\cdot (\mathbf {f} \otimes \mathbf {v} )\right)~{\text{dV}}~.}$ 利用高斯散度定理及恆等式 ${\displaystyle (\mathbf {a} \otimes \mathbf {b} )\cdot \mathbf {n} =(\mathbf {b} \cdot \mathbf {n} )\mathbf {a} }$，可得 ${\displaystyle {{\cfrac {\mathrm {d} }{\mathrm {d} t}}\left(\int _{\Omega (t)}\mathbf {f} ~{\text{dV}}\right)=\int _{\Omega (t)}{\frac {\partial \mathbf {f} }{\partial t}}~{\text{dV}}+\int _{\partial \Omega (t)}(\mathbf {f} \otimes \mathbf {v} )\cdot \mathbf {n} ~{\text{dA}}=\int _{\Omega (t)}{\frac {\partial \mathbf {f} }{\partial t}}~{\text{dV}}+\int _{\partial \Omega (t)}(\mathbf {v} \cdot \mathbf {n} )\mathbf {f} ~{\text{dA}}\qquad \square }}$

錯誤的引用

此定理常被錯誤的引用為只針對物質體積（material volume）的形式，若將只針對物質體積應用於物質體積以外的區域中，就會出現問題。

特別形式

若${\displaystyle \Omega }$不隨時間改變，則${\displaystyle \mathbf {v} _{b}=0}$，且恆等式化簡為以下的形式

${\displaystyle {\cfrac {\mathrm {d} }{\mathrm {d} t}}\int _{\Omega }f~{\text{dV}}=\int _{\Omega }{\frac {\partial f}{\partial t}}~{\text{dV}}~,}$

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${\displaystyle f}$和${\displaystyle y}$和${\displaystyle z}$無關，且${\displaystyle \Omega (t)}$為${\displaystyle y-z}$平面的單位方塊，且有${\displaystyle a(t)}$及${\displaystyle b(t)}$的極限，雷諾傳輸定理會簡化為

${\displaystyle {\cfrac {\mathrm {d} }{\mathrm {d} t}}\int _{a(t)}^{b(t)}f~{\text{dx}}=\int _{a(t)}^{b(t)}{\frac {\partial f}{\partial t}}~{\text{dx}}+{\frac {\partial b(t)}{\partial t}}f(b(t),t)-{\frac {\partial a(t)}{\partial t}}f(a(t),t)~,}$

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
• 萊布尼茲積分律（英語：Leibniz integral rule）

腳註

1. L. Gary Leal, 2007, p. 23.
2. O. Reynolds, 1903, Vol. 3, p. 12–13
3. J.E. Marsden and A. Tromba, 5th ed. 2003
4. H. Yamaguchi, Engineering Fluid Mechanics, Springer c2008 p23
5. T. Belytschko, W. K. Liu, and B. Moran, 2000, Nonlinear Finite Elements for Continua and Structures, John Wiley and Sons, Ltd., New York.
6. Gurtin M. E., 1981, An Introduction to Continuum Mechanics. Academic Press, New York, p. 77.

參考資料

• L. G. Leal, 2007, Advanced transport phenomena: fluid mechanics and convective transport processes, Cambridge University Press, p. 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:Volume 1, Volume 2, Volume 3,

• https://web.archive.org/web/20080327180821/http://www.catea.org/grade/mecheng/mod6/mod6.html#slide1
• http://planetmath.org/reynoldstransporttheorem （頁面存檔備份，存於互聯網檔案館）