雷诺传输定理

要推導的雷諾傳輸定理是：

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

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${\displaystyle \mathbf {x} }$內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，不一定要是流速。函數${\displaystyle \mathbf {f} }$可以是張量、向量或純量函數。注意等式左邊的積分只是時間的函數，因此可以用全微分。

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

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

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${\displaystyle J(\mathbf {X} ,t)=\det}$. 則目前組態及參考組態的積分有以下的關係 ${\displaystyle \int _{\Omega (t)}\mathbf {f} (\mathbf {x} ,t)~{\text{dV}}=\int _{\Omega _{0}}\mathbf {f} ~J(\mathbf {X} ,t)~{\text{dV}}_{0}=\int _{\Omega _{0}}{\hat {\mathbf {f} }}(\mathbf {X} ,t)~J(\mathbf {X} ,t)~{\text{dV}}_{0}~.}$ 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 \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 ${\displaystyle {\cfrac {\mathrm {d} }{\mathrm {d} t}}\left(\int _{\Omega (t)}\mathbf {f} (\mathbf {x} ,t)~{\text{dV}}\right)=\lim _{\Delta t\rightarrow 0}{\cfrac {1}{\Delta t}}\left(\int _{\Omega _{0}}{\hat {\mathbf {f} }}(\mathbf {X} ,t+\Delta t)~J(\mathbf {X} ,t+\Delta t)~{\text{dV}}_{0}-\int _{\Omega _{0}}{\hat {\mathbf {f} }}(\mathbf {X} ,t)~J(\mathbf {X} ,t)~{\text{dV}}_{0}\right)~.}$ 因為${\displaystyle \Omega _{0}}$和時間無關，可得 ${\displaystyle {\begin{aligned}{\cfrac {\mathrm {d} }{\mathrm {d} t}}\left(\int _{\Omega (t)}\mathbf {f} (\mathbf {x} ,t)~{\text{dV}}\right)&=\int _{\Omega _{0}}\left~{\text{dV}}_{0}\\&=\int _{\Omega _{0}}{\frac {\partial }{\partial t}}~{\text{dV}}_{0}\\&=\int _{\Omega _{0}}\left({\frac {\partial }{\partial t}}~J(\mathbf {X} ,t)+{\hat {\mathbf {f} }}(\mathbf {X} ,t)~{\frac {\partial }{\partial t}}\right)~{\text{dV}}_{0}\end{aligned}}}$ 現在，${\displaystyle \det {\boldsymbol {F}}}$的時間導數為 ${\displaystyle {\frac {\partial J(\mathbf {X} ,t)}{\partial t}}={\frac {\partial }{\partial t}}(\det {\boldsymbol {F}})=(\det {\boldsymbol {F}})({\boldsymbol {\nabla }}\cdot \mathbf {v} )=J(\mathbf {X} ,t)~{\boldsymbol {\nabla }}\cdot \mathbf {v} ({\boldsymbol {\varphi }}(\mathbf {X} ,t),t)=J(\mathbf {X} ,t)~{\boldsymbol {\nabla }}\cdot \mathbf {v} (\mathbf {x} ,t)~.}$ 因此 ${\displaystyle {\begin{aligned}{\cfrac {\mathrm {d} }{\mathrm {d} t}}\left(\int _{\Omega (t)}\mathbf {f} (\mathbf {x} ,t)~{\text{dV}}\right)&=\int _{\Omega _{0}}\left({\frac {\partial }{\partial t}}~J(\mathbf {X} ,t)+{\hat {\mathbf {f} }}(\mathbf {X} ,t)~J(\mathbf {X} ,t)~{\boldsymbol {\nabla }}\cdot \mathbf {v} (\mathbf {x} ,t)\right)~{\text{dV}}_{0}\\&=\int _{\Omega _{0}}\left({\frac {\partial }{\partial t}}+{\hat {\mathbf {f} }}(\mathbf {X} ,t)~{\boldsymbol {\nabla }}\cdot \mathbf {v} (\mathbf {x} ,t)\right)~J(\mathbf {X} ,t)~{\text{dV}}_{0}\\&=\int _{\Omega (t)}\left({\dot {\mathbf {f} }}(\mathbf {x} ,t)+\mathbf {f} (\mathbf {x} ,t)~{\boldsymbol {\nabla }}\cdot \mathbf {v} (\mathbf {x} ,t)\right)~{\text{dV}}\end{aligned}}}$ 其中${\displaystyle {\dot {\mathbf {f} }}}$為${\displaystyle \mathbf {f} }$的材料導數，現在材料導數為 ${\displaystyle {\dot {\mathbf {f} }}(\mathbf {x} ,t)={\frac {\partial \mathbf {f} (\mathbf {x} ,t)}{\partial t}}+\cdot \mathbf {v} (\mathbf {x} ,t)~.}$ 因此 ${\displaystyle {\cfrac {\mathrm {d} }{\mathrm {d} t}}\left(\int _{\Omega (t)}\mathbf {f} (\mathbf {x} ,t)~{\text{dV}}\right)=\int _{\Omega (t)}\left({\frac {\partial \mathbf {f} (\mathbf {x} ,t)}{\partial t}}+\cdot \mathbf {v} (\mathbf {x} ,t)+\mathbf {f} (\mathbf {x} ,t)~{\boldsymbol {\nabla }}\cdot \mathbf {v} (\mathbf {x} ,t)\right)~{\text{dV}}}$ 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 ${\displaystyle {{\cfrac {\mathrm {d} }{\mathrm {d} t}}\left(\int _{\Omega (t)}\mathbf {f} ~{\text{dV}}\right)=\int _{\Omega (t)}{\frac {\partial \mathbf {f} }{\partial t}}~{\text{dV}}+\int _{\partial \Omega (t)}(\mathbf {f} \otimes \mathbf {v} )\cdot \mathbf {n} ~{\text{dA}}=\int _{\Omega (t)}{\frac {\partial \mathbf {f} }{\partial t}}~{\text{dV}}+\int _{\partial \Omega (t)}(\mathbf {v} \cdot \mathbf {n} )\mathbf {f} ~{\text{dA}}\qquad \square }}$

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。

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。

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。

• 積分符號內取微分
• 萊布尼茲積分律

• 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 .

• 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 （页面存档备份，存于）