# 雷諾傳輸定理

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

## 通用型式

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

## 針對流體塊的形式

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

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

## 特別形式

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

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

### 在一維下的詮釋及簡化

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

## 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,