雷诺传输定理

雷诺传输定理也称为莱布尼兹-雷诺传输定理或雷诺输运定理，是以积分符号内取微分闻名的莱布尼兹积分的三维推广。

雷诺传输定理得名自奥斯鲍恩·雷诺（1842–1912），用来调整积分量的微分，用来推导连续介质力学的基础方程。

考虑在时变的区域$\Omega (t)$积分$\mathbf {f} =\mathbf {f} (\mathbf {x} ,t)$，其边界为$\partial \Omega (t)$，考虑上式对时间的微分：

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

若要求上述积分的导数，会有两个问题，$\mathbf {f}$的时间相依性，及因$\Omega$动态的边界而增加或减少的空间，雷诺传输定理提供了必要的框架。

通用型式

雷诺传输定理可表为以下形式是：

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

其中$\mathbf {n} (\mathbf {x} ,t)$为向外的单位法向量，$\mathbf {x}$为区域中的一点，也是积分变数，${\text{dV}}$ 及${\text{dA}}$是位于$\mathbf {x}$的体积元素及表面元素，$\mathbf {v} _{b}(\mathbf {x} ,t)$为面积元素的速度而非流速。函数$\mathbf {f}$可以是张量、向量或标量函数。注意等式左边的积分只是时间的函数，所以采用全微分符号。

针对流体块的形式

在连续介质力学中，此定理常用在没有物质进来或离开的流体块或固体中。若$\Omega (t)$为一流体块，则存在速度函数$\mathbf {v} =\mathbf {v} (\mathbf {x} ,t)$及边界元素符合下式

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

上式在替代后，可以得到以下的定理

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

针对一流体块的证明

令$\Omega _{0}$为区域$\Omega (t)$的参考组态，令其运动及形变梯度为 $\mathbf {x} ={\boldsymbol {\varphi }}(\mathbf {X} ,t)~;\qquad \implies \qquad {\boldsymbol {F}}(\mathbf {X} ,t)={\boldsymbol {\nabla }}_{\circ }{\boldsymbol {\varphi }}~.$ 令$J(\mathbf {X} ,t)=\det[{\boldsymbol {F}}(\mathbf {X} ,t)]$. 则目前组态及参考组态的积分有以下的关系 $\int _{\Omega (t)}\mathbf {f} (\mathbf {x} ,t)~{\text{dV}}=\int _{\Omega _{0}}\mathbf {f} [{\boldsymbol {\varphi }}(\mathbf {X} ,t),t]~J(\mathbf {X} ,t)~{\text{dV}}_{0}=\int _{\Omega _{0}}{\hat {\mathbf {f} }}(\mathbf {X} ,t)~J(\mathbf {X} ,t)~{\text{dV}}_{0}~.$ That this derivation is for a material element is implicit in the time constancy of the reference configuration: it is constant in material coordinates. 针对体积积分的微分定义为 ${\cfrac {\mathrm {d} }{\mathrm {d} t}}\left(\int _{\Omega (t)}\mathbf {f} (\mathbf {x} ,t)~{\text{dV}}\right)=\lim _{\Delta t\rightarrow 0}{\cfrac {1}{\Delta t}}\left(\int _{\Omega (t+\Delta t)}\mathbf {f} (\mathbf {x} ,t+\Delta t)~{\text{dV}}-\int _{\Omega (t)}\mathbf {f} (\mathbf {x} ,t)~{\text{dV}}\right)~.$ 将上式转换为对参考组态的积分，可得 ${\cfrac {\mathrm {d} }{\mathrm {d} t}}\left(\int _{\Omega (t)}\mathbf {f} (\mathbf {x} ,t)~{\text{dV}}\right)=\lim _{\Delta t\rightarrow 0}{\cfrac {1}{\Delta t}}\left(\int _{\Omega _{0}}{\hat {\mathbf {f} }}(\mathbf {X} ,t+\Delta t)~J(\mathbf {X} ,t+\Delta t)~{\text{dV}}_{0}-\int _{\Omega _{0}}{\hat {\mathbf {f} }}(\mathbf {X} ,t)~J(\mathbf {X} ,t)~{\text{dV}}_{0}\right)~.$ 因为$\Omega _{0}$和时间无关，可得 ${\begin{aligned}{\cfrac {\mathrm {d} }{\mathrm {d} t}}\left(\int _{\Omega (t)}\mathbf {f} (\mathbf {x} ,t)~{\text{dV}}\right)&=\int _{\Omega _{0}}\left[\lim _{\Delta t\rightarrow 0}{\cfrac {{\hat {\mathbf {f} }}(\mathbf {X} ,t+\Delta t)~J(\mathbf {X} ,t+\Delta t)-{\hat {\mathbf {f} }}(\mathbf {X} ,t)~J(\mathbf {X} ,t)}{\Delta t}}\right]~{\text{dV}}_{0}\\&=\int _{\Omega _{0}}{\frac {\partial }{\partial t}}[{\hat {\mathbf {f} }}(\mathbf {X} ,t)~J(\mathbf {X} ,t)]~{\text{dV}}_{0}\\&=\int _{\Omega _{0}}\left({\frac {\partial }{\partial t}}[{\hat {\mathbf {f} }}(\mathbf {X} ,t)]~J(\mathbf {X} ,t)+{\hat {\mathbf {f} }}(\mathbf {X} ,t)~{\frac {\partial }{\partial t}}[J(\mathbf {X} ,t)]\right)~{\text{dV}}_{0}\end{aligned}}$ 现在，$\det {\boldsymbol {F}}$的时间导数为 ${\frac {\partial J(\mathbf {X} ,t)}{\partial t}}={\frac {\partial }{\partial t}}(\det {\boldsymbol {F}})=(\det {\boldsymbol {F}})({\boldsymbol {\nabla }}\cdot \mathbf {v} )=J(\mathbf {X} ,t)~{\boldsymbol {\nabla }}\cdot \mathbf {v} ({\boldsymbol {\varphi }}(\mathbf {X} ,t),t)=J(\mathbf {X} ,t)~{\boldsymbol {\nabla }}\cdot \mathbf {v} (\mathbf {x} ,t)~.$ 因此 ${\begin{aligned}{\cfrac {\mathrm {d} }{\mathrm {d} t}}\left(\int _{\Omega (t)}\mathbf {f} (\mathbf {x} ,t)~{\text{dV}}\right)&=\int _{\Omega _{0}}\left({\frac {\partial }{\partial t}}[{\hat {\mathbf {f} }}(\mathbf {X} ,t)]~J(\mathbf {X} ,t)+{\hat {\mathbf {f} }}(\mathbf {X} ,t)~J(\mathbf {X} ,t)~{\boldsymbol {\nabla }}\cdot \mathbf {v} (\mathbf {x} ,t)\right)~{\text{dV}}_{0}\\&=\int _{\Omega _{0}}\left({\frac {\partial }{\partial t}}[{\hat {\mathbf {f} }}(\mathbf {X} ,t)]+{\hat {\mathbf {f} }}(\mathbf {X} ,t)~{\boldsymbol {\nabla }}\cdot \mathbf {v} (\mathbf {x} ,t)\right)~J(\mathbf {X} ,t)~{\text{dV}}_{0}\\&=\int _{\Omega (t)}\left({\dot {\mathbf {f} }}(\mathbf {x} ,t)+\mathbf {f} (\mathbf {x} ,t)~{\boldsymbol {\nabla }}\cdot \mathbf {v} (\mathbf {x} ,t)\right)~{\text{dV}}\end{aligned}}$ 其中${\dot {\mathbf {f} }}$为$\mathbf {f}$的材料导数，现在材料导数为 ${\dot {\mathbf {f} }}(\mathbf {x} ,t)={\frac {\partial \mathbf {f} (\mathbf {x} ,t)}{\partial t}}+[{\boldsymbol {\nabla }}\mathbf {f} (\mathbf {x} ,t)]\cdot \mathbf {v} (\mathbf {x} ,t)~.$ 因此 ${\cfrac {\mathrm {d} }{\mathrm {d} t}}\left(\int _{\Omega (t)}\mathbf {f} (\mathbf {x} ,t)~{\text{dV}}\right)=\int _{\Omega (t)}\left({\frac {\partial \mathbf {f} (\mathbf {x} ,t)}{\partial t}}+[{\boldsymbol {\nabla }}\mathbf {f} (\mathbf {x} ,t)]\cdot \mathbf {v} (\mathbf {x} ,t)+\mathbf {f} (\mathbf {x} ,t)~{\boldsymbol {\nabla }}\cdot \mathbf {v} (\mathbf {x} ,t)\right)~{\text{dV}}$ 或者 ${\cfrac {\mathrm {d} }{\mathrm {d} t}}\left(\int _{\Omega (t)}\mathbf {f} ~{\text{dV}}\right)=\int _{\Omega (t)}\left({\frac {\partial \mathbf {f} }{\partial t}}+{\boldsymbol {\nabla }}\mathbf {f} \cdot \mathbf {v} +\mathbf {f} ~{\boldsymbol {\nabla }}\cdot \mathbf {v} \right)~{\text{dV}}~.$ 利用以下的恒等式 ${\boldsymbol {\nabla }}\cdot (\mathbf {v} \otimes \mathbf {w} )=\mathbf {v} ({\boldsymbol {\nabla }}\cdot \mathbf {w} )+{\boldsymbol {\nabla }}\mathbf {v} \cdot \mathbf {w}$ 可得 ${\cfrac {\mathrm {d} }{\mathrm {d} t}}\left(\int _{\Omega (t)}\mathbf {f} ~{\text{dV}}\right)=\int _{\Omega (t)}\left({\frac {\partial \mathbf {f} }{\partial t}}+{\boldsymbol {\nabla }}\cdot (\mathbf {f} \otimes \mathbf {v} )\right)~{\text{dV}}~.$ 利用高斯散度定理及恒等式 $(\mathbf {a} \otimes \mathbf {b} )\cdot \mathbf {n} =(\mathbf {b} \cdot \mathbf {n} )\mathbf {a}$，可得 ${{\cfrac {\mathrm {d} }{\mathrm {d} t}}\left(\int _{\Omega (t)}\mathbf {f} ~{\text{dV}}\right)=\int _{\Omega (t)}{\frac {\partial \mathbf {f} }{\partial t}}~{\text{dV}}+\int _{\partial \Omega (t)}(\mathbf {f} \otimes \mathbf {v} )\cdot \mathbf {n} ~{\text{dA}}=\int _{\Omega (t)}{\frac {\partial \mathbf {f} }{\partial t}}~{\text{dV}}+\int _{\partial \Omega (t)}(\mathbf {v} \cdot \mathbf {n} )\mathbf {f} ~{\text{dA}}\qquad \square }$

错误的引用

此定理常被错误的引用为只针对物质体积（material volume）的形式，若将只针对物质体积应用于物质体积以外的区域中，就会出现问题。

特别形式

若$\Omega$不随时间改变，则$\mathbf {v} _{b}=0$，且恒等式化简为以下的形式

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

不过若用了不正确的雷诺传输定理，无法进行上述的简化。

在一维下的诠释及简化

此定理是积分符号内取微分的高维延伸，有些情形下可以简化为积分符号内取微分。假设$f$和$y$和$z$无关，且$\Omega (t)$为$y-z$平面的单位方块，且有$a(t)$及$b(t)$的极限，雷诺传输定理会简化为

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

上述是由积分符号内取微分来的表示式，但x及t变数已经对调。

相关条目

• 积分符号内取微分
• 莱布尼兹积分律

参考资料

• 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 .

外部链接

• 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，存于互联网档案馆）