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

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

若要求上述积分的导数，会有两个问题，${\displaystyle \mathbf {f} }$的时间相依性，及因${\displaystyle \Omega }$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^[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${\displaystyle \mathbf {n} (\mathbf {x} ,t)}$为向外的单位法向量，${\displaystyle \mathbf {x} }$为区域中的一点，也是积分变数，${\displaystyle {\text{dV}}}$ 及${\displaystyle {\text{dA}}}$是位于${\displaystyle \mathbf {x} }$的体积元素及表面元素，${\displaystyle \mathbf {v} _{b}(\mathbf {x} ,t)}$为面积元素的速度而非流速。函数${\displaystyle \mathbf {f} }$可以是张量、向量或标量函数^[4]。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${\displaystyle \Omega (t)}$为一流体块，则存在速度函数${\displaystyle \mathbf {v} =\mathbf {v} (\mathbf {x} ,t)}$及边界元素符合下式

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

上式在替代后，可以得到以下的定理^[5]

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

针对一流体块的证明

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

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

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

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${\displaystyle f}$和${\displaystyle y}$和${\displaystyle z}$无关，且${\displaystyle \Omega (t)}$为${\displaystyle y-z}$平面的单位方块，且有${\displaystyle a(t)}$及${\displaystyle b(t)}$的极限，雷诺传输定理会简化为

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

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（英语：Leibniz integral rule）