雷诺传输定理

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 Ω (t) 积分 f =f (x,t)，其边界为 ∂ Ω (t)，考虑上式对时间的微分： （公式略）

若要求上述积分的导数，会有两个问题， f 的时间相依性，及因 Ω 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： （公式略）

其中 n (x,t) 为向外的单位法向量， x 为区域中的一点，也是积分变数， dV 及 dA 是位于 x 的体积元素及表面元素， v _(b)(x,t) 为面积元素的速度而非流速。函数 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 Ω (t) 为一流体块，则存在速度函数 v =v (x,t) 及边界元素符合下式 v ^(b)· n =v · n.

上式在替代后，可以得到以下的定理 （公式略）

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 Ω 不随时间改变，则 v _(b)=0，且恒等式化简为以下的形式 （公式略）

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 Ω (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 （公式略）

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。

In differential calculus, the Reynolds transport theorem (also known as the Leibniz–Reynolds transport theorem), or simply the Reynolds theorem, named after Osborne Reynolds (1842–1912), is a three-dimensional generalization of the Leibniz integral rule. It is used to recast time derivatives of integrated quantities and is useful in formulating the basic equations of continuum mechanics.

Consider integrating f = f(x,t) over the time-dependent region Ω(t) that has boundary ∂Ω(t), then taking the derivative with respect to time: d/dt∫ _(Ω (t))f dV.

If we wish to move the derivative into the integral, there are two issues: the time dependence of f, and the introduction of and removal of space from Ω due to its dynamic boundary. Reynolds transport theorem provides the necessary framework.

General form

The Reynolds transport theorem can be expressed as follows: d/dt∫ _(Ω (t))f dV=∫ _(Ω (t))(∂ f)/(∂ t) dV+∫ _(∂ Ω (t))(v _(b)· n)f dA

in which n(x,t) is the outward-pointing unit normal vector, x is a point in the region and is the variable of integration, dV and dA are volume and surface elements at x, and vb(x,t) is the velocity of the area element (not the flow velocity). The function f may be tensor-, vector- or scalar-valued. Note that the integral on the left hand side is a function solely of time, and so the total derivative has been used.

Form for a material element

In continuum mechanics, this theorem is often used for material elements. These are parcels of fluids or solids which no material enters or leaves. If Ω(t) is a material element then there is a velocity function v = v(x,t), and the boundary elements obey v _(b)· n =v · n.

This condition may be substituted to obtain: d/dt(∫ _(Ω (t))f dV)=∫ _(Ω (t))(∂ f)/(∂ t) dV+∫ _(∂ Ω (t))(v · n)f dA.

A special case

If we take Ω to be constant with respect to time, then vb = 0 and the identity reduces to d/dt∫ _(Ω)f dV=∫ _(Ω)(∂ f)/(∂ t) dV.

as expected. (This simplification is not possible if the flow velocity is incorrectly used in place of the velocity of an area element.)

Interpretation and reduction to one dimension

The theorem is the higher-dimensional extension of differentiation under the integral sign and reduces to that expression in some cases. Suppose f is independent of y and z, and that Ω(t) is a unit square in the yz-plane and has x limits a(t) and b(t). Then Reynolds transport theorem reduces to （公式略）

which, up to swapping x and t, is the standard expression for differentiation under the integral sign.