Reynolds transport theorem

Reynolds transport theorem can be expressed as follows:{{cite book |authorlink=L. Gary Leal |first=L. G. |last=Leal |year=2007 |title=Advanced transport phenomena: fluid mechanics and convective transport processes |publisher=Cambridge University Press |isbn=978-0-521-84910-4 |page=23 }}{{cite book |authorlink=Osborne Reynolds |first=O. |last=Reynolds |year=1903 |title=Papers on Mechanical and Physical Subjects |volume=3, The Sub-Mechanics of the Universe |publisher=Cambridge University Press |location=Cambridge |pages=12–13 }}{{cite book |authorlink=Jerrold E. Marsden |first=J. E. |last=Marsden |authorlink2=Anthony Tromba |first2=A. |last2=Tromba |year=2003 |title=Vector Calculus |edition=5th |publisher=W. H. Freeman |location=New York |isbn=978-0-7167-4992-9 }}

$$\backslash frac\{d\}\{dt\}\backslash int\_\{\backslash Omega(t)\}\; \backslash mathbf\{f\}\backslash ,dV\; =\; \backslash int\_\{\backslash Omega(t)\}\; \backslash frac\{\backslash partial\; \backslash mathbf\{f\}\}\{\backslash partial\; t\}\backslash ,dV\; +\; \backslash int\_\{\backslash partial\; \backslash Omega(t)\}\; \backslash left(\backslash mathbf\{v\}\_b\backslash cdot\backslash mathbf\{n\}\backslash right)\backslash mathbf\{f\}\backslash ,dA$$

in which {{math|n(x,t)}} is the outward-pointing unit normal vector, {{math|x}} is a point in the region and is the variable of integration, {{math|dV}} and {{math|dA}} are volume and surface elements at {{math|x}}, and {{math|v_b(x,t)}} is the velocity of the area element (not the flow velocity). The function {{math|f}} may be tensor-, vector- or scalar-valued.{{cite book |first=H. |last=Yamaguchi |title=Engineering Fluid Mechanics |location=Dordrecht |publisher=Springer |year=2008 |page=23 |isbn=978-1-4020-6741-9 }} Note that the integral on the left hand side is a function solely of time, and so the total derivative has been used.

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 {{math|Ω(t)}} is a material element then there is a velocity function {{math|1=v = v(x,t)}}, and the boundary elements obey

$$\backslash mathbf\{v\}\_b\backslash cdot\backslash mathbf\{n\}=\backslash mathbf\{v\}\backslash cdot\backslash mathbf\{n\}.$$

This condition may be substituted to obtain:{{cite book |authorlink=Ted Belytschko |first=T. |last=Belytschko |first2=W. K. |last2=Liu |first3=B. |last3=Moran |year=2000 |title=Nonlinear Finite Elements for Continua and Structures |publisher=John Wiley and Sons |location=New York |isbn=0-471-98773-5 }}

$$\backslash frac\{d\}\{dt\}\backslash left(\backslash int\_\{\backslash Omega(t)\}\; \backslash mathbf\{f\}\backslash ,dV\backslash right)\; =\; \backslash int\_\{\backslash Omega(t)\}\; \backslash frac\{\backslash partial\; \backslash mathbf\{f\}\}\{\backslash partial\; t\}\backslash ,dV\; +\; \backslash int\_\{\backslash partial\; \backslash Omega(t)\}\; (\backslash mathbf\{v\}\backslash cdot\backslash mathbf\{n\})\backslash mathbf\{f\}\backslash ,dA.$$

{{math proof| title = Proof for a material element | proof =

Let {{math|Ω₀}} be reference configuration of the region {{math|Ω(t)}}. Let

the motion and the deformation gradient be given by

$$\backslash mathbf\{x\}\; =\; \backslash boldsymbol\{\backslash varphi\}(\backslash mathbf\{X\},\; t),$$

$$\backslash boldsymbol\{F\}(\backslash mathbf\{X\},t)\; =\; \backslash boldsymbol\{\backslash nabla\}\backslash boldsymbol\{\backslash varphi\}.$$

Let {{math|1=J(X,t) = det F(X,t)}}. Define

$$\backslash hat\{\backslash mathbf\{f\}\}(\backslash mathbf\{X\},\; t)\; =\; \backslash mathbf\{f\}(\backslash boldsymbol\{\backslash varphi\}(\backslash mathbf\{X\},\; t),\; t).$$

Then the integrals in the current and the reference configurations are related by

\int_{\Omega(t)} \mathbf{f}(\mathbf{x},t)\,dV

= \int_{\Omega_0} \mathbf{f}(\boldsymbol{\varphi}(\mathbf{X},t),t) \, J(\mathbf{X},t) \,dV_0

= \int_{\Omega_0} \hat{\mathbf{f}}(\mathbf{X},t) \, J(\mathbf{X},t) \, 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. The time derivative of an integral over a volume is defined as

$$\backslash frac\{d\}\{dt\}\backslash left(\; \backslash int\_\{\backslash Omega(t)\}\; \backslash mathbf\{f\}(\backslash mathbf\{x\},t)\backslash ,dV\backslash right)\; =\; \backslash lim\_\{\backslash Delta\; t\; \backslash to\; 0\}\; \backslash frac\{1\}\{\backslash Delta\; t\}\; \backslash left(\backslash int\_\{\backslash Omega(t\; +\; \backslash Delta\; t)\}\; \backslash mathbf\{f\}(\backslash mathbf\{x\},t+\backslash Delta\; t)\backslash ,dV\; -\; \backslash int\_\{\backslash Omega(t)\}\; \backslash mathbf\{f\}(\backslash mathbf\{x\},t)\backslash ,dV\backslash right).$$

Converting into integrals over the reference configuration, we get

$$\backslash frac\{d\}\{dt\}\; \backslash left(\; \backslash int\_\{\backslash Omega(t)\}\; \backslash mathbf\{f\}(\backslash mathbf\{x\},t)\; \backslash ,\; dV\backslash right)\; =$$

\lim_{\Delta t \to 0} \frac{1}{\Delta t} \left(\int_{\Omega_0} \hat{\mathbf{f}}(\mathbf{X},t+\Delta t)\, J(\mathbf{X},t+\Delta t)\,dV_0 - \int_{\Omega_0} \hat{\mathbf{f}}(\mathbf{X},t)\, J(\mathbf{X},t)\, dV_0\right).

Since {{math|Ω₀}} is independent of time, we have

$$\backslash begin\{align\}$$

\frac{d}{dt}\left( \int_{\Omega(t)} \mathbf{f}(\mathbf{x},t)\,dV\right)

&= \int_{\Omega_0} \left(\lim_{\Delta t \to 0} \frac{ \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)\,dV_0 \\

&= \int_{\Omega_0} \frac{\partial}{\partial t}\left(\hat{\mathbf{f}}(\mathbf{X},t)\, J(\mathbf{X},t)\right)\,dV_0 \\

&= \int_{\Omega_0} \left( \frac{\partial}{\partial t}\big(\hat{\mathbf{f}}(\mathbf{X},t)\big)\, J(\mathbf{X},t)+ \hat{\mathbf{f}}(\mathbf{X},t)\,\frac{\partial}{\partial t}\big(J(\mathbf{X},t)\big)\right) \,dV_0.

\end{align}

The time derivative of {{mvar|J}} is given by:{{cite book |authorlink=Morton Gurtin |last=Gurtin |first=M. E. |year=1981 |title=An Introduction to Continuum Mechanics |publisher=Academic Press |location=New York |page=77 |isbn=0-12-309750-9 }}

$$\backslash begin\{align\}$$

\frac{\partial J(\mathbf{X},t)}{\partial t} &= \frac{\partial}{\partial t}(\det\boldsymbol{F}) \\

&= (\det\boldsymbol{F}) \operatorname{tr}\left(\boldsymbol{F}^{-1} \frac{\partial\boldsymbol F}{\partial t}\right)\\

&= (\det\boldsymbol{F}) \operatorname{tr}\left(\frac{\partial \boldsymbol{X}}{\partial \boldsymbol{\varphi}} \frac{\partial}{\partial t}\left(\frac{\partial\boldsymbol{\varphi}}{\partial\boldsymbol{X}}\right)\right)\\

&= (\det\boldsymbol{F}) \operatorname{tr}\left(\frac{\partial \boldsymbol{X}}{\partial \boldsymbol{\varphi}} \frac{\partial}{\partial \boldsymbol{X}}\left(\frac{\partial\boldsymbol{\varphi}}{\partial t}\right)\right) \\

&= (\det\boldsymbol{F}) \operatorname{tr}\left( \frac{\partial}{\partial \boldsymbol{x}}\left(\frac{\partial\boldsymbol{\varphi}}{\partial t}\right)\right) \\

&= (\det\boldsymbol{F})(\boldsymbol{\nabla} \cdot \mathbf{v}) \\

&= J(\mathbf{X},t)\,\boldsymbol{\nabla} \cdot \mathbf{v}\big(\boldsymbol{\varphi}(\mathbf{X},t),t\big) \\

&= J(\mathbf{X},t)\,\boldsymbol{\nabla} \cdot \mathbf{v}(\mathbf{x},t). \end{align}

Therefore,

$$\backslash begin\{align\}$$

\frac{d}{dt}\left( \int_{\Omega(t)} \mathbf{f}(\mathbf{x},t)\,dV\right)

&= \int_{\Omega_0} \left( \frac{\partial}{\partial t}\left(\hat{\mathbf{f}}(\mathbf{X},t)\right)\,J(\mathbf{X},t)+ \hat{\mathbf{f}}(\mathbf{X},t)\,J(\mathbf{X},t)\,\boldsymbol{\nabla} \cdot \mathbf{v}(\mathbf{x},t)\right) \,dV_0 \\

&= \int_{\Omega_0} \left(\frac{\partial}{\partial t}\left(\hat{\mathbf{f}}(\mathbf{X},t)\right)+ \hat{\mathbf{f}}(\mathbf{X},t)\,\boldsymbol{\nabla} \cdot \mathbf{v}(\mathbf{x},t)\right)\,J(\mathbf{X},t) \,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)\,dV.

\end{align}

where $\backslash dot\{\backslash mathbf\{f\}\}$ is the material time derivative of {{math|f}}. The material derivative is given by

$$\backslash dot\{\backslash mathbf\{f\}\}(\backslash mathbf\{x\},t)\; =\; \backslash frac\{\backslash partial\; \backslash mathbf\{f\}(\backslash mathbf\{x\},t)\}\{\backslash partial\; t\}\; +\; \backslash big(\backslash boldsymbol\{\backslash nabla\}\; \backslash mathbf\{f\}(\backslash mathbf\{x\},t)\backslash big)\backslash cdot\backslash mathbf\{v\}(\backslash mathbf\{x\},t).$$

Therefore,

$$\backslash frac\{d\}\{dt\}\backslash left(\; \backslash int\_\{\backslash Omega(t)\}\; \backslash mathbf\{f\}(\backslash mathbf\{x\},t)\backslash ,dV\backslash right)\; =$$

\int_{\Omega(t)} \left( \frac{\partial \mathbf{f}(\mathbf{x},t)}{\partial t} + \big(\boldsymbol{\nabla} \mathbf{f}(\mathbf{x},t)\big) \cdot\mathbf{v}(\mathbf{x},t) + \mathbf{f}(\mathbf{x},t)\,\boldsymbol{\nabla} \cdot \mathbf{v}(\mathbf{x},t) \right) \,dV,

or,

$$\backslash frac\{d\}\{dt\}\backslash left(\; \backslash int\_\{\backslash Omega(t)\}\; \backslash mathbf\{f\}\backslash ,dV\backslash right)$$

= \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)\,dV.

Using the identity

$$\backslash boldsymbol\{\backslash nabla\}\; \backslash cdot\; (\backslash mathbf\{v\}\backslash otimes\backslash mathbf\{w\})\; =\; \backslash mathbf\{v\}(\backslash boldsymbol\{\backslash nabla\}\; \backslash cdot\; \backslash mathbf\{w\})\; +\; \backslash boldsymbol\{\backslash nabla\}\backslash mathbf\{v\}\backslash cdot\backslash mathbf\{w\},$$

we then have

$$\backslash frac\{d\}\{dt\}\backslash left(\; \backslash int\_\{\backslash Omega(t)\}\; \backslash mathbf\{f\}\backslash ,dV\backslash right)$$

= \int_{\Omega(t)} \left(\frac{\partial \mathbf{f}}{\partial t} + \boldsymbol{\nabla} \cdot (\mathbf{f}\otimes\mathbf{v})\right)\,dV.

Using the divergence theorem and the identity {{math|1=(a ⊗ b) · n = (b · n)a}}, we have

$$\backslash begin\{align\}$$

\frac{d}{dt}\left( \int_{\Omega(t)} \mathbf{f}\,dV\right)

&= \int_{\Omega(t)}\frac{\partial \mathbf{f}}{\partial t}\,dV + \int_{\partial \Omega(t)}(\mathbf{f}\otimes\mathbf{v})\cdot\mathbf{n}\,dA \\

&= \int_{\Omega(t)}\frac{\partial \mathbf{f}}{\partial t}\,dV + \int_{\partial \Omega(t)}(\mathbf{v}\cdot\mathbf{n})\mathbf{f}\,dA.

\end{align}

Q.E.D.

}}

If we take {{math|Ω}} to be constant with respect to time, then {{math|v_b {{=}} 0}} and the identity reduces to

$$\backslash frac\{d\}\{dt\}\backslash int\_\{\backslash Omega\}\; f\backslash ,dV\; =\; \backslash int\_\{\backslash Omega\}\; \backslash frac\{\backslash partial\; f\}\{\backslash partial\; t\}\backslash ,dV.$$

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

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

$$\backslash frac\{d\}\{dt\}\backslash int\_\{a(t)\}^\{b(t)\}\; f(x,t)\backslash ,dx\; =\; \backslash int\_\{a(t)\}^\{b(t)\}\; \backslash frac\{\backslash partial\; f\}\{\backslash partial\; t\}\backslash ,dx\; +\; \backslash frac\{\backslash partial\; b(t)\}\{\backslash partial\; t\}\; f\backslash big(b(t),t\backslash big)\; -\; \backslash frac\{\backslash partial\; a(t)\}\{\backslash partial\; t\}\; f\backslash big(a(t),t\backslash big)\; \backslash ,,$$

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

{{Portal|Mathematics}}

• {{annotated link|Leibniz integral rule}}

{{clear}}

{{Reflist}}

• Osborne Reynolds, Collected Papers on Mechanical and Physical Subjects, in three volumes, published circa 1903, now fully and freely available in digital format: [https://archive.org/details/papersonmechanic01reynrich Volume 1], [https://archive.org/details/papersonmechanic02reynrich Volume 2], [https://archive.org/details/papersonmechanic03reynrich Volume 3],
• {{cite web |title=Module 6 - Reynolds Transport Theorem |work=ME6601: Introduction to Fluid Mechanics |publisher=Georgia Tech |url=http://www.catea.org/grade/mecheng/mod6/mod6.html#slide1 |archivedate=March 27, 2008 |archiveurl=https://web.archive.org/web/20080327180821/http://www.catea.org/grade/mecheng/mod6/mod6.html#slide1 }}
• {{Cite web |title=Reynolds transport theorem |url=https://planetmath.org/reynoldstransporttheorem |access-date=2024-04-22 |website=planetmath.org}}

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