Reynolds transport theorem

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 $\partialΩ(t)$ , then taking the derivative with respect to time:

{\frac {d}{dt}}\int _{\Omega (t)}\mathbf {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[edit]

Reynolds transport theorem can be expressed as follows:^[1]^[2]^[3]

{\frac {d}{dt}}\int _{\Omega (t)}\mathbf {f} \,dV=\int _{\Omega (t)}{\frac {\partial \mathbf {f} }{\partial t}}\,dV+\int _{\partial \Omega (t)}\left(\mathbf {v} _{b}\cdot \mathbf {n} \right)\mathbf {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 $v b (x, t)$ is the velocity of the area element (not the flow velocity). The function $f$ may be tensor-, vector- or scalar-valued.^[4] 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[edit]

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

\mathbf {v} _{b}\cdot \mathbf {n} =\mathbf {v} \cdot \mathbf {n} .

This condition may be substituted to obtain:^[5]

{\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 {v} \cdot \mathbf {n} )\mathbf {f} \,dA.

Proof for a material element

Let $Ω 0$ be reference configuration of the region $Ω(t)$ . Let the motion and the deformation gradient be given by

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

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

Let $J (X, t) = det F (X, t)$ . Define

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

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

{\begin{aligned}\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}.\end{aligned}}

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

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

Converting into integrals over the reference configuration, we get

{\frac {d}{dt}}\left(\int _{\Omega (t)}\mathbf {f} (\mathbf {x} ,t)\,dV\right)=\lim _{\Delta t\rightarrow 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 $Ω 0$ is independent of time, we have

{\begin{aligned}{\frac {d}{dt}}\left(\int _{\Omega (t)}\mathbf {f} (\mathbf {x} ,t)\,dV\right)&=\int _{\Omega _{0}}\left(\lim _{\Delta t\rightarrow 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{aligned}}

The time derivative of $J$ is given by:^[6]

{\begin{aligned}{\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{aligned}}

Therefore,

{\begin{aligned}{\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{aligned}}

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

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

Therefore,

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

{\frac {d}{dt}}\left(\int _{\Omega (t)}\mathbf {f} \,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)\,dV.

Using the identity

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

we then have

{\frac {d}{dt}}\left(\int _{\Omega (t)}\mathbf {f} \,dV\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 $(a \otimes b) \cdot n = (b \cdot n) a$ , we have

{\begin{aligned}{\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.\qquad \square \end{aligned}}

A special case[edit]

If we take $Ω$ to be constant with respect to time, then $v b = 0$ and the identity reduces to

{\frac {d}{dt}}\int _{\Omega }f\,dV=\int _{\Omega }{\frac {\partial f}{\partial 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[edit]

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

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

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

References[edit]

^ Leal, L. G. (2007). Advanced transport phenomena: fluid mechanics and convective transport processes. Cambridge University Press. p. 23. ISBN 978-0-521-84910-4.
^ Reynolds, O. (1903). Papers on Mechanical and Physical Subjects. Vol. 3, The Sub-Mechanics of the Universe. Cambridge: Cambridge University Press. pp. 12–13.
^ Marsden, J. E.; Tromba, A. (2003). Vector Calculus (5th ed.). New York: W. H. Freeman. ISBN 978-0-7167-4992-9.
^ Yamaguchi, H. (2008). Engineering Fluid Mechanics. Dordrecht: Springer. p. 23. ISBN 978-1-4020-6741-9.
^ Belytschko, T.; Liu, W. K.; Moran, B. (2000). Nonlinear Finite Elements for Continua and Structures. New York: John Wiley and Sons. ISBN 0-471-98773-5.
^ Gurtin, M. E. (1981). An Introduction to Continuum Mechanics. New York: Academic Press. p. 77. ISBN 0-12-309750-9.

External links[edit]

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,
"Module 6 - Reynolds Transport Theorem". ME6601: Introduction to Fluid Mechanics. Georgia Tech. Archived from the original on March 27, 2008.
http://planetmath.org/reynoldstransporttheorem

[LGLp23-1] Leal, L. G. (2007). Advanced transport phenomena: fluid mechanics and convective transport processes. Cambridge University Press. p. 23. ISBN 978-0-521-84910-4.

[OR14-2] Reynolds, O. (1903). Papers on Mechanical and Physical Subjects. Vol. 3, The Sub-Mechanics of the Universe. Cambridge: Cambridge University Press. pp. 12–13.

[MTp23-3] Marsden, J. E.; Tromba, A. (2003). Vector Calculus (5th ed.). New York: W. H. Freeman. ISBN 978-0-7167-4992-9.

[4] Yamaguchi, H. (2008). Engineering Fluid Mechanics. Dordrecht: Springer. p. 23. ISBN 978-1-4020-6741-9.

[BLM-5] Belytschko, T.; Liu, W. K.; Moran, B. (2000). Nonlinear Finite Elements for Continua and Structures. New York: John Wiley and Sons. ISBN 0-471-98773-5.

[Gurtin-6] Gurtin, M. E. (1981). An Introduction to Continuum Mechanics. New York: Academic Press. p. 77. ISBN 0-12-309750-9.

[1]

[2]

[3]

[4]

[5]

[6]

Reynolds transport theorem

General form[edit]

Form for a material element[edit]

A special case[edit]

Interpretation and reduction to one dimension[edit]

See also[edit]

References[edit]

External links[edit]

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