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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:

If we wish to move the derivative within 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

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

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

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

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

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 byLet

*J*(**X**,*t*) = det(**F****X**,*t*). DefineThen the integrals in the current and the reference configurations are related by

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

Converting into integrals over the reference configuration, we get

Since Ω

_{0}is independent of time, we haveThe time derivative of

**J**is given by:^{[6]}Therefore,

where is the material time derivative of

**f**. The material derivative is given byTherefore,

or,

Using the identity

we then have

Using the divergence theorem and the identity (

**a**⊗**b**) ·**n**= (**b**·**n**)**a**, we have

## A special case

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

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

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

## See also

## Notes

**^**L. G. Leal, 2007, p. 23.**^**O. Reynolds, 1903, Vol. 3, p. 12–13**^**J.E. Marsden and A. Tromba, 5th ed. 2003**^**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.

## References

- Leal, L. G. (2007).
*Advanced transport phenomena: fluid mechanics and convective transport processes*. Cambridge University Press. ISBN 978-0-521-84910-4. - Marsden, J. E.; Tromba, A. (2003).
*Vector Calculus*(5th ed.). New York: W. H. Freeman. ISBN 978-0-7167-4992-9. - Reynolds, O. (1903).
*Papers on Mechanical and Physical Subjects*. Vol. 3, The Sub-Mechanics of the Universe. Cambridge: Cambridge University Press.`|volume=`

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## External links

- 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