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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:
General form
Reynolds transport theorem can be expressed as follows:[1][2][3]
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
Let Ω0 be reference configuration of the region Ω(t). Let the motion and the deformation gradient be given by
Let J(X,t) = det F(X,t). Define
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 Ω0 is independent of time, we have
The time derivative of J is given by:[6]
Therefore,
Therefore,
Using the identity
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 vb = 0 and the identity reduces to
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
See also
- Leibniz integral rule – Differentiation under the integral sign formula
References
- ^ 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
- 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.
- "Reynolds transport theorem". planetmath.org. Retrieved 2024-04-22.