From Leibniz integral rule to Reynolds' transport theorem
(2019) , 6 pages
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- Abstract
- Evaluating the derivative of an integral over a moving volume can be performed with the help of Reynolds' transport theorem, which is a generalisation of Leibniz integral rule. Most continuum or fluid mechanics textbooks provide a demonstration of Reynolds' transport theorem by having recourse to a blend of schematics and elementary calculus (e.g. Kundu et al. 2012). In this working note, a rather rigorous approach to the demonstration is presented, whose starting point is the introduction of a new coordinate system transforming the moving volume into a fixed one. The crux of the line of argument lies in the evaluation of the Jacobian of the coordinate transformation and the selection of the instant at which time derivatives are calculated. A graphical illustration and the treatment of a few special cases help grasp the significance of the theorem. Nothing is novel herein. In fact, some inspiration was sought in Aris (1962). It is noteworthy, however, that the style of the mathematical developments is intended to be familiar to those interested in fluid mechanics.
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Citations
Deleersnijder, E. (2019). From Leibniz integral rule to Reynolds’ transport theorem. https://hdl.handle.net/2078.5/63660