Reynolds Transport Theorem

The Reynolds transport theorem, also called simply the Reynolds theorem, is an important result in fluid mechanics that's often considered a three-dimensional analog of the Leibniz integral rule. Given any scalar quantity associated with a moving fluid, the general form of Reynolds transport theorem says

Here,  is the convective derivative,  is the usual gradient,  denotes the material volume at time , and denotes the velocity vector.

Because of its relation to the Leibniz rule, the Reynolds transport theorem is sometimes called the Leibniz-Reynolds transport theorem.

Worth noting is the large number of variants of Reynolds transport theorem present in the literature. Indeed, the formula is extremely general and can be applied to a variety of contexts in vastly many circumstances. As such, different literature will inevitably have equations which often look different than the above equation in both appearance and complexity.

REFERENCES:

Gray, D. "Why the Balance Principle Should Replace the Reynolds Transport Theorem." 2008. http://tinyurl.com/q324mw3.

Leal, L. G. Advanced Transport Phenomena: Fluid Mechanics and Convective Transport Processes. New York: Cambridge University Press, 1999.