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 B(x,t) associated with a moving fluid, the general form of Reynolds transport theorem says

 D/(Dt)[int_(V_m(t))B(x,t)dV]=int_(V_m(t))[(partialB)/(partialt)+del ·(Bu)dV].

Here, D/Dt is the convective derivative, del is the usual gradient, V_m(t) denotes the material volume at time t, and u 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.

See also

Convective Derivative, Gradient, Leibniz Integral Rule, Scalar Field, Velocity Vector, Volume, Volume Element, Volume Integral

This entry contributed by Christopher Stover

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Gray, D. "Why the Balance Principle Should Replace the Reynolds Transport Theorem." 2008., L. G. Advanced Transport Phenomena: Fluid Mechanics and Convective Transport Processes. New York: Cambridge University Press, 1999.

Cite this as:

Stover, Christopher. "Reynolds Transport Theorem." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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