Reynolds transport theorem

In differential calculus, the Reynolds transport theorem (also known as the Leibniz–Reynolds transport theorem), or in short Reynolds' theorem, is a three-dimensional generalization of the Leibniz integral rule which is also known as differentiation under the integral sign. The theorem is named after Osborne Reynolds (1842–1912). It is used to recast 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.

Reynolds' transport theorem can be expressed as:

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. Note that the integral on the left hand side is a function solely of time, and so the total derivative has been used.

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:

If we take Ω to be constant with respect to time, then v_b = 0 and the identity reduces to

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