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Reynolds transport theorem

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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.

Contents

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:

d d t ∫ Ω ( t ) f d V .

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.

General form

Reynolds' transport theorem can be derived as:

d d t ∫ Ω ( t ) f d V = ∫ Ω ( t ) ∂ f ∂ t d V + ∫ ∂ Ω ( t ) ( v b ⋅ n ) f d A

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 vb(x,t) is the velocity of the area element – so not necessarily 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.

Form for a material element

In continuum mechanics this theorem is often used for material elements, which 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

v b ⋅ n = v ⋅ n .

This condition may be substituted to obtain

d d t ( ∫ Ω ( t ) f d V ) = ∫ Ω ( t ) ∂ f ∂ t d V + ∫ ∂ Ω ( t ) ( v ⋅ n ) f d A .

A special case

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

d d t ∫ Ω f d V = ∫ Ω ∂ f ∂ t d V .

as expected. This simplification is not possible if an incorrect form of the Reynolds transport theorem is used.

Interpretation and reduction to one dimension

The theorem is the higher-dimensional extension of Differentiation under the integral sign and should reduce 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

d d t ∫ a ( t ) b ( t ) f ( x , t ) d x = ∫ a ( t ) b ( t ) ∂ f ∂ t d x + ∂ b ( t ) ∂ t f ( b ( t ) , t ) − ∂ a ( t ) ∂ t f ( a ( t ) , t ) ,

which is the expression given on Differentiation under the integral sign, except that there the variables x and t have been swapped.

References

Reynolds transport theorem Wikipedia
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