Reynolds transport theorem

In differential calculus, the Reynolds transport theorem (also known as the Leibniz–Reynolds transport theorem), or simply the Reynolds theorem, named after Osborne Reynolds (1842–1912), is a three-dimensional generalization of the Leibniz integral rule. It is used to recast time derivatives of integrated quantities and is useful in formulating the basic equations of continuum mechanics.

Consider integrating over the time-dependent region that has boundary , then taking the derivative with respect to time:
$\frac\int_ \mathbf\,dV.$
If we wish to move the derivative into the integral, there are two issues: the time dependence of , and the introduction of and removal of space from due to its dynamic boundary. Reynolds transport theorem provides the necessary framework.

General form

Reynolds transport theorem can be expressed as follows:
$\frac\int_ \mathbf\,dV = \int_ \frac\,dV + \int_ \left(\mathbf_b\cdot\mathbf\right)\mathbf\,dA$
in which is the outward-pointing unit normal vector, is a point in the region and is the variable of integration, and are volume and surface elements at , and is the velocity of the area element (''not'' the flow velocity). The function 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. These are parcels of fluids or solids which no material enters or leaves. If is a material element then there is a velocity function , and the boundary elements obey
$\mathbf_b\cdot\mathbf=\mathbf\cdot\mathbf.$
This condition may be substituted to obtain:
$\frac\left(\int_ \mathbf\,dV\right) = \int_ \frac\,dV + \int_ (\mathbf\cdot\mathbf)\mathbf\,dA.$

(\mathbf, t) = \mathbf(\boldsymbol(\mathbf, t), t).
Then the integrals in the current and the reference configurations are related by
$\int_ \mathbf(\mathbf,t)\,dV = \int_ \mathbf(\boldsymbol(\mathbf,t),t) \, J(\mathbf,t) \,dV_0 = \int_ \hat(\mathbf,t) \, J(\mathbf,t) \, dV_0.$

That this derivation is for a material element is implicit in the time constancy of the reference configuration: it is constant in material coordinates. The time derivative of an integral over a volume is defined as
$\frac\left( \int_ \mathbf(\mathbf,t)\,dV\right) = \lim_ \frac \left(\int_ \mathbf(\mathbf,t+\Delta t)\,dV - \int_ \mathbf(\mathbf,t)\,dV\right).$

Converting into integrals over the reference configuration, we get
$\frac \left( \int_ \mathbf(\mathbf,t) \, dV\right) = \lim_ \frac \left(\int_ \hat(\mathbf,t+\Delta t)\, J(\mathbf,t+\Delta t)\,dV_0 - \int_ \hat(\mathbf,t)\, J(\mathbf,t)\, dV_0\right).$

Since is independent of time, we have
$\begin \frac\left( \int_ \mathbf(\mathbf,t)\,dV\right) &= \int_ \left(\lim_ \frac \right)\,dV_0 \\ &= \int_ \frac\left(\hat(\mathbf,t)\, J(\mathbf,t)\right)\,dV_0 \\ &= \int_ \left( \frac\big(\hat(\mathbf,t)\big)\, J(\mathbf,t)+ \hat(\mathbf,t)\,\frac\big(J(\mathbf,t)\big)\right) \,dV_0. \end$

The time derivative of is given by:
$\begin \frac &= \frac(\det\boldsymbol) \\ &= (\det\boldsymbol) \operatorname\left(\boldsymbol^ \frac\right)\\ &= (\det\boldsymbol) \operatorname\left(\frac \frac\left(\frac\right)\right)\\ &= (\det\boldsymbol) \operatorname\left(\frac \frac\left(\frac\right)\right) \\ &= (\det\boldsymbol) \operatorname\left( \frac\left(\frac\right)\right) \\ &= (\det\boldsymbol)(\boldsymbol \cdot \mathbf) \\ &= J(\mathbf,t)\,\boldsymbol \cdot \mathbf\big(\boldsymbol(\mathbf,t),t\big) \\ &= J(\mathbf,t)\,\boldsymbol \cdot \mathbf(\mathbf,t). \end$

Therefore,
$\begin \frac\left( \int_ \mathbf(\mathbf,t)\,dV\right) &= \int_ \left( \frac\left(\hat(\mathbf,t)\right)\,J(\mathbf,t)+ \hat(\mathbf,t)\,J(\mathbf,t)\,\boldsymbol \cdot \mathbf(\mathbf,t)\right) \,dV_0 \\ &= \int_ \left(\frac\left(\hat(\mathbf,t)\right)+ \hat(\mathbf,t)\,\boldsymbol \cdot \mathbf(\mathbf,t)\right)\,J(\mathbf,t) \,dV_0 \\ &= \int_ \left(\dot(\mathbf,t)+ \mathbf(\mathbf,t)\,\boldsymbol \cdot \mathbf(\mathbf,t)\right)\,dV. \end$
where $\dot$ is the material time derivative of . The material derivative is given by
$\dot(\mathbf,t) = \frac + \big(\boldsymbol \mathbf(\mathbf,t)\big)\cdot\mathbf(\mathbf,t).$

Therefore,
$\frac\left( \int_ \mathbf(\mathbf,t)\,dV\right) = \int_ \left( \frac + \big(\boldsymbol \mathbf(\mathbf,t)\big) \cdot\mathbf(\mathbf,t) + \mathbf(\mathbf,t)\,\boldsymbol \cdot \mathbf(\mathbf,t) \right) \,dV,$
or,
$\frac\left( \int_ \mathbf\,dV\right) = \int_ \left( \frac + \boldsymbol \mathbf\cdot\mathbf + \mathbf\,\boldsymbol \cdot \mathbf\right)\,dV.$

Using the identity
$\boldsymbol \cdot (\mathbf\otimes\mathbf) = \mathbf(\boldsymbol \cdot \mathbf) + \boldsymbol\mathbf\cdot\mathbf,$
we then have
$\frac\left( \int_ \mathbf\,dV\right) = \int_ \left(\frac + \boldsymbol \cdot (\mathbf\otimes\mathbf)\right)\,dV.$

Using the divergence theorem and the identity , we have
$\begin \frac\left( \int_ \mathbf\,dV\right) &= \int_\frac\,dV + \int_(\mathbf\otimes\mathbf)\cdot\mathbf\,dA \\ &= \int_\frac\,dV + \int_(\mathbf\cdot\mathbf)\mathbf\,dA. \end$
Q.E.D.

A special case

If we take to be constant with respect to time, then and the identity reduces to
$\frac\int_ f\,dV = \int_ \frac\,dV.$
as expected. (This simplification is not possible if the flow velocity is incorrectly used in place of the velocity of an area element.)

Interpretation and reduction to one dimension

The theorem is the higher-dimensional extension of differentiation under the integral sign and reduces to that expression in some cases. Suppose is independent of and , and that is a unit square in the -plane and has limits and . Then Reynolds transport theorem reduces to
$\frac\int_^ f(x,t)\,dx = \int_^ \frac\,dx + \frac f\big(b(t),t\big) - \frac f\big(a(t),t\big) \,,$
which, up to swapping and , is the standard expression for differentiation under the integral sign.

See also

*

References

External links

* Osborne Reynolds, Collected Papers on Mechanical and Physical Subjects, in three volumes, published circa 1903, now fully and freely available in digital format
Volume 1Volume 2Volume 3

*
* {{Cite web , title=Reynolds transport theorem , url=https://planetmath.org/reynoldstransporttheorem , access-date=2024-04-22 , website=planetmath.org
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