In differential calculus, the 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. It is used to recast 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: If we wish to move the derivative within 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: 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. 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.
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 This condition may be substituted to obtain: Let be reference configuration of the region. Let the motion and the deformation gradient be given by Let. Define Then the integrals in the current and the reference configurations are related by 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 Converting into integrals over the reference configuration, we get Since is independent of time, we have The time derivative of is given by: Therefore, where is the material time derivative of. The material derivative is given by Therefore, or, Using the identity we then have Using the divergence theorem and the identity, we have
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 which, up to swapping and, is the standard expression for differentiation under the integral sign.
