Reynolds transport theorem


Reynolds transport theoremMathworldPlanetmath [1] is a fundamental theorem used in formulating the basic laws of fluid mechanics. We will enunciate and demonstrate in this entry the referred theorem. For our purpose, let us consider a fluid flow, characterized by its streamlines, in the Euclidean vector space (3,) and embedded on it we consider, a continuum body occupying a volume 𝒱 whose particles are fixed by their material (Lagrangian) coordinatesPlanetmathPlanetmath 𝐗, and a region where a control volume 𝔳 is defined whose points are fixed by it spatial (Eulerian) coordinates 𝐱 and bounded by the control surface 𝔳. An arbitrary tensor field of any rank is defined over the fluid flow according to the following definition.

Definition 1.

We call an extensive tensor property to the expression

Ψ(𝐱,t):=𝔳ψ(𝐱,t)ρ(𝐱,t)𝑑v, (1)

where ψ(x,t) is the respective intensive tensor property.

Theorem’s hypothesis

The kinematics of the continuum can be described by a diffeomorphism χ which, at any given instant t[0,), gives the spatial coordinates 𝐱 of the material particle 𝐗,


Indeed the above sentence corresponds to a change of coordinates which must verify


J being the JacobianMathworldPlanetmathPlanetmath of transformation and Fij the Cartesian components of the so-called strain gradientMathworldPlanetmath tensor 𝐅.

Reynolds transport theorem 1.

The material rate of an extensive tensor property associate to a continuum body B is equal to the local rate of such property in a control volume v plus the efflux of the respective intensive property across its control surface v.


By taking on Eq.(1) the material time derivative,


since t(dv)=0 (𝐱 fixed) on the first integral and by applying the Gauss-Green divergence theoremMathworldPlanetmathPlanetmath on the second integral at the left-hand side. Finally, by substituting Eq.(1) on the first integral at the right-hand side, we obtain

Ψ˙=Ψt+𝔳ψρ𝐯𝐧𝑑a, (2)

endorsing the theorem statement. ∎


  • 1 O. Reynolds, Papers on mechanical and physical subjects-the sub-mechanics of the Universe, Collected Work, Volume III, Cambridge University Press, 1903.
Title Reynolds transport theorem
Canonical name ReynoldsTransportTheorem
Date of creation 2013-03-22 16:00:48
Last modified on 2013-03-22 16:00:48
Owner perucho (2192)
Last modified by perucho (2192)
Numerical id 7
Author perucho (2192)
Entry type Theorem
Classification msc 53A45