# Reynolds transport theorem

## Introduction

[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 $(\mathbb{R}^{3},\lVert\cdot\rVert)$ and embedded on it we consider, a continuum body $\mathscr{B}$ occupying a volume $\mathscr{V}$ whose particles are fixed by their material (Lagrangian) coordinates $\mathbf{X}$, and a region $\Re$ where a control volume $\mathfrak{v}$ is defined whose points are fixed by it spatial (Eulerian) coordinates $\mathbf{x}$ and bounded by the control surface $\partial\mathfrak{v}$. 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

 $\displaystyle\Psi(\mathbf{x},t):=\int_{\mathfrak{v}}\psi(\mathbf{x},t)\rho(% \mathbf{x},t)dv,$ (1)

where $\psi(\mathbf{x},t)$ is the respective intensive tensor property.

## Theorem’s hypothesis

The kinematics of the continuum can be described by a diffeomorphism $\chi$ which, at any given instant $t\in[0,\infty)\subset\mathbb{R}$, gives the spatial coordinates $\mathbf{x}$ of the material particle $\mathbf{X}$,

 $\displaystyle\mathscr{V}\times[0,\infty)\rightarrow\mathfrak{v}\times[0,\infty% ),\qquad t\mapsto t,\qquad\mathbf{X}\mapsto\mathbf{x}=\chi(\mathbf{X},t).$

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

 $\displaystyle J=\bigg{|}\frac{\partial{x}_{i}}{\partial{X}_{j}}\bigg{|}\equiv% \big{|}{F_{ij}}\big{|}\neq{0},\qquad F_{ij}:=\frac{\partial{x}_{i}}{\partial{X% }_{j}},$

$J$ being the Jacobian of transformation and $F_{ij}$ the Cartesian components of the so-called strain gradient tensor $\mathbf{F}$.

###### Reynolds transport theorem 1.

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

###### Proof.

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

 $\displaystyle\frac{D\Psi}{Dt}=\dot{\Psi}=\dot{\overline{\int_{\mathfrak{v}}% \psi\rho\;dv}}=\dot{\overline{\int_{\mathscr{V}}\psi\rho{J}dV}}=\int_{\mathscr% {V}}\dot{\overline{\psi\rho{J}}}dV=\int_{\mathscr{V}}(\dot{\overline{\psi\rho}% }J+\psi\rho\dot{J})dV=$
 $\displaystyle\int_{\mathscr{V}}\Big{\{}J\Big{[}\frac{\partial}{\partial{t}}(% \psi\rho)+\mathbf{v}\!\cdot\!\nabla_{x}(\psi\rho)\Big{]}+\psi\rho\;(J\nabla_{x% }\!\cdot\!\mathbf{v})\Big{\}}dV=\int_{\mathscr{V}}\Big{\{}\Big{[}\frac{% \partial}{\partial{t}}(\psi\rho)\Big{]}+\big{[}\mathbf{v}\!\cdot\!\nabla_{x}(% \psi\rho)+(\psi\rho)\nabla_{x}\!\cdot\!\mathbf{v}\big{]}\Big{\}}(JdV)$
 $\displaystyle=\int_{\mathfrak{v}}\frac{\partial}{\partial{t}}(\psi\rho)dv+\int% _{\mathfrak{v}}\nabla_{x}\!\cdot\!(\psi\rho\;\mathbf{v})dv=\frac{\partial}{% \partial{t}}\int_{\mathfrak{v}}\psi\rho\,dv+\int_{\partial\mathfrak{v}}\psi% \rho\,\mathbf{v}\!\cdot\!\mathbf{n}\,da,$

since $\partial_{t}(dv)=0$ ($\mathbf{x}$ fixed) on the first integral and by applying the Gauss-Green divergence theorem 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

 $\displaystyle\dot{\Psi}=\frac{\partial\Psi}{\partial{t}}+\int_{\partial% \mathfrak{v}}\psi\rho\,\mathbf{v}\!\cdot\!\mathbf{n}\,da,$ (2)

endorsing the theorem statement. ∎

## References

• 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 ReynoldsTransportTheorem 2013-03-22 16:00:48 2013-03-22 16:00:48 perucho (2192) perucho (2192) 7 perucho (2192) Theorem msc 53A45