The Reynolds Transport Theorem

For completion purposes, I feel obliged to discuss the Reynolds transport theorem. Although I would like to derive the fluid flow equations from scratch, the Reynolds transport theorem provides an avenue for a simple way to derive them. Henceforth, I decided to use the many ways of deriving the conservation equations, whether in integral or differential form.

I will base the current derivation on the text by James A. Fay Introduction to Fluid Mechanics. I believe it is an excellent text on fluid mechanics that focuses on the essential physics of fluid flows.

We first have to distinguish between a material volume and a control volume. A material volume is a volume of fluid that contains the same fluid as it moves and deforms in time.


A control volume is a fixed volume in space where the fluid passes through.


This is tightly linked to the previous discussions on the material derivative and its connection with the Lagrangian and Eulerian views. A material volume is part of a Lagrangian description whereas a control volume is part of the Eulerian description.

Now let us consider and "extensive" property B whose "intensive" property is b. For example, mass is an extensive property, whereas the density is the corresponding intensive property. An extensive property describes a specific part of the fluid (e.g. the mass is different for different volumes of the same fluid) while the intensive property is intrinsic (e.g. the density is the same for different volumes of the same fluid). In simple terms, an intensive property is the extensive property per unit mass.

The Reynolds transport theorem can be thought of as the integral form of the material derivative. It mainly relates the rate of change of an extensive property of a given material volume to the rate of change of the corresponding intensive property.

The total amount of property B in a given material volume is therefore

\begin{align} B = \int_{\mathcal{V}}{\rho b {\rm d} \mathcal{V}} \end{align}

As the material volume moves around, the quantity B inside M changes due to external forces or internal reactions for example. Therefore, it is convenient to compute the time rate of change of B

\begin{align} \frac{\mathrm{d}B}{\mathrm{d}t} = \frac{\partial B}{\partial t} + \dot{B}_{\mathrm{out}} - \dot{B}_{\mathrm{in}} \end{align}

Eq. 2 means that the rate of change of the quantity B in the material volume is equal to the rate of change of B within the fixed control volume plus the net flowrate of the quanity B through the control surface. The RHS of Eq. 2 can be expressed as follows

\begin{align} \frac{\partial B}{\partial t} = \frac{\partial }{\partial t}\int_{\mathcal{V}}{\rho b {\rm d} \mathcal{V}} \end{align}


\begin{align} \dot{B}_{\mathrm{out}} - \dot{B}_{\mathrm{in}} = \int_{\mathrm{C.S.}}{\rho \mathbf{V} \cdot \mathbf{n}\mathrm{d}\mathcal{S}} \end{align}


There is an alternative way of deriving the Reynolds transport theorem, however, it makes use of the continuity equation which we have not derived yet. So this will be postponed to a later post.