# Reynolds Transport Theorem Notational Issue

1. Oct 14, 2009

Perhaps someone can clear up some confusion for me. EQs (1) and (2) are synonymous, but use different notations for the derivative.

In my textbook, the RTT is written in the following form:

Given some fluid property 'B,' let $\beta$ be the intensive property $\beta = d\, B/d\, m$. Then the rate of change of B of the system can be written in terms of a control volume:

$$\frac{d}{dt}(B_{sys}) = \frac{d}{dt}(\int_{cv}\beta\rho\,dV) + \int_{cs}\beta\rho(\mathbf{v}\cdot\mathbf{n})\, dA \,\,\,\,\,\,\,\,\,(1)$$

But, I have seen it elsewhere using partials on the RHS and material derivative on the LHS:

$$\frac{D}{Dt}(B_{sys}) = \frac{\partial}{\partial{t}}(\int_{cv}\beta\rho\,dV) + \int_{cs}\beta\rho(\mathbf{v}\cdot\mathbf{n})\, dA\,\,\,\,\,\,\,\,(2)$$

So my questions are :

i.) Comparing the left-hand-sides of (1) and (2) I presume that $d/dt \equiv D/Dt$ ?

ii.) Why are we using partials, $\partial/\partial{t}$, in (2) and not in (1) ?

iii.) With regard to the material derivative, what is it exactly? Am I correct in saying that with respect to fluid motion, EQs (1) and (2) tell us about the change in a fluid property 'B' as we follow the system upstream?

Last edited: Oct 14, 2009
2. Oct 14, 2009