Short answer
The two formulations are indeed equivalent, but not in a trivial way: deriving one from the other requires a change of perspective that is not obvious if you are not used to it.
To apply Reynolds Theorem, you always need to consider a material volume (meaning, a moving volume defined by some portion of the fluid), but if you have a control volume $V$ (a fixed volume through which the fluid flows), you can think about the material volume that coincide with $V$ at time $t$, and apply Reynold's theorem to it
Detailed answer
To explain the topic in full detail, I will have to talk about three things:
- Leibniz integral rule, the mathematical fondation of Reynolds Theorem
- The original formulation of Reynolds Theorem
- The conservation equation, a consequence of Reynolds Theorem for a control volume
Leibniz integral rule
https://en.wikipedia.org/wiki/Leibniz_integral_rule
Let $V(t)$ be a time-dependant volume of border $\partial V(t)$
We consider the field $g(M,t)$, that also depends on time
Let $G(t)=\iiint_{V(t)}g(M,t)dV$
That means $G(t)$ is the integral of $g$ over all the volume $V(t)$, volume that depends on time.
Leibniz integral rule says:
$$
\frac{ \partial G }{ \partial t } = \iiint_{V(t)}\frac{ \partial g }{ \partial t }(M,t)dV + \iint_{\partial V(t)} g(M,t) \vec{v_{b}}(M,t)\cdot \vec{dS}
$$
Here, $\vec{v_{b}}(M,t)$ has nothing to do with a fluid: it is just the speed of the border at position $M$ and at time $t$
Let's get give name to the 3 terms:
- the left-hand side of the equation is the total variation
- the first right term is the local variation
- the last term is the convective term
This equation looks pretty similar to Reynolds Transport Theorem, clearly.
But before writing it down, I have to clarify the physical interpretation of each one of these terms.
Reynolds transport theorem
https://en.wikipedia.org/wiki/Reynolds_transport_theorem
Now, let's study a fluid.
This fluid has a velocity at each point, $\vec{v}(M,t)$
Let's choose an extensive property of the fluid denoted by the letter $g$ (it can be $\rho$ for mass, $u$ for energy, or even the constant $1$ for volume ...)
To apply Leibniz integral rule, we have to chose a 3D region that changes with time.
We could really take whatever region we want, but to give a physical meaning to the total variation, the clever idea is to chose a region that moves with the fluid: a material volume.
That way, we know the integral of $g$ over the region is the quantity of $g$ in a specific system, namely the portion of fluid that you chose.
In that setup, we can write:
$$
\frac{ D G }{ D t } = \iiint_{V(t)}\frac{ \partial g }{ \partial t }(M,t)dV + \iint_{\partial V(t)} g(M,t) \vec{v}\cdot \vec{dS}
$$
Here, the term $\frac{ D G }{ D t }$ represents the variation of $g$ in the system, and I wrote it with the capital letter $D$ to show it is a variation for a moving (or material) system. It is a quite common convention (see here )
So to make sense of Leibniz integration rule, we had to chose a material volume.
But to study a volume that does not move, how could we do ?
The continuity equation
https://en.wikipedia.org/wiki/Continuity_equation
I will use the name "conservation equation" since it is really what it is about
Let's study the same property $g$ of a fluid, but now what we want is
$$
\frac{d}{dt} \iiint_{V} g(M,t)
$$
with $V$ a fixed volume.
Hopefully, we can use the previous theorem with a smart trick.
Chose a time $t_{0}$
I will define $\tilde V(t)$ as:
- $\tilde V(t_{0})=V$
- $\tilde V(t)$ is a material volume that follows the particles of fluid.
In other words $\tilde V$ is the material volume that coincides with $V$ at time $t_{0}$
If we define $\tilde G(t)=\iiint_{\tilde V(t)}g(M,t)dV$, we can then use Reynolds transport theorem to write:
$$
\frac{ D \tilde G }{ D t }(t_{0}) = \iiint_{V}\frac{ \partial g }{ \partial t }(M,t)dV + \iint_{\partial V} g(M,t) \vec{v}\cdot \vec{dS}
$$
Now, if I note $P$ the variation of $g$ in the system of the material volume, I can rearrange the terms to get
$$
\frac{d}{dt}\iiint_{V}g(M,t)dV = P - \iint_{\partial V}g(M,t) \vec{v} \cdot \vec{dS}
$$
Which gives me the variation of $g$ in my control volume !
And the last equation is exactly the general form of a conservation law:
For example if $g=\rho$ and there is no source (meaning the mass in the material volume is constant), we get:
$$
\vec{\nabla} \cdot (\rho \vec{v}) = - \frac{ \partial \rho }{ \partial t }
$$