# Reynolds transport theorem and steady flow

Consider the Reynolds Transport Theorem in the following form: $$\frac{d}{dt}\int_V{\rho}\boldsymbol{v} \ dV = \int_V{\frac{\partial(\rho\boldsymbol{v})}{\partial t}} \ dV + \int_S{(\rho\boldsymbol{v})(\boldsymbol{v}\boldsymbol{n})} \ dS$$ where $\boldsymbol{n}$ is the normal vector, $\rho$ density, and $\boldsymbol{v}$ velocity. $S$ and $V$ denote the surface and volume a fixed control volume.

For a steady flow of an incompressible fluid $\frac{d}{dt}\int_V{\rho}\boldsymbol{v} \ dV$ is zero. Since $\frac{\partial(\rho\boldsymbol{v})}{\partial t} = \rho\frac{\partial\boldsymbol{v}}{\partial t}+\boldsymbol{v}\frac{\partial\rho}{\partial t}$, this terms should also be zero with the given conditions. This implies $\int_S{(\rho\boldsymbol{v})(\boldsymbol{v}\boldsymbol{n})} \ dS = 0$. Which can not be generally true.

Where is flaw in this argumentation?

EDIT

The following example came up in the discussion: Consider a simple pipe with a 90 degree bend, constant $A$ and influx velocity $\vec{v}_{in}$ (something like this: http://thepipefitting.com/wp-content/uploads/2014/08/90-degree-elbow-drawing.jpg).

$$\int_S{(\rho\boldsymbol{v})(\boldsymbol{v}\boldsymbol{n})} \ dS = \int_{S_I} \begin{pmatrix}0\\v_{in}\end{pmatrix}\left[ \begin{pmatrix}0\\v_{in}\end{pmatrix} \begin{pmatrix}0\\-1\end{pmatrix}\right]dS_I + \int_{S_{II}}{ \begin{pmatrix}v_{out}\\0\end{pmatrix}\left[ \begin{pmatrix}v_{out}\\0\end{pmatrix} \begin{pmatrix}1\\0\end{pmatrix}\right]}dS_{II} = \rho A \begin{pmatrix}v_{out}^2\\-v_{in}^2\end{pmatrix}$$ which does not evaluate to $\begin{pmatrix}0\\0\end{pmatrix}$ since $v_{in}$ is a non-zero velocity.

• This is a pretty common confusion. For a detailed explanation, see this answer Commented Jun 25, 2023 at 12:28

The flaw is that, in steady flow, $\frac{d}{dt}\int_V{\rho}\boldsymbol{v} \ dV$ does not necessarily have to be zero. This quantity represents the time rate of change within the moving control volume. Each subsequent moving control volume might exhibit the exact same time rate of change. As reckoned by a stationary observer, this would mean that at any given spatial location, nothing would be changing with time.