# Reynolds Transport Theorem for Rotating Pipe

I have a problem that I'm trying to solve for an engineering application. Maybe I have the wrong assumptions, or expectations of the underlying physics; but maybe one of you can help clear that up. The problem is this:

The pipe will be rotating along the axis specified with a certain angular velocity. Additionally, the cross sectional area A2 < A1. I've tried to derive a formula showing the relationship between the mass flow rate, angular velocity, and the radial force exerted on the pipe to no avail. All of the results I've attained indicate that there is no relationship between the radial force and the angular velocity. I believe that there is, I just haven't figured out how to solve it.

The most promising approach I've taken was to use the angular momentum version of RTT for the steady state problem:

$$\sum Torques = \int \rho(\vec r \times \vec v)(\vec v \cdot n)dA$$

But I'm interested in the radial force that the fluid exerts on the pipe. If the angular velocity is constant then the torque should be zero, and this does not help me because I cannot then use the torque definition to solve for the forces on the pipe:

$$T = \vec r \times \vec F$$

And even if the torque was non-zero, the torque is not dependent on the radial force, so it does not help there either.

I'm trying to validate CFD results with an analytical approximation, so maybe this is not a problem that RTT can help with. I'm not sure. I never worked on a problem like this during undergrad, so I need some help with this one! Maybe I need to solve for the pressure gradient, which should depend on the angular velocity, and use that to determine the flow velocities - but again I'm unsure!

If you want force in radial direction then write force balance equation in the radial direction and not in the tangential direction. Momentum flux in the radial direction is $\int_{A_1}dA~\rho(\mathbf{v_1}\cdot\mathbf{n})\mathbf{v_1}$ which is destroyed at the bend where the fluid takes a 90 degree turn. The force exerted by the pipe on the fluid (equal and opposite to force exerted on the pipe by the fluid) is equal to the aforementioned momentum flux $\approx\rho A_1v_1^2$.
• You see, mass flux through the pipe (which carries momentum) determines the angular velocity as well as the radial force ($v_1$ and $v_2$ are connected by continuity equation). Therefore knowing angular velocity you may determine $v_2$ from which $v_1$ is determined and from which radial force is determined. – Deep Apr 14 '18 at 6:18
• Ok, so I can determine $v_2$ using $\omega r$ or something else? – denbjornen505 Apr 15 '18 at 2:52