# Can I treat fluid a rigid body calculating Moment of inertia? If not, how would I calculate it?

I wanted to set out calculating how much moment of inertia a bottle of water had so as to see how fast a bottle had to spin to create a non-zero constant angular velocity in the water.

As I understand, concepts of angular momentum and linear momentum (moment of momentum) applies to points, and how moment of inertia can be found when many particles are clumped together as a rigid body.

To my knowledge other than due to hydrostatic pressure differences, the water body will deform due to viscosity of the particles, in which angular velocity of the container will be passed on to the particles on the side. Is the difference small enough that I could still go on to treat the entire fluid as having a rigid body Rotation?

e.g. Having the Navier-Stokes equation of motion for fluid flow reduced to $$G = g - a$$.

## 1 Answer

No, you can't the MMOI of a "static" fluid for a rotating scheme.

If you slowly bring a rotating container of fluid up to speed you will notice the fluid will "ride-up" bringing a lot of the mass further away from the rotating axis. This will increase the MMOI dramatically.

Even though all the fluid might be moving with the same rotational velocity in a steady-state condition, the shape of the fluid will be different enough to make this sort of idealization, not a good one.

If you have a good idea of the distribution of fluid mass, then you can try to do the integral to find the MMOI of the deformed fluid using the volume of the revolution process from calculus.

For a fluid which not all parts rotate with the same rate, there is still a total angular momentum involved which you can calculate. And if you know the rotational speed of the container you can divide the two to estimate an effective MMOI for the fluid.