Using Navier Stokes for rigid body with constant angular speed

[SmileyBG313]

I've seen several examples of using Navier Stokes in a rotating container where gravity is purely in the Z direction. These solutions generally used cylindircal coordinate systems.

I wanted to attempt this problem where the gravity vector does not point purely in the Z direction. (ie g=g(r,theta,z) with components g_r, g_theta and g_z).

It seemed simple enough, but when I got near the end and did a units check, I realized I had a problem. basically, it came from the fact that g_theta had units of degrees, but needed ft/sec^2 (or m/s^2 which ever your preference) as both g_r and g_z had.

Here is a brief outline of what I did (which mirrors the problems I've seen.)

Assuming steady state, rigid body and incompressible fluid, I started with

-(gradient of Pressure vector)+ density*(gravity vector) = density*(accleration vector)

And wrote out the components in terms of the cylindrical coordinates.

[here the previous problems assumed that g_theta and g_r were 0 and g_z=-g. I am keeping them as variables]

Also, in the acceleration, the z and theta components ar 0, while the r compontent is centrifigal acceleration (-omega^2 * r)

since the sides are equal, we can collect terms for each of the unit vectors and these each become there own equations.

[this yeilds what each of the partial derivatives of pressure are]

since P=p(r,theta,z) then

dP = (partial of p with respect to r)*dr + (partial of p with respect to z)*dz + (partial of p with respect to theta)*dtheta

**(this may have been where my error is)

since these partial dreivatives have already been solved, they are plugged in and the integrals are taken, integrating from a reference point p1=p(r1,theta1,z1) to an arbitrary point.

solvinging and assuming values for the reference point at the appex of the surface gives you workable equations. By assuming that the the surface has a constant pressure across it, the equations can be solved so that z=z(r,theta).

[also, it is neccisary to compare the volumes of the fluid before and during rotation to solve for the apex point]

If you like, I can attach a file that is more indepth. I appreciate any help that can be given.

 

[anathema]

Thank you for sharing your approach to solving the problem of a rotating container with a non-uniform gravity vector. It is always interesting to see different methods and perspectives in tackling a scientific problem.

From your description, it seems that your equations and approach are correct. However, as you mentioned, the issue lies in the units of your g_theta component. This is a common mistake in many scientific calculations, and it is crucial to always double check the units in order to ensure the accuracy of our calculations.

In your case, the g_theta component should have units of radians per second squared, not degrees. This can be easily converted by multiplying by a conversion factor of (pi/180)^2. Once this correction is made, your equations should give you the correct results.

Thank you for pointing out this issue and for sharing your approach. I hope this helps and good luck with your further investigations.

 

[m2003]

I find your attempt to apply Navier Stokes for a rigid body with constant angular speed in a non-traditional manner to be intriguing. It is commendable that you are thinking outside of the box and trying to solve a problem that has not been addressed before.

However, as you have discovered, there are some issues with your approach. The units check that you performed is a crucial step in any scientific calculation and it is good that you caught the error before proceeding further.

I would suggest that you revisit your equations and see if you can find a way to incorporate the non-traditional gravity vector without creating any unit inconsistencies. It may require a different approach or a different set of equations altogether.

Additionally, it would be helpful to see the more in-depth file that you have mentioned. This may provide more insight into your thought process and potential areas where improvements can be made.

Overall, I applaud your effort and encourage you to continue exploring this problem. As scientists, we must always be open to new ideas and approaches, and your attempt to apply Navier Stokes in a non-traditional manner is a testament to that mindset. Keep up the good work and I wish you success in your future endeavors.