Using Navier Stokes for rigid body with constant angular speed

  • #1
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 breif 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.
 

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