Help with Modeling a Liquid Vortex. (Related to General Fusion) I want to model liquid lead swirling in a sphere.  This is connected to General Fusion’s fusion machine.  A 55 million dollar, Jeff Bezos funded, 60 person company trying to change the world with cheap, clean, fusion energy.  Here is the problem:

I do not think there is an analytical solution (to navier-stokes) for this.  I looked at Oseen-Lamb vortices and Rankine Vortices.  Also, I don’t think GF published their solution.  I want to answer the following questions (in this order):


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*Is there a canned solution to this? (I don’t think so)

*Which situation do you model? (Steady State)

*What is the right coordinate system to use?  

*How do you write the boundary conditions for this?

*Can you arrive at math that is solvable? 

*Does the centripetal force overcome the surface tension/inter molecular forces? 

*What is the shape of the air cavity?

*What is the minimum speed needed to maintain the air cavity?

*Do they continuously inject and pull during compression?


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I am going try solving it.  I will post what I have on here.  Help appreciated.
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Earlier version of this question:
I am working through a problem, modeling liquid lead spinning in a 1 meter diameter sphere. Looking for some help. This is a model of General Fusion's machine. Here is a picture of what I am trying to model:

The reactor is a liquid lithium/lead fluid being swirled around a steel chamber, with an air cavity in the center. 14 pistons strike anvils which sit in holes along it's outer walls. This creates a pressure wave which compresses a cavity in the center. At present this cavity is air filled.  Here are the properties’ of the liquid lead (roughly):
•   Density: 10,000 Kg/M^3
•   Temperature: 673 Kelvin
•   Viscosity: 0.18 N*S/M^2
•   Lead Velocity (at Wall, Estimate): < 4 M/S
•   Air cavity: 0.4 meters diameter
My gut tells me that air/lead surface tension will also be needed. What I want to understand is the shape of the cavity in the center. I imagine a vortex like water draining from a bathtub.

Except that this is drained from both top and bottom. My plan is to start with the two dimensional case, Navier-stokes equation, incompressable, steady state, in cylindrical coordinates:

By making this incompressable, can't I remove the other terms on the left hand side of the equation? Yes/No?
If so, I would continue to simplify.

I would start the process of separating the variables, but I need to deal with this pressure term. How do I eliminate it? Or do I try to solve Pressure as a function of Theta, R and Z? Any other issues you see?
Other options:
After solving this situation, I intend to add in the Z-direction, to see why fluid tilts as it spins around. I know that fluid moves fast as it gets closer to a drain. I also see that using Bernoullis' equation I can find that there is a pressure drop near the center of a vortex. If anyone has another approach to modeling this vortex, I would appreciate it.

Would love for any assistance here.
Edit: Looking at two option for modeling this vortex.
1. Rankine: The Rankine is a very simple model of the vortex. It has a center where the rotation rises linearly. It passes a critical radius, where there is the highest rotation. After this it decays 1/r. This is an analytical solution to the Navier-Stokes. I put this into excel and plotted it.



*Lamb-Oseen: This is a Rankine vortex, which decays with time due to viscosity. I was thinking that this (running backwards) could be a way to estimate the starting of a vortex.



Both of these equations do not help me with the shape of the cone. Can anyone else recommend an analytical solution of the Navier-Stokes for this?
 A: 
Is there a canned solution to this? (I don’t think so)

No, probably not. There are very few cases in which there is an analytic solution for the Navier-Stokes equations. For pretty much all cases, you have to do it numerically.

Which situation do you model? (Steady State)

This is only a guess, but probably once the air has filled the sphere, you should be able to start the simulation. I'd model the air as a boundary condition so that you only worry about the liquid lead.

What is the right coordinate system to use? 

Spherical.

How do you write the boundary conditions for this?

Mixed: Dirichlet boundaries for mass and von Neumann for velocity.

Can you arrive at math that is solvable? 

If you mean, Is this problem analytically solvable? then the answer is likely no (see my answer to #1). If that's not what you mean, then I don't know what you mean.

Does the centripetal force overcome the surface tension/inter molecular forces? 

I imagine that it can, but if you're modeling this as a fluid (i.e., using Navier-Stokes), then you shouldn't care about this (it's a fluid, we ignore atomic interactions because of the difference in scales).

What is the shape of the air cavity?

Looks like a prolate spheroidal.

What is the minimum speed needed to maintain the air cavity?

I think you'd have to use a two-fluid model to determine this. I haven't worked on a two-fluid model, but it looks like it's a rather tough thing to do (probably even more so if you're trying to write your own code, taking someone else's code is just too easy--if you do that, make sure you do not treat it as a black box that gives you data, understanding how & what it does is key to understanding any result it gives you).

Do they continuously inject and pull during compression?

It looks like the plasma flows through the pipes at a continuous rate, but that's a spec you'd have to get from General Fusion.
