Fluid Mechanics PDE's and existence of solutions in inviscid, axisymmteric case

[AlexGreen]

I'm writing a major paper on fluid flow in vortices (think tornado) and the solutions don't generally exist at all. Truly original research is not required, but I want to give a rigorous proof of the Navier-Stokes eq, starting from the basic continuity equation, explain their assumptions (continuous flow, no quantum behavior), and then go on to illustrate an analytically solvable case where we assume axisymmetric flow and inviscid flow. Both these conditions reduce the complexity considerably, but the few papers published i can find still use rather sophisticated dimensional analysis to solve the equations or even more powerful (and less intelligible to me, since i haven't seen them yet) Lie algebra symmetry groups. I need some suggestions on where to find more analysis of exact solutions for fluid flow in a vortex. Or how to modify my paper so I can actually make at least SOME calculations, on my own, in a day or two. This thing has to be written and in by wednesday or thursday at the latest. I appreciate any advice or discussion.

Also, I really want to use lagrangian dynamics or hamiltonian dynamics to solve these problems but it seems that calculating the kinetic energy field is very difficult, and I'm not sure whether the potential energy field would be anything besides the gravity potential or whether it would simply be too difficult to calculate because of some other factors i am unaware of. Both would obviously be necessary to use either dynamic analysis.

 

[gtoguy97]

I'm not sure if this is possible, but I was wondering if there is any way to use numerical methods such as finite element analysis or finite volume analysis to solve the problem. I know that these methods are typically used for problems where the governing equations are known, but I'm not sure if they can be used in this case. If so, can you point me to some resources that would explain how to use these methods?Finally, I am aware of the fact that computer simulations are often used to study fluid flow in vortices, but I am not sure if I have the time to learn and use such a complicated method. Do you have any suggestions on how I might be able to include computer simulations into my paper within the given time limit? Thank you for your help!