Validity of Navier-Stokes at Extreme Mach Nos. (M = 100,000)

[K41]

So, speaking theoretically, if you could accelerate a fluid to extreme Mach number at sea level, then at what point does the Navier Stokes number lose its validity? What equations would you then use to model this potentially extreme momentum?

I presume based on the fact that Newton's Law's are an approximation to the classical world that only really fail at extreme velocities approaching the speed of light, how close to those velocities do we need to be before we need to worry about the validity of the equations and can we apply corrections to N-S to correct for this?

Finally, is it theoretically possible to collect a group of neutrino's for instance, compact them so that they can regarded as a continuum (very low Knudsen number) and therefore model this from using what we've just discussed?

 

[jedishrfu]

Here's a paper that discusses the Navier-Stokes equation and its limitations and extensions:

http://web.mit.edu/ngh-group/pubs final website/2-28_DFD05paper.pdf

 

[K41]

That is more to do with small scale phenomena. I was looking more specifically at extreme velocity. But you are correct in thinking that inevitably at higher velocities, smaller scales must be found. There must surely be a limit to this though.

 

[Mech_Engineer]

Navier–Stokes equations balance momentum in the flow, and are utilized in the "continuum flow" regime. At very high supersonic speeds for high altitudes continuum flow mechanics start to break down and are replaced by "free molecular" flow due to reduced collisions between gas molecules. I'm by no means an expert in this area, but I know a practical example of free molecular flow is the a space vehicle re-entering the atmosphere.

My feeling is at very high hypersonic speeds and low altitudes continuum flow equations will start to break down due to the temperature and pressure delta across the shock boundary (if that boundary is hot enough to "induce" free molecular flow-like conditions in the shock's wake). It might be Navier-Stokes works if utilized in concert with some "creative" boundary conditions, but when you're talking M > 100,000 it's anyone's guess as to what's going on there... The speed you're describing is over 1/10 the speed of light, my guess is travel through an atmosphere at that speed would be like setting off a nuclear bomb in front of the vehicle.

 

[bigfooted]

Here is a nice introduction to relativistic fluid dynamics. Practical applications are I think mainly in the area of plasma physics:

http://mathreview.uwaterloo.ca/archive/voli/2/olsthoorn.pdf

For high Knudsen numbers, you need to solve the Boltzmann equation, I think you can derive the Navier-Stokes equations from it, so you can see it as a generalization of the N-S equations.