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 Dr. Tsimento May 21, 2001 08:32

Presure range of the Navier Stokes Equations

For which Pressure Range are the Navier Stokes equations valid? Are they valid in mTorr range?

 Ibrahim Mustafa May 22, 2001 01:14

Re: Presure range of the Navier Stokes Equations

Navier Stokes Equations are independent of the pressure range. Please look at how those equations are derived.

 Jim Park May 22, 2001 09:17

Re: Presure range of the Navier Stokes Equations

My texts derive the NS equations under the assumption of a continuum medium (individual molecular effects can be averaged). For a gas, this implies pressures greater than some minimum. The governing parameter is the Knudsen number, which relates the 'average' distance between molecules (the mean free path) to physical dimensions of the flow situation (duct sizes, boundary layer thickness, tube diameter, etc). Higher Knudsen numbers imply deviations from the assumption of continuum flow. Often, Kn < 0.01 is used as the cutoff. Low pressures and high mean free paths are correlated, so the pressure does matter.

For flows with large Mean Free Path, things get a bit more complicated. A nice basic understanding can be gained from the little book by Present, "Kinetic Theory of Gases", McGraw-Hill. It may be out of print. Check your institution's library for current texts.

 Jim Park May 22, 2001 09:20

Re: Presure range of the Navier Stokes Equations

Opps!

Forgot to change my e-mail address. This should be posted with the correct address.

 Barney May 22, 2001 18:49

Re: Presure range of the Navier Stokes Equations

Fundamental Equations of Fluid Mechanics (Hoffmann,Chiang,Siddiqui,Papadakis)

--------------------------------------------------

Navier-Stokes Kn <= 0.03

Modified Navier-Stokes 0.03 <= Kn <= 0.2 --------------------------------------------------

 Dave May 22, 2001 21:02

Re: Presure range of the Navier Stokes Equations

You mean p ~ 1 mTorr = 1/760 Atm ?

Gas, liquid?

It seems probable NS eqs are good, but you should give T or density in addition, to know if continuum hipothesis is good. I recommend a Book in Kinetic Theory, or Gas Dynamics or some Fluid Dynamics text book if it explains NS eqs. derivation from scratch (I mean, not the Euler type derivation considering a small fluid box, which assume continuum, but from kinetic theory).

Cheers,

Dave

 Dave May 22, 2001 21:07

Re: Presure range of the Navier Stokes Equations

PS. Additionally, for liquids you can have cavitation effects when pressure gets small (then P<Pvap, liquid becomes gas and bubbles appear, very common in boat propellers). This effect is not predicted obviously by NS equations...they need to be modified to account for that. I said. Cheers again,

Dave

 I. Dotsikas May 23, 2001 10:12

Re: Presure range of the Navier Stokes Equations

Hi,

Your pressure range should be above 10 Torr. From this range you can be sure that your fluid behaves accord the laws of continuum mechanics and you have a Mexwell distribution of the velocity molekules. If your pressure range is bolow this pressure, then your calculations are false. The more you deminish the pressure the wrongest are your calculations. In this range your modell equations (Navier Stokes) are wrong (not valid) and you must use something like Monte Carlo.

Donīt try to spring from the five meter tower to the children swimming pool. It will work (the computer gives you always an answer) but you might get hurt.

Jannis

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