Re: [Phys-l] Bernoulli's equation

["LaMontagne, Bob" <RLAMONT@providence.edu>]

Wikipedia (Google "Bernoulli's Equation") also accounts for the polytropic coefficient in it's exposition of various forms of  Bernoulli's Equation. The equations are presented with more sophisticated fonts that are easier to read than most.

Bob at PC

________________________________________
From: phys-l-bounces@carnot.physics.buffalo.edu [phys-l-bounces@carnot.physics.buffalo.edu] On Behalf Of John Denker [jsd@av8n.com]
Sent: Wednesday, December 01, 2010 2:18 PM
To: Forum for Physics Educators
Subject: Re: [Phys-l] Bernoulli's equation

On 12/01/2010 10:53 AM, Espinosa, James wrote:

> Bernoulli's equation does not require that the fluid be
> incompressible;     [1]

True.

And that's a good thing, since there are no incompressible fluids.

> what it requires is that the density along a
> streamline be constant.         [2]

Huh?  Statement [2] directly contradicts statement [1].  The
density cannot remain constant if the pressure is changing
(given that all fluids are compressible).

> This is why the equation can be used for
> gases, such as air, so long as it is not being compressed along a
> streamline.  This is why the equation can be used for subsonic
> flight, though not for supersonic flight.

That's also untrue, multiple times over.

First of all, the /first order/ version of Bernoulli equation --
which is the version most commonly encountered by non-experts --
fails for airspeeds well within the subsonic regime.  This is
well known to pilots;  it explains the difference between EAS
(equivalent airspeed) and CAS (calibrated airspeed).

Secondly, the suggested explanation for why it fails is wrong.
The real reason it fails is that the first-order equation is (and
always was) valid to first order ONLY.  It correctly accounts for
the first-order variations in pressure, density, temperature, et
cetera ... but doesn't account for any of those things to second
order or higher.

Yes, I know lots of pilot-training books refer to this issue
as the "compressibility correction" but that is a terrible
misnomer.  The problem with the first-order version has nothing
to do with a fluid that is "incompressible" or "not being compressed
along a stream line."  I've said it before and I'll say it again:
there are no incompressible fluids.

The problem (and the solution) depend much more directly on the
adiabatic exponent γ than on the compressibility κ.

As previously mentioned, all this is explained in detail at
  http://www.av8n.com/physics/bernoulli.htm

That includes the unrestricted version (not restricted to small
ΔP) as well as the first-order and second-order approximations.
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