Jose wrote:
fredfighter wrote:

Please show us your arithmetic. Suppose a 1500 lb airplane is
flying horizontally at 120 mph at 5000 feet above MSL. What
are the vertical and horizontal components of the momentum
of that aircraft?

That is, on a microscopic scale, no. The wing is constantly
freefalling, then being bounced back up by impact with air molecules.
Averaged over all the molecules, yes, the net is zero (the wing flies)
but on a microscopic scale, the wing is in constant brownian motion.
This implies momentum transfer, and following the momentum on a
microscopic scale is instructive.

OK, show us your arithmetic.

First, do you agree that air is made of individual molecules separated
by a lot of space compared to the size of the molecules themselves?

Yes.

Then do you accept that a wing is in freefall during all the (very brief
but very numerous) time in between molecular collisions? (If not, what
holds it up when it is not in contact with any air molecules?)

Yes.


If so, then during the time it is in freefall, it acquires a downward
velocity. Small, no doubt, but nonzero.

Sometimes it does, sometimes it does not. I'll allow as the vertical
component of velocity decreases during that time, for a positive up
coordinate system and a plane in (macroscopic) level flight.

Do you agree that in each collision momentum is transferred to the
air molecule that is equal and opposite to the momentum transferred
to the wing?

The next molecular impact
pushes it back up. On the average they will sum to a net zero vertical
motion. Is this the arithmetic you want to see?

No, I want you to calculate the horizontal and vertical componenets
of momentum for the example I gave, or any other reasonable example
of a fixed wing airplane in horizontal flight.

...

That's the shortcut. Where does the Bernoulli effect come from - on a
molecular level? That's what I'm addressing. The Bernoulli effect is a
shortcut for doing the calculation in bulk (where it makes the most
sense if you want a numerical answer) but it all comes from molecular
collisions.

I agreed quite some time ago that the theoretical basis for
macroscopic gas laws is to be found in statistical mechanics.

On a macroscopic level, the vertical component of momentum of the
wing is zero. Therefor on a macroscopic level, the sum of the
momenta transferred to the air molecules, integrated over all of
the air molecules must also be zero by Newton's third law.

Right?

For an airplane in straight level flight there is no net momentum
transfer in the vertical direction, between the air and the airplane,
just like there is no net vertical force acting on the airplane.

--

FF