Not sure how applicable this is "at the molecular level", but a heuristic (at
least in the limit of incompressible fluids) explanation is nice, and doesn't
rely on the mathematics, or at least on the notion of streamlines and the like.
Consider a pipe of cross-sectional area A1, which connects to a section of
smaller cross-sectional area A2. Continuity (which students can understand at a
gut-level, at least for incompressibles) says that the speed in the vicinity of
A2 must be higher than that near A1. For this to have happened, the linear
momentum of a given fluid parcel is higher at A2 than at A1, and so the pressure
at A1 has to have been higher than at A2.
Reversing the flow, or simply connecting back to a pipe section of larger
cross-section argues for a higher pressure at the larger cross-section again.
/**************************************
"The four points of the compass be logic, knowledge, wisdom and the unknown.
Some do bow in that final direction. Others advance upon it. To bow before the
one is to lose sight of the three. I may submit to the unknown, but never to the
unknowable." ~~Roger Zelazny, in "Lord of Light"
***************************************/
________________________________
From: William Robertson <wrobert9@ix.netcom.com>
To: Forum for Physics Educators <phys-l@carnot.physics.buffalo.edu>
Sent: Wed, November 17, 2010 2:04:20 PM
Subject: [Phys-l] question about Bernoulli
I have for some time strived to explain the Bernoulli effect in terms
of what's happening at the molecular level. We all know the
mathematical explanation that leads to higher velocities being
associated with lower pressures, but I want something that does not
rely on the mathematics. In other words, what are the molecules doing
that leads to the Bernoulli effect?
For the purposes of discussion, consider a syringe with no
constriction at the end--basically a plunger inside a cylinder that is
open on one end. When you push the plunger, the air inside moves
faster. Were you to poke a hole in the side of the syringe while
depressing the plunger, would the
outside air be pushed into the plunger because of Bernoulli? If we
agree that this is an example of the Bernoulli effect alone, and not
something else, then I would like to present three possible
explanations of what’s going on at the molecular level. I’ll try to
provide potential problems associated with each explanation, and would
appreciate comments on the explanations. Of course if you feel none of
them are correct, alternative explanations are welcome.
Explanation 1.
Before you push on the plunger, the air molecules inside and outside
the syringe are in random motion and exert equal pressures inside and
outside the cylinder. When you push on the plunger, you are causing
the molecules to have an increased velocity parallel to the sides of
the cylinder. The only way this would reduce the pressure on the
inside of the cylinder, and perpendicular to the walls of the
cylinder, is if in pushing the air inside, the plunger also somehow
causes a reduced component of velocity of the air molecules
perpendicular to the walls of the cylinder. This would not happen with
air molecules that are not enclosed in a cylinder (changing motion in
one direction doesn’t affect motion in a perpendicular direction), so
is it possible that through interaction with the walls, the column of
air inside the cylinder somehow becomes “collimated,” with reduced
velocity component perpendicular to the walls? Another problem with
this explanation is that it does not account for lower pressure with
increased speed, unless there is an increase in the “collimation” of
the air with increased speed of the plunger.
Explanation 2.
When you speed up a volume of air, you are effectively increasing the
distance between the molecules. The analogy given is that of a line of
cars moving at a certain speed, so the time difference between two
successive cars is 1 second. Now double the speed of the cars. The
time difference between successive cars is still one second (as the
explanation goes), but the cars are now farther apart. If the cars are
farther apart, then there is a lower car density. If molecules behave
in this way, then increasing the speed of the molecules would decrease
density and lead to a lower pressure.
But do air molecules really "spread out" as they move faster? In
setting up Bernoulli mathematically, we use the equation of continuity
and talk about equal volumes of fluid in different places, so it’s not
clear that there is any increase in distance between molecules that
are moving faster.
Explanation 3--Entrainment.
The basic argument here is that fast moving molecules "drag along"
molecules near the surface in question. So with the syringe, the
faster moving molecules near the sides of the syringe drag along
molecules closer to the sides of the syringe (there is a stationary
layer right next to the sides) and create an area of lower density.
This lower density means lower pressure. What is the dragging
mechanism here? If we're talking about an ideal gas, can this happen?
And if the answer is that this doesn't apply to an ideal gas, then
does that mean Bernoulli doesn't apply when friction and turbulence
are at an absolute minimum?
For further consideration, think about how each of these explanations
apply to what happens when you crack a window while driving. There is
a Bernoulli effect (or is it something else?) that causes objects
inside the car to be pushed out the window. That air rushing by your
car is certainly not collimated--the motion of your car does not alter
the perpendicular component of motion of the air outside your car.
That seems to eliminate explanation 1. The molecules of air rushing by
your car are not any farther apart as a result of the motion of your
car, so that seems to eliminate explanation 2. Explanation 3 is still
in play in this example.
To summarize, I'm looking for an explanation of Bernoulli based on
what the molecules are doing, not based on simply mathematics. I want
to do this in situations that are Bernoulli alone (possible?) rather
than Coanda or Magnus effects. Any help is appreciated.