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Re: [Phys-l] question about Bernoulli



Again, there is no argument from me about the physics of the flow in terms of pressure, density, temperature, etc., as covered by the Bernoulli equation.

There are flow equations that traffic engineers can apply to situations where three lanes converge to two on a limited access highway. They will apply accurately to an ensemble of cars travelling down the road and will consistently give the same results for the overall flow of traffic when the same conditions apply. An individual driver, however, does not experience what the ensemble experiences. He sees the space between his car and the one in front of him and responds accordingly. If the car ahead stops, he stops. If the car ahead speeds up or starts up from rest, he does likewise after a short delay for reaction time. The individual driver does not experience the parameters that the engineers equations deal with. When the reduction of lanes is about to occur a few miles down the road, the driver has no idea that there is a constriction ahead - he only knows that the traffic around him is slowing down and at some point he is forced to stop. Only after a mile or so of stop and go driving does the driver see the three lanes reduce to two.

That is what I meant by my comment in a previous posting that an individual molecule does not know that it is moving from a wide pipe into a narrower one. It only knows that the symmetry of its collisions with the molecules around it is changing - it only experiences the results of the laws of physics by banging into other molecules in a purely local way.

My own way of visualizing this is that molecules entering the region of a restriction in a pipe will encounter, by necessity, a greater number of molecules colliding with it upstream of itself at the constriction and fewer from downstream - simply because the ones that make it into the constriction are moving freely downstream while the ones upstream that encounter the constriction have a significant fraction bounced back upstream. (JM's "squeezing") In an established equilibrium flow, the details of how this was set up are no longer important, there is a higher density upstream and a lower density downstream of the constriction. The asymmetry of the collisions will result in an increase in the component of velocity downstream. If the average energy of all the molecules remains constant during the multitude of collisions, an increase in the downstream speed component of the molecules in the pipe has to be accompanied by a decrease in the perpendicular components (JD's v2x + v2y + v2z = E). In essence, the temperature decreases for the gas in the constriction and the collisions with the walls have less force per molecule (as well as a density effect). Leaving the constriction results in the opposite changes. Molecules encountering collisions have many more directions to be deflected to before striking a wall so the average downstream speed decreases. I am not sure that my scenario avoids macroscopic assumptions any more than other approaches I have objected to on this thread, but it's the limit to which I can envision the microscopic happenings.

I once wrote a simulation for this in FORTRAN. I started with a large number of molecules in a box all with the same speed but traveling in random directions in 3D. I then let them collide with each other randomly and soon a Boltzmann distribution of speeds developed. Once this part of the simulation was secure, I put the molecules in a pipe with a restriction followed by a widening of the pipe. The molecules were all given the same initial KE but with a slightly higher value to the right for the component parallel to the pipe. I had the molecules leaving one end of the pipe reappear back on the other end. After a long run to establish an equilibrium flow, the molecules moved faster through the constriction as expected. What disappointed me was that I didn't learn anything I didn't already know about the microscopic details of the process in the simulation versus a real flow, The mechanism (on a molecular level) for the increased speed and reduced pressure was not revealed regardless of how I reduced and presented the data about individual molecules. It was the old thing that the better a simulation gets the less sense it makes to run it versus looking at the real process.

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, November 24, 2010 12:40 AM
To: Forum for Physics Educators
Subject: Re: [Phys-l] question about Bernoulli

On 11/23/2010 06:31 PM, LaMontagne, Bob wrote:
However, these arguments unfortunately all boil down to "the
pressure is lower because it has to be".

That is an utter travesty of what has been said.

It would be more fair to say that if the fluid did not speed up,
there would be a host of consequences. It is easy to observe
that these consequences do not occur. It is also easy to rule
them out on firm theoretical grounds. I'm talking about consequences
such as unbounded accumulation of fluid somewhere in the pipe.

Similarly, if the pressure did not decrease there would be a host
of consequences. It is easy to observe that these consequences
do not occur. It is also easy to rule them out on firm theoretical
grounds. I'm talking about consequences such as gross violations of
the conservation of momentum.