Chronology Current Month Current Thread Current Date
[Year List] [Month List (current year)] [Date Index] [Thread Index] [Thread Prev] [Thread Next] [Date Prev] [Date Next]

Re: [Phys-l] question about Bernoulli



I think one thing we can say with a good bit of certainty is that the pressure reduction is not in any case the result of some kind of "preferential reduction of the perpendicular velocity components" and that it is not even easily related to molecular velocity at all in the case of liquids.

Here's what I *think* we have established:

1. The pressure field is set up during the transient phase that results in satisfying the continuity equation by the procedure described here:

https://carnot.physics.buffalo.edu/archives/2010/11_2010/msg00778.html

2. Now, because the process is adiabatic, the equation of state is barotropic and the density and temperature are both monotonically increasing functions of pressure. (i.e., less squeezing => lower density and temperature) as described here:

https://carnot.physics.buffalo.edu/archives/2010/11_2010/msg00762.html

3. Thus, the temperature will generally be lower in the low pressure regions. For a gas this will translate fairly directly into lower molecular velocities (in all directions), but they are not the *only* factor in the pressure reduction. For a liquid, the translation is less direct and the pressure reduction is mostly associated with the (very small) density reduction.

John Mallinckrodt
Cal Poly Pomona

On Nov 27, 2010, at 10:37 PM, William Robertson wrote:

Maybe people misunderstand what I'm looking for. The conversation
below gets halfway there. My comments interspersed.

Bill



On Nov 23, 2010, at 9:22 AM, John Denker wrote:

On 11/23/2010 08:27 AM, LaMontagne, Bob wrote:

the component of velocity of the molecules perpendicular to the
pipe walls - was it reduced?

Yes.

This is a major step. Certainly if the component of velocity of the
molecules perpendicular to the pipe walls is reduced, that would
explain a lower pressure. Any lay person paying attention could
understand that. It makes sense because if all we concentrate on in
the Bernoulli equation is pressure and velocity, we must assume that
height and density are constant.

and by what mechanism was it reduced?

Particle/particle collisions.

Particle-particle collisions cause this. Great. Is it possible to look
at what happens to molecules, as an aggregate, as they enter a
constriction? Is there some mechanism, non-mathematical, that explains
why the component of velocity of these molecules perpendicular to the
walls is reduced? It's not a question about individual molecules, but
a question about what happens to all of them, in a statistical sense.
We can look at individual collisions among molecules and between
molecules and a constricting container, subject to a pressure
difference, that might result in the molecules as a whole in the
narrow part of the container having a reduced velocity component
perpendicular to the walls.

I understand the energy arguments. I understand that pressure
gradients must result in differences in velocity. I understand that
there must be a pressure difference for the acceleration of the
molecules, which absolutely means the pressure is lower in one part of
the flow. Just hoping for a picture of what's happening to these
molecules, individually and/or collectively, that results in a lesser
perpendicular velocity component in the are of flow where they're
moving faster. And here I'm assuming that a lower perpendicular
velocity component is the answer. In my original post, I stated
various explanations for Bernoulli, two at least that included a
change in density (in one case locally and in one case globally). I
would love to be able to rule out the density argument, at least, but
that's why I'm asking everyone here to look at it.



Each collision can be visualized in terms of points on a sphere
in six-dimensional phase space. Ignoring the mass for simplicity,
we have:
v1x^2 + v1y^2 + v1z^2 + v2x^2 + v2y^2 + v2z^2 = E = constant

Each collision will cause the system to random-walk on the
sphere. If every collision were one-dimensional (i.e. zero
impact parameter) we would not have a random walk, but since
the impact parameters are random we do. The energy will soon
be distributed over all the accessible states.
_______________________________________________
Forum for Physics Educators
Phys-l@carnot.physics.buffalo.edu
https://carnot.physics.buffalo.edu/mailman/listinfo/phys-l

_______________________________________________
Forum for Physics Educators
Phys-l@carnot.physics.buffalo.edu
https://carnot.physics.buffalo.edu/mailman/listinfo/phys-l