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pressure at an intangible boundary



cliff parker wrote:

Pressure equals force per unit area. Seems like a
surface is an necessity.

People seem to be struggling with this.
It is important to realize that air is a
fluid, so it has pressure everywhere.
Pressure is not something that happens only
when the fluid hits a tangible surface.

Yes, pressure is force per unit area.
But we can get a more sophisticated notion
of pressure if we restate that as momentum
flow across the area.

Force is just momentum flow. (The _net_ force
is the momentum flow from A to B minus the
momentum flow from B to A.)

Pressure is a particular part of the momentum
flow, namely the flow in the x-direction of
the x-component of momentum, plus flow in the
y-direction of the y-component, plus flow in
the z-direction of the z-component.

(The other parts of the momentum flow, such as
the flow in the x-direction of the y-component
of momentum, are perfectly real but aren't
called pressure. They have to do with viscosity.
All these terms collectively are called _stress_.
Pressure is one part of the stress. For now
let's assume zero viscosity, and talk only about
pressure.)

Anything that carries momentum from parcel A to
parcel B contributes to the pressure. The figure
http://www.monmouth.com/~jsd/physics/gif48/mom-flow.gif
shows two different ways this could happen.

On the left, we have some gas in a box. The pressure
in the top of the box is less than the pressure in
the bottom of the box, because a gravitational field
is acting on the fluid. Gravity is causing a downward
force, which in equilibrium is balanced by an upward
force due to the pressure gradient. We can see how
the momentum is being transferred, because there are
two places where the blue particle hits the red
particle, transferring momentum at the boundary
between top and bottom. The boundary is intangible,
but it is still a boundary.

You can, if you want, put a tangible boundary there,
i.e. a piston. Then the blue particle transfers
momentum to the piston and the piston immediately
transfers it to the red particle. This doesn't
change the story. Momentum transfer is the key
idea, and that happens with or without the piston.

Now we turn to the scenario on the right side of
the figure. It's the same, except that the two
particles don't collide -- they just fly past each
other. The momentum transfer is the same!!!! It
is still true that the pressure in the bottom is
larger because of gravity, and it is still true
that momentum is being transfered from bottom to
top because of the pressure gradient.

In this case, you do not have the option of
installing a tangible piston. The boundary
is necessarily intangible. But we can still
talk about momentum flowing across the boundary.

This notion of "flow" is central to what we
mean by conservation; see
http://www.monmouth.com/~jsd/physics/conservative-flow.htm

Momentum flow is how you derive the equations
of fluid dynamics:
http://www.monmouth.com/~jsd/physics/euler-flow.htm

All this generalizes beautifully to D=3+1 spacetime;
see Misner/Thorne/Wheeler Chapter 5.

This posting is the position of the writer, not that of Euler, Navier,
or Stokes.

This posting is the position of the writer, not that of SUNY-BSC, NAU or the AAPT.