Is this a correct interpretation of pressure?

So I am told that pressure = Force per Area --> F/A..

When considering the units of Force I find that force = kg * m/s^2

When considering the units of Area I find that area = m^2

Thus the units of Pressure = kg/s^2 *1/m = d^2(mass)/dt^2 * 1/m

If I interpret this correctly that means that if one had a long glass tube

(picture below):

=========================================================================

=========================================================================

And we considered a segment of the tube

                  this is the segment:


===============|===================================|=====================

===============|===================================|=====================

and we assumed that there was matter flowing through the entire tube:

===============|===================================|=====================

* * * * * *** * * ** * * * * * * ** * * * * * * ** * * *** *

===============|===================================|=====================

Then at any given moment: the pressure in this particular segment could be given by:

The change in the rate of mass flow/length of segment:

= d^2(mass in segment)/dt^2 *1/length

Is this correct?

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@007 I love how the mass is completely non-uniform. – Isopycnal Oscillation May 4 '13 at 2:21
Thanks! Good OP' – frogeyedpeas May 4 '13 at 2:29
Stupid autocorrect... I meant ASCII art not OP – frogeyedpeas May 4 '13 at 2:30

New answer

What you've done here is just dimensional analysis. But you've gone a little too far. In particular, just because two things have the same dimensions doesn't mean that they are equal. If you want to expand $F/A$ a little more, you can choose your favorite from $$F = \frac{d p}{dt} = \frac{d}{dt}(m\, v) = m\, \frac{d}{dt} v = m\, a$$ Here, $p$ is the momentum, $m$ is some amount of mass you're keeping track of, $v$ is its velocity, and $a$ the acceleration.

Now, pressure is really defined as being the force on a unit of area. So if you don't choose an area, you can't define pressure. There might be situations where there is some momentum change in a volume without the matter directly hitting a surfce. For example, with an electromagnetic force. So there could be a net force on a volume. But again, you can't call anything a pressure unless you select an area to measure it.

Old answer (from before reading your clarifications)

Pressure is the force per unit area applied perpendicular to the surface of an object. (Or at least the area over which you're imagining measuring the pressure.)

If I understand you correctly, you're interested in the pressure on the surface of the tube. Is this correct? But you're referring to the mass flowing past the tube. That is, it's flowing parallel to the surface. So, in your simplified model where the matter is flowing straight, there would be zero pressure. Of course, in real life, the matter would probably having some internal motions as well, which means it would bounce off the walls of the tube, exerting pressure.

In short, we would need more information to calculate the pressure.

For your particular setup, you still haven't told us about any interactions between particles of the matter. From what you've told me, I can still assume that the matter particles all have the same velocity (but are scattered at different positions). Now, as you know, force is proportional to acceleration. Since the velocity is constant, there's no acceleration, so there must be no force, which means there's no pressure.

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I mean the pressure of a slice cut perpendicular to the walls of the tube, so not the walls themselves but rather an imaginary wall in the middle... (Can pressure be extended to exist for entire blocks of volume?) – frogeyedpeas May 4 '13 at 2:25
I meant for the diagram to make it appear that the matter was moving at different speeds... Such that they tend to clump as well as dissipate in different areas... – frogeyedpeas May 4 '13 at 2:44