I understand the meaning of pressure on a surface to be force acting per unit area.
But when it comes to understanding pressure in the context of fluids in motion or at rest I think I am having some trouble grasping the concept.
When we say the pressure of a gas in a container (which may be of any arbitrary blobby shape) is say 2 Pascals, what does it really mean?
1) Does it mean the pressure of the gas on ANY small area considered on the surface of the container == 2 pascals?
2) Or Does it mean the pressure on a thin small wafer (possible imaginary?) placed anywhere inside the gas == 2 pascals?
I am really confused!
To add to my problem of understand pressure, is a sentence from Chorin and Marsden's a Mathemtical introduction to fluid dynamics when they define an ideal fluid.
*"Lets us define an ideal fluid with the following property: For any motion of the fluid there
is a function p(x,t) called the pressure such that if S is a surface in the fluid with a chosen unit normal n, the force of stress exerted across the surface S per unit area at x in S at time t is p(x,t) n i.e.
force across S per unit area=p(x,T)n"*
Here is my problem: Suppose we have an ideal fluid and a time and position varying pressure function as above. For a thin small wafer placed at x at time t, there are two equal and oppsite forces acting across its surface i.e. p(x,t)n and -p(x,t)n since the surface has 2 forces in the direction of the two normals to it.
This reasoning means the force on the thin small wafer is zero. Which means that the force at any point the the fluid at any time is zero. Which is totally counterintuitive. Why would they define an ideal fluid like that?
Best Answer
I think the wafer thought experiment is a little self-defeating, the reason being that the force exists equally on both sides of the wafer. Maybe you'll say, well that's not a problem, because there's still a measurable force between the two surfaces. But that force also exists on the edges, so we would change the problem of defining/measuring pressure in a gas to that of measuring it in a solid. It doesn't make headway to answer the question.
My definition:
Consider a rigid container that contains a perfect vacuum in the inside. This container must be placed within the gas in question, and it must not be moving relative to the gas. The pressure of the gas is then defined as the force per unit area on any flat surface of this container. In the case of an infinitely flat right cylinder, this can be directly measured by compressive force between the two flat circular sides.
Impractical? Yes. However, I think this answers the question. There is an objective physical concept of a perfect vacuum. Without introducing that, there can never be any definition of absolute pressure. Otherwise, who's to say what zero pressure is?