It is important to realize that in fluids (including water and air)
there is 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 just one contribution to the momentum flow. There are
other contributions to the the momentum flow, notably from the
shear viscosity and the bulk viscosity. We will not
have much to say about viscosity, except to say: (a) It is
trickier than you might imagine to give a simple, non-mathematical
way of distinguishing pressure from bulk viscosity effects;
the simple explanations are not correct, and the correct explanations
are not simple. (b) There are many practical situations where
the effects of viscosity are small compared to other effects,
especially when we are interested in large objects moving quickly
through ordinary not-especially-viscous fluids.
Temporarily, for simplicity, let’s assume the viscosity is very small,
and talk only about pressure.
Figure 1 shows two different ways that momentum can be
carried from parcel A to parcel B. Both ways contribute to the
pressure.
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.
In either case, we have one particle that turns around when it
collides with the bottom of the box; this is how pressure acts
on the bottom surface. Also in either case we have one particle that
turns around near the top of the box due to gravity; at the top
of the box there is very little pressure. In either case the
whole process can be summarized as a transfer of momentum from the
gravitational field to the bottom of the box.
This notion of "flow" is central to what we mean by conservation; see
reference 1.
Momentum flow is how you derive the equations of fluid dynamics;
see reference 2.
All this generalizes beautifully to D=3+1 spacetime;
see reference 3 Chapter 5.
