Figure 1: Pressure = Momentum Flow
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.