Understand Pressure in Fluids: Conceptual Guide

[gazeem]

I'm looking for a conceptual understanding of pressure in fluid.

According to what I've gathered, in static fluid, the pressure at any point in the fluid depends on the depth within the fluid, because there is more fluid weighing down on an object the deeper into the fluid it is. However, for some reason, the forces due to pressure are acting in every direction, which means at a given depth, the net force acting on an object would be the pressure*SurfaceArea at the bottom of the object - the pressure*surfaceArea at the top of the object, and horizontal forces would cancel out because they're going in opposite directions and there's no difference in pressure horizontally through a fluid.

A question I have though:
a) If Pascal's principle is true that pressure is evenly distributed throughout a fluid, what does this really mean if at different heights within a fluid, there is a different amount of pressure? Is the pressure being exerted on a particular point of the side of the fluid container greater than the pressure exerted on a lower point on the side of the fluid container?

Further, what I don't understand then is pressure in fluid in motion. Regarding a (steady-flow, nonviscous) fluid flowing through a horizontal tube, my questions are:

b) How is the fluid moving in a direction? How come the horizontal forces due to pressure in this case don't cancel out horizontally like they do in static fluid? Or is the pressure referred to in things like Bernoulli's equation referring to the pressure the fluid exerts on the tube and not the pressure within the fluid?
c) Is the pressure the fluid exerts on the top wall of the tube less than the force it exerts on the bottom wall of the tube? Or does something about the fact that it is moving cause the pressure to be equal throughout.

Even if you don't explicitly answer all my questions, if someone could provide maybe an evident gap in my understand that's causing my misunderstandings I would greatly appreciate it, thank you.

 

[russ_watters]

A question I have though:
a) If Pascal's principle is true that pressure is evenly distributed throughout a fluid, what does this really mean if at different heights within a fluid, there is a different amount of pressure?

The basic answer to both questions/examples is that "fluid pressure is the same in all directions" is only approximately true/applicable. Real fluids are made of molecules of finite size and mass and the force above is smaller than the force below by the weight of the molecule.

For fluids in motion, the assumption requires zero frictional pressure loss in the flow.

 

[dRic2]

@russ_watters fluid mechanics assumes the fluid to be viewed as a continuum of matter. Navier-Stokes equations fail to the describe the motion of a fluid at the atomic scale as far as I know. In the assumption of continuum mechanics, pressure is a scalar quantity thus it has to be independent of direction. If we are considering a single molecule then what does pressure even mean ?

Not an expert but I'll try my best to answer.

According to wikipedia:

Pascal's law (also Pascal's principle or the principle of transmission of fluid-pressure) is a principle in fluid mechanics given by Blaise Pascal that states that a pressure change at any point in a confined [/u]incompressible fluid[/u] is transmitted throughout the fluid such that the same change occurs everywhere.

Here we are talking about changes in pressure, not the value of pressure itself. This is a consequence of thinking the fluid as incompressible.

b) How is the fluid moving in a direction?

It depends. If the fluid is moving at constant velocity then the resultant force acting on it must vanish. If there is no viscous drag you don not need a pressure gradient to move the fluid Just like you do not need to push a rock in free space. You need a pressure gradient because in the real world there is always viscous friction which opposes the motion of the fluid.

c) Is the pressure the fluid exerts on the top wall of the tube less than the force it exerts on the bottom wall of the tube?

Technically yes. But most of the time you neglect this little contribution (especially for little tubes).

 

[Chestermiller]

The reason that the pressure is the same horizontally at a given depth as vertically: Picture squeezing a water balloon from above and below. What happens sideways?

Pressure is the same in all directions at a given point in a fluid. It is present throughout the fluid, both at the walls and internally.