# Relationship between velocity and pressure of a fluid in motion

I'm confused about the relationship between the velocity and the pressure of a fluid in motion. According to Bernoulli's equation, mathematically when the velocity increases the pressure has to decrease and vice versa because of the conservation of energy. But why doesn't the pressure law (P = F / A) prove this? I mean, for fluids in motion, once the area of a cross section decreases the velocity increases, thus the pressure should increase. Unless the force changes too (decreases).

Any rational explanation about it?

• I also added an answer, you see right. – enbin Nov 21 '19 at 14:34

## 3 Answers

I believe the confusion can be resolved first by realizing that directions are important in your question. And second by understanding the microscopic origin of pressure.

The first observation then concerns the fact that force is a vectorial quantity and the definition of pressure you give is not general enough. So the force that should be taken into account is the one done perpendicularly to the surface $A$. Alternatively you can consider an area $A$ as a vector too, then the general definition would include $\cos\theta$, where $\theta$ would be the angle between the force and area vector.

For the second point, intuitively, think that pressure is produced by particle collisions which exert a force on a surface (by delivering momentum). Having said that, Bernoulli's equation concerns mostly pressure due to height differences and perpendicular velocities (mostly), no regard to external forces or directions. So a way to think about it is the following. Think about a flat horizontal surface (perhaps a house roof), under normal conditions particles in the air fly in all directions and collide with the roof producing a certain pressure (let's say atmospheric pressure). When the wind starts blowing strongly then most particles in the air will have a horizontal velocity so vertical collisions will be reduced, namely less particles will exert force on the roof since most of them are flying horizontally. The consequence is that the force is reduced, hence the pressure on the roof is reduced (so if the wind is hard enough to reduce the pressure to a value below the pressure within the house (ignoring how it is attached) it will get blown.) So indeed more velocity is less pressure at a macroscopic level for directions that are perpendicular.

• Your example helped. – Stephen Alexander May 15 '18 at 10:06
• Unfortunately, the roof example isn't correct. "particles in the air will have a horizontal velocity so vertical collisions will be reduced, namely less particles will exert force on the roof since most of them are flying horizontally" doesn't follow a-priori. It depends on how the air was accelerated: If it was speed up via a nozzle, then some of the vertical motion was converted to horizontal motion by the nozzle walls and associated shaped flow. (The air also cools in the process) But most wind wasn't accelerated that way, and stays both the same pressure and temperature as static air. – Bob Jacobsen May 15 '18 at 14:50
• Should I add the roof is in the middle of an empty field? And that the wind is strictly flowing horizontally in a laminar flow? I believe you are missing the point of the fundamental principles in play which I think the OP was looking for. If under the conditions stated above you still believe you can prove my example fails, feel free to open a chat to discuss. – ohneVal May 15 '18 at 15:10

For a fluid to speed up, the pressure behind it has to be more than the pressure ahead of it. That difference provides the needed net force to accelerate the fluid.

Hence the pressure in the narrow throat of a Venturi has to be less than at the entrance.

To slow a fluid down, the pressure in front has to be more than the pressure behind. Ditto.

So why doesn’t your P = F/A idea work?

It’s because the letter “F” means different things at different places. Having the same letter doesn’t mean they”re the same thing or same number; only physics determines that.

• Doesn't a fluid move from a high pressure to a low pressure one? – Stephen Alexander May 15 '18 at 10:08
• @StephenAlexander thanks for catching that, I had a comparison backwards. Fixed. – Bob Jacobsen May 15 '18 at 13:39

It is assumed that when the fluid flows at Q flow rate in a uniform diameter pipe, the pressure required upstream of the pipe is p (small letter p). Assuming that the diameter of section A downstream of the pipe decreases, the flow of the fluid is hindered. If the flow rate is still Q, then the pressure upstream of the pipe must be increased to P (capital P). In this way, the pressure is large (the upstream pressure is larger than that of section A), the velocity is small (the upstream velocity is smaller than that of section A) and the velocity is large (the section A is larger than the upstream velocity) and the pressure is small (the section A is smaller than the upstream pressure).