# Why does pressure decrease as velocity increases?

Bernoulli's principle states that as velocity increase pressure decreases. But higher the velocity, greater is the temperature and pressure must be high. Can you explain the situation in both the cases?

• I think you should remember that Bernoulli's principle is only valid in inviscid incompressible flows (a situation only applicable far away from boundaries). As such no pressure gradient is required to drive the flow to counteract viscous forces. In such case why would a higher velocity lead to higher temperatures and pressures? At higher velocity, molecules are spaced further apart (i.e. their mean free path increases) and subsequently the pressure decreases. – nluigi Mar 7 '16 at 11:45
• It's very simple. The only way the velocity of a parcel of gas can increase is by being pulled by lower pressure (or pushed by higher). So there's a direct relationship between velocity and pressure. (A violation of this rule is if something else accelerates the gas, as in the case of a conductive plasma in a magnetic field.) – Mike Dunlavey Mar 7 '16 at 15:17
• "higher the velocity, greater is the temperature": may you confuse average velocity (= flow) with variance of the velocity around the average (= temperature) ? – Fabrice NEYRET Mar 7 '16 at 21:29

In a river segment, if I suddenly make the flow faster, then by conservation of the rate (amount of water crossing a vertical cut per second) the water level must decrease.

In 3D the pressure plays the role of the water level.

Energetically speaking, both play the role of potential energy, that can exchange energy with kinetics energy. (stably, or oscillately (=waves)).

Your premise "but higher the velocity, greater is the temperature and pressure must be high" is wrong.

To see why, you must recognize that the velocity field $\mathbf v$ that you calculate in fluid dynamics refers to a "fluid particle", which is composed of maybe some $10^{10\pm 5}$ molecules.

On the other hand, the (local) temperature, pressure, density and other thermodynamic parameters refer to these subsystems of fluid particles. The relation $kT\sim mu^2$ applies to r.m.s. speed of the molecules in a fluid particle and not to the CM motion of the whole fluid particle (the $\mathbf v$ field).

We can say that the $\mathbf v$ field is a manifestation of a kind of "ordered" motion, i.e. useful for extracting mechanical work. On the other hand, pressure, like temperature, is a manifestation of "disordered" motion. Therefore, it is really more intuitive (at least to me) that it goes like Bernoulli says: if the ordered motion diminishes, this energy has to go somewhere, and therefore it is reasonable that it reappears as disordered motion.

To get a good intuition of Bernoulli's theorem, I suggest Feynman.

First of all, understand that Bernoulli's equation talks about point pressure and the velocity of the fluid at that particular point. It is the conversion of one form of energy to another. If a fluid flow is restricted, the velocity of the fluid reduces, thereby the kinetic energy decreases and the equivalent amount of energy is increased as the pressure energy, which therefore increases the point pressure of the fluid.

The second case is different. It talks about the pressure gradient (difference in pressure between two points along a fluid flow) and the velocity of the fluid. If the pressure gradient is increased, the velocity of the fluid increases owing to the increase in driving force.

Hope this helps :)