There is widespread confusion (among learners) about the Bernoulli effect, and I think
a large part of this confusion is owing to muddling up cause and effect. So
I am providing this question and answer to see if others find it helpful.
Let's consider laminar flow (i.e. smooth flow in layers) of a fluid with
pressure $P$, density $\rho$, flow speed $v$, gravitational
potential $\phi$. Then Bernoulli's theorem is that,
when heat conduction and work against viscous forces is negligible then
along any given streamline,
$$
u + \frac{P}{\rho}+ \frac{1}{2} v^2 + \phi = \mbox{const.}
$$
where $u$ is the internal energy per unit mass of fluid when the fluid is not
moving and is at zero height. (The $u$ term can be dropped for an incompressible
fluid).
The Bernoulli theorem is often presented with statements along the lines of
"when the velocity is larger, the pressure drops owing to the Bernoulli
effect". You might see a diagram showing a pipe of diminishing diameter
with pressure sensors such as manometers attached. The accompanying text
asserts that the pressure drop is owing to the increase in flow velocity.
Hence my question:
In the Bernoulli effect, is it that the increased flow velocity causes the
drop in pressure, or is it the other way around: the drop in pressure causes
the increased flow velocity?
Best Answer
The answer is that the pressure change is the cause and the flow velocity is the effect.
Note that the Bernoulli theorem applies to the changes of $P$, $\rho$, $u$, $\phi$ and $v$ along a streamline. It is not a statement about pressure gradients in some other direction. Let's take a case where $u$ and $\phi$ are independent of position. If the pressure in some region is higher, then it makes sense that the fluid will slow on moving into that region, since it has to do work on the fluid ahead to push it along. Similarly, on moving from a higher to a lower pressure region the flow will speed up because pressure forces from the higher pressure region do more work (for a given distance moved) than the work expended in pushing against the fluid ahead, so the fluid's kinetic energy will increase.
When we consider Newton's second law we say the force is the cause and the acceleration is the effect. We should say the same about fluid flow.
A thoughtful physicist might reply that since Bernoulli's theorem applies to a steady state of flow it is not possible to separate cause and effect; the theorem merely asserts that both properties appear together and one cannot have one without the other. I feel that nonetheless it is clearer to say that the pressure difference is the cause and a change in flow velocity is the effect, because this makes sense of the process happening at each fluid element: it is being accelerated exactly as Newton's second law dictates.
Let's consider flow down a pipe of diminishing diameter. As the fluid enters the region of diminishing diameter, the flow is restricted at first so, before steady state conditions are achieved, there is liable to be a slowing of the flow into region ahead of the restriction as the density there builds up. This will also increase the pressure. The increase in pressure is caused by the arrival of more fluid than leaves. This results in a pressure difference between this region and regions further along, with the result that fluid leaving the region accelerates and flows more quickly through the restricted diameter. Eventually there is a steady state of flow and Bernoulli's theorem can be applied, and my statements about cause and effect also apply.
There is a very nice demonstration experiment where one hangs two curling pieces of card or paper from adjacent horizontal rods. The investigator (or victim) is invited to blow air between the cards and the intuition is that one might expect them to move apart, but in fact they move together. Why? It is because the air pressure behind the cards stays fixed, while the air flow set up by the blowing leads to a lower pressure in the moving air between the cards. It seems amazing that this blowing will lower rather than raise the pressure. But I think what happens here is that there is an initial transient phase in which there is an increased pressure while the air initially between the cards is accelerated away, and then the steady flow at lower pressure sets in after that.
When air flows more rapidly over the top than the bottom of an aerofoil, this is owing to (rather than the cause of) the pressure variation along the streamlines. There is a lower pressure above than below the aerofoil, and therefore a lift force and also a greater acceleration of air along the flow lines that pass over the wing. The interesting question now becomes, how does that pressure difference come about in the first place? To answer this one gives thought to the movement of the whole fluid from before to after the aerofoil (in a model where the aerofoil is fixed and the fluid flows) and it becomes apparent that in steady state the flow pattern must be as just described and therefore the pressure variation must be as just described. But this 'therefore' is a reasoning or deductive 'therefore', not a cause-and-effect 'therefore'.
(By the way, the only forces on the wings of an aeroplane are the forces acting right there on the wing. These are pressure forces and viscous forces. It is not good physics to say that movement of air after the wing has passed by is the 'cause' of lift because there is no action at a distance in physics. The cause of the lift is the interaction right at the wing surface.)
Added note
This is an added note to react to comments on the relation between force and acceleration.
Forces come about via interactions such as the electromagnetic interaction. When molecules in a fluid push on one another, it is owing mostly to close collisions. The electron clouds come close and each repels the other. Each electron cloud is a source of an electromagnetic field which interacts with charges. For each field event where a charge is present and consequently experiences a force, the events which act as source of the field lie in the past light cone. This helps to clarify that the acceleration is indeed a response to a situation that came about owing to past events. The acceleration is not the cause of the force which is acting.