Bernoulli's principle

Question

In steady flow, the air that flows faster causes a lower pressure, the lower pressure causes it to move faster, or exactly both at the same time?

Also how is it that we're talking about the pressure of the fluid, if in reality that term for pressure comes from a pressure that's outside the immediate system (according to the derivation; see below).

$$dW=p_1A_1ds_1 - p_2A_2ds_2$$

• $dW$ is work done on the system
• $p_1A_1ds_1$ is work caused by a pressure that's outside the fluid and in the same direction as the flow

• $p_2A_2ds_2$ is work caused by a pressure at the other ending of the system that opposes the direction of the flow.

• If this is not immediately clear, I'm considering a closed pipe and my system is an element of fluid within that pipe.

One last thing: how do you solve any relativistic effect for the Bernoulli equation? Like, suppose I'm in a car with a certain velocity, is the pressure inside my car greater or less than the pressure outside? Well that depends on the frame of reference, etc.

EDIT: The car is not sealed.

Answer 1

Confusion abounds in conversation relating to Bernoulli's principle, so I will do my best to dispel some of the more pernicious misconceptions.

(1) Bernoulli's principle, in general, applies only to a single streamline, with unique stagnation properties. While many aerodynamic/hydrodynamic flows (e.g. irrotational flows) do exhibit uniform stagnation pressure everywhere, this is generally not to be expected. Case in point, even though the static pressure just inside and just outside your car window are equal, the stagnation properties will be very different (higher outside, substantially lower within the confines of the car, where the flow is a swirling welter of separated flow). Faster velocity does not always mean lower pressure, contrary to what many are led to believe. You must look at the flow as a whole, especially the upstream conditions.

(2) Stagnation properties do in fact depend on the reference frame, while static properties do not. That is to say, stagnation properties are relative. In the stationary (ground) reference frame the ambient air is not moving, so the static and stagnation properties are identical. However, in the car's reference frame, the flow far upstream is moving rather quickly, so it's stagnation pressure, temperature, and density will necessarily be higher than the corresponding static values.

(3) Pressures, or more accurately, pressure gradients cause accelerations in an ideal flow, not the other way around; forces cause accelerations, not vice versa. This is a big source of confusion when talking about aerodynamic lift, vorticity distributions, and the Biot-Savart Law, but just keep in mind the difference between a physical principle and a merely useful mathematical tool/concept. The phenomena are coupled via conservation of mass and conservation of linear momentum, but ultimately the imposed pressure field causes the observed velocity distribution.

(4) It's best not to construe Bernoulli's principle as an expression of conservation of energy, as technically it's not (it completely neglects internal energy). No, Bernoulli's principle is best understood as an integrated expression of the conservation of linear momentum, $F=ma$, where nominal force is replaced with force per unit volume (which is equal to the negative local pressure gradient), and mass is replaced by mass per unit volume (which is the fluid density). In these terms, Newton's Second Law becomes:

$\frac{Force}{Unit Volume}=\left(\frac{Mass}{Unit Volume}\right)(Acceleration)$

which gives us

$-\frac{dP}{ds}=\rho v\frac{dv}{ds}$

which is rearranged as

$\frac{dP}{ds}+\rho v\frac{dv}{ds}=0$,

or

$\frac{dP}{\rho}+vdv=0$,

which when integrated gives us the classic incompressible Bernoulli Eqn.:

$\boxed{P+\frac{1}{2}\rho v^2=P_0}$,

The best reference on this topic (like most topics concerning fluid mechanics) is the NCFMF series entry entitled "Pressure Fields and Fluid Acceleration," hosted by the legend himself, Ascher H. Shapiro. He clears up a lot of the confusion surrounding this concept in a readily intelligible and fascinating way.

https://www.youtube.com/watch?v=8VrTpLa4qbM

Answer 2

As an answer to:

... flow hits the top bump in the wing, it wants to keep a linear trajectory, and because the wing has curvature this will then create some kind of partial vacuum, and then this causes the pressure gradient. Am I right?

Yes. The Bernoulli effect is simply a moving fluid stream carrying away the surrounding fluid, leaving a low pressure zone around it. If an object's surface is nearby it will be sucked onto the stream (my apology to Prof. Julius Sumner Miller, the atmosphere will push it on). The air travels faster over the top of an earoplane wing, relatively speaking, because it has further to go over the top in the same time . It is held on the surface by the Bernoulli effect & deflected downwards, sucking the wing up (sorry Julius, the atmosphere pushes it up).