Imagine two bodies of fluid separated by a horizontal frictionless, and slightly flexible membrane, with the top body of fluid moving with a finite velocity to, say, the left, while the bottom body of fluid is stationary. Assume for simplicity that there is no gravity, or else that the whole system is in free fall.
According to the Bernoulli's principle the pressure of the moving fluid acting on the top of the membrane would be less than that exerted by the stationary fluid underneath. (That is a common (at least part) explanation given for the lift exerted on an aerofoil shaped wing of an aircraft.) Therefore for a stationary observer, A, the slightly flexible membrane would be observed to flex upwards.
However, consider a second observer, B, moving with uniform velocity to the left, at exactly the same speed as the top body of fluid. To observer B, the top body of fluid would appear to be stationary, and the bottom body of fluid would appear to be moving to the right. So for observer B, the application of Bernoulli's principle would lead to the conclusion that the pressure of the moving fluid acting on the bottom of the membrane would be less than that exerted by the stationary fluid above the membrane. So to observer B, the membrane would be observed to flex downwards.
How can it be the case that the two observers, A and B, moving with constant velocity relative to each other, would observe the membrane flex in opposite directions?
The Bernoulli equation is frame-dependent. So, the pressure drop depends on the frame of reference. I think that the correct way is to use the work-energy theorem, which is frame-independent.
There is no paradox. What you have described are two different problems. If you consider the same problem by observers A and B, you will obtain the same result.
Namely, the forms of basic laws in the absolute and inertial relative frames of references are the same, with an exception that in the absolute frame of references the absolute velocity is used, and in the relative frame of reference the relative velocity should be used. It is important to note that static pressures are the same in two frames of references.
If we assume in the described problem that the total pressure is the same in the top and bottom body fluid, then the static pressure in the resting fluid corresponds to the total pressure, and the static pressure in moving fluid equals total pressure minus dynamic pressure. If we are speaking about the same problem, the observer B will measure the same static pressures, and he will observe the same membrane flex (upwards).
In your description of the problem, observer B does not observe the same situation as observer A since the static pressures are different for two observers.
The moving fluid determines a pressure decrease that acts on the membrane while the fluid is moving relative to the membrane, so both observers would see the membrane flex upward, as only the top fluid body is moving relative to the membrane. It has nothing to do with the observer - moving or not, so there is no paradox.
To really understand the source of aerodynamic forces, generation of lift and drag, it is the best to set aside for a moment Bernoulli and Newton. Imagine a stationary object submerged in a fluid. The surface of the object is subjected to fluid static pressure, created by the bombardment of the surface with the fluid molecules as per kinetic molecular theory, molecules moving randomly with an average speed determined by the absolute temperature. The forces generated by the static pressure cancels as the pressure is uniformly distributed along the surface of the object(neglecting gravitational effects).
Imagine the same object, but now fluid moving relative to the object, - the random average molecular speed is having a component induced by the local macroscopic fluid movement. This change in molecular speed is determining changes in the molecular bombardment of the surface and determining changes in the pressure distribution around the object. As the pressure distribution is no longer constant, the forces generated do no longer cancel out. This is much better way to 'visualise' the source of aerodynamic forces, closer to the actual physical phenomena.
Sure, Bernoulli and Newton are still valid, but as expressed, both theories are confusing in explaining aerodynamic forces, demonstrating again that they are models of the real world, and we surely should keep in mind that 'map is not the territory', we should not confuse models of reality with reality itself.
In response to Dusan's answer, I thought that specifying that the membrane was frictionless would mean that the actual motion of the membrane would not be relevant. It is difficult to imagine how the movement of the membrane would affect the behaviour of the system.
If you recall the pressure differential-based derivation of the BEq, the first thing that you'd note is that the equation can only be applied between two points within the same fluid. The two points could be anywhere within the fluid if the flow is irrotational. The points must lie on the same streamline if the flow is rotational. Once you separate the fluids using a membrane, we no longer have points in the same fluid, and BEq is no longer valid.
- Badges
- Science topic
- Similar topics
- Engineering
- Control Theory
- Control Systems
- Control Systems Engineering