In which frame (with respect to which object) velocity is taken in Bernoulli equation?

Question

Duplicate: Bernoulli's equation and reference frames

why we need to take velocity of air in tunnel with respect to train & velocity of train with respect to earth (inertial frame) (here we are comparing pressure difference between air inside and outside train)?

A train with cross-sectional area $𝑆_𝑡$ is moving with speed $𝑣_𝑡$ inside a long tunnel of cross-sectional area $𝑆_0 (𝑆_0 = 4𝑆_𝑡)$. Assume that almost all the air (density $\rho$) in front of the train flows back between its sides and the walls of the tunnel. Also, the air flow with respect to the train is steady and laminar. Take the ambient pressure and that inside the train to be $𝑝_0$. If the pressure in the region between the sides of the train and the tunnel walls is $𝑝$, then $𝑝_0 − 𝑝 = \frac{7} {2𝑁} 𝜌𝑣_𝑡^2$ . The value of 𝑁 is _______.

Answer: 9

Solution: https://youtu.be/RXmdziWqI6I?t=540

Question Source: JEE Advanced 2020

Such thing is also applied to find pressure difference above and below airplane wings by NCERT. Do we need to take velocities of fluid with respect to moving object? Question is kept for future readers.

Answer 1

A key step in deducing Bernoulli’s equation from Euler’s equation is that the fluid flow is steady, i.e.

$$\frac{∂v}{∂t} = 0$$

However, when the fluid flow is steady according to an observer in one frame of reference, the flow will not, in general, be steady according to observers in frames that are in motion with respect to the first frame. That is, the validity of Bernoulli’s equation in one frame of reference does not, in general, imply that it is valid in other frames of reference. Bernoulli’s equation is NOT relativistically invariant.

In the present example, the flow of air is steady according to an observer at rest with respect to the moving train. So, Bernoulli’s equation holds for this observer, and (s)he correctly predicts that the air pressure inside the train is higher than that just outside the train.

However, an observer at rest with respect to the rail on which the train is driving detects a pulse of wind as the train passes by. The velocity of the air (at a particular point in the “rail” frame) is not steady, and Bernoulli’s equation does not apply in this frame. This observer does not think that the air pressure inside the train is lower than that outside the train, even though the velocity of air inside the train is higher than that outside the train in the “rail” frame. The “rail” observer must make a more complicated analysis in which the moving train continually encounters new regions of “stagnant” air, whose inertia results in forces on the air in the train (through the open window) that increase the pressure of the air in the train compared to that outside, in agreement with the observer inside the train. Solution video in question explains other things.

Details from Source

Video explanation of above matter.

Answer 2

I seem to understand your problem. I feel u are having difficulty in understanding how is V = 4/3.Vtrain. If so then it is very simple. See this relationship is achieved by equation of continuity and in that we have to consider the velocity of fluid w.r.t pipe. Thus :- V. 3A( as out of 4A ,A Is blocked by train) = Vtrain. 4A And talking about Bernoulli's theoram, it is based on energy conservation which can be applied from any from and got the correct answer ( you can check it for yourself)? I hope u have understood if not please revert back