Application of Bernoulli's theorem

Question

My teacher told me the following examples of Bernoulli's principle. I didn't understand how Bernoulli's principle is applied in these examples.

1. It is dangerous to stand on edge of platform when train is approaching.
2. During storms the roof of houses are blown off.

I have trouble finding the areas of low pressure and high pressure in the above example. Where are they located? I have trouble thinking the uses of Bernoulli's principle. When is Bernoulli's principle used and why?

I have trouble figuring out how was Bernoulli's principle was used for the above example to show these example true as Bernoulli's principle states that the sum of pressure energy, kinetic energy and Potential Energy per unit volume is constant for streamline flow of ideal fluid.

Answer 1

First a brief summary on what Bernoulli principle and the corresponding equation state in the simple case of air at equilibrium. As you mentioned, Bernoulli's effect tells you that energy is conserved along a streamline. Thus the sum of kinetic energy, potential energy and internal energy remains constant and an increase in the speed of the fluid – implying an increase in both its dynamic pressure $\rho v^2/2$ and kinetic energy – occurs with a simultaneous decrease in (the sum of) its static pressure, potential energy and internal energy.

Another simple view is the following: a pressure gradient between two points on a streamline causes a net force, a 'push' from the higher pressure area to the lower pressure one. Thus, by Newton's 2nd Law, acceleration of air is caused by pressure gradients. Air is accelerated in direction of the velocity if the pressure goes down. Thus the decrease of pressure is the cause of a higher velocity. [See also Weltner, Klaus; Ingelman-Sundberg, Martin, Misinterpretations of Bernoulli's Law]

Let's now focus on your two examples:

1. The passing train causes an increase in the velocity of the air in proximity of the platform edge. This temporary effect, lowers the pressure experienced in front of a person standing there, i.e., it creates a difference of pressure between the back and the front of the person, pushing her forward.

To be more precise, the pressure of the air is $p_A$ before the approaching train passes. Since everything is at equilibrium, there is no gradient between the air in front and behind the edge of the platform. During the transit, the air between the person and the train experiences an increased velocity, and therefore a lower pressure, e.g. $p'=p_A-\Delta p$. This $\Delta p$ can be calculated using Bernoulli's equation. Its effect is that of a net force, pushing the person from the edge of the platform towards the tracks.

2. The same effect causes an house roof to blow off when there is strong wind. Indeed, when the wind kicks up, the pressure on top of the roof (outside the house) drops, and the pressure gradient between inside (higher pressure, stationary air) and outside (low pressure, high speed) works as a net force pushing the roof upwards.

In both cases one could argue that the air could flow in the surroundings of the place where the phenomenon takes place, restoring equilibrium. But in fact, in both cases, the velocities are such that there isn't time for air to flow in and out of the house to cause the corresponding decrease in the interior pressure. That's why you get a pressure gradient across the roof.