Bernoulli Equation and Continuity Equation for Air Flow

Question

In textbooks, Bernoulli Equation and Continuity Equation are typically used with Fluids. It's unclear how they can be applied to air flows.
Once thing I noticed is that since $rho$ is very small for air, so we typically ignore the term ρgy in Bernoulli equation.
In the example of Pitot tubes below, I can understand the incoming air flow is brought to stop so $v_2=0$. But why is $p_1$ atmosphere pressure? Because of natural gas law and density of the air?
https://en.wikipedia.org/wiki/Pitot_tube
Another question about the above example is that seems that we cannot use continuity equation. As $A_1$ and $A_2$ are the same. But $v_1$ is not zero but $v_2$ is zero. This is fundamentally because air is compressible?
Any guideline on applying Bernoulli Equation and Continuity Equation for air flow?

Bernhard · Answer

When we talk about fluid in fluid dynamics, it applies to both liquids and gasses. So this works for both the continuity equation and the Bernoulli equation. For incompressible liquids indeed the density drops out of the continuity equation, but in all other cases you  include them.
For your example, I assume point 1 is outside the pitot tube. You didn't specify where you put it exactly, so, you can put it well in front of the airplane, where the air is not disturbed by the plane. So in that case, there is no other option than to select the atmospheric pressure (which depends on weather conditions as well as altitude).
I don't know practical pitot tube are engineered. But I suppose they put them in a place on the flying object where the pressure is not disturbed to much, and they actually measure a pressure difference.
Of course you can use the continuity equation too, but that alone is not enough to derive the velocity. You will need the Navier-Stokes equations too. If you do the analysis correctly with the right assumptions, I guarantee that you will arrive at the same solution as with using the Bernoulli equation. This is left as an exercise to the reader.

Bernoulli Equation and Continuity Equation for Air Flow

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