Suction caused by an open window of a plane

So I know this question has been asked before, and I have seen other posts. The reason I want to post a new one is because there is something about the Bernoulli's principle I want to understand.

Is it correct to say in this case:

$$P_1+\frac{1}{2}\rho v_1^2+ \rho gh_1=P_2+\frac{1}{2}\rho v_2^2+ \rho gh_2$$

that the term $$\rho gh$$ would go away since they are at the same height? So

$$P_1+\frac{1}{2}\rho v_1^2=P_2+\frac{1}{2}\rho v_2^2$$

Now, since it's a moving plane, due to the frame of reference, we can say the air inside the plane is moving and outside isn't (or assume there is barely any wind). Is my thinking correct, where the left side of the equation is for the air inside the plane?

$$P_1+\frac{1}{2}\rho v_1^2=P_2$$

But, inside the plane, the air is pressurised, so the pressure is higher than outside. Now, we added another positive term to it? I know something is wrong, but I don't know where.

The change in pressure would be

$$P_2-P_1=\frac{1}{2}\rho v_1^2$$

and the "suction force" would be

$$F=\Delta P*A=\frac{1}{2}\rho v_1^2*A$$

where $$A$$ is the area of the open window?

PS: I know doors or windows can't be opened mid air. I just want to know if my understanding is correct!

• Myth busters fun: thescottishsun.co.uk/video/travel/… Jan 17, 2022 at 20:42
• Well done for using MathJax. Piece of an advice: the $*$ character is reserved for convolution operation, rather use $\cdot$ (\cdot) to indicate multiplication. Jan 17, 2022 at 20:44

You are misunderstanding Bernoulli. The quantity $$P+\frac 12 \rho v^2$$ is a constant along streamlines. The Bernoulli value can jump from one streamline to another when they are separated by a surface with with vorticity. Contrary to popular claims, the pressure in a jet of rapidly moving air has the same pressure as the stationary air through which it moves because the jet is bounded by a surface with vorticity.