So my question is relatively simple. First, consider a bottle modeled as a cylinder with length $L$ and cross-section $A$. You fill it to the brim with water on the ground. Take it up on an airplane and open it. The pressure change will cause the water to leak.
How high will the water rise after opening it on the plane? I figured I'd just have to equate pressures before and after and extract the difference in volume
But Bernoulli's equation is only applicable to incompressible fluids so what do I use?
Being water an incompressible fluid, you van apply Bernoulli's equation as you suggested in the question: $$P_1+1/2\rho v_1^2+\rho g h_1=P_2+1/2 \rho v_2^2+\rho g h_2$$