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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?

an4s
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ViktorV
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    Water is incompressible. – Sam Jan 19 '20 at 11:50
  • But that doesn't make any sense, the water does not flow over on the ground but it DOES flow over in the plane. Hence it must have expanded when the atmospheric pressure is lowered. The bottle itself does not expand or contract – ViktorV Jan 19 '20 at 13:19
  • Have you conducted an experiment giving this result? – Sam Jan 19 '20 at 17:15
  • If you’ve actually performed this experiment, have you controlled for temperature changes? – amd Jan 19 '20 at 18:12

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Being water an incompressible fluid, you van apply Bernoulli's equation as you suggested in the question: $$P_1+1/2\rho v_1^2+\rho g h_1=P_2+1/2 \rho v_2^2+\rho g h_2$$

Matteo
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  • Thanks for the answer! but this confuses me, if the water leaks out at a lower atmospheric pressure, that must mean the fluid expands doesn't it? Hence the density must lower – ViktorV Jan 19 '20 at 12:00
  • No, $\rho$ is costant. – Matteo Jan 19 '20 at 12:06
  • But that doesn't make any sense, the water does not flow over on the ground but it DOES flow over in the plane. Hence it must have expanded when the atmospheric pressure is lowered. The bottle itself does not expand or contract – ViktorV Jan 19 '20 at 13:18
  • @ViktorVandenBergh: The answer (+1) shows with Bernoulli's equation, that there is no reason why a fluid with constant density cannot move when a pressure gradient is imposed. For example, steady flow is attained when net pressure force is balanced by viscous stress.

    The experiment however involves another phenomenon as well -- a force impulse causing acceleration. At the higher altitude as the container is opened there is an net force acting on fluid parcels near the surface (the pressure above is lower than the pressure in the fluid below) and these parcels must accelerate (F = ma).

     – RRL Jan 19 '20 at 21:17