I don't think you can use Bernoulli's equation to solve, as most requirements are not met (dm/dt is not constant, air is not incompressible. You may use its differential form, though:
[tex]\frac{dp}{\rho} + d(\frac{v^2}{2}) + g dz = 0[/tex]
Now you'll have to make a few hypothesis, such as the processes is not turbulent (which it is, however, though it would be almost impossible to accurately describe this phenomenon without this consideration, so your answer will be physically wrong), the air is an ideal gas (or find a good equation of state, though ideal gas law is good).
The specific mass of air in both situations are different, but remember that they must equal when in equilibrium.
The differential equation you will have to solve will be a combination of that differential form of Bernoulli's law and the fact that
[tex]\frac{dm}{dt} = V \frac{d\rho}{dt} = Av[/tex]
Interesting problem, by the way.
