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Wouldn't scale and viscosity play a role? Seriously, imagine a river vs a capillary tube. Also how many dimensions? And forces involved? Is that a blockage between 2 and 3? Are the walls breakable? How will the fluid hold air? Are the lines into structure 5 lower than the walls? Is this in a vacuum?
you may be overthinking it
There is no mention of any fluid involved, just a faucet. So lets think inside of the box and assume we have some form of 2d-gravity and it is going to rain a newtonian fluid? I think most surface area on the top is draining into 5. If it snows the whole sheet can turn white and the problem is gone, too.
Also, these structures are all 2 dimensional.
It does not specify what they are being filled with. They are all already filled with air.