In Flight, to prevent the plane from diving, the pilot puts it upside down before flipping it back over just before landing. This is supposed to be loosely based on flight 261 from Alaska Airlines which unfortunately crashed without survivor after attempting a similar desperate manoeuvre. Let's just focus on the inverted flight part. Can a commercial airliner (50+ passengers) fly inverted for more than a few seconds before aerodynamics and/or propulsion problems arise? What are these specific problems?
Absolutely. As a matter of fact, one of the first commercially successful aircraft (the Boeing 707) did a barrel roll during a demonstration flight. In case you don't believe my claim, here is a video to back up my claim:
(EDITORIAL NOTE: I take exception to the part where the NatGeo writers state "something the aircraft had never been designed to do" because a perfect barrel roll is a constant 1G maneuver (generally with anything between .5G to 3G, well within most aircraft design parameters), and doesn't stress the aircraft in any way, although this particular execution was probably not per any aerobatics manual due to the actual fight going on.)
Basic aerodynamics doesn't really care what the orientation of the aircraft is as long as you are able to balance the forces. The basic forces you have to deal with are weight, lift, thrust and drag. In an inverted position, you may think that the combination of lift and weight would be detrimental to flight, however you can tilt the angle of attack of the wing so that even in an inverted position you can get the lift to be opposite of the "top" surface of the wing. Otherwise an aircraft would have a hard time descending.
The Smithsonian Air & Space museum has a nice web page that explains how inverted flight works. Aircraft are basically the same on a meta level. The thing to keep in mind, that many subsystems of the aircraft may not operate well for extended inverted flight. However, (and keep in mind I have not seen the movie in question) most systems will maintain operation for at least some time due to residual fluids and such in the oil and fuel systems. Depending on the aircraft make and model, the fuel and oil lines should provide at least 30 seconds to a minute of uninterrupted operation.
For aircraft that do fly inverted, you can check out any airshow and see all manner of aircraft fly inverted for extended periods of time. Jets and propeller aircraft have well understood physics. You can learn more about aerodynamics at this page if that helps. If you look at this image:
All you have to do is get the angle of attack to be negative for the right-side up wing (thus if the wing is upside down, it's creating a net up force against gravity). So again, it's quite possible and doesn't violate any flight dynamics.
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Airbus aircraft will not allow inverted flight with the controls in normal law. If you hold full side stick in an airbus it will stop the roll at a pre determined limit. It doesn't matter how quickly you move the side stick. (Roll rate is also governed).
In fact you can apply full side and full back stick with max power and the aircraft will climb out in an ascending spiral. Both pitch and roll are limited by the flight envelope computer.
Other types will allow inverted flight. Boeing and MD aircraft do not have the same flight envelope systems as airbus.
As mentioned in other answers the issue would be fuel, oil and hydraulic systems are not designed for any sort of prolonged negative g
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