Why does the ink flow out of the fountain pen when the plane slowly gains height?

Answer 1

Pressure.

That's the short answer.

Back when people used fountain pens as a matter of course, aeroplane cabins had variable pressure throughout the flight. Pens, shampoos, potato chip bags, etc. expand, pop, or leak at high altitudes is because of low air pressure.

The air inside a sealed container holds the pressure of the environment when it was closed. If this container travels up, to Denver or up in an aeroplane, the air pressure --barometric pressure-- of the environment becomes lower than it was closer to sea level. The container expands in dimensions or leaks in an attempt to equalize the pressure.

These days people use ball point pens and they're pens don't leak. ;)

Actually, these days commercial aeroplanes have better seals and climate control systems. However bleeding fountain pens may still be a problem because the pressure during the flight will decrease to that of a mountain city.

According the Boeing (plane manufacturer) "The cabin air system in today's jetliners is designed to provide a safe, comfortable cabin environment at cruising altitudes that can reach upwards of 40,000 feet. At those altitudes, the cabin must be pressurized to enable passengers and crew to breathe normally. By government regulation, the cabin pressure cannot be less, at maximum cruise altitude, than the equivalent of outside air pressure at 8,000 feet."

However, the container/luggage hold is outside the climate control system. Therefore, shampoo in your luggage may still ooze. Hint: puff the air out of the bottle when you pack, and put in a sealed ziplock-type bag in a luggage pocket.