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Is it physically possible to increase the pressure inside a water diving bell so that it exceeds the pressure outside it while submerged by allowing water into it?

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Not that I'm aware of, no.

The interface between the air pocket and the surface of the water should be approximately the same pressure. If they were not at the same pressure, then once would be pushing against the other with a greater force, and the interface would move until the forces equalized (therefore the pressures approximately equalize).

Adding more water of the same pressure to the bell would not increase the air pressure. To get that water in, you would either need to let air out, or pressurize the water to greater than the surroundings. If you add in pressurized water, you may temporarily increase the pressure of the air.

So pumping in more water would work as long as you keep the pump going, but letting in more water implies a natural flow; which wouldn't happen when the air and water are the same pressure.

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