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Fight to the death on the SHIELD Helicarrier
No BFR.
Who's winning?

David Banner starts off the stronger and his absorption powers extend to inorganic and 'inert' materials so he should be able to weld Shaw to the spot and then slide a rail into him or something.

Although Shaw can absorb and output far more power than DB (well, DB showed a limit, Shaw didn't), he doesn't start off as stacked and he needs heavy attacks to leech power from.

So, all in all, Shaw can't kill DB but can be killed by DB.

David Banner was Hulk's dad in 2003 Hulk, right?

Hulk Daddy turns into water and drowns Shaw.

That would probably work, yeah.

It's on the Helicarrier though.

the fight is restricted to the OP's arena? Even if that's so, you have to assume that they would eventually destroy it and end up somewhere with water.