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In this XKCD What If, an indestructible hairdryer outputs 2 GW of power, turning the earth under it into gas:

Periodic explosions of gas beneath the box launch it into the air, and it starts fires and forms a new lava pool where it lands.

At 20 GW:

The box begins to be tossed around by the powerful updrafts it’s creating.

Later it outputs 11 PW, turning the surface of an entire lake into plasma and launching itself into space:

a hair dryer launching itself into space

Is this really accurate, though?

The energy is being emitted isotropically, so the energy itself shouldn't cause any net thrust in any direction, right?

If the object were surrounded by a fluid on all sides, would there be net force upward? Due to creating convection currents that push upwards against it? Does this depend on the density of the object?

Or does the thrust upward only exist when it's on a boundary between air and earth/water? The liquid/solid material below the box is greatly increasing in volume due to the phase changes, while the air above is not changing in volume as much, so the gases and plasma are rushing upward overall and carrying the box with it?

But that effect would drop off with height, leading to the case of it being surrounded by air on all sides again.

In what conditions does the superhot object launch upward into space, and in what conditions does it melt its way into the center of the earth?

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1 Answer

In the final part of the XKCD article the box containing hairdrier is floating on a lake, and the power it's producing boils the water and causes a steam explosion. It's the steam explosion that flings the box into space. As you've anticipated in your question, the motion is because it's environment is assymetric. Once the hairdrier has been launched away from the water surface the force reduces with distance just as in any explosion.

If the hairdrier was surrounded by fluid on all sides then as long as the depth of the fluid was effectively infinite, i.e. bigger than the range of the hairdrier's heating effect, then there would be no net motion.

Rock contains a lot of water and dissolved gas, which is (one reason) why volcanos explode. So I'd guess that even well away from surface water the box is unlikely to just quietly melt its way to the core. However I doubt the sort of blast you see in volcanos would provide enough force to launch the box into orbit.

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I agree with this answer, but I'm not so sure there'd be no net motion in the deep fluid case. The heated surroundings would rise due to buoyancy, and maybe that would carry the box upward... –  Nathaniel Jul 13 '13 at 8:37
    
I guess this indestructible box is mostly air, so it's going to have a lot of buoyancy –  endolith Jul 15 '13 at 3:00
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