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Geodynamic thermal runaway with melting

We consider the penetration of a solid medium by a foreign body which is large enough for frictional heating to melt the medium and maintain a thin liquid layer ahead of the body. This study is motivated by the possibility of the Earth's core having been formed by liquid iron diapirs melting their way through the solid, deformable mantle. Our principal results are the existence of a critical size for the body for the motion to be maintained under gravity and the ease with which an immiscible liquid body can penetrate at constant velocity compared to a solid one.