Thermal evolution of a metal drop falling in a less dense, more viscous fluid

The initial state of terrestrial planets was partly determined, during accretion, by the fall of metal drops in a liquid magma ocean. Here we perform systematic numerical simulations in two-dimensional cylindrical axisymmetric geometry of these falling dynamics and associated heat exchanges at the scale of one single drop for various initial sizes and ambient viscosities. We explore Reynolds number in the range 0.05-48, viscosity ratios in the range 50-4000, Weber number in the range 0.04-5, and Peclet number in the range 70-850. We show that heat exchange between the two phases occurs predominantly at the front section of the drop. Our systematic, parametric study shows that the thermal boundary layer thickness, the depth and time for equilibration, the Nusselt number, and the magma ocean volume affected by thermal exchanges all scale as power laws of the Peclet number. Because of drop distortions, these scaling laws deviate from the classical balances considering only heat diffusion through a laminar thermal boundary layer. Finally, when considering a temperature-dependent viscosity of the ambient fluid, we show that a low-viscosity layer surrounds the drop, which influences the thermal evolution of nondeformable, low-Reynolds-number drops only and decreases the breakup distance for some limited breakup modes.