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 2D 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 exchanges between the two phases occurs predominantly at the front section of the drop. Our systematic, parametric study exhibits shows that the thermal boundary layer thickness, the depth and time for equilibration, the Nusselt number, and the magma ocean volume affected by thermal echanges, 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 non-deformable, low Reynolds number drops only, and decreases the breakup distance for some limited breakup modes.