8The latest stages of planetary accretion involved large impacts between differen-9 tiated bodies, hence large scale melting events. Consequently, the iron brought 10 by the impactors sank within a deep magma ocean, before reaching the proto-11 core. Yet the fluid dynamics of this process remains poorly known. Here, we 12 report numerical simulations of the sinking dynamics of an initially spherical 13 liquid iron drop within a molten silicate phase, up to its possible fragmentation. 14 We consider a 2D cylindrical axisymmetric geometry. We vary the viscosity of 15 the molten silicates in the range of 0.05 Pa.s to 100 Pa.s and the initial radius of 16 the iron drop in the range of 1mm to 350 mm. Hence, we investigate Reynolds 17 number in the range of [0.027 -85600] and Weber number in the range of [0.073 -18 7480]. Our numerical model constrains the morphology, dynamics and stability 19 of the iron drop as a function of the dimensionless Weber and Reynolds numbers 20 as well as of the viscosity ratio between the molten silicates and the liquid iron 21 drop. In particular, we show that the maximal stable drop radius and the crit-22 ical Weber number are monotonically increasing functions of the magma ocean 23 viscosity. The momentum boundary layer thickness depends mainly on the drop 24 radius and slightly on the magma ocean viscosity. Increasing the viscosity of 25 the silicate phase prevents oscillations of the iron phase and limits the exchange 26 surface. Oppositely, increasing the initial radius of the iron drop enhances its 27 deformation and increases its relative exchange surface. Above the critical We-28 ber number, we confirm that the fragmentation of the liquid iron occurs within 29 a falling distance equal to 3.5-8 times the drop initial radius in the explored 30 range of moderate Weber number, and we describe a variety of fragmentation 31 regimes. Consequences for Earth's formation models are briefly assessed.32
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 temperaturedependent 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.
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