Settling behaviour of particles in Rayleigh-Benard convection

Our numerical study evaluates the settling rate of solid particles, suspended in a highly 
vigorous, finite Prandtl number convection of a bottom heated fluid. We explore a broad 
range of model parameters, covering particle types appearing in various natural systems, 
and focus in particular on crystals nucleating during the cooling of a magma ocean. The 
motion of inertial particles within thermal convection is non-trivial, and under idealized 
conditions of spherical shaped particles with small Reynolds number it follows the 
Maxey-Riley equation (Maxey and Riley, 1983). Two scaling laws exist for the settling 
velocities in such system: for particles with small but finite response time, the Stokes' 
law is typically applied. For particles with a vanishing response time, a theoretical model 
was developed by Martin and Nokes (1989), who also validated their prediction with analogue 
experiments. 

We develop a new theoretical model for the settling velocities. Our approach describes 
sedimentation of particles as a random process with two key constituents: i) transport 
from convection cells into slow regions of the flow, and ii) the probability of escaping 
slow regions if a particle enters them. By quantifying the rates of these two processes, 
we derive a new equation that bridges the gap between the above mentioned scaling laws. 
Moreover, we identify four distinct regimes of settling behaviour and analyze the lateral 
distribution of positions where particles reach the bottom boundary. Finally, we apply our 
results to the freezing of a magma ocean, making inferences about its equilibrium vs 
fractional crystallization. The numerical experiments are performed in 2D cartesian geometry 
using the freely available code CH4 (Calzavarini, 2019).

References:
Maxey, M. R. and Riley, J. J.(1983): Equation of motion for a small rigid sphere in a nonuniform flow. 
Physics of Fluids, 26(4), 883-889.

Martin, D and Nokes, R (1989): A fluid-dynamic study of crystal settling in convecting magmas. 
Journal of Petrology, 30(6), 1471-1500.

Calzavarini, E (2019): Eulerian–Lagrangian fluid dynamics platform: The ch4-project. Software Impacts, 1, 100002.