Settling behaviour of particles in Rayleigh-Benard convection

<p>Our numerical study evaluates the settling rate of solid particles, suspended in a highly <br>vigorous, finite Prandtl number convection of a bottom heated fluid. We explore a broad <br>range of model parameters, covering particle types appearing in various natural systems, <br>and focus in particular on crystals nucleating during the cooling of a magma ocean. The <br>motion of inertial particles within thermal convection is non-trivial, and under idealized <br>conditions of spherical shaped particles with small Reynolds number it follows the <br>Maxey-Riley equation (Maxey and Riley, 1983). Two scaling laws exist for the settling <br>velocities in such system: for particles with small but finite response time, the Stokes' <br>law is typically applied. For particles with a vanishing response time, a theoretical model <br>was developed by Martin and Nokes (1989), who also validated their prediction with analogue <br>experiments. </p><p>We develop a new theoretical model for the settling velocities. Our approach describes <br>sedimentation of particles as a random process with two key constituents: i) transport <br>from convection cells into slow regions of the flow, and ii) the probability of escaping <br>slow regions if a particle enters them. By quantifying the rates of these two processes, <br>we derive a new equation that bridges the gap between the above mentioned scaling laws. <br>Moreover, we identify four distinct regimes of settling behaviour and analyze the lateral <br>distribution of positions where particles reach the bottom boundary. Finally, we apply our <br>results to the freezing of a magma ocean, making inferences about its equilibrium vs <br>fractional crystallization. The numerical experiments are performed in 2D cartesian geometry <br>using the freely available code CH4 (Calzavarini, 2019).</p><p>References:<br>Maxey, M. R. and Riley, J. J.(1983): Equation of motion for a small rigid sphere in a nonuniform flow. <br>Physics of Fluids, 26(4), 883-889.</p><p>Martin, D and Nokes, R (1989): A fluid-dynamic study of crystal settling in convecting magmas. <br>Journal of Petrology, 30(6), 1471-1500.</p><p>Calzavarini, E (2019): Eulerian–Lagrangian fluid dynamics platform: The ch4-project. Software Impacts, 1, 100002.</p>