Dual Reciprocity Boundary Element Method for studying thermal flow in cooling Magma Oceans

Earth's early history is marked by a giant impact with a Mars-sized object which lead to the formation of the moon. This impact event likely led to a substantial amount of melting of the Earth's interior. Subsequent cooling of the Earth involved extensive crystallization in this "magma ocean" over a relatively short period of time. While chemical evidence from ancient sources provides some clues on the rate of cooling, computational models of such phenomena are sparse. Modeling the crystal settling behavior requires solving a coupled system of partial differential equations, specifically the Stokes flow equation coupled with the heat equation through an advection term. Our work uses the dual reciprocity boundary element method (DRBEM) to model such a system. DRBEM extends on the boundary element method (BEM) to solve the heat equation for a multiparticle system in an infinite suspension fluid while avoiding the expensive rediscretization inherit in other methods. In DRBEM, terms arising from the material time derivative of temperature are expanded in a series of function, known as radial basis functions. By modeling this system we are able to simulate thermal interactions among an arbitrary number of crystals settling in a magma ocean. The types of interactions include the enhanced cooling of the magma due to an advecting matrix of cold crystal particles. We are also able to observe the interactions of the crystals' thermal profiles. As a crystal settles, it leaves behind a trail of lower temperature suspension fluid, which then interacts with surrounding particles.