An operator splitting scheme for coupling macroscopic transport and grain growth in a two-phase multiscale solidification model: Part I – Model and solution scheme

We present a two-phase multiscale solidification model, which accounts for coupled heat, mass, solute and momentum transfer and notably describes the motion of globular equiaxed grains. On the microscopic scale the model takes into account nucleation, finite diffusion in both solid and liquid phases, and resolves the diffusion-controlled phase change by a solute balance at the solid–liquid interface coupled with a thermodynamic equilibrium condition for the interface. The main idea of the paper is the solution algorithm of the multiscale model. We propose and present in detail an operator splitting method that resolves the global conservation equations in two stages. In the first solution stage we resolve and integrate globally only the convection terms. In the second solution stage we resolve and integrate locally the contributions of the grain growth kinetics, i.e. the phase change and the microscopic diffusion. We analyze this splitting method based on a scaling analysis that shows that the justification of the method stems from the separation of time scales of convection, diffusion and phase change. We discuss this scale separation in detail and define a test case that is presented and analyzed in a companion paper.