Ein Zweiskalenmodell für fest-flüssig Phasenübergänge mit dendritischer Mikrostruktur und Konvektion

点击翻译按钮获取中文摘要

In solidification processes, as e.g. the casting of metals, there often occurs a specific dendritic (tree-like) microstructure of the solid-liquid interphase. Models describing the evolution of single or few dendrites are available, but the small scale (typical size) of the dendritic structures prevents their application in a numerical simulation of e.g. a whole workpiece in a casting process. In this project we develop and analyze two-scale models for solidification processes with dendritic microstructure in mixtures of two components. The model is derived from a phase field model including convection, depending on a small parameter e describing the scale of the microstructure. Using techniques from homogenization theory a limit problem for e > 0 is constructed. This problem consists of macroscopic differential equations for heat and fluid flow and of microscopic problems modeling the evolution of single dendritic crystals. A mathematical analysis includes a proof of the existence and uniqueness of solutions as well as an estimate for the difference of the solutions of both models. The reliability and applicability of the two-scale model shall be studied with the help of numerical simulations for selected model problems.