C. Eck
Germany
On Homogenization and Two Scale Models for Liquid--Solid Phase Transitions with Dendritic Microstructure
Ch. Eck and P. Knabner Institute for Applied Mathematics, University Erlangen--Nurnberg, Germany
In liquid-solid phase transition problems often a specific dendritic
microstructure of the phase interface is observed.
It is generated by an instability of a `smooth' phase interface
in an undercooled melt with respect to small perturbations.
The mathematical modeling and numerical solution of the process on a
microscopic scale is now well understood. Suitable models consist of
differential equations for heat and solute transport supplemented by either
a sharp interface model including surface energy and kinetic undercooling
or a phase field model for the evolution of the unknown phase interface.
However, for computations on a macroscopic scale the direct calculation
using these models is not suitable, since it requires the resolution
of the entire microstructure with a very fine grid.
In the lecture a two scale model for liquid--solid phase transitions
of binary material is presented. The model is based on a formal
homogenization of a sharp interface model with `fast' heat and `slow'
solute diffusion. The homogenization limit consists of a macroscopic
heat diffusion equation and, for each point of the domain under
consideration, of a local cell problem describing the solute diffusion
on the microscopic scale and the evolution of the phase
interface. The formal derivation of the model is presented,
the problem of justification is addressed and first numerical examples
are shown.
