Properties of stochastic permeability

When modeled at macroscopic length scales, the complex dendritic network in the solid-plus-liquid region of a solidifying alloy (the mushy zone") has been modeled as a continuum based on the theory of porous media. The most important property of a porous medium is its permeability, which relates the macroscopic pressure gradient to the throughput of fluid flow. Knowledge of the permeability of the mushy zone as a function of the local volume-fraction of liquid and other morphological parameters is thus essential to successfully modeling the flow of interdendritic liquid during alloy solidification. Permeability is usually treated as a deterministic function of parameters that can be calculated by the model (e.g., local solid fraction, dendrite arm spacing). However, recent results show that the length scales that must be resolved are too small for the assumption of deterministic behavior to be valid, and investigators must confront the stochastic behavior of the permeability field. We describe early work to investigate the spatial structure of the stochastic permeability at these small scales, with a view to develop a comprehensive treatment of stochastic permeability to enable improved modeling.