Global theory of steady deep-cellular growth in directional solidification

Abstract

The present paper is concerned with the global asymptotic theory of steady deep-cellular growth in directional solidification of binary mixtures. We consider the two-dimensional model with nonzero isotropic surface tension and obtain the global uniformly valid asymptotic solutions for the steady state of the system in the limit of the Péclet number ε→0; ε is defined as the ratio of the radius of the cell's tip and mass diffussion length. The whole physical space is divided into the outer region and root region; the solutions in each subregion are solved, respectively, and matched with each other in the intermediate region. The results show that given growth conditions and material properties, the global solutions for steady state of the system contain two free parameters: the Péclet number and asymptotic width parameter λ(0), which are related to the geometry of cellular structure: the cell tip radius and primary spacing. One of the most important conclusions drawn from this analysis is that the steady-state solutions of cellular growth have a complicated structure with three internal layers in the root region; for given (ε,λ(0)), there exists a discrete set of the global steady-state solutions subject to the quantization condition that are profoundly affected by the surface tension. Each eigenvalue calculated from this quantization condition determines the total length of the finger described by the corresponding global steady-state solution.