Patterns of convection in solidifying binary solutions

Abstract

During the solidification of two-component solutions a two-phase mushy layer often forms consisting of solid dendritic crystals and solution in thermal equilibrium. Here, we extend previous weakly nonlinear analyses of convection in mushy layers to the derivation and study of a pattern equation by including a continuous spectrum of horizontal wave vectors in the development. The resulting equation is of the Swift-Hohenberg form with an additional quadratic term that destroys the up-down symmetry of the pattern as in other studies of nonBoussinesq convective pattern formation. In this case, the loss of symmetry is rooted in a nonBoussinesq dependence of the permeability on the solid-fraction of the mushy layer. We also study the motion of localized chimney structures that results from their interactions in a simplified one-dimensional approximation of the full pattern equation.

Finite bandwidth, nonlinear convective flow in a mushy layer

“…The only other investigation of convection pattern in a mushy layer, which took into account a continuous spectrum of horizontal convective modes, was that due to Keating et al (2011). These authors derived a simplified model which took into account a continuous spectrum of modes and was based on the assumptions of those adopted in the model due to Amberg and Homsy (1993).…”

“…There have been a number of studies of the problem of nonlinear convective flow in mushy layers during the solidification of binary alloys (Amberg and Homsy 1993, Anderson and Worster 1995, Schulze and Worster 1999, Chung and Chen 2000, Worster 2000, Roper et al 2008, Guba and Worster 2010, Keating et al 2011, Zhong et al 2012, Rees Jones and Worster 2013, Wells et al 2013. For weakly nonlinear steady convection in such layers such as those studied by Amberg and Homsy 1993, Anderson and Worster 1995, Chung and Chen 2000and Roper et al 2008 authors considered solutions in the form of rolls and hexagons and used a discrete modal approach to determine such solutions.…”

“…However, a wider class of such solutions can be obtained by taking into account the continuous finite bandwidth of the convective modes as was done before for thermal convective rolls (Newell and Whitehead 1969) and in the case of thermal convection with a boundary imperfection Riley 1989, Riahi 1996). Keating et al (2011) developed a highly simplified model to study convection in a mushy layer by including a continuous spectrum of horizontal wave vectors and mainly focused on the motion of localized chimney structures in a one-dimensional approximation of the original flow system. In these flow systems as well in other nonlinear systems (Chomaz andCouairon 2000, Lombardi et al 2011) a more realistic wave packet form of the disturbances are taken into account to determine the stability of the basic state system.…”