Weak convection, liquid inclusions and the formation of chimneys in mushy layers

We present a numerical study of steady convection in a two-dimensional mushy layer during solidification of a binary mixture at a constant speed V. The mushy layer is modelled as a reactive porous medium whose permeability is a function of the local solid fraction. The flow in the liquid region above the mushy layer is modelled using the Stokes equations (i.e. the Prandtl number is taken to be infinite). The calculations follow the development of buoyancy-driven convection as the flow amplitude is increased to the level where the solid fraction is driven to zero at some point within the mushy region. We show that this event cannot occur before the local buoyancy-driven volume flux exceeds the solidification rate V. Further increases in the flow amplitude lead to the formation of a region with negative solid fraction, indicating the need to switch from the Darcy approximation to the Stokes flow approximation. These regions ultimately become what are known as chimneys. We exhibit solutions which give the detailed structure of the temperature, solute, flow and solid fraction fields within the mushy layer. A key finding of the numerics is that these fledgling chimneys emerge from the interior of the mushy layer, rather than eating their way down from the top of the layer, as the amplitude of the steady convection is increased. We discuss some qualitative features of the resulting liquid inclusions and, in the light of these, reassess the interfacial conditions between mushy and liquid regions.