Localisation of mushy-layer convection by background flow*

Under certain conditions, the directional solidification of a binary alloy leads to the development of a dendritic mushy region. Convective instability in the mush initiates the formation of chimneys (freckles). We examine the r\^ole of a weak and slowly varying background flow in the localisation of mushy-layer convection. Both two and three dimensional perturbations to a two dimensional background flow are considered in the near-eutectic limit, where the solid fraction in the mush is small. We find that three-dimensional disturbances are localised at places where there is upflow in the mush. We present amplitude equations to describe the evolution of both the 2D and 3D disturbances.