Pore-scale mushy layer modelling
Equations describing mushy systems, in which solid and liquid are described as a single continuum, have been extensively studied. Most studies of mushy layers have assumed them to be ‘ideal’, such that the liquid and solid were in perfect thermodynamic equilibrium. It has become possible to simulate flows of passive porous media at the pore scale, where liquid and solid are treated as separate continua. In this contribution, we study the simplest possible mushy layers at the pore scale, modelling a single straight cylindrical pore surrounded by a cylindrical annulus representing the solid matrix. Heat and solute can be exchanged at the solid–liquid boundary. We consider harmonic temperature and concentration perturbations and examine their transport rates due to advection and diffusion and the melting and solidification driven by this transport. We compare the results of this numerical model with those of a one-dimensional ideal mushy layer and with analytical solutions valid for ideal mushy layers for small temperature variations. We demonstrate that for small values of an appropriately defined Péclet number, the results of the pore-scale model agree well with ideal mushy layer theory for both transport rates and melting. As this Péclet number increases, the temperature and concentration profiles with radius within the pore differ significantly from constant, and the behaviour of the pore-scale model differs significantly from that of an ideal mushy layer. Some effects of mechanical dispersion arise naturally in our pore-scale model and are shown to be important at high Péclet number.
