M. Grae Worster

Instabilities of the liquid and mushy regions during solidification of alloys

The solidification of melts can be profoundly influenced by convection. In alloys, compositional convection can be driven by solute gradients generated as one component of the alloy is preferentially incorporated within the solid, even when the thermal field is stabilizing. In this paper, two modes of compositional convection during solidification from below are uncovered using a linear-stability analysis : one, which we shall call the ‘mushy-layer mode’, is driven by buoyant residual fluid within a mushy layer, or porous medium, of dendritic crystals; the other, which we shall call the ‘ boundary-layer mode ’ is associated with a narrow compositional boundary layer in the melt just above the mush-liquid interface. Either mode can be the first to become unstable depending on the thermodynamical and physical properties of the alloy. The marginally stable eigenfunctions suggest that the boundary-layer mode results in fine-scale convection in the melt above the mushy layer and leaves the interstitial fluid of the mushy layer virtually stagnant. In contrast, the mushy-layer mode causes perturbations to the solid fraction of the mushy layer that are indicative of a tendency to form chimneys, which are vertical channels of reduced or zero solid fraction that have been observed experimentally. Particular attention is focused on the mushy-layer mode and its dependence upon the thermodynamical properties of the alloy. The results of this analysis are used to make a number of interpretations of earlier experimental studies such as the observations that some systems are less prone to form chimneys and that the regions of melt in these systems evolve to supersaturated conditions, while the melt evolves to unsaturated conditions once chimneys have formed. In addition, good quantitative agreement is found between the results of the linear-stability analysis and the experimental results of Tait & Jaupart (1992) for the onset of the mushy-layer mode of convection.

Solidification of an alloy from a cooled boundary

We Present a mathematical model for the region of dendritic or cellular growth which often forms during the solidification of alloys. The model treats the region of mixed phase (solid and liquid) as a continuum whose properties vary with the local volume fraction of solid. It is assumed that transports of heat and of solute are by diffusion alone, and the model is closed by a condition of marginal equilibrium. Results are obtained for the unidirectional solidification of an alloy from a plane wall. The spatial variations of solid fraction are highly suggestive of the types of morphology that can occur, and a wealth of different structures are found as the physical parameters are varied. Although the model ignores gravity entirely, the results can be applied to the solidification from below of an alloy which is initially less dense than its eutectic. Predictions for the growth rate of the mixed-phase region agree well with existing experimental measurements of ice growing from aqueous salt solutions.

Natural convection in a mushy layer

Governing equations for a mushy layer are analysed in the asymptotic regime R m [Gt ] 1, where R m is an appropriately defined Rayleigh number. A model is proposed in which there is downward flow everywhere in the mushy layer except in and near localized chimneys, which are characterized by having zero solid fraction. Upward, convective flow within the chimneys is driven by compositional buoyancy. The radius of each chimney is determined locally by thermal balances within a boundary layer that surrounds it. Simple solutions are derived to determine the structure of the mushy layer away from the immediate vicinity of chimneys in order to demonstrate the gross effects of convection upon the solidification within the layer.

Interfacial conditions between a pure fluid and a porous medium: implications for binary alloy solidification

The single-domain Darcy–Brinkman model is applied to some analytically tractable flows through adjacent porous and pure-fluid domains and is compared systematically with the multiple-domain Stokes–Darcy model. In particular, we focus on the interaction between flow and solidification within the mushy layer during binary alloy solidification in a corner flow and on the effects of the chosen mathematical description on the resulting macrosegregation patterns. Large-scale results provided by the multiple-domain formulation depend strongly on the microscopic interfacial conditions. No satisfactory agreement between the single- and multiple-domain approaches is obtained when using previously suggested conditions written directly at the interface between the liquid and the porous medium. Rather, we define a viscous transition zone inside the porous domain, where the Stokes equation still applies, and we impose continuity of pressure and velocities across it. This new condition provides good agreement between the two formulations of solidification problems when there is a continuous variation of porosity across the interface between a partially solidified region (mushy zone) and the melt.

Natural convection during solidification of an alloy from above with application to the evolution of sea ice

We describe a series of laboratory experiments in which aqueous salt solutions were cooled and solidied from above. These solutions serve as model systems of metallic castings, magma chambers and sea ice. As the solutions freeze they form a matrix of ice crystals and interstitial brine, called a mushy layer. The brine initially remains conned to the mushy layer. Convection of brine from the interior of the mushy layer begins abruptly once the depth of the layer exceeds a critical value. The principal path for brine expelled from the mushy layer is through ‘brine channels’, vertical channels of essentially zero solid fraction, which are commonly observed in sea ice and metallic castings. By varying the initial and boundary conditions in the experiments, we have been able to determine the parameters controlling the critical depth of the mushy layer. The results are consistent with the hypothesis that brine expulsion is initially determined by a critical Rayleigh number for the mushy layer. The convection of salty fluid out of the mushy layer allows additional solidication within it, which increases the solid fraction. We present the rst measurements of the temporal evolution of the solid fraction within a laboratory simulation of growing sea ice. We show how the additional growth of ice within the layer aects its rate of growth.

Premelting dynamics in a continuum model of frost heave

Frost heave is the process by which the freezing of water-saturated soil causes the deformation and upward thrust of the ground surface. We describe the fundamental interactions between phase change and fluid flow in partially frozen, saturated porous media (soils) that are responsible for frost heave. Water remains only partially frozen in a porous medium at temperatures below

Convection and crystallization in magma cooled from above

0\,^\circ

Weakly nonlinear analysis of convection in mushy layers during the solidification of binary alloys

C owing both to the depression of the freezing temperature at curved phase boundaries and to interfacial premelting caused by long-range intermolecular forces. We show that while the former contributes to the geometry of fluid pathways, it is solely the latter effect that generates the forces necessary for frost heave. We develop a simple model describing the formation and evolution of the ice lenses (layers of ice devoid of soil particles) that drive heave, based on integral force balances. We determine conditions under which either (i) a single ice lens propagates with no leading frozen fringe, or (ii) a single, propagating ice lens is separated from unfrozen soil by a partially frozen fringe, or (iii) multiple ice lenses form.

Solidification of colloidal suspensions

Calculations are presented for the cooling from above of melts in the Di-An system in which the kinetics of crystallization are incorporated and play a dominant role. The results indicate that even with no initial superheat whatsoever, convection plays an important role in the cooling of magma chambers and allows substantial internal cooling, crystallization and differentiation. The calculations show, in agreement with observations, that in magma bodies hundreds of meters thick, crystallization occurs predominantly in the interior or at the floor, even though heat is lost only from the roof. The ratio of the final thickness of the layer formed at the floor to that formed at the roof increases as the overall size of the chamber increases, owing to the effects of convection.

Stability of ice-sheet grounding lines

We consider the solidification of a binary alloy in a mushy layer and analyse the system near the onset of buoyancy-driven convection in the layer. We employ a neareutectic approximation and consider the limit of large far-field temperature. These asymptotic limits allow us to examine the rich dynamics of the mushy layer in the form of small deviations from the classical case of convection in a horizontal porous layer of uniform permeability. Of particular interest are the effects of the asymmetries in the basic state and the non-uniform permeability in the mushy layer, which lead to transcritically bifurcating convection with hexagonal planform. We obtain a set of three coupled amplitude equations describing the evolution of small-amplitude convecting states in the mushy layer. These equations are analysed to determine the stability of and competition between two-dimensional roll and hexagonal convection patterns. We find that either rolls or hexagons can be stable. Furthermore, hexagons with either upflow or downflow at the centres can be stable, depending on the relative strengths of different physical mechanisms. We determine how to adjust the control parameters to minimize the degree of subcriticality of the bifurcation and hence render the system globally more stable. Finally, the amplitude equations reveal the presence of a new oscillatory instability.