Interactions between solidification and compositional convection in mushy layers

Mushy layers are ubiquitous during the solidification of alloys. They are regions of mixed phase wherein solid crystals are bathed in the melt from which they grew. The matrix of crystals forms a porous medium through which the melt can flow, driven either by external forces or by its own buoyancy in a gravitational field. Buoyancy-driven convection of the melt depends both on temperature gradients, which are necessary for solidification, and on compositional gradients, which are generated as certain components of the alloy are preferentially incorporated in the solid phase and the remaining components are expelled into the melt. In fully liquid regions, the combined action of temperature and concentration on the density of the liquid can cause various forms of double-diffusive convection. However, in the interior of mushy regions the temperature and concentration are thermodynamically coupled so only single-diffusive convection can occur. Typically, the effect of composition on the buoyancy of the melt is much greater than the effect of temperature, and thus convection in mushy layers in driven primarily by the computational gradients within them. The rising interstitial liquid is relatively dilute, having come from colder regions of the mushy layer, where the liquidus concentration is lower, and can dissolve the crystal matrix through which it flows. This is the fundamental process by which chimneys are formed. It is a nonlinear process that requires the convective velocities to be sufficiently large, so fully fledged chimneys (narrow channels) might be avoided by means that weaken the flow. Better still would be to prevent convection altogether, since even weak convection will cause lateral, compositional inhomogeneities in castings. This report outlines three studies that examine the onset of convection within mushy layers

Formation of Chimneys in Mushy Layers: Experiment and Simulation

In this fluid dyanmics video, we show experimental images and simulations of chimney formation in mushy layers. A directional solidification apparatus was used to freeze 25 wt % aqueous ammonium chloride solutions at controlled rates in a narrow Hele-Shaw cell (1mm gap). The convective motion is imaged with schlieren. We demonstrate the ability to numerically simulate mushy layer growth for direct comparison with experiments

Instability of radially spreading extensional flows. Part 1. Experimental analysis

We present laboratory experiments that show that fingering patterns can emerge when circular interfaces of strain-rate-softening fluids displace less viscous fluids in extensionally dominated flows. The fingers were separated by regions in which the fluid appeared to be torn apart. Initially, the interface had a large dominant wavenumber, but some of the fingers progressively merged so that the number of fingers gradually declined in time. We find that the transition rate to a lower wavenumber during this cascade is faster the larger is the discharge flux of the displacing fluid. At late times, depending on the discharge flux, the pattern either converged into an integer wavenumber or varied stochastically within a finite range of wavenumbers, implying convergence to a fractional wavenumber. In that stage of the evolution we find that the average wavenumber declines with the discharge flux of the displacing fluid.Israel Science Foundation (grant no. 1368/16

On the thermodynamic boundary conditions of a solidifying mushy layer with outflow

AbstractThe free-boundary problem between a liquid region and a mushy layer (a reactive porous medium) must respect both thermodynamic and fluid dynamical considerations. We develop a steady two-dimensional forced-flow configuration to investigate the thermodynamic condition of marginal equilibrium that applies to a solidifying mushy layer with outflow and requires that streamlines are tangent to isotherms at the interface. We show that a ‘two-domain’ approach in which the mushy layer and liquid region are distinct domains is consistent with marginal equilibrium by extending the Stokes equations in a narrow transition region within the mushy layer. We show that the tangential fluid velocity changes rapidly in the transition region to satisfy marginal equilibrium. In convecting mushy layers with liquid channels, a buoyancy gradient can drive this tangential flow. We use asymptotic analysis in the limit of small Darcy number to derive a regime diagram for the existence of steady solutions. Thus we show that marginal equilibrium is a robust boundary condition and can be used without precise knowledge of the fluid flow near the interface.This research began as a project between D. Conroy and M.G.W. at the Geophysical Fluid Dynamics Program: Woods Hole Oceanographic Institution (2006). We gratefully acknowledge helpful discussions with T. Schulze.This is the accepted manuscript for a paper Journal of Fluid Mechanics, Volume 762, January 2015, R1 (12 pages) © 2014 Cambridge University Press, DOI: 10.1017/jfm.2014.65

Solidification of a binary alloy: finite-element, single-domain simulation and new benchmark solutions

A finite-element simulation of binary alloy solidification based on a single-domain formulation is presented and tested. Resolution of phase change is first checked by comparison with the analytical results of Worster (1986) for purely diffusive solidification. Fluid dynamical processes without phase change are then tested by comparison with previous numerical studies of thermal convection in a pure fluid (de Vahl Davis 1983, Mayne et al. 2000, Wan et al. 2001), in a porous medium with a constant porosity (Lauriat & Prasad 1989, Ni et al. 1997) and in a mixed liquid-porous medium with a spatially variable porosity (Ni et al. 1997, Zabaras & Samanta 2004). Finally, new benchmark solutions for simultaneous flow through both fluid and porous domains and for convective solidification processes are presented, based on the similarity solutions in corner-flow geometries recently obtained by Le Bars & Worster (2006). Good agreement is found for all tests, hence validating our physical and numerical methods. More generally, the computations presented here could now be considered as standard and reliable analytical benchmarks for numerical simulations, specifically and independently testing the different processes underlying binary alloy solidification

Controls on microstructural features during solidification of colloidal suspensions

We present a mathematical model of the directional freezing of colloidal suspensions. Key ingredients of the model are the disjoining forces between the colloidal particles and the solidified suspending fluid, flow of the suspending fluid towards the solidification front through an accumulating layer of particles, and flow through microscopic films of unfrozen liquid separating particles from the freezing front. Our model predicts three dif- ferent modes of solidification leading to different microstructures: dendritic formations; laddered structures of ice spears and lenses; a frozen fringe, from which transverse ice lenses can form. It explains why different researchers have reported the existence of ice lensing with and without the pre-existence of frozen fringes. Our quantitative predictions are encapsulated within a universal, dimensionless phase diagram showing which mi- crostructure is to be expected under which operating conditions, and we show that these predictions are consistent with previous experimental studies as well as new experiments that we present here.Innovation Foundation for Doctor Dissertation in Northwestern Poltechnical University (CX201703) National Natural Science Foundation of China (Grant No. 51701155) British Council China Scholarship Counci