Convection In Mushy Layers Research Articles

Mushy-layer convection

Complex physical processes that affect the solidification of multicomponent fluids have implications for materials science and geophysics.

Inertial effects on steady hydromagnetic mushy layer convection

Inertial effects on steady hydromagnetic mushy layer convection

Nonlinear mushy-layer convection with chimneys: stability and optimal solute fluxes

AbstractWe model buoyancy-driven convection with chimneys – channels of zero solid fraction – in a mushy layer formed during directional solidification of a binary alloy in two dimensions. A large suite of numerical simulations is combined with scaling analysis in order to study the parametric dependence of the flow. Stability boundaries are calculated for states of finite-amplitude convection with chimneys, which for a narrow domain can be interpreted in terms of a modified Rayleigh number criterion based on the domain width and mushy-layer permeability. For solidification in a wide domain with multiple chimneys, it has previously been hypothesized that the chimney spacing will adjust to optimize the rate of removal of potential energy from the system. For a wide variety of initial liquid concentration conditions, we consider the detailed flow structure in this optimal state and derive scaling laws for how the flow evolves as the strength of convection increases. For moderate mushy-layer Rayleigh numbers these flow properties support a solute flux that increases linearly with Rayleigh number. This behaviour does not persist indefinitely, however, with porosity-dependent flow saturation resulting in sublinear growth of the solute flux for sufficiently large Rayleigh numbers. Finally, we consider the influence of the porosity dependence of permeability, with a cubic function and a Carman–Kozeny permeability yielding qualitatively similar system dynamics and flow profiles for the optimal states.

Finite-sample-size effects on convection in mushy layers

AbstractWe report theoretical and experimental investigations of the flow instability responsible for mushy-layer convection with chimneys, drainage channels devoid of solid, during steady-state solidification of aqueous ammonium chloride. Under certain growth conditions a state of steady mushy-layer growth with no flow is unstable to the onset of convection, resulting in the formation of chimneys. We present regime diagrams to quantify the state of the flow as a function of the initial liquid concentration, the porous-medium Rayleigh number, and the sample width. For a given liquid concentration, increasing both the porous-medium Rayleigh number and the sample width drove a transition from a weakly convecting chimney free state to a state of mushy-layer convection with fully developed chimneys. Increasing the concentration ratio stabilized the system and suppressed the formation of chimneys. As the initial liquid concentration increased, onset of convection and formation of chimneys occurred at larger values of the porous-medium Rayleigh number, but the critical cell widths for chimney formation are far less sensitive to the liquid concentration. At the highest liquid concentration, the mushy-layer mode of convection did not occur in the experiment. The formation of multiple chimneys and the morphological transitions between these states are discussed. The experimental results are interpreted in terms of a previous theoretical analysis of finite amplitude convection with chimneys, with a single value of the mushy-layer permeability consistent with the liquid concentrations considered in this study.

Nonlinear steady hydromagnetic convection in mushy layers during alloy solidification

We consider the problem of nonlinear steady convection in a horizontal mushy layer, which is subjected to a vertical magnetic field, during alloy solidification. Under near-eutectic approximation and the limit of large far-field temperature, we determine the finite-amplitude solutions to the weakly nonlinear problem by using a perturbation technique. We obtain specific information about convective flow solutions in the form of rolls, squares, rectangles, down-hexagons with down-flow at the cells’ centers and up-flow at the cells’ boundaries and up-hexagons with up-flow at the cells’ centers and down-flow at the cells’ boundaries. The stability of these solutions is also investigated with respect to arbitrary disturbances. We found, in particular, that only up-hexagons or supercritical rolls can be stable. No subcritical non-hexagonal pattern convection was found to exist. The subcritical domain of stable hexagons and the supercritical domain of stable rolls were found to be enhanced by the magnetic field as well as by the non-uniformity of the permeability of the medium, but the magnetic field narrows down the supercritical domain for the stable hexagons. The magnetic field appears to reduce the tendency for chimney formation at the nodes of up-hexagons, while such a tendency is enhanced by the magnetic field at the nodes of down-hexagons.

Localisation of convection in mushy layers by weak background flow

It is known that freckles form at the sidewalls of directionally solidified materials. We present a weakly nonlinear analysis of the effects of a weak and slowly varying background flow formed by non-axial thermal gradients on convection near onset in a mushy layer. We find that in the two-dimensional case, the onset of mush convection occurs away from the walls. However if three-dimensional disturbances are allowed, the onset occurs near the walls of the container confining the mush. We derive amplitude equations governing this behaviour and simulate their evolution numerically.

Patterns of convection in solidifying binary solutions

During the solidification of two-component solutions a two-phase mushy layer often forms consisting of solid dendritic crystals and solution in thermal equilibrium. Here, we extend previous weakly nonlinear analyses of convection in mushy layers to the derivation and study of a pattern equation by including a continuous spectrum of horizontal wave vectors in the development. The resulting equation is of the Swift–Hohenberg form with an additional quadratic term that destroys the up-down symmetry of the pattern as in other studies of non-Boussinesq convective pattern formation. In this case, the loss of symmetry is rooted in a non-Boussinesq dependence of the permeability on the solid-fraction of the mushy layer. We also study the motion of localized chimney structures that results from their interactions in a simplified one-dimensional approximation of the full pattern equation.

Interactions between steady and oscillatory convection in mushy layers

We study nonlinear, two-dimensional convection in a mushy layer during solidification of a binary mixture. We consider a particular limit in which the onset of oscillatory convection just precedes the onset of steady overturning convection, at a prescribed aspect ratio of convection patterns. This asymptotic limit allows us to determine nonlinear solutions analytically. The results provide a complete description of the stability of and transitions between steady and oscillatory convection as functions of the Rayleigh number and the compositional ratio. Of particular focus are the effects of the basic-state asymmetries and non-uniformity in the permeability of the mushy layer, which give rise to abrupt (hysteretic) transitions in the system. We find that the transition between travelling and standing waves, as well as that between standing waves and steady convection, can be hysteretic. The relevance of our theoretical predictions to recent experiments on directionally solidifying mushy layers is also discussed.

On weakly nonlinear convection in mushy layers during solidification of alloys

We consider the problem of weakly nonlinear buoyant convection in horizontal mushy layers with permeable mush–liquid interface during the solidification of binary alloys. We analyse the effects of several parameters of the problem on the stationary modes of convection in the form of either a hexagonal pattern or a non-hexagonal pattern such as rolls, rectangles and squares. No assumption is made on the thickness of the mushy layer, and a number of simplifying assumptions made in previous theoretical investigations of the problem are relaxed here in order to study the problem based on a more realistic model. Using both analytical and numerical methods, we determine the steady solutions for the nonlinear problem in a range of the Rayleigh number R near its critical value. Both the nonlinear basic state and variable permeability of the present problem favour hexagon-pattern convection. The results of the analyses and computations indicate that depending on the range of values of the parameters, bifurcation to hexagonal or non-hexagonal convection can be either supercritical or subcritical. However, among all the computed solutions in the particular range of values of the parameters that are most relevant to those of the experiments, only convection in the form of down-hexagons with downflow at the cell centres and upflow at the cell boundaries, was found to be realizable, in the sense that its amplitude increases with R.

On oscillatory modes of nonlinear compositional convection in mushy layers

We consider the problem of nonlinear compositional convection in horizontal mushy layers during the solidification of binary alloys. Under a near-eutectic approximation and the limit of large far-field temperature, we determine a number of weakly nonlinear oscillatory solutions, and the stability of these solutions with respect to arbitrary disturbances is then investigated. The present investigation is an extension of the problem of the oscillatory modes of convection, which was investigated by [D.N. Riahi, On nonlinear convection in mushy layers. Part 1. Oscillatory modes of convection, J. Fluid Mech. 467 (2002) 331–359] in the absence of the main permeability parameter K 1 and very recently by [P. Guba, M.G. Worster, Nonlinear oscillatory convection in mushy layers, J. Fluid Mech. 553 (2006) 419–443] for two-dimensional cases in the presence of K 1 , to include the effects of K 1 , over a range of values of the other parameters, and for both two- and three-dimensional motion. It was found, in particular, that the results reported in [D.N. Riahi, On nonlinear convection in mushy layers. Part 1. Oscillatory modes of convection, J. Fluid Mech. 467 (2002) 331–359; P. Guba, M.G. Worster, Nonlinear oscillatory convection in mushy layers, J. Fluid Mech. 553 (2006) 419–443] are recovered if K 1 is zero or sufficiently small. In such cases two-dimensional solutions in the form of simple-travelling rolls are mostly the only stable and preferred solutions. However, as in the more realistic cases, if K 1 is not sufficiently small, then such solutions are replaced by preferred and stable three-dimensional solutions, which are mostly simple-travelling waves in the form of rectangles, squares or hexagons.

On Mixed Oscillatory and Steady Modes of Nonlinear Convection in Mushy Layers

We consider the problem of mixed oscillatory and steady modes of nonlinear compositional convection in horizontal mushy layers during the solidification of binary alloys. Under a near-eutectic approximation and the limit of large far-field temperature, we determine a number of two- and three-dimensional weakly nonlinear mixed solutions, and the stability of these solutions with respect to arbitrary three-dimensional disturbances is then investigated. The present investigation is an extension of the problem of mixed oscillatory and steady modes of convection, which was investigated by Riahi (J Fluid Mech 517: 71–101, 2004), where some calculated results were inaccurate due to the presence of a singular point in the equation for the linear frequency. Here we resolve the problem and find some significant new results. In particular, over a wide range of the parameter values, we find that the properties of the preferred and stable solution in the form of particular subcritical mixed standing and steady hexagons appeared to be now in much better agreement with the available experimental results (Tai et al., Nature 359:406–408, 1992) than the one reported in Riahi (J Fluid Mech 517:71–101, 2004). We also determined a number of new types of preferred supercritical solutions, which can be preferred over particular values of the parameters and at relatively higher values of the amplitude of convection.

Nonlinear Buoyant Convection in Mushy Layers during Alloy Solidification

Nonlinear Buoyant Convection in Mushy Layers during Alloy Solidification

Nonlinear oscillatory convection in mushy layers

We study the problem of nonlinear development of oscillatory convective instability in a two-dimensional mushy layer during solidification of a binary mixture. We adopt the near-eutectic limit, making the problem analytically tractable using standard perturbation techniques. We consider also a distinguished limit of large Stefan number, which allows a destabilization of the system to an oscillatory mode of convection. We find that either travelling waves or standing waves can be supercritically stable, depending strongly on the sensitivity of permeability of the mushy layer to variations in the local solid fraction: mushy-layer systems with relatively weak sensitivity are more likely to select travelling waves rather than standing waves in the nonlinear regime. Furthermore, the decrease in permeability is found to promote the subcritical, and hence more unstable, primary oscillatory states. In addition to mapping out the location of different stable oscillatory patterns in the available parameter space, we give the detailed spatio-temporal structure of the corresponding thermal, flow and solid-fraction fields within the mushy layer, as well as the local bulk composition in the resulting eutectic solid.

On nonlinear convection in mushy layers. Part 2. Mixed oscillatory and stationary modes of convection

This paper presents part 2 of a study of nonlinear convection in horizontal mushy layers during the solidification of binary alloys. Part 1 dealt with only the oscillatory modes of convection (Riahi, J. Fluid Mech. vol. 467, 2002, pp. 331–359). In the present paper we consider the particular range of parameters where the critical values of the scaled Rayleigh number just above its lowest subcritical value where convection is possible. Such a preferred solution has properties mostly in agreement with the experimental results due to Tait et al. (Nature, vol. 359, 1992, pp. 406–408) in the sense that the flow is downward at the cell centres, upward at the cell boundaries and there is some tendency for channel formation at the cell nodes.

On nonlinear convection in mushy layers Part 1. Oscillatory modes of convection

We consider the problem of nonlinear convection in horizontal mushy layers during the solidification of binary alloys. We analyse the oscillatory modes of convection in the form of two- and three-dimensional travelling and standing waves. Under a near-eutectic approximation and the limit of large far-field temperature, we determine the solutions to the nonlinear problem by using a perturbation technique, and the stability of two- and three-dimensional solutions in the form of simple travelling waves, general travelling waves and standing waves is investigated. The results of the stability and the nonlinear analyses indicate that supercritical simple travelling rolls are stable over most of the studied range of parameter values, while supercritical standing rolls can be stable only over some small range of parameter values, where the simple travelling rolls are unstable. The results of the investigation of the onset of plume convection and chimney formation leading to the occurrence of freckles in the alloy crystal indicate that the chimney of the plume can be generated internally or near the lower boundary of the mushy layer. The roles of a Stefan number, a permeability parameter and a concentration ratio on the flow instability in both linear and nonlinear regimes are also determined.

Onset of plume convection in mushy layers

A weakly nonlinear analysis is employed to investigate the onset of plume convection in the mushy layer of a binary solution directionally solidified from below. An improved mathematical model including the constant pressure condition at the melt/mush interface is applied to analytically analyse the nonlinear behaviour of the convection. Results show that, due to the consideration of the constant pressure condition at the interface, the present analytical results are much better in comparison with the experiments than those obtained by previous studies, in which the no-vertical-flow condition at the interface was considered. It is also shown that the bifurcation to three-dimensional hexagon convection, corresponding to the onset of plume convection, is subcritical. For the case of small concentration ratio [Cscr ] (equation (5b)) the channel of the plume is generated at the top of the mush and grows downwards into the mush. For the case of large [Cscr ], the channel may be generated at the interior of the mush and grow upwards to the top of the mush. The possible parameter regime in which the flow is of a stable down-centre hexagon is discussed.

CONVECTION IN MUSHY LAYERS

▪ Abstract As a molten alloy or any multi-component liquid is cooled and solidified the growing solid phase usually forms a porous matrix through which the residual liquid can flow. The reactive two-phase medium comprising the solid matrix and residual liquid is called a mushy layer. Buoyancy forces, owing primarily to compositional depletion as one or more of the components of the alloy are extracted to form the solid phase, can drive convection in the layer. In this review, I present an account of various studies of buoyancy-driven convection in mushy layers, paying particular attention to the complex interactions between solidification and flow that lead to novel styles of convective behavior, including focusing of the flow to produce chimneys: narrow, vertical channels devoid of solid. I define an ‘ideal’ mushy layer and argue that chimneys are an inevitable consequence of convection in ideal mushy layers. The absence of chimneys in certain laboratory experiments is explained in terms of nonideal effects.

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

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.