Numerical solution to the Saffman-Taylor finger problem with kinetic undercooling regularisation

The Saffman-Taylor finger problem is to predict the shape and,in particular, width of a finger of fluid travelling in a Hele-Shaw cell filled with a different, more viscous fluid. In experiments the width is dependent on the speed of propagation of the finger, tending to half the total cell width as the speed increases. To predict this result mathematically, nonlinear effects on the fluid interface must be considered; usually surface tension is included for this purpose. This makes the mathematical problem suffciently diffcult that asymptotic or numerical methods must be used. In this paper we adapt numerical methods used to solve the Saffman-Taylor finger problem with surface tension to instead include the effect of kinetic undercooling, a regularisation effect important in Stefan melting-freezing problems, for which Hele-Shaw flow serves as a leading order approximation when the specific heat of a substance is much smaller than its latent heat. We find the existence of a solution branch where the finger width tends to zero as the propagation speed increases, disagreeing with some aspects of the asymptotic analysis of the same problem. We also find a second solution branch, supporting the idea of a countably infinite number of branches as with the surface tension problem.

Crystal growth from undercooling melts under strong gradients of temperatures generating in short-duration microgravity

Crystal growth of bulk CdTe in short-duration microgravity is performed by the unidirectional cooling method. The largest growth grains in microgravity samples are 4X2mm. The cooling profiles indicate undercooling melts in microgravity. Cooling melt samples in microgravity generate strong gradient of temperature due to stop thermal convections. Temperature distribution in the melt is calculated by the one-dimensional equation of heat conduction, and about 100 K-undercooling is considered to occur at the cooling surface.

Bubble extinction in Hele-Shaw flow with surface tension and kinetic undercooling regularisation

We perform an analytic and numerical study of an inviscid contracting bubble in a two-dimensional Hele-Shaw cell, where the effects of both surface tension and kinetic undercooling on the moving bubble boundary are not neglected. In contrast to expanding bubbles, in which both boundary effects regularise the ill-posedness arising from the viscous (Saffman-Taylor) instability, we show that in contracting bubbles the two boundary effects are in competition, with surface tension stabilising the boundary, and kinetic undercooling destabilising it. This competition leads to interesting bifurcation behaviour in the asymptotic shape of the bubble in the limit it approaches extinction. In this limit, the boundary may tend to become either circular, or approach a line or "slit" of zero thickness, depending on the initial condition and the value of a nondimensional surface tension parameter. We show that over a critical range of surface tension values, both these asymptotic shapes are stable. In this regime there exists a third, unstable branch of limiting self-similar bubble shapes, with an asymptotic aspect ratio (dependent on the surface tension) between zero and one. We support our asymptotic analysis with a numerical scheme that utilises the applicability of complex variable theory to Hele-Shaw flow.

The effect of surface tension and kinetic undercooling on a radially-symmetric melting problem

The addition of surface tension to the classical Stefan problem for melting a sphere causes the solution to blow up at a finite time before complete melting takes place. This singular behaviour is characterised by the speed of the solid-melt interface and the flux of heat at the interface both becoming unbounded in the blow-up limit. In this paper, we use numerical simulation for a particular energy-conserving one-phase version of the problem to show that kinetic undercooling regularises this blow-up, so that the model with both surface tension and kinetic undercooling has solutions that are regular right up to complete melting. By examining the regime in which the dimensionless kinetic undercooling parameter is small, our results demonstrate how physically realistic solutions to this Stefan problem are consistent with observations of abrupt melting of nanoscaled particles.

Corner and finger formation in Hele-Shaw flow with kinetic undercooling regularisation

We examine the effect of a kinetic undercooling condition on the evolution of a free boundary in Hele--Shaw flow, in both bubble and channel geometries. We present analytical and numerical evidence that the bubble boundary is unstable and may develop one or more corners in finite time, for both expansion and contraction cases. This loss of regularity is interesting because it occurs regardless of whether the less viscous fluid is displacing the more viscous fluid, or vice versa. We show that small contracting bubbles are described to leading order by a well-studied geometric flow rule. Exact solutions to this asymptotic problem continue past the corner formation until the bubble contracts to a point as a slit in the limit. Lastly, we consider the evolving boundary with kinetic undercooling in a Saffman--Taylor channel geometry. The boundary may either form corners in finite time, or evolve to a single long finger travelling at constant speed, depending on the strength of kinetic undercooling. We demonstrate these two different behaviours numerically. For the travelling finger, we present results of a numerical solution method similar to that used to demonstrate the selection of discrete fingers by surface tension. With kinetic undercooling, a continuum of corner-free travelling fingers exists for any finger width above a critical value, which goes to zero as the kinetic undercooling vanishes. We have not been able to compute the discrete family of analytic solutions, predicted by previous asymptotic analysis, because the numerical scheme cannot distinguish between solutions characterised by analytic fingers and those which are corner-free but non-analytic.

Saffman-Taylor fingers with kinetic undercooling

The mathematical model of a steadily propagating Saffman-Taylor finger in a Hele-Shaw channel has applications to two-dimensional interacting streamer discharges which are aligned in a periodic array. In the streamer context, the relevant regularisation on the interface is not provided by surface tension, but instead has been postulated to involve a mechanism equivalent to kinetic undercooling, which acts to penalise high velocities and prevent blow-up of the unregularised solution. Previous asymptotic results for the Hele-Shaw finger problem with kinetic undercooling suggest that for a given value of the kinetic undercooling parameter, there is a discrete set of possible finger shapes, each analytic at the nose and occupying a different fraction of the channel width. In the limit in which the kinetic undercooling parameter vanishes, the fraction for each family approaches 1/2, suggesting that this selection of 1/2 by kinetic undercooling is qualitatively similar to the well-known analogue with surface tension. We treat the numerical problem of computing these Saffman-Taylor fingers with kinetic undercooling, which turns out to be more subtle than the analogue with surface tension, since kinetic undercooling permits finger shapes which are corner-free but not analytic. We provide numerical evidence for the selection mechanism by setting up a problem with both kinetic undercooling and surface tension, and numerically taking the limit that the surface tension vanishes.

An equivalent undercooling model to account for flow effect on binary alloy dendrite growth rate

A theoretical analysis is carried out to observe the influence of important flow parameters such as Nusselt number and Sherwood number on the tip speed of an equiaxed dendrite growing in a convecting alloy melt. The effect of thermal and solutal transfer at the interface due to convection is equated to an undercooling of the melt, and an expression is derived for this equivalent undercooling in terms of the flow Nusselt number and Sherwood number. Results for the equivalent undercooling are compared with corresponding numerical values obtained by performing simulations based on the enthalpy method. This method represents a relatively simple procedure to analyze the effects of melt convection on the growth rate of dendrites. (C) 2013 Elsevier Ltd. All rights reserved.

Effect of shrinkage induced flow on binary alloy dendrite growth: An equivalent undercooling model

This paper presents a theoretical model for studying the effects of shrinkage induced flow on the growth rate of binary alloy dendrites. An equivalent undercooling of the melt is defined in terms of ratio of the phase densities to represent the change in dendrite growth rate due to variation in solutal and thermal transport resulting from shrinkage induced flow. Subsequently, results for dendrite growth rate predicted by the equivalent undercooling model is compared with the corresponding predictions obtained using an enthalpy based numerical method for dendrite growth with shrinkage. The agreement is found to be good. Published by Elsevier Ltd.

Undercooling and Rapid Solidification of Nb-Si Eutectic Alloys Studied by Long Drop Tube

Niobium-silicide alloys have great potential for high temperature turbine applications. The two-phase Nb/Nb5Si3 in situ composites exhibit a good balance in mechanical properties. Using the 52 in drop tube, the effect of undercooling and rapid solidification on the solidification process and micro-structural characterization of Nb-Si eutectic alloy was studied. The microstructures of the Nb-Si composites were investigated by optics microscope (OM), X-ray diffraction (XRD) and scanning electron microscope (SEM) equipped with X-ray energy dispersive spectrometry (EDS). Up to 480 K, deep undercooling of the Nb-Si eutectic samples was successfully obtained, which corresponds to 25% of the liquidus temperature. Contrasting to the conventional microstructure usually found in the Nb-Si eutectic alloy, the microstructure of the undercooled sample is divided into the fine and coarse regions. The most commonly observed microstructure is Nb+Nb5Si3, and the Nb3Si phase is not be found. The change of coarseness of microstructure is due to different cooling rates during and after recalescence. The large undercooling is sufficient to completely bypass the high temperature phase field.

Undercooling and Rapid Solidification of Nb-Si Eutectic Alloys Studied by Long Drop Tube

Niobium-silicide alloys have great potential for high temperature turbine applications. The two-phase Nb/Nb5Si3 in situ composites exhibit a good balance in mechanical properties. Using the 52 in drop tube, the effect of undercooling and rapid solidification on the solidification process and micro-structural characterization of Nb-Si eutectic alloy was studied. The microstructures of the Nb-Si composites were investigated by optics microscope (OM), X-ray diffraction (XRD) and scanning electron microscope (SEM) equipped with X-ray energy dispersive spectrometry (EDS). Up to 480 K, deep undercooling of the Nb-Si eutectic samples was successfully obtained, which corresponds to 25% of the liquidus temperature. Contrasting to the conventional microstructure usually found in the Nb-Si eutectic alloy, the microstructure of the undercooled sample is divided into the fine and coarse regions. The most commonly observed microstructure is Nb+Nb5Si3, and the Nb3Si phase is not be found. The change of coarseness of microstructure is due to different cooling rates during and after recalescence. The large undercooling is sufficient to completely bypass the high temperature phase field.

Effect of solute on the growth rate and the constitutional undercooling ahead of the advancing interface during solidification of an alloy and the implications for nucleation

A framework is presented for modeling the nucleation in the constitutionally supercooled liquid ahead of the advancing solid/liquid interface. The effects of temperature gradient, imposed velocity, slope of liquidus, and initial concentration have been taken into account in this model by considering the effect of interface retardation, which is caused by solute buildup at the interface. Furthermore, the effect of solute concentration on the chemical driving force for nucleation has been considered in this model. The model is used for describing the nucleation of Al-Si and Al-Cu alloys. It was found that the solute of Si has a significant impact on the chemical driving force for nucleation in AI-Si alloys whereas Cu has almost no effect in Al-Cu alloys.

Asymptotic and numerical results for a model of solvent-dependent drug diffusion through polymeric spheres

A model for drug diffusion from a spherical polymeric drug delivery device is considered. The model contains two key features. The first is that solvent diffuses into the polymer, which then transitions from a glassy to a rubbery state. The interface between the two states of polymer is modelled as a moving boundary, whose speed is governed by a kinetic law; the same moving boundary problem arises in the one-phase limit of a Stefan problem with kinetic undercooling. The second feature is that drug diffuses only through the rubbery region, with a nonlinear diffusion coefficient that depends on the concentration of solvent. We analyse the model using both formal asymptotics and numerical computation, the latter by applying a front-fixing scheme with a finite volume method. Previous results are extended and comparisons are made with linear models that work well under certain parameter regimes. Finally, a model for a multi-layered drug delivery device is suggested, which allows for more flexible control of drug release.