horizontal solidification of average and low prandtl fluids in microgravity

Applied Numerical Mathematics 56 (2006) 682–694www.elsevier.com/locate/apnum

Horizontal solidification of average and low Prandtl fluidsin microgravity

M.M. Cerimele, D. Mansutti, F. Pistella ∗

Istituto per le Applicazioni del Calcolo, Viale del Policlinico 137, 00161 Roma, Italy

Available online 21 July 2005

Abstract

We compare the results of the numerical simulations of the horizontal Bridgman solidification (no-growth case)of Succinonitrile (SCN) and Silicon liquid samples. We approach several different conditions: the closed and theopen top crucible in a microgravitational environment where low amplitude and low frequency g-jittering is con-sidered.

The mathematical model here adopted describes the flow of the liquid phase as an incompressible Newtonianfluid, the heat transport phenomena and the evolution of the phase front. The stream-function/vorticity formulationof the flow of the liquid phase and the front-fixing treatment of the moving phase front are used. The numericalapproximation is based on a second order ENO scheme combined with a second order time scheme.

Interestingly we find that, in the case of SCN, thermocapillarity induces a secondary oscillatory flow in theaverage Prandtl melt that is not influenced by g-jittering; furthermore, it accelerates the flow and causes the curvingof the phase front. Within the Silicon sample, the low Prandtl melt does not undergo such a bifurcation and thephase front keeps flat in each case considered. In the open top crucible also the Silicon melt flow is faster than inthe closed top case. In general, in the tests here considered, g-jittering appears to be uninfluencial as the only effectobserved is an oscillatory behaviour of the streamfunction characterized by the same amplitude and frequency.© 2005 IMACS. Published by Elsevier B.V. All rights reserved.

Keywords: Phase-transition; Front-fixing; Oscillatory flows

* Corresponding author.E-mail addresses: [email protected] (M.M. Cerimele), [email protected] (D. Mansutti), [email protected]

(F. Pistella).

0168-9274/$30.00 © 2005 IMACS. Published by Elsevier B.V. All rights reserved.doi:10.1016/j.apnum.2005.06.001

M.M. Cerimele et al. / Applied Numerical Mathematics 56 (2006) 682–694 683

1. Introduction

The directional solidification of semi-conductor crystals by the Bridgman technique provides a typicalexample of free and moving boundary problem. In [10,7,9] it has been demonstrated that a satisfactoryanalysis of liquid/solid (L/S) phase transitions has to take into account the coupling effects of heat andmass transfer, gravity, melt flow and solid deformation fields. Actually, in artificial crystal growth, itis quite known that the characteristics of the final product are dependent on the occurrence of timeoscillatory instabilities in the melt flow that are the main cause of the appearance of dishomogeneities inthe crystal.

Numerical simulation of these processes allows to predict the behaviour of the melt, flow configura-tions, temperature and pressure distributions, the deformations of the solid and the motion and the shapeof the phase front. These results are very useful for applications because physical experiments are quitecomplicated due to the high temperature at which phase changes occur. For example, in artificial crystalgrowth, the prediction of the critical conditions for the onset of the secondary time oscillations in themelt flow allows growers to avoid them by properly selecting the growth conditions. An extensive liter-ature starting from [5] does exists on numerical simulations devoted to this issue; in particular, for lowPrandtl number melts (e.g. Pr = 0.015), the paper by Zhou and Zebib [18] provides the characterizationof several periodic flow configurations and a satisfactory reference list.

The present contribution should be considered in this context as it describes the results of a researchin progress on the numerical study of Bridgman horizontal solidification processes. With this technique,initially, the crystal grows as a device moves on the crucible from one extremum to the other by taking theheat out and carrying along the solidification front. During this process, in the melt, convective motionsset in due to the buoyancy force and to the horizontal temperature gradient. Thermocapillarity at the topfree surface of the crucible enhances or counteracts these effects. Then a modulation of thermocapillarityversus buoyancy (or vice-versa) is sufficient in order to control the stability of the melt motion and avoidthe dangerous time oscillations. As a consequence, in this field experiments on space labs or in generalmicrogravitational conditions have been and are being developed [10].

As far as the mathematical modeling is concerned, for a pure material, as the phase interface is asufficiently regular curve, differential models arising from the continuum mechanics conservation lawscombined with either front-tracking or front-fixing techniques produce results with about equal perfor-mances, from both viewpoints, accuracy and efficiency [15,11].

In [1], we proposed a front-fixing method for the simulation of L/S phase changes of a pure material.The Navier–Stokes equations within the Boussinesq approximation are adopted to cover the momentumand mass balance in the liquid phase. The equation for energy balance within the Fourier hypothesisis solved in both phases, liquid and solid. The classical Stefan condition for the release of the latentheat provides the extra equation for the description of the evolution of the moving L/S interface. Thestream-function/vorticity formulation is chosen in order to meet easily the mass conservation law. Thenumerical treatment is based upon a second order ENO (essentially non-oscillatory) scheme combinedwith a second order time approximation. The discrete equations are solved by a splitting technique.Second order interpolation and classical geometry arguments are the tools to follow the progression ofthe solidification front. We extended the original version of the model by using a time dependent non-dimensional form, that allows to handle the phase transition process also when the liquid region is verynarrow [4]. We adopted the mentioned model and numerical method for reproducing the data from apopular melting experiment of a pure gallium slab [6] and, in [2], we proposed a modified numerical test

684 M.M. Cerimele et al. / Applied Numerical Mathematics 56 (2006) 682–694

that provided an interpretation of the numerous mismatchings between the experiment and the relatednumerical simulations in literature.

Then we applied the extended approach to the horizontal Bridgman solidification of Succinonitrile(SCN) within the experimental set-up proposed by Yeoh et al. in [17] and faced both full gravity andmicrogravity conditions [3]; there we found that the process appears clearly sensitive to the intensity ofthe gravity field and then it is intrinsically suitable for physical experimentation on a space lab. So wehave continued this investigation and, now, gather the last results in the present paper, where, in the sameexperimental set-up, we compare the solidification process of an average Prandtl fluid (SCN) versus alow Prandtl fluid (Silicon) in both cases when they undergo the effects of a constant microgravity fieldand g-jittering (oscillatory microgravity field), including also thermocapillarity at the fluid/air interface.

The structure of this paper is the following: the next section is devoted to the presentation of themathematical model whereas the numerical procedure is described in Section 3, the numerical tests aredetailed in Section 4 and conclusions are drawn in Section 5.

2. Mathematical formulation

2.1. Governing equations

The mathematical formulation here considered is suited to treat both solidification and meltingprocesses. We exemplify it by considering a specific problem, that is the solidification of a liquid materialin a Bridgman apparatus.

Within this technique, used to grow artificial crystals from melt, the material in liquid phase is placed ina horizontal cylindrical crucible; the crystal grows as a device moves on the crucible from one extremumto the other by taking the heat out and carrying along the solidification front. During this process aconvective flow starts within the melt due to the horizontal temperature gradient.

The unsteady Navier–Stokes equations for incompressible non-isothermal fluids are adopted to repre-sent the melt dynamics:(

∂v∂t

+ (v · ∇)v)

= −∇p + μ∇2v − ρL

[1 − α(TL − Tp)

]g, (1)

∇ · v = 0, (2)

ρLcL

(∂TL

∂t+ (v · ∇)TL

)= kL∇2TL, (3)

where v is the velocity field, p is the pressure, TL is the temperature of the melt, g is the accelerationdue to gravity, μ is the dynamic viscosity, α is the volumetric thermal expansion coefficient, cL is thespecific heat and kL is the thermal conductivity of the melt. The above equations are built on the basisof the Boussinesq approximation: ρL is the density at the melting point; within the buoyancy term, thedensity ρ is supposed to vary according to the linear law ρ = ρL · [1 − α(TL − Tp)], where Tp is themelting temperature.

In the solid phase only heat diffusion occurs, which is governed by the equation

ρScS

∂TS = kS∇2TS, (4)
∂t


where TS , ρS , cS and kS are respectively temperature, density, specific heat and thermal conductivity ofthe solid.

On the phase front the equation

ρL�Hv · n = −kL

∂TL

∂n+ kS

∂TS

∂n(5)

holds, where �H is the latent heat characteristic of the material, the derivatives are meant to be withrespect to a prefixed normal to the front and n is the corresponding normal unit vector. Eq. (5) is theStefan condition [14] which expresses the balance of the energy exchanged between the two phaseswhose difference has to be equal to the latent heat.

2.2. Initial and boundary conditions

The above equations are completed by the following boundary conditions:

• a condition on the melt velocity field is required for the momentum equation (1); we impose

v = 0 (no-slip)

at the walls of the crucible (impermeability) and at the phase front.At the top of the crucible we consider two different experimental settings, that is either the case ofopen top in presence of thermocapillary effects and the case of closed top. The last one is still de-scribed by the no-slip condition (see above). The open top case is presented here within the restrictionof flat (horizontal) free-surface and in two dimensions; hereby we impose

μ∂u

∂y= −γ

∂TL

∂x,

v = 0,

where x and y are the coordinates respectively along the horizontal and the vertical direction, u andv are respectively the horizontal and vertical components of the velocity field of the melt and γ isdefined as γ = − ∂σ

∂TLwith σ the surface tension coefficient.

• the energy equations for the melt (3) and the solid (4) require conditions on the temperature fields ortheir normal derivative on the whole boundary. As we suppose that the material is pure, the phase-change temperature is a sharp value, Tp , just mentioned in the previous section. Then we impose

TL = TH (TH � Tp, superheating)

at the appropriate vertical wall;

TL = TS = Tp (phase-change temperature)

at the phase boundary;

TS = TC (TC < Tp, subcooling)

at the other vertical wall; eventually, we suppose that the horizontal crucible walls (open top caseincluded) are conducting and impose linear temperature profile.


Concerning the initial conditions, we observe the Bridgman growth starting from an initial time t0when a portion of material is already in solid phase and the remaining is melt at rest. Conducting tem-perature profile is imposed in the whole sample.

2.3. The (ψ − ω − T ) model and its non-dimensional form

The study of the Bridgman process in the longitudinal vertical plane of symmetry of the crucible isusually adopted as a simpler model to investigate the problem.

We formulate the two-dimensional problem via the stream function variable in order to overcomethe difficulty of meeting the mass conservation law within the melt flow. Then we solve the vorticity-stream function model where the momentum equation (1) is substituted by the vorticity transport equationobtained by applying the curl operator.

Let us call D2 and (x, y) respectively the two-dimensional rectangular domain (the section of thecrucible) and the space coordinates. Be l and h respectively the length and the height of D2. Let uschoose the origin O of the reference frame at the lower left vertex of D2 and the coordinate axes parallelto the edges of the rectangle.

For the sake of generality we transform the model in a non-dimensional form by adopting the listed

characteristic quantities: l0, the thickness of the initial strip of molten material, as reference length,l20ρLcL

kL

as reference time, and TH − Tp , the horizontal temperature difference, as reference temperature. In thefollowing we shall not change the notation and it will be clear from the context when the variables are tobe meant dimensional or non-dimensional.

Now taking into account the relations among the velocity v = (u, v), the vorticity ω and the streamfunction ψ ,

u = ∂ψ

∂y, v = −∂ψ

∂x,

ω = ∂v

∂x− ∂u

∂y

the non-dimensional conservative form of the equations for the melt appears to be

∂ω

∂t+ ∂

∂x

(∂ψ

∂yω

)− ∂

∂y

(∂ψ

∂xω

)= Pr∇2ω + Gr Pr2 ∂TL

∂x, (6)

∇2ψ = −ω, (7)∂TL

∂t+ ∂

∂x

(∂ψ

∂yTL

)− ∂

∂y

(∂ψ

∂xTL

)= ∇2TL. (8)

These equations hold in D2L, the domain of the liquid, for t � 0, Gr = gα(TH −Tp)l30ρ2

L

μ2 is the Grashofnumber and Pr = μcL

kLis the Prandtl number of the melt.

The non-dimensional form of the equation for the energy in the solid phase is

∂TS

∂t= k̄S

k̄L

∇2TS, (9)

where k̄L and k̄S are respectively the thermal diffusivity of the liquid and of the solid phases (k̄L =kL , k̄S = kS ). Eq. (9) holds in D2, the domain occupied by the solid, for t � t0.

ρLcL ρScS S


By defining the Marangoni number Ma = γ (TH −Tp)l0

μk̄L, the non-dimensional form of the thermocapillar-

ity condition appears:

∂u

∂y= −Ma

∂TL

∂x. (10)

The non-dimensional form of the equation for the interface (5) allows the introduction of the non-dimensional parameter St, the Stefan number, that is defined as St = cL(TH −Tp)

�H; then Eq. (5) becomes:

v · n = St

(∂TL

∂n− kS

kL

∂TS

∂n

). (11)

2.4. Coordinate transformation

In the two-dimensional domain D2, let us call C the phase boundary, C = {(x, y): x = f (y, t), y ∈[0, h/ l0]} (notice that f (y, t0) = l0). In order to handle the numerical difficulties caused by the movingfront, we adopt a front-fixing technique and introduce changes of coordinates: in the liquid phase

TL : (x, y) ∈ D2L �→ (ξ, η) ∈ [0,1] × [0, h/ l0],

ξ = x

f (y, t), η = y,

and in the solid phase

TS : (x, y) ∈ D2S �→ (ξ, η) ∈ [0,1] × [0, h/ l0],

ξ = x − f (y, t)

l/ l0 − f (y, t), η = y.

The numerical treatment of the Stefan condition (11) in the new coordinate system will be morestraightforward, as the phase boundary is reduced to a vertical line.

3. Numerical procedure

We solve Eqs. (6)–(9) and (11) and the related initial and boundary conditions by a numerical methodbased on a second order ENO (essentially non-oscillatory) scheme [8]. As concerning the liquid phase, itis essential to underline that accuracy and stability of the solution depend strongly on the approximationadopted for the convective terms, because the flow structure may be very complex (see [4]). For a betterdescription of the convective terms, the ENO schemes use a local adaptive set of points chosen accordingto the order of the method and the upwinding criteria and spread as much as possible along each coordi-nate direction towards the regions where the solution is smooth, in order to obtain the greatest amount ofinformation. As a result these methods are characterized by good resolution of high gradient fields andreach satisfactory accuracy even by using relatively coarse grids.

The discretization of the diffusive terms in the energy equations and the Poisson equation for thestream function is accomplished via second order centered differences.

Then we develop the calculation with a splitting technique. Starting from the initial condition, ateach time step Eqs. (6)–(9), accordingly transformed by means of TL and TS , are integrated separately.


Moreover in the liquid phase the treatment of both momentum and energy equations is accomplishedby splitting convective and diffusive terms: a TVD Runge–Kutta scheme of second order is used forthe time integration of the convective terms and an Euler scheme is used for the time integration of thediffusive terms. The generalized Newton method is used to solve the discretized Poisson equations forthe stream-function. Finally, for the evaluation of the updated phase front position, the current liquid andsolid temperature fields provide the normal derivatives appearing in (11) that are approximated to thesecond order of accuracy.

Concerning the numerical stability restrictions, we have considered the results of the analysis of theadopted ENO scheme applied to a generic conservation equation (see [12]) and of the explicit schemefor a diffusion equation; as a consequence the two following limitations on the time step �t hold:

�t

( |u|max

�x+ |v|max

�y

)� λ < 1,

�t � �x2�y2

2r(�x2 + �y2),

with |u|max and |v|max the maximum of the absolute value of the Cartesian components of the transportvelocity, λ an arbitrary constant smaller than unity and r the specific diffusion coefficient. We are awarethat for our model a stability analysis including the effects of the factors in the differential operators dueto the coordinate transformations TL and TS would be necessary. However the highly non-linear characterof phase transition problems due to the presence of the moving boundary is such that in practice timesteps largely below the numerical stability threshold are required.

The numerical approach here described has been already tested in [1,4].

4. Simulation tests

Our aim, here, is to describe and compare the characteristics of a horizontal Bridgman solidificationprocess in no-growth conditions in the cases of an average Prandtl melt, SCN, and a low Prandtl melt,Silicon.

For the sake of confidence in our methodology, we shall remind first our numerical simulation ofa known physical experiment with SCN in a closed crucible in full gravity conditions developed byYeoh et al. [17] (test A). Then we present the simulation of the same experiment with both selectedmaterials, SCN and Silicon, in a microgravitational environment (test A1) even in presence of g-jittering,the oscillatory microgravitational field holding on a space platform (test A2). We repeated the same testsfor the open top crucible including the thermocapillary effects (respectively test B1 and test B2).

In the Yeoh et al. physical experiment the heating and cooling jackets of the Bridgman furnace werekept at l = 0.0525 m apart and at temperatures respectively of TH = 350.64 K and TC = 326.39 K. Wesimulated the process in the ampoule core cell limited by the two jackets that, in the no-growth case, arefixed. It is experimentally seen that out of that region the melt is at rest. The height of the cell is fixed ath = 0.006 m. In each test we suppose that at initial time the thickness of the molten strip of material isl0 = 4.2 × 10−2.

Then, in the numerical tests that follow, the microgravity field is evaluated μg = 9.811 × 10−6 m s−2,corresponding to a typical spacecraft environment; the g-jittering field, which has to be added when itapplies, is chosen to be low amplitude and low frequency, that is 10−6 µg cos(2π0.1t).


Fig. 1. Test A. Streamlines and reference velocity values at equilibrium.

We remind that the relevant physical parameters for SCN have the following values: Tp = 331.38 K,ρL = 980 kg m−3, ρS = 1016 kg m−3, kL = 0.223 W m−1 K−1, kS = 0.225 W m−1 K−1, cL = 2000J kg−1 K−1, cS = 1955 J kg−1 K−1, α = 0.00081 K−1, �H = 46500 J kg−1; the Prandtl, Grashof, Ste-

fan and Marangoni numbers respectively result Pr = 23.04, Gr = 1677.3 × 103, St = 0.92 and Ma =326.47 × 103.

For test A we report in Fig.1 a result that we already showed in [3]: the streamlines and the velocityreference values in the region close to the phase front that compare quite well with the plot and dataprovided by Yeoh et al. in [17]. We accomplished this test forcing the temperature profile at the horizontalboundaries to be the same of the one at equilibrium in the experiment. The space grid adopted was[41(melt) + 13(solid)] × 41.

In view of the new tests we developed a mesh refinement analysis. The results are summarized inFig. 2, where the maximum values of the streamfunction, ψmax, versus time are graphed in the caseof four different space meshes: [41 + 13] × 61, [61 + 19] × 91, [81 + 25] × 121, [121 + 37] × 181 (insquare brackets the sum of the grid point numbers along (x) in the melt and in the solid domains isindicated). By comparing the plots we reached the conclusion that the space mesh [61 + 19] × 91 isan admissible compromise for accuracy without exceeding in computing time; we chose this space gridfor the following tests. In the context of the numerical simulations of this kind of processes in finitedifferences or in finite volumes [13], the considered space grids are really quite coarse and this fact hasto be taken as an advantage allowed by the numerical scheme adopted that provides accurate numericalsolutions as well.

In Fig. 3, we show the plots of ψmax in the four tests A1, A2, B1 and B2 with SCN. We can see clearlythat melt flow converges to a steady state in the case A1; in case A2, ψmax undergoes the oscillationsof the gravity field and oscillates with the same amplitude (O(10−13)); in both cases B1 and B2, whenthermocapillarity comes into play, the melt flows faster and converges to a quasi-periodic regime (withtwo frequencies). Obviously, in these last two cases, it is thermocapillarity that induces such oscillatory


Fig. 2. Mesh refinement analysis for Test A. Plot of ψmax (m s−2) versus t (s) for different space grids:A = [41 + 13] × 61,B = [61 + 19] × 91, C = [81 + 25] × 121,D = [121 + 37] × 181.

Fig. 3. SCN in microgravity. Plots of ψmax (m s−2) versus t (s).


Fig. 4. Test A1 for SCN. Streamlines (top) and isothermal lines (bottom) at thermodynamical equilibrium.

Fig. 5. Test B1 for SCN. Streamlines (top) and isothermal lines (bottom) at thermodynamical equilibrium.

behaviour that is much more significant than the g-jittering effect as the amplitude of oscillations is fiveorders of magnitude higher (O(10−8)).

In Fig. 4, streamlines and isothermal lines at thermodynamical equilibrium are plotted for the constantmicrogravitational field. The convection flow is very weak so that the phase front appears flat. For test A2we got qualitatively the same plots.

In test B1, although the presence of time oscillations of the streamfunction ψmax, thermodynamicalequilibrium is reached as well. In Fig. 5, streamlines and isothermal lines at equilibrium are plotted forthe constant microgravitational field. Qualitatively the same plots are obtained when g-jittering is added(test B2). In these cases the phase front is curved as the melt flow is no longer centre symmetric.

We have obtained quite different results when we tested Silicon whose physical parameters havethe following values: Tp = 1683 K, ρL = 2420 kg m−3, ρS = 2300 kg m−3, kL = 64 W m−1 K−1, kS =22 W m−1 K−1, cL = 1000 J kg−1 K−1, cS = 2300 J kg−1 K−1, α = 10−4 K−1, �H = 1800 × 103 J kg−1;the Prandtl, Grashof, Stefan and Marangoni numbers respectively result Pr = 1.096 × 10−2, Gr =16765.8 × 103, St = 1.077 × 10−2 and Ma = 0. Also in this case we have repeated the four tests abovedescribed, A1, A2, B1 and B2, keeping the dimensions of the ampoule core cell but adjusting the temper-ature of the heating and cooling jackets around Silicon’s melting temperature; we chose TH = 1702.4 Kand TC = 1669.59 K. In Fig. 6 we have summarized the numerical results by means of the plot of the


Fig. 6. Silicon in microgravity. Plots of ψmax (m s−2) versus t (s).

Fig. 7. Test B1 for Silicon. Streamlines (top) and isothermal lines (bottom) at thermodynamical equilibrium.

evolution of the maximum value of the streamfunction ψmax in each case. From this plot we deduce thatthermocapillarity in the open top crucible still makes the melt flow faster doubling the magnitude of ψmax

although, we notice, the increase of velocity is much lower than for SCN. Furthermore we observe thatthermocapillarity in Silicon does not cause the onset of an oscillatory secondary flow. As concerningg-jittering effects, we see that the consequent time oscillations of ψmax are characterized by the samefrequency and amplitude of the same order. Then we looked at the phase-front shape and the melt flowconfiguration and observed that for Silicon they are the same in each test. In Fig. 7 we show the stream-


lines and the phase front shape, and the isothermal lines for test B1, that for SCN it is the most criticalcase. On the contrary for Silicon, also in this case, the melt flow configuration keeps the central sym-metry and the phase interface remains flat; moreover the convection flow of the melt is so weak that thedistribution of the isothermal lines appears to be the one typical of a conducting process.

5. Conclusions

We have simulated numerically the approach to the thermodynamical equilibrium during a horizontalBridgman solidification process (no-growth case) for two molten materials, SCN and Silicon, respec-tively characterized by average and low Prandtl numbers. We have seen that they exhibit quite a differentbehaviour as we briefly summarize.

The open top crucible provides the most critical case (tests B1 and B2) in particular for SCN asthermocapillarity induces significative time oscillations in the melt and curved phase front either withconstant microgravity and with g-jittering; we stress this phenomenon as we know that it might beeligible to be the cause of imperfections in the final product during crystal growth [16].

For both materials, the considered g-jittering appears to be absolutely unessential versus the melt flowconfiguration, the phase front shape and the temperature distribution other than for inducing time os-cillations in the ψmax with the same characteristics of the g-jittering acceleration component; as theseoscillations are very weak, in the case of SCN in open top crucible, they do not interact with the oscilla-tions due to thermocapillarity.

We conclude reminding that in any case the Bridgman solidification technique is suitable for micro-gravitational applications, in particular when the crucible has closed top, as melt flow convection is centresymmetric and the phase front keeps flat, both aspects contributing to a uniform phase transition processtowards the production of structurally homogeneous solid samples.

For SCN and Silicon we have in progress the investigation of the effect of larger amplitude and higherfrequency g-jittering.

Acknowledgements

This work has been developed within the project (2002/2003) on liquid/solid phase transition fundedby the Italian Space Agency (ASI) (subcontract between IAC/Dipartimento di Scienza e Ingegneria delloSpazio, Univ. di Napoli) and within the institutional project “Environmental Dynamics” funded by CNR.

References

[1] M.M. Cerimele, D. Mansutti, F. Pistella, Simulation of the melt flow in a horizontal Bridgman growth via a front-fixingmethod, in: R.W. Lewis, J.T. Cross (Eds.), Numerical Methods in Thermal Problems, Pineridge Press, Swansea, 1997.

[2] M.M. Cerimele, D. Mansutti, F. Pistella, A front-fixing method for flows in liquid/solid phase change with a benchmarktest, in: CD-Rom Proceedings of ECCOMAS 2000, Barcelona, 11–14 September 2000.

[3] M.M. Cerimele, D. Mansutti, F. Pistella, Front-fixing modeling of directional solidification in microgravity, in: A.M. Anile,V. Capasso, A. Greco (Eds.), Progress in Industrial Mathematics at ECMI 2000, Mathematics in Industry, vol. 1, Springer,Wien, 2002, pp. 197–203.


[4] M.M. Cerimele, D. Mansutti, F. Pistella, Numerical modeling of liquid/solid phase transitions: Analysis of a Galliummelting test, Comput. Fluids 31 (2002) 437–451.

[5] M.J. Crochet, F.T. Geyling, Numerical simulation of horizontal Bridgman growth of gallium arsenide crystal, J. CrystalGrowth 65 (1983) 166–172.

[6] C. Gau, R. Viskanta, Melting and solidification of a pure metal on a vertical wall, J. Heat Transfer 108 (1986) 174–181.[7] M.E. Glicksman, S.R. Coriell, G.B. McFadden, Interaction of flows with the crystal–melt interface, Annu. Rev. Fluid

Mech. 18 (1986) 307–335.[8] A. Harten, S. Osher, Uniformly high-order accurate non-oscillatory schemes I, SIAM J. Numer. Anal. 24 (1987) 279–309.[9] H.E. Huppert, The fluid mechanics of solidification, J. Fluid Mech. 212 (1990) 209–240.

[10] D.T.J. Hurle, Convective transport in melt growth system, J. Crystal Growth 65 (1983) 124–132.[11] M. Lacroix, V.R. Voller, Finite difference solutions of solidification phase change problems: Transformed versus fixed

grids, Numer. Heat Transfer B 17 (1990) 25–41.[12] C. Shu, S. Osher, Efficient implementation of essentially non oscillatory shock-capturing schemes II, J. Comput. Phys. 83

(1989) 32–78.[13] F. Stella, M. Giangi, Melting of pure metal on vertical wall: numerical simulation, Numer. Heat Transfer A 38 (2000)

193–208.[14] A. Visintin, Models of Phase Transition, Birkhäuser, Boston, 1996.[15] R. Viswanath, Y. Jaluria, A comparison of different solution methodologies for melting and solidification problems in

enclosures, Numer. Heat Transfer Part B 24 (1993) 77–105.[16] M.G. Worster, Interfaces on all scales, in: P. Ehrhard, et al. (Eds.), Interactive Dynamics of Convection and Solidification,

Kluwer Academic, Dordrecht, 1992.[17] G.H. Yeoh, G. de Vahl Davis, E. Leonardi, H.C. de Groh III, M.A. Yao, A numerical and experimental study of natural

convection and interface shape in crystal growth, J. Crystal Growth 173 (1997) 492–502.[18] H. Zhou, A. Zebib, Oscillatory convection in solidifying pure metals, Numer. Heat Transfer Part A 22 (1992) 435–468.

horizontal solidification of average and low prandtl fluids in microgravity

Documents