TRANSPORT PHENOMENA IN CRYSTAL GROWTH UNDER STRONG MAGNETIC FIELDS-APPLICATION TO THE TRAVELING HEATER METHOD

V. Kumar1, S. Dost2* & F. Durst1

1 Institute for Fluid Mechanics (LSTM), Friedrich-Alexander University, Erlangen, Germany2* Crystal Growth Laboratory, University of Victoria, Victoria, BC, Canada V8W 3P6

E-mail: sdost@me.uvic.ca, Tel: 1 250 721 8898, Fax: 1 250 721 6294

ABSTRACTThe article presents a 3-D numerical simulation studyfor the transport phenomena occurring in growth ofsingle crystal semiconductors under strong magneticfields. The magnetic body force components along withthe gravitational body force act on the points of theliquid zone and consequently suppress the convectiveflow occurring in the liquid solution zone. However, athigh magnetic fields the magnetic body forcecomponents that also depend on flow velocity presentnumerical challenges particularly in terms ofconvergence of iterations. In this article a 3-D finitevolume-based computer code developed in-house isimplemented using a novel numerical approach toremedy such difficulties. As an application, theTravelling Heater Method was selected for the growthof CdTe crystals under a static vertical magnetic field.

INTRODUCTIONThe application of applied magnetic field in crystalgrowth is of great interest in minimizing the adverseeffect of natural convection on the growth process andcrystal quality. However, the use of high magnetic fieldsin numerical simulations exhibits great challenges interms of numerical stability and convergence. Forinstance, in the THM growth of CdTe single crystalsgiven in [1] the numerical model, which used acommercial finite volume–based package (CFX), leadto an interesting phenomenon. The maximum velocityof the flow field was given as a function of the

Hartmann number ( Ha = B rc σ / µ( )1/ 2 with rc the radius

of the growth cell), and similar to what was observed in[2-4], it was shown that the maximum velocity obeys alogarithmic law. In addition, the flow field exhibitedthree, distinct behaviour: a stable region up to thecritical value of 8.0 kGauss (Ha less than 160), atransitional region with B from 8.0 to some value above12.0 kGauss (Ha from 160 to 250 (approximately)), andan unstable region with higher field levels (Ha>250).Flow velocity decreases with the increasing magneticfield in the stable region, but increases in theintermediate and unstable regions. Within the stableregion, the relation of the maximum flow intensity andHartmann number obeys a power law of

Umax ∝ Ha−5 /4. The optimum magnetic field level in

this system was about 8.0 kGauss (Ha = 160) for auniform growth; flattest interface. Such results are verysignificant for experimentalists for designing theirexperimental set ups properly. A similar behaviour inthe velocity field was also observed numerically in theelectroepitaxial growth of GaAs in [5]. The question ofwhether such behaviour of the velocity field in crystalgrowth under magnetic fields is physical or just

numerical, still remains to be proven experimentally. Inthe absence of such experimental data, it appears to bedue to the way the equations are solved numerically.

In [1,5], numerical simulations could not beperformed for magnetic fields higher than 12.0 kGauss(Ha = 250) due to numerical instability and convergenceproblems in iterations. Even for the region of Ha = 160-250, many numerical innovative techniques were usedto keep the system numerically stable, and it is possiblethat such transient and unstable behaviour in this level(Ha = 160-250) were of numerical nature.

A similar problem, the THM growth of a SiGesystem, was considered in [6], and the field equationswere solved using a finite element-based commercialpackage (FIDAP). Numerical simulations could only beconducted up to a 2.0 kGauss level (about Ha = 20 withrespect to diameter), and at higher magnetic field levelsnumerical difficulties were encountered leading tostrong convection and unstable fluid flow, and non-convergence in iterations. Results presented in [6]predicted a similar behaviour of the fluid velocity givenin [1] since the simulation was in the stable region of[1]; Ha = 20 less than 160. However the power lawobtained in [6] for the stable region was different than

that of [1], i.e. Umax ∝ Ha−7 /4. Their optimum magnetic

field level for their system was about 2.0 kGauss (Ha =20) for a flatter interface.

As seen, when a flow problem is simulated understrong magnetic fields, numerical instability andproblems in convergence of iterations may occur. Thismay lead to either conflicting or in complete results.This is mainly is due to the following. The magneticforce components are not only functions of the squareof the magnetic field intensity but also depend on thevelocity components. These magnetic force terms,which are on the right-hand side of the momentumequations, are usually treated as source terms incomputer codes. This is usually the source of such non-convergence or instability. In this article we introduce aninnovative numerical technique to handle suchnumerical difficulties. As an application, the TravellingHeater Method (THM) was selected for the growth ofCdTe crystals under a static vertical magnetic field.

The governing equations are solved numerically.Evolution of the growth interface is simulated with thehelp of a moving grid algorithm in a block-structuredfinite-volume code. The convergence rate of iterationsis significantly improved by treating a part of themagnetic body force terms implicitly in the iterationloop. Consequently, the magnetic field as high as 15.0kGauss (Ha = 290) did not lead to any convergenceproblems and unstable flows. Simulation results arepresented for the flow, concentration, temperature, andelectric potential fields in the liquid solution. Transport

structures were stable all the way up to Ha = 290.Application of magnetic field suppresses convection inthe solution. The behaviour of the velocity field given in[1], and also implied in [6], was not observed up to a15.0 kGauss (Ha = 290) magnetic field level, insteadthe velocity field continued to decrease monotonicallywith the increasing magnetic field level.

SIMULATION MODELIn both [1] and [6], contribution of the induced electricfield in the magnetic body force terms in the momentumequations was not taken into account. In the presentwork we simulate the THM system considered in [26]for the growth of CdTe single crystals under highmagnetic fields, and the contribution of the inducedelectric field is included in the analysis.

Fig.1 THM growth system of CdTe: (a) a schematic viewof the THM crucible, and (b) an interior view of thenumerical grid.

Fig.1. (a) presents a schematic view of the THM systemused in [1]. The solid phase represents the source(feed) and the substrate (seed). The outer and innerradii of the THM crucible are 1.5 and 1.3~cm,respectively. The crucible is made of silica to withstandthe high temperature in the furnace. Initially, the heightsof the source ( H1 ), solution ( H 2 ) and substrate ( H 3 )

are 3.0, 2.3 and 0.7 cm, respectively. Thickness of thecrucible wall is 0.2 cm. The three-dimensional block-structured grid containing 36 blocks was created to mapthe THM growth system shown in Fig. 1(a). An insideview of the numerical grid is presented in Fig. 1 (b). Theactual grid consists of approximately 133,000 controlvolumes (CV). The O-grid topology is adopted tomaintain the quality of the mesh which is used toconform the cylindrical shaped geometry. The cruciblein the present set-up is mapped with 21 outer blocksand additional 15 inner blocks are created to mapsource, solution and substrate. Thermophysicalproperties of the CdTe THM growth are presented inTable 1 and are taken from [1].

Movement of the growth (substrate/liquid solution)and dissolution (liquid solution/source) interfaces issimulated with a moving grid technique. The block-structured finite-volume numerical code is adapted totrack the moving interfaces between any two blocks ofarbitrary topologies. Interface tracking methods involvemovement of the numerical grid which is requiredbecause of the motion of the boundary between thesolid and liquid phases and/or recreation of theinterface. Different blocks can be computed on different

processors in a simulation and the data communicationbetween blocks on different processors is performed bymessage passing interface (MPI) library. Tracking ofsuch a complex geometry becomes quite challengingbecause of the complex structured grid. Quality of thegrid is maintained during the simulations to avoid highlyskewed cells within the numerical domain.

To account for the movement of the grids due totracking of interface, the conservation equation can betransformed by the following ``Arbitrary Lagrangian-Eulerian'' approach where the base vector is kept fixed;the grid movement results in additional convectivefluxes in the conservation equations. This gridmovement has to be taken into account when the liquid-solid interface moves during the growth process.

Table 1. Thermo-physical properties of THM system [1]Property Material Symbol Value Unit

Density solutionsolidsilica

ρ5620.05680.02220.0

kg / m3

Thermalconductivity

solutionsolidsilica

λ5.73.83.1

W / mK

Specificheat

solutionsolidsilica

cp372.0160.0770.0

J / kgK

Electricconductivity

solution

solidσ E

2.0 × 105

3.6 × 103 Ω−1m−1

Dynamicviscosity

solution µ1.81 × 10−3 Ns / m2

Thermalexpansioncoefficient

solution βT 1.2 × 10−4 1 / K

Solutalexpansioncoefficient

solution βC -0.056

Latent heatof fusion

solution ΔH 2.09 × 105 J / kg

Prandtlnumber

solution Pr 0.06

Table 2. Variables in the generic transport equationConservedquantity

Φ ΓΦ QΦ

Mass 1 0 0Momentum U j

µ−∂P

∂x j+ ρg jβT (T − Tref )

+ρg jβC (C − Cref ) + ε jqr JqBrEnergy cp,lT λl 0

Solutemass

C D 0

Governing EquationsIn the present model, it was assumed that the liquidsolution is incompressible and Newtonian, and theinduced magnetic field is small compared to the appliedmagnetic field and therefore can be neglected. Maxwell

equations are then reduced to a single equation: theelectric charge balance. The thermomechanicalbalance equations, namely the continuity, balance ofmomentum, balance of energy, and mass transport arepresented in integral forms for an arbitrary control-volume (CV) of volume ΔV and surface area ΔS :Continuity

∂ρ

∂tΔV∫ dV + ρUiΔS∫ dSi = 0 (1)

Momentum

∂

∂tΔV∫ (ρU j )dV + ρUiΔS∫ U jdSi

= µ∂U j

∂xiΔS∫ dSi −∂P

∂x jΔV∫ dV

+ ρg jΔV∫ {βT (T − Tref ) + βC (C − Cref )}dV

+ {σ Eε jqrΔV∫ (−∂φ

∂xq+ εqmnUmBn )

Jq

Br}dV

(2)

Energy∂

∂tΔV∫ (ρcp,lT )dV + ρUiΔS∫ cp,lTdSi = λl∂T

∂xiΔS∫ dSi (3)

Mass transport∂

∂tΔV∫ (ρC)dV + ρUiΔS∫ CdSi = ρD∂C

∂xiΔS∫ dSi (4)

Electric charge balance

∂φ

∂tΔV∫ dV = ε ijkU jΔS∫ BkdSi (5)

Within the Boussinesq approximation, the solutiondensity is varied only in the source terms of the

momentum equations. In equations (2) and (5), ε ijk

denotes the well-known permutation symbol. In thecase of moving grids, the volume and the surface areaof the control volume are not constant in time andhence the first terms of the left hand side can bemodified according to the Leibniz rule

∂

∂tΔV∫ (ρΦ)dV =d

dtρΦdV − ρUi

g

ΔS∫ΔV∫ ΦdSi (6)

whereΦ represents any transport variables. By theapplication of the Leibniz rule, the integral of rate ofchange of a quantity in a fixed shape control volume ischanged into two parts: the rate of change of thequantity with the deformation of the control volume andthe convective fluxes arising due to the movement ofthe control volume boundaries. By using the Leibnizrule in Eqs. (1) - (4), a generic transport equation for atransport variable Φ can be written as:d

dtρΦdV + ρ

ΔS∫ΔV∫ (Ui −Uig )ΦdSi

= ΓΦ

∂Φ

∂xiΔS∫ dSi + QΦΔV∫ dV(7)

where ΓΦ and QΦ represent respectively a material

coefficient and the source term for the transportquantity Φ (their values are given in Table 2).

In the solid phase, the field equations, Eqs. (1)-(4),reduce to the energy balance in the form:

d

dtρcp,sTdVΔV∫ − ρUi

gcp,sTdSiΔS∫ = λs∂T

∂xidSiΔS∫ (8)

Boundary and Interface ConditionsThe required boundary and interface conditions for thecomputational domain are as follows. For the velocityfield, the Stokes no-slip boundary condition is applied,i.e. Ui = 0 for i = 1, 2, 3 on all boundaries. At the side

vertical wall of the liquid zone, no mass flux condition isused, which reduces to the zero gradient boundarycondition in concentration:∂C

∂n= 0 (9)

where n is the normal direction to the wall. It isassumed that the electric current, heat flux, electricpotential, and temperature are continuous across thesidewall:

σ E

∂φ

∂n l

= σ E

∂φ

∂n s

, λ∂T

∂n l

= λ∂T

∂n s

,

φl = φs , and Tl = Ts (10)

At the growth and dissolution interfaces, Dirichlet typeboundary conditions are used, i.e.T = Tg , C = Cg , T = Td , and C = Cd (11)

The growth interface Tg and dissolution interface Tdtemperatures were set to 985 K and 991 K,

respectively. The concentrations Cg and Cd are the

equilibrium values determined according to thefollowing equation [1]:

C = 0.209 exp(8.2 −8607.29

T) (12)

At the growth and dissolution interfaces the assumptionof continuous induced electric current is used, i.e., thecondition in Eq. (10)1.

In order to estimate the fluxes due to gridmovement at an interface, the interfacial undercoolingis neglected and therefore the interfacial energybalance or Stefan-condition is applied (see e.g. [7] fordetail):

ρlΔHUiI +

12ρl (1 −

ρlρs)2 (Ui

I )3 = λs∂T

∂xi s

− λl∂T

∂xi l

(13)

Since the liquid and solid densities at the interfacetemperature are close to each other, the influence ofdensity change on interfacial velocity is negligible andtherefore the second term on the left hand side of theabove equation can be neglected. It should beemphasized that the interface velocity computed fromthe Stefan condition should not be directly utilized in the

above mentioned conservation equations (for Uig

)

otherwise mass, momentum, species and energysources or sinks will be generated in the respectiveequations (see [8-10]). Due to the discretization errors,for instance in case of incompressible flows, the gridflux term may not cancel out the unsteady term to give

out divergence free velocity, i.e. ρUiS∫ dSi = 0 . This

inconsistency arising from the discretization of theequations introduces constraints on the size of the timestep of integration and can be avoided by taking a so-called space conservation law (SCL) into account. Thiswas pointed out initially in [11], and later in [8,9] haveshown the necessity of solving space conservation lawfor arbitrary shaped domains in case of movingboundaries. The space conservation law has thefollowing form:

d

dtdV

ΔV∫ − Uig

ΔS∫ dSi = 0 (14)

The interfacial velocity from Stefan balance (UI ,i ) is

only utilized to recreate numerical grids for the domain.For the crucible outer boundaries (top, bottom and

side walls) the following thermal boundary condition isapplied:

−λs∂T

∂n= h(T − Tf ) (15)

where h = 220W / m2K is the heat transfer coefficientmeasured from experiments and the furnacetemperature profile was taken from [1].

Numerical methodologyIn the previous section, conservation equations arepresented in the Cartesian form, however, solution ofthe equations is sought on curvilinear body-fitted grids.Hence, the terms involving gradients and divergenceoperators were transformed from the Cartesian to thecurvilinear coordinates. The coupling between thevelocities and mass in the momentum field equationswas treated with the SIMPLE algorithm of [12] where apressure-correction equation is solved to correct bothpressure and velocity fields. The discretized continuityequation needs the velocity of fluid at cell faces and ifthis is carried out by a linear interpolation of velocity incase collocated grids, a non-physical checkerboardvelocity and pressure fields and therefore velocityinterpolation proposed in [13] were taken into account.The term involving the grid velocities in the transportequations needs special care. This is due to the factthat the treatment of change of volume due to the gridmovement may easily give rise to numerical source orsink terms in the continuity equation, which may thencause a convergence problem in the pressurecorrection equation. Furthermore, in case of highmagnetic fields, the convergence rate may getdeteriorated significantly due to the strong negativesource terms. Therefore, the source terms in themomentum equations requires a special treatment(details can be found in [14]).

Solution ApproachAt first, only the steady-state energy equation is solvedin order to obtain a suitable initial condition for the

temperature field. With the temperature field obtainedfrom the steady-state simulations, the time dependentmomentum, energy and mass transport equation aresolved. The second order time implicit scheme isadopted for the time integration. The implicit timeschemes are inherently stable even for large values ofCourant number and therefore the scheme helps tochoose larger time steps than the explicit schemes. Alarge number of computations were performed forvarious strengths of the magnetic field varying from 2.0to 15.0 kGauss. In each case the simulations wereperformed for approximately 3500 time steps to obtaininitial flow and thermal fields. Once, the flow field iscomputed the tracking of the growth and dissolutioninterfaces were started with the help of the moving gridalgorithm for about 500-600 time steps. Within eachtime step, the residuals for each equation were broughtdown to 5—6 orders of magnitude. As mentionedearlier, the SIMPLE algorithm was adopted for thepressure velocity coupling in the momentum equations.After every time step, the computational grids arerecreated with the help of an algebraic grid generator.The movement of a block influences all its immediateneighboring blocks and therefore the gridsmanagement becomes a little bit complex.

(a) (b) (c)

(d) (e) (f)

(h) (i) (j)

Fig.2 Computed temperature (a-c), flow (d-f), andconcentration (h-j) fields in the vertical plane at 2.0,6.0, and 10.0 kGauss field levels.

RESULTS AND DISCUSSIONSimulation results are presented in this section.Temperature, flow and concentration fields computed atseveral magnetic field levels are presented in thevertical plane of the liquid zone in Fig.2, and in thehorizontal plane of the liquid zone in Fig.3. Asexpected, the common characteristic of all thesetransport structures is the development of three-dimensional structures (non-axisymmetric) onset of thegrowth process in spite of the initially imposed

axisymmetric boundary conditions. This shows theimportance of using three-dimensional models foraccurate predictions.

Due to the transient nature of the problem (due tothe constant movement of the heater) the transportstructures are always unsteady, but the isothermsshown in Fig.3a-c may still provide an insight about theevolution of the growth interface. As can be seen, asthe applied magnetic field intensity increases, isothermsare becoming more concave towards to the liquid zone,giving rise to a favorable condition for growth of crystalswith uniform composition and less inclusions. It appearsthat the magnetic intensity level somewhere between6.0 to 8.0 kGauss is optimum, above which the growthinterface becomes more concave and is unfavorable forhigh quality crystal growth. This will be seen clearlylater from the computed interface shapes. This level ofmagnetic field is called “optimum” since it leads to themost favorable growth interface (slightly concavetowards the liquid zone) for better crystals.

(a) (b) (c)

(d) (e) (f)

(h) (i) (j)

Fig.3 Computed temperature (a-c), flow (d-f), andconcentration (h-j) fields in the horizontal plane at2.0, 6.0, and 10.0 kGauss field levels.

The computed optimum magnetic field level in thepresent study, which is about 6.0-8.0 kGauss, agreeswith those computed in [1] and [6]. However, as can beseen from Fig.2a-j and Fig.3a-j, all the fields(temperature, concentration and flow) becomesmoother and more axisymmetric with the increasingmagnetic field level. There is no unstable or strong flowdevelopment above the optimum magnetic fieldintensity level, contrary to what observed in both [1] and[6]. This shows that, within the limits of assumedsimplifications and assumptions in the model for theTHM system considered here, such unstable andstrong transport structures observed in [1] and [6] areperhaps numerical. The numerical treatment presentedin the present study eliminated such instabilitiesobserved earlier and allowed us to carry out

computations up to 15.0 kGauss (Ha = 290) level,without any convergence problems. However, to thebest of our knowledge, the physical existence of suchunstable and strong transport structures in such a THMsystem remains to be proven experimentally.

Fig.4 The maximum flow velocity as function of theapplied magnetic field.

Fig. 5 The shape of the growth interface at differentmagnetic field levels.

As seen from Fig.2d-f and Fig3h-j, the flow fieldcontinues to be suppressed with the increasingmagnetic field intensity. In order to show thisquantitatively, the maximum velocity in the liquid zonewas computed and plotted versus applied magneticfield in Fig.4. As expected the maximum velocity obeysa logarithmic laws as also predicted in [1] and [6], butcontrary what has been observed in [1] and [6], the flowfield continues to be suppressed (up to Ha = 290magnetic field level).

Evolution of the growth interface is presented inFig.5 for five levels of the magnetic field. As expectedfrom the computed temperature field (Figs.2 and 3), themagnetic field level of about from 6.0 to 8.0 kGaussleads to the most favorable growth interface. This is

sufficiently close to the results of [1], considering theabsence of the effect of induced electric field in [1].

CONCLUSIONSThe article presented a 3-D numerical simulation studyfor the THM growth of CdTe single crystalsemiconductors under strong magnetic fields. A 3-Dfinite volume-based computer code developed in-houseis implemented using a novel numerical approach toremedy such difficulties. The convergence rate ofiterations is significantly improved by treating a part ofthe magnetic body force terms implicitly in the iterationloop. Consequently, the magnetic field as high as 15.0kGauss did not lead to any convergence problems.

Simulation results presented for the flow,concentration, and temperature fields in the liquidsolution show that, in spite of the initially assumed-axisymmetric boundary conditions, three dimensionaltransport structures develop as soon as the growthprocess begins. There is an optimum magnetic fieldlevel for which the growth interface becomes slightlyconcave towards the liquid zone, giving rise to afavorable condition for growth crystal with uniformcrystal composition and less inclusions. The optimummagnetic field level observed in this study agrees withthose of the literature.

However, the computed transport structuresbecome more suppressed and smoother with theincreasing magnetic field level, and are stable and donot become stronger even above this optimummagnetic field level, up to 15 kGauss (Ha=290).Application of the magnetic field suppresses the flowvelocity in the solution, and the magnitude of themaximum flow velocity decreases monotonically withincreasing magnetic field intensity. However, theverification of the simulation results of this study, andalso others of the literature, remains to be provenexperimentally.

Acknowledgement: The work has been carried outduring Professor Dost’s study leave at LSTM, Friedrich-Alexander University, Erlangen, Germany, and thesupport provided is gratefully acknowledged. The workwas also supported financially through Dr. Dost’sNSERC discovery grant.

NOMENCLATURER1 Outer radius,R2 Inner radius

U

maxmaximum velocity

T temperatureC solute concentration in the solutiont timeUi the velocity component in the xi -direction

P pressureg j gravity vector

D solute diffusion coefficientsφ electrical potential

Jq electric current density vector

Bk magnetic field vector

Uig

grid velocity

Subscriptsl liquid solutions solid phaseg growth interface

d dissolution interfaceref reference values

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