Inertial e ects in the rotationally driven melt motion …lopez/pdf/ZAMP_TSLW00.pdfThis article treats the melt motion driven by the rotations of the crystal and crucible about their

Z. angew. Math. Phys. 51 (2000) 267–2890044-2275/00/020267-23 $ 1.50+0.20/0c© 2000 Birkhauser Verlag, Basel

Zeitschrift fur angewandteMathematik und Physik ZAMP

Inertial effects in the rotationally driven melt motionduring the Czochralski growth of silicon crystals with astrong axial magnetic field

G. Talmage, S.-H. Shyu, J. M. Lopez and J. S. Walker

Abstract. This article treats the melt motion driven by the rotations of the crystal and crucibleabout their common vertical axis during the Czochralski growth of silicon crystals with a strong,uniform, vertical magnetic field produced by a solenoid around the crystal growth furnace. Sincemolten silicon is an excellent electrical conductor, the interaction parameter N and the Hart-mann number Ha are both large for typical magnetic field strengths, so that composite singularperturbation techniques for N 1 and Ha 1 are appropriate. An inertialess solution, whichalso assumed that N Ha3/2, was presented in a previous paper. In the inertialess solution,the largest gradient of the azimuthal velocity vθ and the largest secondary flow with radial andaxial velocities vr and vz both occur inside an interior layer with an O(Ha−1/2) radial thicknessat the vertical cylinder directly beneath the periphery of the crystal, where the growth interfacemeets the free surface. For all current experimental studies, the assumption that N Ha3/2

is not satisfied. The appropriate assumption is that N = O(Ha3/2), and inertial effects are notnegligible inside the interior layer. An intersection region, which is formed by the intersectionof the interior layer and a Hartmann layer with an O(Ha−1) axial thickness adjacent to thecrystal-melt interface and free surface, is intrinsically coupled to the interior layer. This arti-cle treats inertial effects in the interior layer and intersection region for N = O(Ha3/2) andHa 1. Non-linear governing equations were derived and solved numerically. A fourth-orderAdams–Bashforth–Moulton predictor-corrector method was used to solve the transport equa-tions for the primary azimuthal velocity and for the secondary-flow vorticity. Poisson equations,which govern the stream functions for both the secondary flow and the electric current density,were solved using a matrix diagonalization technique. The effects of inertia on the melt motionare discussed. This type of study provides for a fuller understanding of the melt motion, withoutwhich defect-free crystals will be difficult to grow on a consistent basis.

Keywords. Magnetic Czochralski crystal growth, inertial effects, rotationally driven flow, asymp-totic analysis.

I. Introduction

The ultimate goal of modeling the growth of single crystals is to establish a processby which defect-free crystals can be grown. Generally, large single crystals grownfrom semiconducting materials have non-uniform properties at both the macroscale

268 G. Talmage, S.-H. Shyu, J. M. Lopez and J. S. Walker ZAMP

and microscale. At the macroscale, these compositional inhomogeneities involvelongitudinal and lateral segregation. At the microscale, the non-uniformities areassociated with fluctuations in dopant or contaminant concentrations. As a rule,the performance of devices (such as large-scale integrated circuits) suffers whencrystals with non-uniform properties are used. Yet, there are instances in which theincorporation of a contaminant may have beneficial consequences. As an example,consider a silicon (Si) Czochralski crystal puller. The melt from which the crystalis grown is contained in a silicon-oxide crucible. During the crystal-growth process,oxygen is eroded from the crucible walls. While much of the oxygen is evaporatedacross the free surface of the melt, some oxygen is incorporated into the growingcrystal [1]. A small amount of oxygen in a wafer sliced from an Si crystal preventswarpage during the preparation of integrated circulits [2].

There are many aspects to any crystal-growth problem, and the significanceof the hydrodynamic behaivor of the melt on the spatial distribution of the com-ponents (including contaminants) was clearly identified early on [3]. Since theCzochralski crystal pulling technique is one of the most widely used methods forgrowing cryltals [4], the effort presented here will concentrate on this technique.See Fig. 1 for an illustration of a Czochralski crystal puller. In the Czochralskicrystal pulling technique there are three distinct driving forces associated with thehydrodinamic behavior of the melt: centrifugal pumping (or forced convection),bouyancy, and Marangoni (or thermocapillary) convection. Each of these threedriving forces has an impact on the amount and uniformity of the individual com-ponents found in the crystal. In order to create the melt, the charge is heatedby placing a graphite heater around the cylindrical crucible. The temperaturedistribution generated by the heater is not axisymmetric. Therefore, to obtaina more symmetrical temperature distribuition, the crucible may be rotated [5].The rotation of the crystal and/or the crucible produces centrifugal pumping. Intheir review article that focused on effects of forced convection, Pimputkar andOstrach [6] cite work in which the melt is assumed to be isothermal. Other effortshave concentrated on isolating the effects of buoyancy-driven flows [7] and of ther-mocapillary convection [8], which is due to the surface-tension variation along thefree surface. Under certain conditions, each of the driving forces can cause distincttypes of flow patterns, which may have deleterious effects on the homogeneity ofthe crystal. Muller and Ostrogorsky [9] provide a review of each of these drivingmechanism and their effect on the melt motion.

Furthermore, the flow regime can range from steady laminar to unsteady, pe-riodic laminar to turbulent. By 1972, it was known that if the flow is not steady,then striations, which are defects in the crystal, appear [10].

One approach to obtaining higher quality crystals is to control convection with-in the melt [11]. Since a magnetic field influences the motion of a conducting fluidand the melt is electrically conducting, it has been suggested that the motion ofthe melt can be controlled by growing a crystal in the presence of a magneticfield [1]. Therefore, our work focuses on the effect of a magnetic field on the bulk

Vol. 51 (2000) Inertial effects in the rotationally driven melt motion 269

eW

W

Heater

Crucible

Crystal

Free Surface

Melt

r = a

z = b

r = 1

r

z

bR

R

aR

Fig. 1. A θ = constant section dor a Czochralski crystal puller with a uniform, axial magneticfield, B. The cylindrical coordinates r and z are normalized with the inside radius of the crucible.

motion, with the understanding that it is the bulk motion that plays a major rolein determining the quality of the crystal.

Any melt motion across magnetic-field lines produces an electric current whichinteracts with the magnetic field to produce an electromagnetic (EM) body force,which opposes the melt motion. Even a weak magnetic field provides enough EMdamping to eliminate turbulence and unsteadiness in most of the melt motion,thus eliminating many micro-defects in the crystal. The EM damping is pro-portional to the strength of the magnetic field squared [12]. Both transverse orhorizontal magnetic fields [1, 13] and axial or vertical magnetic fields [13, 14] havebeen considered. Both magnetic-field configurations produce melt motions whichhave a deleterious effect on the crystal [15]. As a consequence of the problemsassociated with both the transverse and axial magnetic-field orientations, a “con-figured” magnetic-field orientation has been proposed [1, 15]. In spite of the factthat purely transverse or purely axial magnetic-field orientations do not producecrystals of the quality initially envisioned, their effects on the bulk flow warrantfurther investigation, because there is little hope of understanding the bulk flowin a complex magnetic-field configuration without a full understanding of the bulkflow in simpler magnetic-field configurations. This effort will focus on axial mag-netic field configurations, with the understanding that they will be of limited usein the actual Czochralski crystal growth process but will provide insights into themechanisms that control the melt motion.

Flows of electrically conducting fluids in the presence of a magnetic field aretermed magnetohydrodynamic (MHD) flows. An MHD flow can differ significantlyfrom the equivalent ordinary hydrodinamic flow. For single-phase MHD flows,


there are several key parameters, among the most important for this effort are:

• the interaction parameter, N , which represents the ratio of the EM body forceto the “inertial force”;• the Hartman number, Ha, which represents the square root of the ratio of the

EM body force to the “viscous force”;• the Reynolds number, Re, which represents the ratio of the “inertial force” to

the “viscous force”; and• the magnetic Reynolds number, Rm, which represents the ratio of the “induced”

magnetic field produced by the electric currents in the melt to the appliedmagnetic field produced by the magnet outside the crystal-growth furnace.

For many terrestrial applications with relatively small characteristic length scales,Rm is sufficiently small that the induced magnetic field is negligible in compari-son to the applied magnetic field. This is the case for crystal-growth problems.Consequently, Rm will not explicitly appear in the final form of the governingequations. The remaining three parameters are not independent: choosing anytwo specifies the third. Here, the relative importance of the EM, inertial, andviscous effects are determined by the order of magnitudes of N and Ha. Thevalue of N is generally very large for crystal-growth processes with typical mag-netic field strengths. However, N 1 may not imply that inertia terms can beneglected, as will be shown presently. For Ha 1, which is the case for crystalgrowth, at least two distinct regions of flow evolve: the Hartmann layer, whichplays a role similar to the hydrodinamic boundary layer, and the core region inwhich viscous effects are negligible. Under certain circumstances, which are oftenfound in crystal-growth problems, a third region of flow emerges: the interior lay-er, has an O(Ha−1/2) dimensionless thickness and often involves large O(Ha1/2)dimensionless velocities [16]. An interior layer lies along magnetic field lines andseparates two core regions. In the melt zone of a Czochralski crystal puller withan axial magnetic field, an interior layer separates the inner core, which lies underthe crystal, and the outer core which lies under the free surface [12]. This regionis the most sensitive to small disturbances, as was shown for homopolar devices[17]. In the Czochralski crystal-growth process, the interior layer is a result of adiscontinuity in the azimuthal velocity, and the O(Ha1/2) axial velocity is drivenby axial variation of the azimuthal velocity.

Current experimental studies indicate that the assumption that N Ha3/2 isnot satisfied. A more appropriate assumption is that N = O(Ha3/2), and inertialeffects are not negligible inside the interior layer. Here, unlike reference [12], weconsider the inertial effects on an unsteady, rotationally driven melt motion inthe presence of a uniform axial magnetic field. Buoyancy and thermocapillaryconvection are neglected, and both Ha and N are significantly greater than one.By isolating one mechanism that drives the melt motion from the others, we will bebetter able to understand the effects of that mechanism. In this case, centrifugalpumping. The flow will be assumed to be axisymmetric: derivatives with respect


to the azimuthal coordinate will be neglected, but there is an azimuthal velocitycomponent. The domain of interest will consist of the interior layer and adjacentsolid surface or free surface intersection regions, which have been shown to bethose regions first affected by inertial effects [17]. While this effort concentrateson one particular region in the melt, axisymmetric unsteady global calculationsare currently being performed (see [18] for a review of this approach). The term“global” implies that the entire Czochralski crystal puller, from the furnace to themelt to the crystal, is treated. Our analysis of the interior layer and its adjacentregions can be used in global models and will provide additional insight into thephysics of the melt motion. It must be emphasized that this work if of a theoreticalnature which represents one step towards a better understanding of the growthprocess.

II. Governing equations and boundary conditions

With a uniform, steady, axial magnetic field the dimensionless equations governingan axisymmetric melt motion in which gravitational effects are neglected are:

N−1DvrDt− v2

θ

r= −∂p

∂r+ jθ +Ha−2

(∇2vr −

vrr2

), (1a)

N−1(DvθDt

+vrvθr

)= −jr +Ha−2

(∇2vθ −

vθr2

), (1b)

N−1DvzDt

= −∂p∂z

+Ha−2∇2vz, (1c)

∂vr∂r

+vrr

+∂vz∂z

= 0,∂jr∂r

+jrr

+∂jz∂z

= 0 (1d,e)

jr = −∂φ∂r

+ vθ, jθ = −vr and jz = −∂φ∂z, (1f,g,h)

where DDt = ∂

∂t + vr∂∂r + vz

∂∂z and ∇2 = ∂2

∂r2+ 1

r∂∂r + ∂2

∂z2 . Equations (1a) through(1c) are the radial, azimuthal, and axial momentum equations, respectively. Equa-tion (1d) is the continuity equation, and Eq. (1e) is derived from the charge con-servation equation. Ohm’s law for a moving conductor is given by Eqs. (1f-h).Here, vr, vθ and vz are the velocity components in the radial, azimuthal, and axialdirections, respectively, p is the pressure, jr, jθ and jz are the electric currentdensity components in the radial, azimuthal, and axial directions, respectively, φis the electric potential function, Ha = BR

√σµ and N = σ2B4

%2Ω2 . The density,

dynamic viscosity, and electrical conductivity of the melt are %, µ and σ, respec-tively, B is the applied magnetic-field strength, R is the dimensional radius of thecrucible, and Ω is the angular velocity of the crucible. The nondimensionalizationof these equations is discussed in depth by Hjellming and Walker [12], and the key


I I

I I

z

r a= r 1=

hHa−1h

i f o w

hHa−1h

z 0=

z b=

r 0=

r

Ha− /1 2 Ha− /1 2

Fig. 2. Melt subregions in a θ = constant section. This figure is not to scale.

feature is that the characteristic velocity for vθ is ΩR, while that for vr and vzis ΩR/

√N . Since we consider unsteady flows, we have added the time derivative

with the characteristic time given by√N/Ω.

For the parameter ranges associated with the Czochralski process and typicalvalues of B, Ha 1, and the flow evolves into distinct regions: the side layer (w),outer core (o), interior layer (f , for free shear layer), inner core (i), Hartmann layers(h), and intersection layers (I). These regions and their dimensions are depicted inFig. 2. The melt has a dimensionless radius of one and a dimensionless height ofb. The crystal face is at z = b and extends from r = 0 to r = a. The free surface,also at z = b, lies between r = a and r = 1. This geometrical configuration, whichassumes a flat growth interface and a flat free surface, is termed the Czochralskibulk flow model.

The assumption that N is significantly greater than one indicates that theN−1 D

Dt (vr , vθ, vz) is negligible everywhere, except inside the interior layer at r = a.Therefore, even when inertial effects become important in the interior layer theyremain negligible in the other regions. Since the interior layer is the most sensitiveto small disturbances, we will focus on this region. However, the current whichdrives the flow within the interior layer comes from the intersection layer at r = aand z = b. Therefore, the intersection layer must somehow be taken into account.One approach that automatically combines the interior and intersection layers isto solve the full equations for the entire melt domain. This approach requires


fine meshes and a consequent reduction in the time step. In addition, most ofthe computational resources would be associated with the core regions and theiradjacent Hartmann layers, where the flow remains inertialess and steady long afterinertial effects are observed in the interior layer.

It should be noted that there are researchers, such as Sackinger et al. [19],Brown et al. [20], and Kakimoto et al. [21], who have had success solving themelt motion in the full domain, assuming axisymmetry. These groups have takena comprehensive (or global) approach to modeling Czochralski crystal growth. Inthe works cited, the temperature distribution of the entire system (which includesthe crystal, melt, furnace, and crucible), the velocity field within the melt, and allthe interface shapes are determined, Sackinger et al. and Brown et al. use a finiteelement method in conjunction with Newton’s method to perform a sequence ofsteady-state calculation at different values of the melt depth, in order to simulatea quasi-steady solidification process. Brown et al. consider the effects of an axialmagnetic field on the melt motion. As has previously been demonstrated, an axialmagnetic field reduces the meridonal flow. Kakimoto et al. also use a finite elementmethod together with an iterative procedure which involves a “reduced density”.Kakimoto et al. indicate excellent agreement between their computational andexperimental results. Our purpose, however, is to achieve a better understandingof the physics. This can be accomplished by focusing on the interior-intersectionregions. Therefore, a more appropriate approach would be to use a compositeinterior-intersection layer technique. In this technique, the leading-order termsfrom an asymptotic analysis of the interior layer and the intersection regions arecombined to form the governing equations which are time dependent.

In order to determine the interior-intersection-layer governing equations, anappropriate asymptotic expansion for the interior layer and for the intersectionregions at r = a and z = 0 or b must be established (see [12] for details of theasymptotics). An asymptotic analysis of the inner core shows that φi = Φi(r)and vθi = dΦi(r)

dr , where the subscript i denotes the first non-zero term in theinner core asymptotic expansion, and Φi is a known function that arises during anintegration. The term Φi is related to the O(1) electric potential function in thecrystal. In the outer core, φ0 = c1 + r2

2 and vθo = r, where the subscript o denotesthe first non-zero term in the outer core asymtotic expansion, and c1 is a constant.The electric potential is continuous across the interior layer, and c1 = Φi(a)− a2

2 .The inner and outer core solutions are summarized in Table 1.

Now, consider the interior layer. The continuity of the electric potential acrossthe interior layer suggests that the asymptotic expansion for the electric potentialfunction in the interior layer is

φf = Φi(a) +Ha−1/2φf (ζ, zt) +O(Ha−1),

where the subscript f denotes the interior (free shear) layer, and ζ = Ha1/2(r−a)is the stretched r-coordinate associated with the interior layer. The azimuthal


Layer/Variable Inner core Outer core

Electric potential φi = Φi(r) φ0 = Φi(a) + (r2 − a2)/2

Azimuthal velocity vθi = dΦi(r)/dr vθo = r

Radial velocity vri = 0 vro = 0

Axial velocity vzi = 0 vz0 = 0

Azimuthal electric current density jθi = 0 jθo = 0

Radial electric current density jri = 0 jro = 0

Axial electric current density jzi = 0 jzo = 0

Pressure dpi/dr = v2θi/r po − (r2 − a2)/2

Table 1. Summary of the inner core and outer core variables [12].

velocity is discontinuous at r = a:

vφf →dΦi(a)dr

as ζ → −∞ and vθf → a as ζ →∞,

and, vθ = vθf (ζ, z, t) +O(Ha−1/2) in the interior layer.The orders-of-magnitude are the same for N = O(Ha3/2) as for N Ha3/2.

With ζ = Ha1/2(r − a), the curvature effects appear as higher-order terms in theinterior and intersection layer equations, except for the centrifugal force term. Inthe expansion of the centrifugal force, the first term in that expansion is of orderone. For the electric current density,

jz = Ha−1/2jzf (ζ, z, t) +O(Ha−1), jr = Ha−1jrf (ζ, z, t) +O(Ha−3/2).

If a secondary-flow stream function ψ is defined as

vr =∂ψ

∂z, vz = −Ha1/2 ∂ψ

∂ζ,

then ψ = ψf (ζ, z, t) +O(Ha−1/2),

vr = vrf (ζ, z, t) +O(Ha−1/2), vz = Ha1/2vzf (ζ, z, t) +O(1).

While the dimensionless vz in the interior layer is large, the ratio of the di-mensional axial velocity to the dimensional azimuthal velocity is small, namelyN−1/2Ha1/2 = Ha−1/4 because vz and vθ were normalized by different character-istic velocities. Nevertheless, all inertial effects in an axisymmetric flow arise fromthe secondary-flow velocities vr and vz. Since the secondary-flow stream functionis O(Ha−1) in the inner core and its Hartmann layers, and it is zero to all ordersin the outer core, wall layer, and their Hartmann layers, the O(1) ψ in the interior


layer represents the largest secondary flow and the first to produce inertial effects.The asymptotic expansion for the pressure is:

p = pi(a) +Ha−1/2pf (ζ, z, t) +O(Ha−1),

where pi(a) ensures that the pressure is continuous across the interior layer. Fi-nally, the azimuthal vorticity associated with the secondary flow is O(Ha),

ω = Haωf(ζ, z, t) +O(Ha1/2),

where ωf = ∂vzf /∂ζ.A similar analysis can be made of the lower and upper intersection regions at

r = a, which are both of thickness O(Ha−1/2)×O(Ha−1).∗

Therefore, not only must the r coordinate be stretched (as was done in the inte-rior layer), but the z coordinate must be stretched as well. However, its stretchingfactor is Ha and not Ha1/2. As in the interior layer, the electric potential functionis continuous across the intersection regions, while the azimuthal velocity is discon-tinuous across the upper intersection region. With ∂

∂z = O(Ha), the leading-orderterm for each of the variables is

vr = O(1), vθ = O(1), vz = O(Ha−1/2),

jr = O(1), jθ = O(1), jz = O(Ha−1/2),

ψ = O(Ha−1), ω = O(Ha),

φ = Φi(a)+Ha−1/2φf (ζ, q, t)+O(Ha−1), p = pi(a)+Ha−1/2pf (ζ, q, t)+O(Ha−1),

where q = 0 or b.Knowledge of the leading-order terms in the interior layer and in the intersec-

tion regions are used to determine the governing equations associated with eachof these domains. To establish the interior-intersection-layer composite solutiongoverning equations, the interior layer equations are compared to both intersectionlayers equations, and the highest order terms in each set of equations are retained.For the interior layer and both intersection regions, the pressure is eliminated fromEqs. (1a, c), generating stream function-vorticity equations. Equations (1e, f, h)are combined to form a governing equation in terms of the electric current densitystream function, which will be defined subsequently. Assuming that the flow thatis treated as asisymmetric, that curvature effects and gravity are neglected, and

∗ In some problems in which the temperature distributions or dopant concentrations are treat-ed, there are indeed square O(Ha−1/2) × O(Ha−1/2) regions at r = a for z = 0 and b [22].However, this is not the case in which we treat only the velocity distribution.


that the flow domain is given by −∞ < ζ < ∞, 0 ≤ z ≤ b, the dimensionlessinterior-intersection-layer composite governing equations are:

∂2ψ

∂ζ2 +Ha−1 ∂2ψ

∂z2 = −ω

∂2h

∂ζ2 +Ha−1 ∂2h

∂z2 =∂vθ∂z

,

α

[∂vθ∂T

+∂

∂ζ

(vθ∂ψ

∂z

)− ∂

∂z

(vθ∂ψ

∂ζ

)]= −∂h

∂z+∂2vθ∂ζ2 +Ha−1∂

2vθ∂z2 ,

α

[∂ω

∂T+

∂

∂ζ

(ω∂ψ

∂z

)− ∂

∂z

(ω∂ψ

∂ζ

)]+ α−1 ∂v

2θ

∂z=∂2ψ

∂z2 +∂2ω

∂ζ2 +Ha−1 ∂2ω

∂z2 ,

where the vorticity ω in the composite layer is defined to be ω = Ha−1/2 ∂vz∂ζ −

Ha−1 ∂vr∂z , and the electric current density stream function h in the composite layer

is related to the electric current density by jr = Ha−1 ∂h∂z and jz = −Ha1/2 ∂h

∂ζ .Here, α = N−1Ha3/2 and T = Ha1/2t.

Furthermore, to reduce the number of unknown parameters, the independent

and dependent variables can be redefined with α∗ =√ba

[dΦi(a)dr − a

]2α as:

z∗ =z

b, ζ∗ =

ζ√b, and T ∗ =

1a√b

[dΦi(a)dr

− a]2T

vθ = a+[dΦi(a)dr

− a]Vθ, ψ =

b

a

[dΦi(a)dr

− a]2

Ψ,

ω =1a

[dΦi(a)dr

− a]2ω∗, h =

[dΦi(a)dr

− a]H. ∗

The dependent variables are functions of ζ∗, z∗ and T ∗, with −∞ < ζ∗ <∞ and0 ≤ z∗ ≤ 1. All dependent variables and time are scaled by the jump in vθ acrossthe interior layer of the square of this jump.

∗ For purposes of completeness,

vr =1a

[dΦi(a)dr

− a]2Vr , vz

√b

a

[dΦi(a)dr

− a]2Vz ,

jr =1b

[dΦi(a)dr

− a]Jr, jz =

1√b

[dΦi(a)dr

− a]Jz,

p = aζ∗ +

√b

a

[dΦi(a)dr

− a]2P, and φ = aζ∗ +

√b

[dΦi(a)dr

− a]

Φ.


Therefore, the dimensionless governing equations for the composite interior-intersection region can be written in terms of the primary azimuthal velocity, Vθ;the secondary-flow stream function and vorticity, ψ and ω, respectively; and theelectric current density stream function, h:

∂2Ψ∂ζ∗2

+ (bHa)−1 ∂2Ψ

∂z∗2= −ω∗, (2a)

∂2H

∂ζ∗2+ (bHa)−1 ∂

2H

∂z∗2=∂Vθ∂z∗

, (2b)

α∗[∂Vθ∂T ∗

+∂

∂ζ∗

(Vθ∂Ψ∂z∗

)− ∂

∂z∗

(Vθ∂Ψ∂ζ∗

)]

= −∂H∂z∗

+∂2Vθ∂ζ∗2

+ (bHa)−1 ∂2Vθ∂z∗2

, (2c)

α∗[∂ω∗

∂T ∗+

∂

∂ζ∗

(ω∗

∂Ψ∂z∗

)− ∂

∂z∗

(ω∗

∂Ψ∂ζ∗

)]+∂V 2

θ

∂z∗+ 2λ

∂Vθ∂z∗

=∂2Ψ∂z∗2

+∂2ω∗

∂ζ∗2+ (bHa)−1 ∂

2ω∗

∂z∗2, (2d)

The parameter λ, which is associated with the centrifugal acceleration in Eq. (2d),is related to the velocity discontinuity at the crystal-face-free-surface intersectionand is defined below. In this approach, an additional paramter, (bHa), appearswhere (bHa) is the Hartmann number based on the melt depth rather than onthe crucible radius — clearly the melt depth is very important for the interiorlayer while the crucible radius is not. The numerical parameter (bHa) mus besufficiently large that there exist distinct Hartmann layers at z∗ = 0 and 1 butnot so large that the steep gradients within the intersection regions are difficultto resolve. An appropriate value for (bHa) is 50. If terms involving (bHa)−1

are removed from Eqs. (2a-d), we recover the governing equations for the interiorlayer.

The value of dΦi(a)dr depends on the electrical properties of the crystal. We treat

the crystal as an electrical insulator, so that

dΦi(a)dr

=a

2(1 + ε) and λ =

2ε− 1

,

where ε is the ratio of the crystal’s angular velocity to the crucible’s. In a moregeneral case, in which the crystal is an electrical conductor, λ = a

/[dΦi(a)dr − a

].


The only effect of non-zero crystal is an electrical conductivity is that the jump invθ across the interior layer increases as crystal conductivity is increased [12]. Theboundary and matching conditions for the composite interior-intersection-regionsolution are:

∂Ψ∂ζ∗

, ω∗ → 0, Vθ → 1−e−bHaz∗+e−bHa(1−z∗), H → −1+e−bHaz∗

+e−bHa(1−z∗),

as ζ∗ → −∞, for 0 ≤ z∗ ≤ 1,

∂Ψ∂ζ∗

, ω∗, Vθ, H → 0, as ζ∗ →∞, for0 ≤ z∗ ≤ 1,

Vθ = H = Ψ =∂Ψ∂z∗

= 0, at z∗ = 0, for −∞ < ζ∗ <∞,

Vθ = 2, H =Ψ =∂Ψ∂z∗

= 0, at z∗ = 1, for ζ∗ < 0,

and∂Vθ∂z∗

= H = Ψ = ω∗ = 0, at z∗ = 1, for ζ∗ > 0.

Since the jump in vθ has been scaled out of every part of the interior-intersection-region boundary value problem except in the parameter λ, the only role of thecrystal’s electrical conductivity is in finding the correct value of λ for each valueof ε. The boundary condition h = 0 at z∗ = 1 for ζ∗ < 0 applies for an electricallyconducting crystal because the adjacent section of the crystal is too small to carryany significant electric current. The role of a non-zero crystal conductivity is toprovide an electrical path which is parallel to the Hartmann layer adjacent to thegrowth interface, i.e., once the radial current is in the Hartmann layer, it can fringeup into the crystal.

We are now in a position to assess why the composite interior-intersection-layerapproach is superior to considering the interior layer alone:

(1) Physically, it is more appropriate to treat the intersection regions withthe interior layer. The interior layer is driven by intersection region which mustaccommodate the change in boundary conditions at z∗ = 1 from Vθ = 2 for ζ∗ < 0to ∂Vθ

∂z∗ = 0 for ζ∗ > 0. In addition, it has been argued that for homopolardevices the nature of the flow within the interior layer is associated with theviscous forces within the intersection regions [17]. These viscous forces reduce thepeak azimuthal velocity sa it is convected toward the boundary. This mechanismrequires the viscous terms Ha−1 ∂2

∂z∗2, which are included in the composite interior-

intersection-layer approach.


(2) Mathematically, if we were to consider the interior-layer governing equa-tions, the terms (bHa)−1 would be neglected in Eqs. (2a-d). Furthermore, for theinertialess case, α∗ = 0 in Eqs. (2c, d), and thus Eqs. (2b, c) decouple from Eqs.(2a, d). Therefore, it is possible to solve for the primary azimuthal velocity andelectric current density stream function analytically. The secondary-flow streamfunction and vorticity are then found numerically. Spatial oscillations of a numer-ical nature are associated with the term ∂V 2

θ

∂z∗ +2λ∂Vθ∂z∗ , as a result of the shear layerbeing under resolved. These oscillations decreased as the mesh was refined, butthey never disappeared in the inertialess case. By including (bHa)−1 ∂2

∂z∗2in each

equation, these spatial oscillation in the z∗ direction are strongly suppressed.(3) Mathematically, there is a singularity at z∗ = 1 in the interior-layer for-

mulation. It may be that this singularity introduces an erroneous solution thatgrwos exponentially as z∗ decreases from one. Furthermore, the method of numer-ically coupling the boundary conditions at z∗ = 0 and z∗ = 1 to the discretizedgoverning equations is standard for the composite solution, but is is not at allclear for the interior-layer solution. Namely, the proper calculation of the elec-tric current density stream function, h, and the primary azimuthal velocity, Vθ,in both H(ζ∗, 0, T ) = −Vθ(ζ∗, 0, T ) and H(ζ∗, 1, T ) = −Vθ(ζ∗, 1, T ) − 2 is notobvious, and depends strongly on the double operators implicit in the governingequations. (Both conditions were established from an asymptotic analysis of theinterior layer.) This issue is eliminated in the composite-solution technique.

III. Numerical method of solution

The governing equations, Eqs. (2a-d), together with the boundary and matchingconditions form a well-posed problem. An analytical solution is not feasible, there-fore the solution will be fully numerical. The system of equations are solved usingsecond-order central differences on a non-uniform mesh together with a fourth-order Adams–Bashforth–Moulton time integration scheme. The stretched mesh isgiven by

ζ∗ = ζ − sζ sin(πζ/ζmax) and z∗ = z − sz sin(2πz)

where ζ = ζmax[−1 + 2(i/imax)] and z = j/jmax, with i = 0, 1, . . . , imax andj = 0, 1, . . . , jmax. Here imax + 1 and jmax + 1 denote the number of nodes inthe radial direction and the vertical direction, respectively. The radial extent ofthe computational domain is given by ζmax. The stretching parameters sζ andsz are set to 3.0 and 0.15 in this study, respectively. This choice of stretchedgrid concentrates the mesh points near the interior layer at ζ∗ = 0 and near theintersection layers at z∗ = 0 and 1, It should be noted that the stretching functionremains one-to-one even with a large stretching parameter, such as three, becausethe range of ζ is restricted.


Discretizing Eqs. (2a-d) in space, results in the system of equations given by:

∂Vθij

∂T= F1(Vθij ,Hij , ω

∗ijΨij),

∂Vω∗ij

∂T= F2(Vθij ,Hij , ω

∗ijΨij),

∇2Hij =∂Vθij∂z

, and ∇2Ψij = −ω∗ij,

where F1 and F2 represent the finite difference approximations for all terms exceptthe time derivatives in Eqs. (2c, d), respectively and T = T ∗/α∗. The Adams–Bashforth–Moulton fourth-order predictor-corrector scheme, is implemented in thefollowing fashion, with the initial conditions based on the inertialess flow andappropriate conditions at time level n,

(1) The predictor step: evaluate

V ∗θij = V nθij +∆T24

[55F1(V nθij ,H

nij , ω

∗nij ,Ψ

nij)− 59F1(V n−1

θij,Hn−1

ij , ω∗n−1ij ,Ψn−1

ij )

+ 37F1(V n−2θij

,Hn−2ij , ω∗n−2

ij ,Ψn−2ij )− 9F1(V n−3

θij,Hn−3


ij )],

for i = 1, 2, . . . , imax − 1 and j = 1, 2, . . . , jmax − 1. A similar equation holds forω∗θij .

(2) Solve ∇2H∗ij =∂V ∗θij∂z

and ∇2Ψ∗ij = −ω∗∗ij for i = 1, 2, . . . , imax − 1 andj = 1, 2, . . . , jmax − 1.

(3) Implement boundary conditions on V ∗θij , ω∗∗ij , H∗ij and Ψ∗ij .

(4) The corrector step: evaluate

V n+1θij

= V ∗θij +∆T24

[9F1(V ∗θij ,H

∗ij , ω

∗∗ij ,Ψ

∗) + 19F1(V nθij ,Hnij , ω

∗nij ,Ψ

nij)

− 5F1(V n−1θij

,Hn−1ij , ω∗n−1

ij ,Ψn−1ij ) + F1(V n−2

θij,Hn−2


ij )]

for i = 1, 2, . . . , imax − 1 and j = 1, 2, . . . , jmax − 1. Again, a similar equationsholds for ω∗n+1

ij .

(5) Solve ∇2Hn+1ij =

∂V n+1θij

∂zand ∇2Ψn+1

ij = −ω∗n+1ij , for i = 1, 2, . . . , imax − 1

and j = 1, 2, . . . , jmax − 1.(6) Implement boundary conditions on V n+1

θij, ω∗n+1

ij , Hn+1ij , and Ψn+1

ij .(7) Proceed to the next time step.The Poisson equations, Steps (2, 5), can be efficiently solved using the matrix

diagonalization method [23] and was recently implemented in parallel [24]. Knowl-edge of both Vθ and ω∗ on the interior grid points are required and given by Steps(1, 4).


Mesh dependency studies were performed on a coarse mesh of size 301 × 21and a fine mesh of size 601 × 41. With both non-uniform distributions of meshpoints, the result were not significantly different. However, on the uniform coarseand fine meshes, with the stretching parameters set to zero, the results were notqualitatively different from that of the non-uniform meshes, but the boundarylayers were not fully resolved on either the coarse of fine uniform mesh. The timestep was 10−5 for the coarser mesh and 5 × 10−6 for the finer mesh. Any largertime step resulted in an exponential growth of the solution. The residuals for thepresented in the next section were O(10−10) or less.

In this problem, ζ∗ → ±∞. A numerical value for infinity must be selected.For the inertialess case, where analytic solutions exist for the primary azimuthalvelocity and electric current density stream function, it is sufficient to use a maxi-mum value of ten. Results were compared for domains with ζ∗max = ±5,±10,±15.Qualitatively, the results were the same for ζ∗max = ±10 and±15, While ζ∗max = ±5had many of the same features, there were some qualitative differences, in partic-ular, near ζ∗ = −5. Numerical studies indicated that these statements are validboth with and without inertial effects.

In the cases presented here, the non-uniform fine mesh with 601×41 nodes wasused, and the infinite boundary, ζ∗ → ±∞, was set to ζ∗max = ±15.

IV. Results

To illustrate the effects of inertia, we consider two cases, both with λ = −0.2857,which corresponds to crystal and crucible rotations in opposite directions with theelectrically insulating crystal rotating at six times the angular rotation rate of thecrucible, i.e. ε = −6. The first case is inertialess with α∗ = 0, and the secondcase has inertial effects with α∗ = 10. As the parameter α∗ increases, we assumethat the interaction parameter decreases and that all other parameters, such asthe Hartmann number or the depth of the melt, remain fixed. In each of the casesconsidered, bHa = 50, and the melt is at the early stages so that b = O(1).

In considering the results, it must be kept in mind that the dependent variableshave been redefined. In particular,

vθ = a+[dΦi(a)dr

− a]Vθ, jr =

1b

[dΦi(a)dr

− a]Jr and jz =

1√b

[dΦi(a)dr

− a]Jz.

Therefore, when Vθ = 0, the fluid undergoes rigid body rotation with vθ = a.The actual direction of the primary azimuthal velocity vθ will depend on the termdΦi(a)dr − a, which for an electrically insulating crystal reduces to a( ε−1

2 ). Withε = −6, the term a( ε−1

2 ) is negative. When Vθ = 2, as it does at z∗ = 1 for ζ∗ < 0,then vθ = −6a = εa, which is speed at which the crystal rotates.

For α∗ = 0, the primary azimuthal velocity contour lines, electric current den-sity streamlines, and secondary-flow streamlines in a meridional plane are shown


0.95

2 0

0.095

z

z

–0.9 –0.1

z

z

Fig. 3(a). Primary azimuthal velocity contour lines for α∗ = 0. Contour interval is 0.095.Fig. 3(b). Electric current density streamlines for α∗ = 0. Contour interval is 0.1.

–0.0148

–0.037

z

z

Fig. 3(c). Secondary flow streamlines for α∗ = 0. Contour interval is 0.0037.

in Figs. 3(a), 3(b), and 3(c), respectively. In Figs. 3(a-c), the radial coordinate,ζ∗, has been scaled by ζ∗max. As a consequence, in these figures ζ∗/ζ∗max is be-tween ±1, and z∗ is between 0 and 1. If ε 6= 1, thre is an axial variation of vθfrom r at the crucible bottom to εr at the crystal-melt interface. The axial vari-ation of the dimensionless, radial centrifugal force, v2

θ/r, drives one or two cellsof secondary flow with radially outward and inward flows where v2

θ/r is large andsmall, respectively. The axial magnetic field has different effects on the secondaryand azimuthal flows. Since axisymmetry precludes an azimuthal electric field, anyradial velocity vr produces an azimuthal electric current density jθ, which in turnproduces the largest possible electromagnetic (EM) body force opposing thsi vr.On the other hand, there is no EM body force opposing vz. Therefore a magnet-ic field elongates secondary-flow circulations axially, while reducing their radialextent.

For the azimuthal flow, if vθ is independent of z, then a radial electric field,Er = −∂φ/∂r, develops which exactly cancels the induced electric field due to vθand the magnetic field, so that there is no radial electric current density and noEM body force opposing vθ. On the other hand, with an axial variation of vθ, Ercannot cancel the induced electric field at every value of z, so that the differencedrives radially outward and inward electric current densities where vθ is larger andsmaller, respectively. Recall that r = 0 at the center of the melt, not at the centerof the interior layer (Fig. 2). This circulation of electric current produces EM


body forces which reduce the larger values of vθ and increase the smaller ones, i.e.,make vθ more axially uniform.

In the outer core of the Czochralski melt, there are no-slip and no-shear con-ditions at z = 0 and z = b, respectively, so that the strong, axial magnetic fieldmakes this melt rotate as a rigid body with the crucible at vθ = r. Since there areno axial variations of vθ, there are no secondary flows and no electric currents inthe outer core. These characteristics are evident in Figs. 3(a), 3(b), and 3(c) forζ∗ > 5, where Vθ = 0, H = 0, and Ψ = 0, respectively.

In the inner region, vθ must vary axially from r at z = 0 to εr at z = b,but the strong magnetic field makes vθ nearly independent of z, except inside theHartmann layers adjacent to the crucible bottom and to the crystal-melt interface.The jump in vθ across each Hartmann layer is proportional to the total O(Ha−1)radial electric current inside the layer, while for an electrically insulating crystalwith φ = (1+ε)r2/2 in the inner core, jr = O(Ha−3) in the inner core. Since we areconsiderating an insulating crystal, the condition of zero net radial current meansthat the jumps in vθ across the two Hartmann layers must be equal, and the innercore rotates as a rigid body with the average angular velocity, i.e., vθ = 1

2(1 + ε)r.In Fig. 3(a), for ζ∗ < −5, Vθ = 1 or vθ = 1

2(1 + ε)a in the core, while Vθ changesfrom 2 or vθ changes to εa across the top intersection region/Hartmann layer, andVθ changes to 0 or vθ changes to a across the bottom intersection region/Hartmannlayer.

For the present results with ε = −6, vθ = r, vθ = −2.5r, and vθ = −6r at thecrucible bottom, in the inner core, and at the crystal-melt interface, respectively.In the inner core, Er exactly cancels the induced electric field associated with theaxially uniform vθ = −2.5r. Since Er is continuous across both Hartmann layers,the increase of vθ to r across the bottom Hartmann layer drives radially outwardelectric current inside this layer, while the decrease of vθ to −6r across the topHartmann layer drives radially inward electric current inside this layer. Theseradially outward and inward currents are evident in Fig. 3(b) for ζ∗ < −5 atz∗ = 0 and z∗ = 1, respectively. The electrical circuit for these radial currents iscompleted by a downward axial current in the inner core, to the left of Fig. 3(b),and by the upward axial current near ζ∗ = 0 in Fig. 3(b).

In the inner core, the radial pressure gradient, ∂p/∂r, exactly cancels the axiallyuniform centrifugal force, v2

θ/r = 6.25r, and this pressure gradient is continuousacross both Hartmann layers. In the top Hartmann layer, v2

θ/r increases from6.25r in the inner core to 36r at the crystal-melt interface, and this increase drivesa radially outward flow inside this layer. In the bottom Hartmann layer, v2

θ/rdecreases from 6.25r in the inner core to zero and then increases slightly to rat the crucible bottom, so that the unbalanced radial pressure gradient drives aradially inward flow inside this layer. These radially outward and inward flows areevident in Fig. 3(c) for ζ∗ < −5 at z∗ = 1 and z∗ = 0, respectively. The circuitfor these radial flows is completed by an upward axial flow inside the inner core,to the left of Fig. 3(c), and by a downward axial flow near ζ∗ = 0 in Fig. 3(c).


The radially outward flow in the top Hartmann layer is larger than the radiallyinward flow in the bottom Hartmann layer, and the difference flows from top tobottom and then back up inside the interior layer, entering the inner core at somelevel and returning through the inner core to the top layer. This top-layer-interior-layer-inner-core secondary-flow circulation is indicated by the streamline leavingthe interior layer near z∗ = 0.6 at ζ∗ = −15 in Fig. 3(c).

In the interior layer for α∗ = 0, the two significant terms in Eq. (1b) are

jr = Ha−2 ∂2vθ∂r2 = Ha−1 ∂

2vθ∂ζ2 . (3)

At each z, vθ must increase from −2.5a for ζ∗ < −5, through −0.75a at ζ∗ = 0,to a for ζ∗ > 5. Therefore, ∂vθ/∂ζ increases from 0 for ζ∗ < −5 to a maximumvalue at ζ∗ = 0, and then decreases back to 0 for ζ∗ > 5. This implies that∂2vθ/∂ζ

2 > 0 for ζ∗ < 0 and ∂2vθ/∂ζ2 < 0 for ζ∗ > 0, so that Eq. (3) indicates

that jr > 0 for ζ∗ < 0 and jr < 0 for ζ∗ > 0. Therefore the electric current densitystreamlines for the upward axial current in Fig. 3(b) must converge as z∗ increasesfrom 0 to 1. For the intersection region at z∗ = 1, the boundary condition on vθchanges abruptly at ζ∗ = 0. For ζ∗ < 0, Vθ = 2 at z∗ = 1, so that there mustbe a radial electric current inside the intersection region to produce the jump inVθ. For ζ∗ > 0, the no-stress condition means that there is no jump in Vθ acrossthe intersection region and no radial electric current. Therefore the current linesin Fig. 3(b) converge to a current sink which is an O(Ha−1) × O(Ha−1) cornerregion at ζ∗ = 0 and z∗ = 1. The convergence of the interior-layer axial current forthe positive and negative values of jr needed to balance the radial gradient of theshear stress in vθ implies that the axial electric field Ez = −∂φ/∂z must increaseas z∗ increases from 0 to 1. Since the O(1) terms in Eq. (1f) in the interior layerare vθ = ∂φ/∂r, the lines of constant vθ must also converge as z∗ increases from 0to 1, as shown in Fig. 3(a). Therefore the transition from vθ = −2.5a for ζ∗ < −5to vθ = a for ζ∗ > 5 occurs gradually over a large ∆ζ∗ near z∗ = 0, but it occursvery abruptly over a very small ∆ζ∗ at ζ∗ = 0 near z∗ = 1.

Secondary flows are driven by axial variations of the centrifugal force, v2θ/r,

which cannot be balanced at every value of z by a radial pressure gradient. Asζ∗ increases from −5 to 5 near z∗ = 1 in the interior layer, v2

θ/r remains close to6.25a until ζ∗ is only slightly below zero. Then v2

θ/r decreases abruptly throught0.5625a at ζ∗ = 0 to 0 at some small positive value of ζ∗, and then increasesquickly to a. Near z∗ = 0, v2

θ/r begins decreasing from 6.25a when ζ∗ is onlyslightly above −5, is quite small for a large range of ζ∗ near 0, and only climbs toa when ζ∗ is close to 5. For virtually every value of ζ∗, v2

θ/r is larger near z∗ = 1than it is near z∗ = 0, so that it drives an additional secondary-flow circulationinside the interior layer with radially outward and inward flows for z∗ > 0.7 andz∗ < 0.7, respectively, as shown in Fig. 3(c).

While ∂v2θ/∂z

∗ > 0 for virtually all values of ζ∗, it is much larger for ζ∗ < 0,where v2

θ/r = 6.25a near z∗ = 1 except at ζ∗ = 0, than it is for ζ∗ > 0, where v2θ/r


0.945

2

0.082

0.0

–0.0035

–0.014z

z

–0.9107

–0.1003

0.0

0.00024

0.00096

z

z

Fig. 4(a). Primary azimuthal velocity contour lines for α∗ = 10. Contour interval is 0.096 forpositive contour lines and is 0.0035 for negative contour lines. Fig. 4(b). Electric current densitystreamlines for α∗ = 10. Contour interval is 0.00024 for positive contour lines and is 0.1013 fornegative contour lines.

–0.01503

–0.03771

8.7 × 10–5

1.0 × 10–6

z

z

Fig. 4(c). Secondary-flow streamlines for α∗ = 10. Contour interval is 2.2 × 10−5 for positivecontour lines and is 0.00378 for negative contour lines.

never exceeds a near z∗ = 1. While the contours in Figs. 3(a) and 3(b) are nearlysymmetric about the ζ∗ = 0 line, the secondary-flow streamlines in Fig. 3(c) areskewed to negative values of ζ∗, so that the vertically downward flow is centerednear ζ∗ = 0. The only evident effect of the small values of ∂v2

θ/∂z∗ for ζ∗ > 0 is

the bulge in the Ψ = −0.0037 contour for ζ∗ > 0.There is a secondary-flow circulation entirely inside the interior layer, and the

interior layer carries the radially aoutward flow from the top Hartmann layer to thebottom layer or inner core. In a formal asymptotic solution for Ha 1 [12], thecirculation inside the interior layer is O(a), while that from the top Hartmann layeris O(Ha−1). One advantage of the present composite solution is that it combinestheO(1) and O(Ha−1) terms into a single solution. In Fig. 3(c) for bHa = 50,the ∆Ψ for the flow from the top Hartmann layer is roughly 0.015, while that forthe circulation inside the interior layer is 0.022. Therefore the distinction betweenthese two circulations in the formal asymptotic solution for Ha 1 is not realisticfor actual values of Ha, and the present composite solution should provide muchmore accurate values.

For α∗ = 10, the primary azimuthal velocity contour lines, electric currentdensity streamlines and secondary-flow streamlines are represented in Figs 4(a),


4(b), and 4(c), resepctively. In these figures, the radial coordinate, ζ∗, has beenscaled by ζ∗max, and z∗ remains unchanged. The contours for Vθ > 0 in Fig 4(a), forH < 0 in Fig. 4(b), and for Ψ < 0 in Fig. 4(c) are very similar to the correspondingcontours in Figs 3. Since α is the characteristic ratio of inertial effects to viscouseffects inside the interior layer, it is moderately surprising that inertial effectsare so small for α∗ = 10. Some preliminary research indicates that a Taylorinstability produces a transition from the present steady axisymmetric flow toan unsteady three-dimensional flow in which waves with axial vorticity propagatein the azimuthal direction inside the interior layer. This instability can not beinvestigated with the present code, which is restricted to unsteady axisymmetricflows. We are currently developing a code for unsteady three-dimensional flows inorder to study the Taylor instability. As we will discuss in the next section, wehave run the present code for α∗ values up to 100, but we do not present resultsfor α∗ > 10 here because these high-α∗ flows are probably unstable with respectto non-axisymmetric disturbances. With fixed values of B, R, b, and the physicalproperties, α∗ is proportional to Ω2. Therefore, as the angular velocities of thecrystal and crucible are increased with a constant ratio ε, the increasing velocitiesproduce only minor inertial effects before the transition to an unsteady, three-dimensional flow. In other words, the inertial effects appear to have little impacton the flow until the onset of an instability, which is sudden. This statement istrue for another rotationally driven MHD flow in which the transition was fromsteady, laminar flow to unsteady, periodic laminar flow [17].

While inertial effects are relatively small for α∗ = 10, the contours for Vθ < 0in Fig. 4(a), for H > 0 in FIg. 4(b), and for Ψ > 0 in Fig. 4(c) show that inertialeffects are not negligible, since contours do not exist for α∗ = 0. Since theseeffects are very small, we had to use much smaller intervals between contours forthese contours than for the contours with the opposite sign in each figure. Herewe use a hat to denote the value of a variable for α∗ = 10 minus its value forα∗ = 0. Qualitatively, the axial flows are essentially parallel to the vθ = constantlines and produce negligible convection of azimuthal momentum, but the values of∂vθ/∂r are large inside the interior layer so that the radial flows produce significantconvection of azimuthal momentum. At every value of z∗, vθ changes from −2.5afor ζ∗ < −5 to a for ζ∗ > 5. For z∗ > 0.7, the radially outward flow convects thesmaller values of vθ, making vθ < 0 for most values of ζ∗. For z∗ < 0.7, the radiallyinward flow convects the larger values of vθ, making vθ > 0 for mos values of ζ∗.Inside the interior layer, the O(1) radial electric field cancels the O(1) inducedelectric field due to vθ, i.e., vθ = ∂φ/∂ζ∗, so that

φ(ζ∗, z∗) =∫ ζ∗

−∞vθ(ζ, z∗)dζ. (4)

The radial convection of azimuthal momentum and the resultant changes inφ produce changes in the contours for Vθ > 0 and h < 0, but these changes areonly evident if the corresponding contours in Figs. 3 and 4 are superimposed.


The evident effects of convection is the cumulative results of vθ < 0 for z∗ > 0.7and of vθ > 0 for z∗ < 0.7 over the range −∞ < ζ∗ < ζ∗0 , where ζ∗0 is roughly 2or 3. Since φ(ζ∗0 , 1) < 0, and φ(ζ∗0 , 0) > 0, jz(ζ∗0 , z

∗) > 0. When this inertiallyinduced additional axial electric current hits the free surface, it splits into a radiallyinward current for ζ∗ < ζ∗0 and a radially outward current for ζ∗ > ζ∗0 . The radiallyoutward current near z∗ = 1 is evident in h > 0 contours in Fig. 4(b), while theadditional radially inward current near z∗ = 1 exists to the left of the h = 0contour in Fig. 4(b), but is not revealed by the much larger contour intervals forh < 0. The radially inward electric current near z∗ = 1 for 0 < ζ∗ < ζ∗0 producesan azimuthal EM body force in the direction of the crucible rotation, and thisforce accelerates the free-surface vθ to 1.049a or Vθ = −0.014 near ζ∗ = ζ∗0 , asshown in Fig. 4(a). The radially outward electric current near z∗ = 1 for ζ∗ > ζ∗0produces an azimuthal EM body force in the direction of the crystal rotation, andthis force decelerates the free-surface back to a for ζ∗ > 5.

For z∗ > 0.7, the region where vθ is close to zero convected radially outward,and for z∗ < 0.7, the region where vθ is close to a is convected radially inward.These convections produce a moderately large negative value of ∂v2

θ/∂z∗ at some

small positive value of ζ∗, so that the larger value of v2θ near z∗ = 0, drives radially

outward flow for z∗ < 0.5, as shown by the contours for ψ > 0 in Fig. 4(c). Thevθ = 1.049a at ζ∗ = ζ∗0 and z∗ = 1 would drive the local secondary flow in theopposite direction but this very small axial variation of the centrifugal force iscompletely overwhelmed by the much larger opposite axial variation occurringcloser to ζ∗ = 0.

V. Discussion

As the angular velocities of the crystal and crucible are increased for a given mag-netic field strength, inertial effects first become significant in the interior layerwhich lies along the magnetic field lines through the outer edge of the crystal. In-side the interior layer, the radial velocity and the radial gradient of the azimuthalvelocity are much larger than they are in any other part of the melt. The radi-al convection of azimuthal momentum leads to two small but significant inertialeffects. The first effect is the development of a weak reverse secondary flow withradially inward and outward flows near the free surface and crucible bottom, re-spectively. The second effect involves changes in the voltages produced by theazimuthal velocity and axial magnetic field. The voltage changes produce a newcirculation of electric current, and the EM body force due to this new currentaccelerates the free-surface azimuthal flow to a velocity which is greater than thatof the crucible.

A primary purpose of this study was to investigate the possibility of a tran-sition from steady to unsteady axisymmetric flow for the rotationally driven flowalone. Langlois [25] treated melt motions with the thermocapillary convection,


buoyant convection, and rotationally driven flow combined. He, as have otherssince, found transitions from steady to unsteady axisymmetric flows, and similartransitions continued to occur when he removed the thermocapillary convection.Since he did not consider the buoyant convection and rotationally driven flow sep-arately, the question remained whether these transitions arise from the buoyantconvection, from the rotationally driven flow or from a coupling between theseflows. We ran our unsteady axisymmetric code for values α∗ up to 100, and thesolution converged to a steady-state one for any initial conditions. We performedthe spatial grid and time step sensitivity studies needed to insure that the codewas predicting the correct transient behavior. Therefore we conclude that thetransition to unsteady axisymmetric flow found by Langlois [25] does not arisefrom the rotationally driven flow alone. Langlois [25] found that increasing thecrucible rotation rate for fixed crystal rotation rate always stabilized the flow, buthe did not investigate increasing both rotation rates with a constant ratio. In afuture study, we will investigate the transition from steady, axisymmetric flow tounsteady, three-dimensional flows for the rotationally driven melt motion with astrong axial magnetic field.

Acknowledgements

This research was supported by the U.S. National Science Foundation under GrantCTS-9321876 and Grant DMS-9512483.

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G. Talmage and S.-H. ShyuDept. of Mech. & Nucl. Eng.The Pennsylvania State UniversityUniversity Park, Pennsylvania 16802(e-mail: [email protected])

J. S. WalkerDept. of Mech. & Indust. Eng.The University of Illinois at UrbanaUrbana, Illinois 61801

J. M. LopezDepartment of MathematicsArizona State UniversityTempe, Arizona 85287

(Received: August 11, 1997; revised March 3, 1998 and January 3, 1999)

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Inertial e ects in the rotationally driven melt motion …lopez/pdf/ZAMP_TSLW00.pdfThis article treats the melt motion driven by the rotations of the crystal and crucible about their