Effects of rotation on a nonaxisymmetric chimney convection during alloy solidification

Journal of Crystal Growth 204 (1999) 382}394

E!ects of rotation on a nonaxisymmetric chimney convection during alloy solidi"cation D.N. Riahi* Department of Theoretical and Applied Mechanics, 216 Talbot Laboratory, University of Illinois at Urbana-Champaign, 104 S. Wright Street, Urbana, IL 61801, USA Received 15 September 1998; accepted 29 March 1999 Communicated by S.R. Coriell

Abstract Nonlinear buoyancy-driven convection in the melt and in cylindrical chimneys within a mushy layer during alloy solidi"cation is investigated under a high-gravity environment, where the rotation axis is inclined at an angle c to the high-gravity vector. In the "nal form of the solidi"ed material these chimneys produce freckles, which are imperfections that reduce the quality of the solidi"ed material. Asymptotic and scaling analyses are applied to a nonaxisymmetric convection in the "nite Prandtl number (P ) melt and in the chimneys. It is found that there are P - and rotationP P dependent ranges where the wall of chimneys may or may not be vertical. There are also c-, P - and rotation-dependent P ranges, where convection in the chimneys can increase or decrease with increasing rotation rate. The results are generally in agreement with available experimental observations.  1999 Elsevier Science B.V. All rights reserved.

1. Introduction Recently, Riahi and Sayre [1] investigated strongly nonlinear compositional convection in a mushy layer adjacent to the solidi"cation front of a solidi"cation system for P "R and steady-state P case under a high gravity ambient where the rotation axis was inclined at an angle c to the highgravity vector which is the resultant vector of those due to normal gravity and averaged centrifugal acceleration terms [2]. Riahi and Sayre considered only axisymmetric convection in in"nite P melt P and did not include the Coriolis force terms in the governing system. They applied asymptotic and

* Tel.: #1-217-333-0679; fax: #1-217-244-5707. E-mail address: [email protected] (D.N. Riahi)

scaling analyses to convection within the mushy layer and in cylindrical chimneys and determined the e!ects of centrifugal force on various #ow features including the vertical #ow velocity in the chimneys. Riahi [3] developed a P -dependent P model under normal gravity condition and zero rotation for axisymmetric convection in the melt during directional solidi"cation of binary alloys. Riahi [4] extended models due to Riahi and Sayre [1] and Riahi [3] to cases where the e!ects of P are P taken into account in a high-gravity environment under weak, moderate and strong centrifugal force, but again Coriolis force e!ects as well as nonaxisymmetric #ow behavior were not taken into account. Riahi [5] investigated strongly nonlinear natural convection in cylindrical chimneys in steady state for P "R and under a high-gravity P condition with inclined rotation. Asymptotic and

0022-0248/99/$ - see front matter  1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 0 2 4 8 ( 9 9 ) 0 0 1 9 3 - 1

D.N. Riahi / Journal of Crystal Growth 204 (1999) 382}394

scaling analyses were applied to weakly nonaxisymmetric convection and e!ects of both centrifugal and Coriolis forces were taken into account. It was found at in"nite P that, for some moderate P values of the rotation rate and cO0, axial convection in the chimneys decreases rapidly with increasing the centrifugal force parameter A above some azimuthally dependent axial level. The axial convection was also found to decrease with increasing the Coriolis parameter ¹ in certain range which also depend on c, ¹ and on the solutal Rayleigh number R. The Coriolis force e!ect was destabilizing for ¹ outside this range of values. The work carried out in Ref. [5] was based on the assumption that the #ow of melt is under certain derived parameter regimes and the radius a of any chimney is an independent parameter with prescribed value. The present paper investigates theoretically, using asymptotic and scaling analyses, the e!ects of centrifugal and Coriolis forces, due to arbitrary inclined rotational constraint, on a nonaxisymmetric chimney convection at "nite values of P which P has not yet been investigated so far. Furthermore, in the present work, following Worster [6], we shall present the results for the case where the derived parameter regime implies that a is a function of the other parameters. Thus, the present study for "nite P and parameter-dependent a is an extension of P that for P "R and prescribed a due to Riahi [5] P and turns out to be important for more realistic nonaxisymmetric convection cases to detect P -deP pendent regimes where chimney convection can either be increased or decreased with increasing the rotation rate. In particular, as is discussed in Section 5, the present results for low P case predicts P the realizable role played by the Coriolis force on the convection in the melt.

2. Governing equations We consider a thin mushy layer adjacent to solidifying surface of a binary alloy melt and of thickness h, where h is the mush}liquid interface elevation from z"0 and the z-axis is assumed to be anti-parallel to the high-gravity vector to be described below. The binary alloy melt of constant

383

composition C and constant temperature ¹ is   solidi"ed at a constant rate < , with the eutectic  temperature ¹ at the position z"0 in a frame  moving with the solidi"cation velocity in the zdirection. A representative "gure is the Fig. 1 given in Ref. [5] and will not be repeated here. The physical model at high gravity is based on assumptions for solidi"cation in a centrifuge [2]. We assume that our solidifying system is placed in a centrifuge rotating at some constant angular velocity X about the centrifuge axis which makes an angle c with respect to z-axis. The centrifuge axis is assumed to be anti-parallel to the earth gravity vector. A representative "gure for solidi"cation system in a centrifuge is the Fig. 2 given in Ref. [5] and will not be repeated here. We consider the Boussinesq approximation form of the equations for momentum, continuity, heat and solute [2,6] in the moving coordinate system whose origin is centered on the solid}mush interface. The governing system of these equations for the solidifying system rotating with the centrifuge basket [2] and translating with the solidifying front at speed < is nondimensionalized in the same way  as described in Ref. [5] which will not be repeated here. In the momentum equations the centrifugal acceleration term is split into an average term, which is superimposed on the gravity term, and a so-called gradient acceleration term [2]. The nondimensional parameter R representing the modi"ed gravity term can then become signi"cantly larger than the corresponding one due to earth's gravity alone for signi"cant rotation rate. Following Refs. [6}8], we consider the mushy layer as a porous medium where Darcy's law holds. Following Ref. [5], we consider the governing equations in a cylindrical coordinate whose axial direction is along the z-axis. We shall consider a nonaxisymmetric convection in a cylindrical chimney, whose axis coincides with the z-axis (Fig. 3 in Ref. [5]), and in the mushy layer in the asymptotic limit of strong compositional buoyancy force, negligible thermal buoyancy and su$ciently large Lewis number i/D, where D and i are solute and thermal di!usivity, respectively. The nondimensional form of the equations for the momentum, continuity, temperature and solute concentrations for the steady #ow of melt in the chimney

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D.N. Riahi / Journal of Crystal Growth 204 (1999) 382}394

#(z sin c cos c sin m#r sin m cos m sin c)mK

are then






1 R ! #u ) u"!HR( P#Sz( )# u P Rz P #H¹u;XK #HASG, (1a)

#(z sin c!r sin c cos c cos m)z( .

(2b)

(1b)

The steady nondimensional form of the equations for the momentum, continuity, temperature and solute concentration in the mushy zone outside the chimneys are

R ! #u ) h" h, Rz

(1c)

(1! )u/P"!R( P#Sz( )#¹(1! )u;XK

R ! #u ) S"0, Rz

(1d)

) u"0,





 

#ASG,

where u"ur( #vmK #wz( is the velocity vector, u is the radial component of the velocity, r( is a unit vector in the radial r-direction, v is the azimuthal component of u, mK is a unit vector in the azimuthal m-direction, w is the axial component of u, z( is a unit vector in the axial z-direction, P the pressure, S is the solute concentration, h is the temperature, P "l/i is the Prandtl number, l is the kinematic P viscosity, R"b*CN i/(<lH) is the solutal E  Rayleigh number, N "(g#XR) is the accelE  eration due to high gravity, N "g corresponds to E normal gravity case while N 'g indicates level of E high gravity, g the acceleration due to normal gravity, R is the perpendicular distance from the center  of gravity of the centrifuge basket to the rotation axis, R is a function of c and R "0 where c"0   degree, H"i/(<P ) is a nondimensional para  meter (H<1 [6]) representing ratio of R in the liquid zone inside the chimneys or above the mushy layer to that in the mushy zone outside chimneys, P is a constant reference value of the permeability  P( ) of the porous medium, is the solid fraction of the mushy zone, ¹"2Xi/(<lH) is the Co riolis parameter, which is square root of the Taylor number, A"b*CXi/(<lH) is the gradient ac celeration parameter, *C"C !C , C is the eu   tectic concentration, b is the expansion coe$cient for solute, XK is a unit vector along the rotation axis de"ned by XK "cos cz( #sin c(cos mr( !sin mmK ) and G is a position vector de"ned by G"(r cos m cos c#r sin m !z sin c cos c cos m)r(

) [(1! )u]"0,



(3a) (3b)



R

R ! #(1! )u ) h" h!S , R Rz Rz

(3c)

R [(1! )(C !S)]#(1! )u ) S"0, P Rz

(3d)

where S "¸/(C *¹) is the Stefan number, R 1 *¹"¹ !¹ , ¹ is the local liquidus temper*  * ature at C"C , C"S*C, C is the speci"c heat  1 per unit volume, ¸ is the latent heat of solidi"cation per unit volume, C "(C !C )/*C is a concentraP   tion ratio, and C is the composition of the solid  phase forming the dentrites. The boundary conditions are given in Ref. [6] and will not be repeated here. Analysis in Ref. [6] indicates that h"S in the mushy zone which is also valid in the present paper. It should also be noted that the velocity vector in Eqs. (3a)}(3d) includes a factor (1! ) in accordance with the original formulation in a mushy layer [6], even though this factor can be ignored since " ";1 [6]. In the next two sections we shall proceed with asymptotic and scaling analyses for the equations given in this section in the asymptotic limit of su$ciently large R, to determine the strongly nonlinear steady state for a nonaxisymmetric #ows in the mushy zone and mainly in the chimneys in two ranges for high and low P values. Only leading P order terms in an asymptotic expansion in R<1 will be dealt with or provided in this paper.

(2a) 3. High Pr analysis Let us designate a(m, z) to be the radius of a chimney under consideration whose axis is assumed to

D.N. Riahi / Journal of Crystal Growth 204 (1999) 382}394

coincide with the z-axis of the coordinate system. It is assumed that a is small (a;1). We assume that the orders of magnitude of r, m and z are, respectively, a, 1 and 1. Assuming the magnitude of (1! )u be of order one in the mushy zone, then Eq. (3a) implies that to the leading term pressure "eld in the mushy zone is una!ected by the #ow velocity and h"S is independent of r and m in this zone. The steady-state form of Eqs. (3c) and (3d) for the r and m independent variables h (z), (z) and w (z) then    imply [(1! )w !1]h "h !S h , (4a)     R  !  (C !h )!h (1! )#(1! )w h "0,  P       (4b) where a prime denotes di!erentiation with respect to z. It is also assumed that C
v)O(u),

(5)

where F can be any dependent variable. Denoting an azimuthal average of any quantity F by



1 p F dm, (6) 1F2" 2p  assuming that any dependent variable and its derivatives are repeated in 2p interval in m, and taking the azimuthal average of the governing equations (Eqs. (1a)}(1d), (3a)}(3d)), we observe that the averaged form of these equations provide qualitatively similar scaling and asymptotic behavior, in the limit of su$ciently large R, to those considered in Ref. [5] under the more restriction

385

averages of the respective quantities in the high P limit of this section. Thus, these results will not P be repeated here. Instead, only the following results, which later will be referred to and are the basis for the high P regime of the Figs. 1}8 presentP edin Section 5, are provided below. Considering the #ow in the chimney described by Eqs. (1a)}(1d) and assuming that S&1 and w<1 hold in the chimney [6], Eq. (1a) then implies that 1w2&HRa,

1u2&HRa, 1t2&HRa,

(8)

where t is the stream function. The analysis presented in this section is for the case where the inertia terms in Eq. (1a) can be, at most, as signi"cant as the viscous terms in Eq. (1a). This assumption together with Eq. (8) then implies that the following range for P should be satis"ed: P P *O(HRa). (9) P This range is classi"ed as the high P range since the P main results become independent of P in this range P as well as in the limiting case of P "R, which can P be seen from the governing equations (Eqs. (1a)}(1d), (3a)}(3d) by setting P "R. P Now, h is the leading order temperature solu tion in the mushy zone outside the chimney and its surrounding boundary layer [6]. Designate h (r, m, z) to be the deviation of h from h . From   Eq. (1c) or Eq. (3c) and the condition 1/a;HR;1/a,

(10)

it is found that h ;1. Integrating the azimuthal  average of the simpli"ed form of Eq. (1c) in r from r"0 to a and following Ref. [6], it is found that 1h 2&1t 2h ln r,    where

(11a)



(7)

? 2pr1w2 dr, (11b)  is the azimuthal average of the vertical #ux in the chimney. In addition, we have the following results:

Hence, the stream function formulation, analysis and the results presented by the Eqs. (5)}(31) in Ref. [5] are also valid qualitatively for the azimuthal

1u2&!1t 2/r as rPa, (12) ? 1(1! )w2&!R1t 2A ln a#¹1F (1! )2, ?   (13a)

v)O(u),

RF (O(F). Rm

2p1t 2" ?

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D.N. Riahi / Journal of Crystal Growth 204 (1999) 382}394

where A ,1!(A/R)z sin c. (13b)  The following results in this paragraph hold near the wall of the chimney. Using Eqs. (8) and (12) in Eq. (13a), we "nd HRaA ln a&!1#¹1F 2, (14)   which hold near the wall of the chimney. Here F is  the axial component of the Coriolis term u;XK . Results (13a), (13b) and (14) hold in a regime, to be referred to as moderate axial velocity regime, provided the following condition does not hold: RH(!1#¹1F 2)<[RA ln a]. (15)   If Eq. (15) holds, then the following results hold in a regime, to be referred to as large axial velocity regime: 1w2&(RHa),

(16)

H&!(ln a/a)A #¹1F 2/(RaH). (17)   It is found that the azimuthal average of the total volume #ux 2p1t 2 in the chimney, due to upward ? #ow, is given by 2pt &2pRH1a2(1#h )j A , ?    provided

(18a)

¹H sin c1a2;1

(18b)

and 2pR1a2 j [b !b (A/R) sin c], 2p1t 2& ?  ¹ sin c  

(18c)

provided ¹H sin c1a2<1,

(18d)

where j (i"1, 2) are order-one parameters which G can depend, in general, on 1a2, c, (A/R) and z. In addition, b and b de"ned by   X X b " (1#h ) dz, b " (1#h )z dz, (18e)       are functions of z and of order one quantities with 0(b (b .  





Before we present the low P analysis in the next P section, we like to make it clear that the di!erence in how the chimney's radius a is treated here as compared to the one treated in Ref. [5]. In Ref. [5] a was assumed as an independent parameter with a prescribed small value so that the relations of the forms (14) and (17), which hold, respectively, for moderate and large axial velocity regimes, simply implied existence of certain relation between H, R, A and ¹ in the parameter space under which the results were valid. Here we follow Ref. [6] and treating a as small quantity but use Eqs. (14) and (17) to determine the dependence of a on H, R, A and ¹ for moderate and large axial velocity regimes, respectively.

4. Low Pr analysis Analysis for this case follows generally in a way somewhat similar to that presented in the previous section and similar assumptions are made leading to Eqs. (4a), (4b), (5) and (6) which do not involve P e!ects. However, for low P regime Eq. (1a) P P implies the following P -dependent scaling: P 1w2&(HRP ), 1u2&a(HRP ), P P 1t2&a(HRP ), (19) P which represent a balance between the inertia, buoyancy and pressure gradient terms in the governing system to the leading orders in the asymptotic limit of su$ciently large R. These scalings are valid, provided P lies in the range P (HR)\;P ;HRa. (20) P From Eq. (1c) or Eq. (3c) and condition 1;(P HR);a\, (21) P we "nd that h ;1 can be possible. Using these  results Eq. (1c) implies azimuthal average form of Eq. (8) in Ref. [5]. Using the procedure described in the previous section the results (11a), (11b), (12), (13a) and (13b) in the previous section and the azimuthal average form of Eqs. (9)}(17) in Ref. [5] follow here as well. The following results in this paragraph hold near the wall of the chimney. Using Eqs. (12) and (19) in

D.N. Riahi / Journal of Crystal Growth 204 (1999) 382}394

Eqs. (13a) and (13b), we "nd

387

condition (24). Using Eqs. (3c), (25) and (26), we "nd

(HRP )A a ln a&!1#¹1F 2. P  

(22)

The result of azimuthal average of Eq. (19) in Ref. [5] is valid again here. Using Eqs. (11a), (19) and (22), we "nd

 

Rh u  &!(HRP )(!1#¹1F 2)/ P  Rr +RA ln a,. 

(23)

R1 2 &RHaP . P Rz

(29)

It is seen that the right-hand side in Eq. (29) is small if a;(RHP )\. (30) P However, the right-hand side in Eq. (29) is not small if

Thus, 1uRh /Rr2 term in the azimuthal average of  Eq. (3d) is negligible if the right-hand side in Eq. (23) is small. For

(RHP )\)0(a);1. (31) P Using the de"nition of the wall of the chimney given by

HRP (!1#¹1F 2)<(RA ln a), P  

[a(z, m), m, z]"0 at r"a(z, m),

(24)

such term must be retained and, thus 1w2&HRa P . P

(25)

Eq. (25) is more restricted than Eqs. (13a) and (13b) since Eq. (25) is under condition (24). Using Eqs. (8), (13b) and (25), we "nd HP &!(HRP )A ln a#¹1F 2/(Ra). P P  

(26)

Using Eqs. (19) and (26) in the the azimuthal average of Eq. (12) in Ref. [5], we obtain 1*P 2&!(a ln a)A #¹1F 2ln a/ P  



[R(P HR)]#(¹/R) 1F 2d P  P #h (A/R)pah (1#cos c)/2,  

(27)

where F is the radial component of the Coriolis  term u;XK . Using Eqs. (22) and (26) in the azimuthal average of Eq. (19) in Ref. [5], we "nd 1h2!h &[!(a ln a)RA #¹1F 2ln a]/    (P HR). P

(28)

Vertical and horizontal advection of solute balance here in this regime where Eq. (24) holds. Results (28) is more restricted than the result of the azimuthal average of Eq. (19) in Ref. [5] since Eq. (28) is under

(32)

Ref. [5] and taking derivative with respect to z of Eq. (32), it follows that R

a/a& . Rz

(33)

If the right-hand-side in Eq. (33) is small, then it indicates that the wall of the chimney can be in the axial direction to the leading order terms in the asymptotic limit of su$ciently large R. Thus the wall of the chimney can be in the axial direction, provided Eqs. (24) and (30) are valid. This result indicates the following interesting conclusion that, based on the scaling of types (26)}(28) and conditions (24) and (30), the wall of the chimney can be in the axial direction to the leading order terms, while such result cannot be concluded if a lies in range of Eq. (31) instead. Applying the scaling of types (22) or (26), using a Polhausen-type method [9], we "nd the following results: 2p1t 2&p1a2(8RHP ) ? P A ; m b ! sin cb  R 






 exp(mj ),  (34a)

provided (HP /R)1a2¹sin c;1 P

(34b)

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D.N. Riahi / Journal of Crystal Growth 204 (1999) 382}394

and 2p1a2R [b !b (A/R)sin c]j , 2p1t 2&   ? ¹ sin c 

(34c)

provided (HP /R)1a2¹ sin c<1. (34d) P Here j is an order-one parameter which can de pend on a, c and (A/R). The parameter j is an  order one parameter which can depend on a, c, (A/R) and z. In addition,



m"

1 if (b !(A/R)b sin c)*0,   !1 if otherwise

(34e)

and b and b are already de"ned in Eqs.   (18a)}(18e). The result w "!2p1t 2N (35)   can then be obtained in direct analogy to that derived in Ref. [5] with N as the number density of the chimneys.

Similar to the high P case presented in the preP vious section, we have here also a moderate axial velocity regime, which is based on Eqs. (13a), (13b) and (22), and a large axial velocity regime which is based on Eqs. (25) and (26).

5. Discussion The results derived in the previous two sections, which are discussed brie#y in this section, deal with the following regimes: high P , represented by Eq. P (9), low P , represented by Eq. (20), moderate axial P velocity, represented either by Eqs. (13a), (13b) and (14) (high P ) or by Eqs. (13a), (13b) and (22) (low P P ), large axial velocity, represented either by Eqs. P (16) and (17) (high P ) or by Eqs. (25) and (26) (low P P ), weak Coriolis force, represented either by Eq. P (18b) (high P ) or by Eq. (34b) (low P ), strong P P Coriolis force, represented either by Eq. (18d) (high P ) or by Eq. (34d) (low P ), and inclined rotation, P P

Fig. 1. Case of high P , moderate axial velocity and weak Coriolis force for R"10, H"10 and A "!0.0005. Orders of magnitude P  of scaled chimney's radius ai(100a), scaled volume #ux si(10t ) and scaled nonazimuthal #ow speed vi(0.01< , < ,(u#w), for ? ? ? c"1503 (i"1) and c"2103 (i"2), versus ¹.

D.N. Riahi / Journal of Crystal Growth 204 (1999) 382}394

represented by sin cO0. Rotational e!ects were found to diminish signi"cantly if sin c"0. For high P limit, which includes #uid cases with P moderate P , the main results are all independent of P P , to the leading order terms, and the results for P both high and low P limits are provided for a given P axial level. The results are given by words mainly for the variations of t and nonazimuthal #ow ? speed < ,(u#w with respect to either ¹ or ? the modi"ed centrifugal parameter A de"ned by  Eq. (13b) and by Figs. 1}8, be referred in this section, which provide order of magnitude dependence of t , < and a on either ¹ or A . ? ?  For high P and inclined rotation, the main reP sults are brie#y as follows. For moderate axial velocity and weak Coriolis force e!ect, the results are presented in the Figs. 1 and 2. The volume #ux increases with ¹ for c(1803 and decreases with increasing ¹ for c'1803. For small "A " and  A (0, t increases with A , while t decreases  ?  ? with increasing A for A '0. For strong Coriolis  

389

force e!ect, the dependence on c is insigni"cant, and the results for < and t remain the same as in ? ? the case c(1803 for weak Coriolis force e!ect as far as variation with ¹ is concerned. However, for small "A " and A '0, < and t increase now with   ? ? A (Fig. 3). For large axial velocity and weak Co riolis force e!ect, the volume #ux and < vary with ? ¹ qualitatively similar to their variations with ¹ in the moderate axial velocity case. For small "A " and  A (0, t and < decrease with increasing A ,  ? ?  while t and < increase with A for A '0 (Fig. 4). ? ?   For strong Coriolis force e!ect, the results are qualitatively similar to the corresponding ones in the weak Coriolis force case for c'1803 and A '0.  For low P limit, the main results depend mostly P on P and are, for inclined rotation and a given P axial level, as follows. For moderate axial velocity and weak Coriolis force e!ect, the results are presented in the Figs. 5 and 6. The volume #ux and < decrease with increasing ¹ for c(1803 and ?

Fig. 2. Case of high P , moderate axial velocity and weak Coriolis force for R"10, H"10 and ¹"1.0. Orders of magnitude of P scaled chimney's radius ai(100a), scaled volume #ux si(100t ) and scaled nonazimuthal #ow speed vi(0.01< ), for c"1503 (i"1) and ? ? c"2103 (i"2), versus A . 

390

D.N. Riahi / Journal of Crystal Growth 204 (1999) 382}394

Fig. 3. Case of high P , moderate axial velocity and strong Coriolis force for R"10, H"10 and ¹"600. Orders of magnitude of P chimeny's radius ai(a), scaled volume #ux si(1000t ) and nonazimuthal #ow speed vi(v ), for c"1503 (i"1), versus A . ? ? 

increase with ¹ for c'1803. For small "A ", t de ? creases with increasing A . For strong Coriolis  force e!ect, t decreases with increasing ¹ (Fig. 7). ? For large axial velocity and weak Coriolis force e!ect, t and < vary with ¹ qualitatively similar ? ? to their variations with ¹ in the moderate axial velocity case. For small "A " and A (0, t and   ? < increase with A until about A "!0.0006, ?   beyond which they decrease with increasing A (Fig. 8). The peak-like behavior of the results  shown in Fig. 8 is due to the competing e!ects of the centrifugal force versus the destabilizing e!ect of the Coriolis force. For A (!0.0006, the cen trifugal force is destabilizing and, thus, convection in the chimneys increases with A . However, for  A '!0.0006, the centrifugal force is stabilizing  and its e!ect increases with A . Thus, convection  now decreases with increasing A . For strong Co riolis force e!ect, t varies with ¹ qualitatively ? similar to the corresponding one for the weak Coriolis force case with c(1803.

The results discussed above, which indicated existence of certain ranges in the (P , ¹, A)-paraP meter space at "nite P and for the inclined rotation P rate where nonaxisymmetric convection in the chimneys can decrease with increasing either Coriolis or centrifugal forces, are in agreement with the experimental "nding [10,11] that inclined rotation with some range of values of the rotation rate can reduce the convection e!ect signi"cantly. Also, the present results for some other ranges of the inclined rotation rate, where chimney convection increases with either ¹ or A, are in agreement with the experimental "nding [12] that rotational e!ects can be destabilizing under certain conditions leading to channel formation in the mushy zone resulting segregates in the solidi"ed material. An interesting aspect of the present results is the way the inclined angle c a!ects the chimney convection. Within certain range controlled by the axial component of the Coriolis force in the (P , ¹, A)-parameter space for small P melt P P

D.N. Riahi / Journal of Crystal Growth 204 (1999) 382}394

391

Fig. 4. Case of high P , large axial velocity and weak Coriolis force for R"10, H"10 and ¹"5.0. Orders of magnitude of scaled P chimney's radius ai(1000a), scaled volume #ux si(1000t ) and scaled nonazimuthal #ow speed vi(0.01< ), for c"1503 (i"1) and ? ? c"2103 (i"2), versus A . 

Fig. 5. Case of low P , moderate axial velocity and weak Coriolis force for R"10, H"10 and A "0.001. Orders of magnitude of P  scaled chimney's radius ai(1000a), scaled volume #ux si(20000t ) and scaled nonazimuthal #ow speed vi(0.1< ), for c"1503 (i"1) and ? ? c"2103 (i"2), versus ¹.

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D.N. Riahi / Journal of Crystal Growth 204 (1999) 382}394

Fig. 6. Case of low P , moderate axial velocity and weak Coriolis force for R"10, H"10 and ¹"0.8. Orders of magnitude of scaled P chimney's radius ai(1000a), scaled volume #ux si(10000t ) and scaled nonazimuthal #ow speed vi(0.1< ), for c"1503 (i"1) and ? ? c"2103 (i"2), versus A . 

Fig. 7. Case of low P , moderate axial velocity and strong Coriolis force for R"10, H"10 and A "10(1.0001). P  Orders of magnitude of chimney's radius ai(a), scaled volume #ux si(1000t ) and nonazimuthal #ow speed vi(< ), for c"1503 (i"1), ? ? versus ¹.

D.N. Riahi / Journal of Crystal Growth 204 (1999) 382}394

393

Fig. 8. Case of low P , large axial velocity and weak Coriolis force for R"10, H"10 and ¹"10.0. Orders of magnitude of scaled P chimney's radius ai(100a), scaled volume #ux si(1000t ) and scaled nonazimuthal #ow speed vi(0.001< ), for c"1503 (i"1), versus A . ? ? 

convection in the chimneys decreases with increasing ¹ if c(1803, while convection in the chimney increases with ¹ if c'1803 keeping all the other parameter values the same as in the c(1803 case. These results agree with some recent experimental studies for small P [13] under centrifugation, P where Coriolis e!ect was found to have such di!erent types of in#uence on the #ow stability depending on the rotation sense of the centrifuge. The present results are also in agreement with some recent computational studies for P "0.02 [14] P under centrifugation, where enhancement of convection was found if centrifuge rotated counterclockwise (sin c(0 here), while convection was reduced if the centrifuge rotated clockwise (sin c'0 here). Finally, it should be noted that an important aspect of the present results is the roles played by the Coriolis and centrifugal forces in both high and low P melt cases. It appears that the Coriolis force P of strong strength can make the radius of the chimney z-dependent in a high P melt, while the centriP

fugal force of moderate strength can make a z-dependent in a low P melt. Suppression or P enhancement of convection in the chimneys can be accomplished by either the centrifugal force or the Coriolis force in particular ranges for A, ¹, H, R and P which are all non-trivial and their relevance P can be tested for particular melt cases in applications.

References [1] D.N. Riahi, T.L. Sayre, Acta Mechanica 118 (1996) 109. [2] W.A. Arnolds, W.R. Wilcox, F. Carlson, A. Chait, L.L. Regel, J. Crystal Growth 119 (1992) 24. [3] D.N. Riahi, Acta Mechanica 127 (1998) 83. [4] D.N. Riahi, in: L.L. Regel, W.R. Wilcox (Eds.), Materials Processing at High Gravity, Plenum, New York, 1997, pp. 169}176. [5] D.N. Riahi, J. Crystal Growth 179 (1997) 287. [6] M.G. Worster, J. Fluid Mech. 224 (1991) 335. [7] P.H. Roberts, D.E. Loper, in: A.M. Soward (Ed.), Stellar and Planetary Magnetism, Gordon and Breach, New York, 1983, pp. 329}349.

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[13] W.J. Ma, F. Tao, Y. Zheng, M.L. Xue, B.J. Zhou, L.Y. Lin, in: L.L. Regel, W.R. Wilcox (Eds.), Materials Processing in High Gravity, Plenum, New York, 1994, pp. 61}66. [14] F. Tao, Y. Zheng, W.J. Ma, M.L. Xue, in: L.L. Regel, W.R. Wilcox (Eds.), Materials Processing in High Gravity, Plenum, New York, 1994, pp. 67}79.