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Steady thermocapillary flows in a two-layer liquid system with flat interfaces
Journal of Crystal Growth 126 (1993) 335—346 North-Holland
~
o’
CRYSTAL GROWTH
Steady thermocapillary flows in a two-layer liquid system with flat interfaces E. Crespo del Arco
a,b, G.P. Extremet 13 and R.L. Sani c “Departamento FIsica Fundamental, UNED, Apdo. 60141, 28080 Madrid, Spain ~‘ IMFM, UM 34 CNRS, 1 Rue Honnorat, F-13003 Marseille, France C Center for Low-Gravity Fluid Mechanics, Department of Chemical Engineering, University of Colorado — Boulder, Boulder,
Colorado 80309, USA Received 28 January 1992; manuscript received in final form 18 August 1992
Steady thermocapillary convection is studied in a system of two flat, superposed layers of immiscible liquids with two fluid—fluid interfaces in a configuration similar to that of an encapsulated crystal growth. The layers are bounded on the sides by isothermal vertical walls maintained at different constant temperatures. A simplified analytical solution is used initially to explore different potential flow regimes in a parameter space of large dimensionality. Then the coupled Navier—Stokes and heat transfer equations are solved numerically with a finite element method via FIDAP, in a rectangular cavity filled with two immiscible liquids in the absence of a gravitational field.
1. Introduction The primary objective of this research is the investigation of the convective flow induced by interfacial forces in a system composed of two layers of immiscible liquids in the absence of gravitational forces. Convective motions driven by a horizontal temperature gradient result from the variation of surface tension with temperature. This configuration has been suggested as a simple model of the encapsulated float zone technique for space processing of electronic materials, It has already been pointed out [1] that the large dimension of the parameter space needed for such a system with two fluid layers makes it difficult to gain an overview of the manifold of possible steady solutions for this problem. Thermocapillary convection in a single layer of a Boussinesq fluid is governed by a relatively small number of dimensionless parameters: the Marangoni number, the capillary number, the Prandtl number and, in the presence of a gravitational field, the Rayleigh number. However, more than ten non-dimensional parameters may be impor0022-0248/93/$06.OO © 1993
—
tant in the two-layer problem. In order to avoid Rayleigh—Taylor instability, the density of the lower layer must be greater than that of the upper layer, but for the other material properties there are not any preferred values. Rasenat et al. [11studied the linear stability of the rest state in two superposed liquids subjected to a vertical gradient of temperature and with one free interface. In their study, convection is due to the buoyancy force (they do not specifically consider surface tension effects between the liquids) and the stability of the linear equations is investigated in a wide range of parameter space. The authors also discuss the results of some related experiments leading to steady convective flows. Nepomnyashchy and Simanovsky [2] study theoretically thermocapillary and buoyancy induced motions in a system of two layers of liquid heated from below (as in ref. [1]) with one free deformable interface. Convective motions driven by a horizontal temperature gradient along the interface between immiscible fluids occur in crystal growth melts. Therefore, there have been a number of studies
Elsevier Science Publishers B.V. All rights reserved
336
E. Crespo del Arco et al.
/ Steady thermocapillary flows in
of simplified two-dimensional models of thermocapillaryflowsinasinglelayerofliquid.Senand Davis [3] consider steady thermocapillary flows in a shallow layer of liquid of vanishing aspect ratio (height to width) with one or two deformable interfaces. Numerical simulations of thermocapillary convection are reported by Zebib et al. [4] in a square cavity with a deformable interface. Numerical simulations of steady thermocapillary flows are also reported by Ben Hadid and Roux [5] in a rectangular cavity with a flat surface and for very high Marangoni numbers. The numerical flows by Rivas [6] are obtained considering different thermal boundary conditions, i.e., a heat fl~ distribution along the horizontal direction which is an appropriate choice for laser melted pools. The flow induced by thermocapillary and buoyancy driven convection was studied experimentally in one layer of liquid by Metzger and Schwabe [7]. Villers and Flatten [8—10]study thermocapillary-buoyancy driven convection in a system of two superposed layers of liquid in presence of a horizontal gradient of temperature. The authors study experimentally the convective flows [8,91 and also obtain the analytical solution for two infinite layers [9,10] with a flat interface between them and a rigid upper horizontal wall. The aim of present work is to analyze the flows that appear in a system of two layers of liquids horizontally heated, with two interfaces, when the motion is due to the combined effect of surface tension on the interface between the liquids and on the upper interface. A simplified linear analytical solution as well as a Galerkin finite element solution via FIDAP [11] is used to obtain steady flows. In the computations, gravitational effect are assumed negligible; this assumption is a reasonable first approximation in the study of the crystal growth processes in a microgravity environment. Preliminary results were reported earlier [12] and other numerical experiments on a two-layer system of liquids in conjunction with a flight experiment are in progress [13]. The analytical solution leads to some interesting results, for example, a significant reduction of the magnitude of the flow in the lower liquid layer by a proper choice of parameters, as shown
two-layer liquid system
V 1
2
4
u 1 Fig. 1. Geometry of the two-layer model: axis and velocity components.
by Doi and Koster [14]. This result is particularly relevant for crystal growth purposes and it is corroborated by numerical calculations [12] in which end effects are included, the basic differences being some structural changes in the overall flow. For example, the appearance of secondary flows near the vertical walls which could lead to a possible origin of instabilities in confined cavities. These corner flows can also be reduced by a proper choice of the Marangoni numbers for the two layers. Under these circumstances, it seems reasonable that a relatively large surface tension force on the upper surface of the encapsulant could be desirable for the processes of crystal growth from melts.
2. Physical modeL and numerical method We consider a system of two viscous, immiscible liquids that fills a rectangular cavity (see fig. 1). The upper liquid fills the region (0
E. Crespo delArco et a!.
/ Steady thermocapilla,y flows in two-layer liquid system
tension that decreases linearly with temperature, on the interface between the liquids,
337
The density is assumed to obey the following equation of state pi=p?{lai(Ti_T0)J.
UlOOlT(TT),
and on the upper surface,
The horizontal velocity is determined by the following boundary and overall mass balances: at
(72o~2T(TT), where ~ ~ assumed constant, are the surface tension temperature coefficients and a~,o20 are the values of the surface tension at some reference temperature T°. Let J’ (U1, l’), the velocity field and its components in the x and z directions. The equations of motion (continuity, momentum and energy) are =
(1) 2J’ç+ge~,
VP, +
(V. V)J7,
p
=
+
— —
(2)
U~ 0,
(8)
=
a “return flow” condition in each layer,
f~U~(z)dz
=
0,
H
dz
=
0,
(9)
0
continuity of velocity components at the interface, z = 0, U~—U~, (10) the shear stress condition on the interface,
v~V aT
aT ~+V.~VT.=x~V2T (3)
1.
0t
f’~2U2(z)
The vertical boundaries of the cavity are maintained at different constant difference temperatures sponding to a temperature ~T.correThe lower rigid wall and the upper surface are assumed insulated,
3. AnalyticaL solution
az
(11)
a~1T~~
and on the upper surface, 02T aT/ax. (12) aU2/az The solution of eqs. (4—7) with the boundary conditions (8—12) is easily found as a function of the dimensionless parameters: Marangoni numbers, ~2
=
Ma 1
In the case of two superposed infinite layers of liquids, the analytical solution can be found. Consider two layers of liquids of height H1 and H2. The continuity, Navier—Stokes and energy equations for the liquids are: au~/ax 0, (4) =
=
o-lT(z1T/L)H~/~lXl,
Ma2 and the ratios of the coefficients of thermal expansions, the viscosities and the thermal diffusivi=
ties, Qa
=
2)2 (p°aH 0 (p aH2)
2LT~/öz2,
1’
(5)
8p~/ax ~ ö
(/L/H)
=
2
ap~/az=—p~(T)g, 0
=
—
~
aT. ax
+x
(
2T.
a 1 ~ ax
Q,L=
(7)
(X/H)~’ will surface also usedefined anotherasMarangoni number theWe upper Ma2 =Q~Q~ Ma2.in
a2T \
+
~).
(~/H)2
(6)
338
E. Crespo del Arco et al.
/ Steady thermocapillary flows in two-layer liquid system
The expression for the velocity U(Z) is given in the appendix. When buoyancy forces are not considered, i.e. Ra1 Ra2 0, the velocity is: =
(J1(Z)
=
1 2(Q~+ x (Z
+
4)
(2 Ma1
—
Ma2Q~Q~)
4) (Z + 1),
4. Numerical method (13)
1 6(Q~+
Numerical computations have been performed with a Galerkin finite element method (FIDAP [11]), using 9-node quadrilateral elements with a graded mesh of 61 X 61 nodes for the velocity and temperature and the corresponding 3-node discontinuous pressure elements. The interfacial
4)
thermal boundary conditions of continuity of temperature and flux are satisfied in this case.
x{[3 Ma2Q~Q~ + 6 Ma2Q~+ 3 Ma1]Z~ —
=
=
(12(Z)
=
at Z 0. This assumption simplifies the analytical solution and is discussed in refs. [9,101.
[4Ma2Q~+ 6 Mai] Z
+
2 Ma1 (14)
where Z is the dimensionless coordinate, Z z/H1, for —H1 z 0 and Z z/H2 for 0 Z H2 (similar for X) and the dimension of the velocity is Xi/Hi. For the same case, i.e. Ra1 Ra2 0, the expression for the dimensionless temperature is easily found: =
=
=
=
Here, surface deformations are not considered, the validity of the assumption of a flat interface in thermocapillary convection is discussed elsewhere [3,61. In the computations the aspect ratio of the cavity was H1 H2 L/4. Two sets of physical parameters, (i) and (ii), have been considered for a range of the Marangoni numbers: (i) Q1~ 1, Q~ 1, corresponding to two liquids with equal thermal conductivity K2 ~1 the bot=
=
=
=
=
tom layer is a low Prandtl number melt, Pr1 0.01, with larger density, Pi 100 P2’ and the top layer a larger Prandtl number encapsulant, Pr2 1 (Ma1 80, Ma2 120, where Ma2 Q~0~Ma2). (ii) Q~ 1, Q~ 100, corresponding to two liquids with 1(2 =i~, Pr 1 = Pr2 = 0.01 and p1 = 100 p2(Ma160, Ma2 120). =
for
—
1
Z
0,
=
1 2 Ma1
T1(x,Z) =X—
-~-~
4
+
8Z3
—
Q +
Ma2Q~Q~ +
=
4
6Z2),
=
(15)
x(3Z for 0Z T2(X,Z)
1, =
x
1 —
___________
QX(Q~
x {[Mai
+
+
4)
Ma2Q~(Q~ + 2)] Z 24
=
=
The spatial accuracy has been tested by mesh refinement studies. For example, the results of using a coarser mesh with 37 x 41 nodes differ by less than 1% in velocity and about 3% in the maximum value of the streamfunction from those of a 61 X 61 nodes mesh (see table 1). (All the
—(2 Ma2Q~+ 3 Ma1)— 18 +(2 Ma1 Ma2Q~Q~)-j-~-j.(16) The general thermal boundary condition on the interface of continuity of flux is not satisfied because the derivative of the temperature is zero —
Table 1 Influence of finite-element grid on the solution for Ma1 = Ma2 =60; the 37x41 and 61x61 grids are graded 1~m~n
_________________________________________ Mesh Uimax U~m~x ~‘ma~ ~ 37 >< 41 61x61
3.41
12.4
0.180
—
1.469
3.42 12.3 0.186 —1.480 _______________________________________________
E. Crespo del Arco et al.
/ Steady thermocapillary flows in two-layer liquid system
_i’ [~jjjj~
I
~iiiii~
339
[~ I~--~ I
U
2
U
1i + Ma1
_______
2 Ma1>M3~2~ 11
2 Ma2
Q14
Ma1<2Ma2
.-
Ma
+_1_\
1
Q}i)
________
—
2
=
2 Ma1
Fig. 2. Schematic U velocity profiles along the vertical direction, Z, and associated flow patterns. Characterization of the regimes I to V.
results reported herein were obtained using the finer mesh 61 X 61.)
only one layer of liquid. In regime (II) for which Ma 1 2Ma2(1 + 1/Q1) the velocity on the upper interface is zero, U2(Z 1) 0 according to (14). When =
In order to compare the solutions obtained with FIDAP with other computations, we performed some calculations of the flow in a single layer of liquid heated horizontally for some cases considered in [5].Thus, using a graded mesh with 33 x 59 nodes, the computed values of the streamfunction and of the horizontal velocity at a reference point, are for Pr 0.015, Ma 5: U(X 2, Z 0) 1.0004 and ~1’max 0.18484 and for Ma 20: U(X= 2, Z = 0) = 2.768 and lPmax = 0.56176. These values are to be compared to the solutions in ref. [51 (31 x 91 graded mesh) for the and Ma = 5: U(X = 2, Z = 0) = 1, ~~~max = 0.2, and same value of the Prandtl number, Pr = 0.015, for Ma=20: U(X=2, Z=0)=2.8, 1I~max=0.6. =
=
=
=
=
4~i2
40
=
A U
+
Ulmax(QX=lOO) U2max (Qx = 100) Ulmax (Qx = 1)
~
U2,max
X
=
30
1/Q~),
the situation corresponds to regime (I) in fig. 2 and the velocity profile is similar to the case with
1)
U1Z 0) -
/ *
0
> 2Ma2(1 +
(Qx
20
5. Results and discussion
Ma1
+
the velocity distribution (III) corresponds to a
=
The types of flows by the analytical solution (13)—(14) arepredicted represented in fig. 2 for positive Ma1 and Ma2, which corresponds to surface tension decreasing with the temperature on both interfaces. When
=
-•-
-
I II
iv
~
~
0
30
60
90
120
150
Fig. 3. Variation versus Ma2 of the U velocity component at the interface between the liquids (U1(Z-0)) and at the upper interface (U2(Z = 1)). Comparison with the computed values of the U at the locations of the extremes for Q~ = 1 and = 1 and 100. Identification of the regimes following Ma2 = Ma2QXQ~.
E. Crespo delArco et a!.
340
/
Steady thermocapil!ary flows in two-layer liquid system
U(Xm~,1)are also displayed for the two sets of parameters (i) and (ii).
flow with two rolls in the upper layer of liquid, 2, and one roll in the lower layer, 1. The flow obtained when 2Ma1 = Ma2 is denoted by (Iv)
In fig. 4, the analytical profile, U(Z), is compared with the numerically calculated U(3,Z) and U(Xmax,Z). For = 100, i.e. the set of conditions (ii), the numerical velocity in the core region is very similar to the analytical result and it is only presented for X = 3. For = 1 there isa slight difference in the profiles when Ma1 = Ma2 60, which becomes more prominent for Ma1 = = 60. The numerically computed horizontal velocities U(X,1) and U(X,0) are displayed in fig. 5. The analytical solution which is independent of X cannot be directly compared with these computed profiles at Z = 0 and Z = 1. In the confined box, the profiles U(X,1) and U(X,0) are not constant and the recirculating
and the horizontal velocity at the liquid—liquid interface is zero, U1(Z = 0) 0, thus suppressing the liquid motion in the lower liquid. Finally, for Ma1 < 4~ì~2, the flow denoted by (V) has two counter rotating rolls in the upper and lower layers of liquids, The numerical results for the finite cavity exhibit similar profiles as shown in fig. 3 for Ma1 = 60. The straight lines represent the analytical velocities at the interface between the liquids U1(Z = 0) and at the upper surface U2(Z = 1). The maximum value of the numerically computed velocity U1(X,Z) at the interface, i.e. Ui,m~= U(Xm~,0) and at the upper surface U2m~ =
=
=
20
40 Qx=1
U
(a)
10
30
Qx=1
(b)
20
10J
~
Qx
~T~OY1
u
= 100
Qx
(c)
= 100
(d)
~____
Fig. 4. Profiles of the analytical velocity U(Z) along the vertical direction and computed solutions for = 1, Ma =60 and four sets of values of and Ma2: Q~= 1, Ma2 = 60 (a); Q~= 1, Ma2 = 120 (b); Q~= 100 Ma2 = 60 (c); Q5 = 100, Ma2 = 120 (d). Computed U solutions at X= 3, U1(3, Z) (x), and at the locations of the extremes, U2(Xmax, Z) (~): Xmax = 3.67 in (a) and Xmax = 3.87 in (b).
E. Crespo delArco et al.
/ Steady thermocapillary flows
~_I62I43~2X~
in two-layer liquid system
341
__
27.0
(c)
U::
Fig. 5. Profiles of the horizontal velocity component U
1
interface U2
for
=
U(X,0) and U2 = U(X,1) along the interface, U1, and along the upper 60, Ma2 = 60 (b); Ma1 = 60, Ma2 = 120 (c).
=
Q~= I and for three values of MaL,.~ndMa2: Ma1 = 60, Ma2 = 15 (a); Ma1 =
flow near the ends, allows the velocity to be non-zero at Z = 1 for regime (II) and Z = 0, for regime (IV) as is illustrated in fig. 8d and fig. 8a, respectively. It is noteworthy that while the agreement of the analytical velocities on the free surfaces, U1 and U2, with the magnitude of the maximal value of the numerical velocities, U(Xm~,0) and U(Xmax,1), is good (see fig. 3), the location of the
maxima, Xmax, varies with the Marangoni number. For Ma1 = Ma2, it moves closer to the cold end of the layer with increasing Marangoni number (see tables 2 and 3). The location of the maximum and the shape of the surface-velocity profile in the vicinity of the maximum (see also fig. 5) are related to the appearance of secondary recirculation zones which result in significant changes in the overall structure of the flow as is
342
E. Crespo delArco et al.
/ Steady thermocapillaryflows in two-layer liquid system
Table 2 Location and value of the maximum horizontal velocity at the middle and upper interfaces for Ma
1 = Ma2 and set (ii)
Ma1 = Ma2
Middle interface
10 20 30 40 60 80
Upper interface
Xi,max
~
U(Z = 0)
X2max
U2max
U(Z = 1)
3.44 3.44 3.50 3.50 3.56 3.62
0.76 1.43 2.02 2.54 3.46 4.26
0.71 1.43 2.14 2.86 4.29 5.71
0.21 2.41 2.63 2.78 2.96 3.08
2.16 4.38 6.64 8.92 13.55 18.21
2.14 4.29 6.43 8.57 12.86 17.14
evident in the streamline contours discussed below. There is a slight difference between the flumerical solutions computed for the two different values of due to the continuity of flux thermal boundary condition at the interface. In the analytical solution this coupling has only been satisfied for the special case of zero flux, The temperature is expected to be very sensitive to from eqs. (15) and (16). The Z dependence of the analytical temperature distribution is presented in figs. 6 and 7. In figs. 6a—6c, the set of physical properties (i) has been used for the numerical calculations. The two sets of conditions (i) and (ii) illustrate the deformation of the isotherms due to the parameter Q~for Ma1 = Ma2 = 60. This parameter only slightly influences the velocity distribution but has a strong effect on the temperature field. The streamlines for the different ratios of Marangoni numbers and the physical properties defined by (i), are presented in fig. 8 for Ma1 = 60
and for both free and rigid upper surface at Z = 1. In fig. 8 the boundary condition at the upper surface 2 corresponds to Ma2 = 120 (a); 60 (b), 30 (c), 15 (d) and Ma2 = 0 (e) (free surface). In fig. 8f, the boundary condition at the upper boundary is a rigid wall. When the surface tension temperature coefficient at the upper interface, Ma2, is large, see figs. 8a and 8b, there is an almost total suppression of the flow in the lower layer as suggested by the infinite layer results and the situation corresponds to (IV) in fig. 2. For Ma1 = 4Ma2, in fig. 8d, the analysis predicts that the horizontal velocity is zero at the top interface, i.e. U2 = 0, situation (II) in fig. 2. When the surface tension force is less important in the upper interface (figs. 8c—8f), the flow is qualitatively similar to the case in which only one layer of liquid is considered and the streamlines of the flow in the lower layer, 1, are comparable to the flows shown in refs. [5,6]. That is, there is a vortex “flywheel” being formed near the cold end, low speeds elsewhere away from interface and a sig-
Table 3
Location
and value of the maximum horizontal velocity at the middle and upper interfaces for set (i)
Ma1
Ma2
Middle interface
Upper interface
Ximax
Uim=
U(Z =0)
X2m=
60 60 60 60 60
120 60 30 15 0
0.68 3.37 3.44 3.50 3.50
3.33 3.42 4.38 4.85 5.25
0.0 4.29 6.43 7.5 8.57
3.87 3.67 3.16
60
Rigid wall
3.37
5.32
0.92
U2m=~
26.27 12.34 5.49 2.395
2.78
—2.22
2.70
—2.29
U(Z
=
30.0 12.86 4.29 0.0
—4.29
1)
E. Crespo delArco et a!.
/ Steady thermocapillary flows in two-layer liquid system
nificant separation zone near the bottom surface. The maximum values of the streamfunction and the velocity in the interface in fig. 8e are IP/Ma = 0.015 and U/Ma = 0.0875 at Ma 1 = 60 (thus, Ma/Pr = 6000) and the corresponding maximal values of the streamfunction and the velocity for Ma/Pr = 6670 are ~P/Ma = 0.016 and U/Ma = 0.095 in ref. [5]. The distribution of streamlines in figs. 8a and 8b shows that the vorticity is maximum in the cold (right) corner near the upper layer, the maximum value is 900 in fig. 8a and
~Ma2=0
343
1
z
QXIOO
0
-1
T(Z)
______________ -
1
0
1
0
1
Ma2=30
Z 0
Ma2 =120
(a) Q~=1OO
-1
~
(7 /•1~ ( j~ -
—
—
(a)Ma1=60.
Ma2
/ / / / / / ( /
—
(b)
I
—
2. 195 the in fig. 8b in dimensionless units ~/H is When surface tension force of the interface more important as in figs. 8d and 8e, the zones of large vorticity occur at the interface between the
Ma 1= 60,
\\
Ma2
=
30
~—~-y/ ( ( ( ) / /
\ ~
I (c)
Fig. 7. Analytical temperature profiles along the vertical and computed isotherm patterns for Ma1 = 60, Ma2 = 60, Q = I, Q~=100(a) and Q~=1(b).
Ma1= 60,
Ma2
=
I
120
Fig. 6. Analytical temperature profile along the vertical direction and computed isotherm patterns for = = 1, for Ma1 = 60 and for three values of Ma2 = 0 (a), 30 (b) and 120 (c).
liquids and at the lower liquid in the proximity of the wall (with maximal values of 108 and 120 in figs. 8d and 8e, respectively), and the vorticity is also important around the flywheel roll. (The streamlines obtained for the set of parameters (ii) are very similar to those obtained for the set (i).) The extension of the recirculation zones near the corners and of the core region can be seen in fig. 8. It is noteworthy that the flows displayed in fig. 8 are similar to the flows obtained in refs. [15,16] with one layer of liquid of the same aspect ratio L/H = 4 and one free surface without surface
344
E. Crespo del Arco et a!.
/ Steady thermocapi!lary flows in two-layer liquid system
tension effects. The latter flow is buoyancy driven and its instability is caused by the successive growth and stretching of the recirculation zones. Thus, in the present case, it might be expected
~
that the recirculation zones would be the cause of instabilities. In summary, the analytical solution gives an excellent approximation to the convective flows
_
cI~ ____________________________________________________ (d) Fig. 8. Streamline patterns for
________________________________________ (f)
= 1 and for various values of Ma 1 and rvla2 with extreme values of !P: (a) Ma1 = 60, 0.1708); (b) Ma Ma2 = 120 (~min= —2.7553, ~ 1 = 60, Ma2 = 60 (~~‘mjn = — 1.4804, ~‘max = 0.1860); (c) Ma1 = 60, Ma2 = 30 (‘t’mix = 07370, ~~‘max= 0.3753); (d) Ma1 = 60, ~2 = 15 (“mi,, = —0.8003, ~~‘max= 0.5723); (e) Ma1 = 60, ~2 = 0(~!’min= —0.8609, ~1’max.= 0.8719); (f) Ma1 = 60, with rigid upper wall (~‘min= — 0.8325, 1(’mxx = 0.6872). =
1’Max =
E. Crespo delArco et al.
/ Steady thermocapil!ary flows in two-layer liquid system
which are characterized by the Marangoni numbers, Ma1 and Ma2, as shown in figs. 2 and 8. The magnitude of the horizontal velocity of the flow numerically computed agrees with the analytical solution as shown in figs. 3, and 4, and the variation of the analytical solution is shown in fig. 5. Instabilities in cavities are expected to be connected with the generation of recirculating regions in the turning region near the end of the
345
and the temperature is T1(x,Z)
=~
1
AU*H~
+
4
Xi Q,~+ 5 Z4 x)—(Q~+fl+ ~[2+2Q~+Qa (Z ~, 20 +4(2K 1
—
QaK2)]
3
cavity (fig.8). The temperature field is strongly dependent on Ma 1 and Ma2, and also on the ratio of thermal diffusivities, Q~(see figs. 6 and 7). crystal growth thesignificantly convective flowFor in the lower liquid purposes, layer can be
z +
—
+
—
reduced using an appropriate choice of the parameters and this result is corroborated by the numerical calculations.
+
12
2Qa
2 z [3Qa 2 —
+
8(2K1
—
QaK2)]
12(2K
+
1
—
QaK2)]
24 (A.2) In the lower layer (2) the velocity is
Qa
Acknowledgements
(12(Z) We acknowledge Dr. P. Bontoux, J.P. Fontaine, E.C.A. acknowledges support from the MEC in J.N. Koster and T. Doi for fruitful discussions. Spain funding a visit toat the Center for GravityforFluid Mechanics University of Low Colorado at Boulder. Also E.C.A. and R.L.S. would like to acknowledge support from the Microgravity Science and Applications Division of NASA. Support from the CNRS (GDR, MFN), the CNES and from the Conseil Regional of PACA (Intern. Exchange Program) in France is also acknowledged.
=
U”
x
1
Q~1
{(
1
+
+
4/3Q~
I
+
—
4 \ 3QM.)
~2~
—6
rL
—
—
—
1 Qa
+121
1 —
Q
2K2 1 —)I 1]
+K2+
Z 1 1
Qp~
2
/ 3K1
2K2
+ +
-~- [~-
-
3
4~ + 1--————K ~ Qa 2)]}~ /2K1
~
—
—~-~--
—~-~--
“1
)j
Qa 2 —+12
(A.3)
The solution of the system (4)—(12) is U*
and the temperature is
L1 =
10 Q~
/ K
Appendix
1(Z)
—
Q~+4
T2(x,Z)=~+ AU*QaH~
X{(Q~+4)Z~+~[2+2Q,~+Qa
( Z5
2
+4(2K1 QaK2)]Z +4[Q~+2Qa+8(2Ki~QaK
/
1
x2Q~4
1+4/3Q~ \ z4 1
/ K1
2K2
—
2)JZ +
~[3Qa —2 + 12(2K1
—
QaK2)l},
(A.1)
10
\Qa
\1
346
E. Crespo delArco et al.
z3 +
+
1
—
—
12
2 +
3
3K
—
—
Q~
Qa
2 +
—
/ Steady ther,nocapil!ary flows in two-layer liquid system
12
4
1
2K2 +
—
Qa
2K1 —
K2
Q~a ,
(A.4) with K1
=
Ra1
—
Ma1/Ra1, K2 = gp~a1(z.IT/L) H~
—
Ma2/Ra2,
=
/-‘~iXi
gp~a2(~iT/L)H~ =
‘
l.L2X2
[2] A.A. Nepomnyashchy and lB. Simanovsky, Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza 3 (1984) 175. [3] A.K. Sen and S.H. Davis, J. Fluid Mech. 121 (1982) 163. [4] A. Zebib, G.M. Homsy and E. Meiburg, Phys. Fluids 28 (1985) 3467. [5] H. Ben Hadid and B. Roux, J. Fluid Mech. 221 (1990) 77. [6] D. Rivas, Phys. Fluids A 3 (1991) 280. [7] J. Metzger and D. Schwabe, Physicochem. Hydrodyn. 10 (1988) 263. [8] D. Villers and J.K. Flatten, Appl. Sci. Res. 45 (1988) 145. [9] D. Villers, PhD Thesis, Université de Mons (1989). [101 D. Villers and J.K. Platten, AppI. Sci. Res. 47(1990)177. [11] M.S. Engelman, FIDAP Users Manual (FDI, Evanston, IL, 1986). [12] E. Crespo del Arco, G.P. Extremet and R.L. Sani, Advan. Space Res. 11(7) (1991) 29. [13] J.P. Fontaine, R.L. Sani and J.N. Koster, in: Proc. AIAA, paper 920689, 30th Aerospace Science Meeting, Reno, NV, 1992.
U * = gp~H~a1A/6,i1.
[14] T. Doi and J.N. Koster, private communication, 1989. [15] J.P. Pulicani, E. Crespo del Arco, A. Randriamampianina, P. Bontoux and R. Feyret, Intern. J. Numer. Methods Fluids 10 (1990) 481.
References
[161 E. Crespo del Arco, A. Randriamampianina and P. Bon-
[1] S. Rasenat, F.H. Busse and I. Rehberg, J. Fluid Mech. 199 (1989) 519.
toux, in: Proc. Conf. on Synergetics, Order and Chaos,
Ed. M.G. Velarde (World Scientific, Singapore, 1988) p. 244.
~
o’
CRYSTAL GROWTH
Steady thermocapillary flows in a two-layer liquid system with flat interfaces E. Crespo del Arco
a,b, G.P. Extremet 13 and R.L. Sani c “Departamento FIsica Fundamental, UNED, Apdo. 60141, 28080 Madrid, Spain ~‘ IMFM, UM 34 CNRS, 1 Rue Honnorat, F-13003 Marseille, France C Center for Low-Gravity Fluid Mechanics, Department of Chemical Engineering, University of Colorado — Boulder, Boulder,
Colorado 80309, USA Received 28 January 1992; manuscript received in final form 18 August 1992
Steady thermocapillary convection is studied in a system of two flat, superposed layers of immiscible liquids with two fluid—fluid interfaces in a configuration similar to that of an encapsulated crystal growth. The layers are bounded on the sides by isothermal vertical walls maintained at different constant temperatures. A simplified analytical solution is used initially to explore different potential flow regimes in a parameter space of large dimensionality. Then the coupled Navier—Stokes and heat transfer equations are solved numerically with a finite element method via FIDAP, in a rectangular cavity filled with two immiscible liquids in the absence of a gravitational field.
1. Introduction The primary objective of this research is the investigation of the convective flow induced by interfacial forces in a system composed of two layers of immiscible liquids in the absence of gravitational forces. Convective motions driven by a horizontal temperature gradient result from the variation of surface tension with temperature. This configuration has been suggested as a simple model of the encapsulated float zone technique for space processing of electronic materials, It has already been pointed out [1] that the large dimension of the parameter space needed for such a system with two fluid layers makes it difficult to gain an overview of the manifold of possible steady solutions for this problem. Thermocapillary convection in a single layer of a Boussinesq fluid is governed by a relatively small number of dimensionless parameters: the Marangoni number, the capillary number, the Prandtl number and, in the presence of a gravitational field, the Rayleigh number. However, more than ten non-dimensional parameters may be impor0022-0248/93/$06.OO © 1993
—
tant in the two-layer problem. In order to avoid Rayleigh—Taylor instability, the density of the lower layer must be greater than that of the upper layer, but for the other material properties there are not any preferred values. Rasenat et al. [11studied the linear stability of the rest state in two superposed liquids subjected to a vertical gradient of temperature and with one free interface. In their study, convection is due to the buoyancy force (they do not specifically consider surface tension effects between the liquids) and the stability of the linear equations is investigated in a wide range of parameter space. The authors also discuss the results of some related experiments leading to steady convective flows. Nepomnyashchy and Simanovsky [2] study theoretically thermocapillary and buoyancy induced motions in a system of two layers of liquid heated from below (as in ref. [1]) with one free deformable interface. Convective motions driven by a horizontal temperature gradient along the interface between immiscible fluids occur in crystal growth melts. Therefore, there have been a number of studies
Elsevier Science Publishers B.V. All rights reserved
336
E. Crespo del Arco et al.
/ Steady thermocapillary flows in
of simplified two-dimensional models of thermocapillaryflowsinasinglelayerofliquid.Senand Davis [3] consider steady thermocapillary flows in a shallow layer of liquid of vanishing aspect ratio (height to width) with one or two deformable interfaces. Numerical simulations of thermocapillary convection are reported by Zebib et al. [4] in a square cavity with a deformable interface. Numerical simulations of steady thermocapillary flows are also reported by Ben Hadid and Roux [5] in a rectangular cavity with a flat surface and for very high Marangoni numbers. The numerical flows by Rivas [6] are obtained considering different thermal boundary conditions, i.e., a heat fl~ distribution along the horizontal direction which is an appropriate choice for laser melted pools. The flow induced by thermocapillary and buoyancy driven convection was studied experimentally in one layer of liquid by Metzger and Schwabe [7]. Villers and Flatten [8—10]study thermocapillary-buoyancy driven convection in a system of two superposed layers of liquid in presence of a horizontal gradient of temperature. The authors study experimentally the convective flows [8,91 and also obtain the analytical solution for two infinite layers [9,10] with a flat interface between them and a rigid upper horizontal wall. The aim of present work is to analyze the flows that appear in a system of two layers of liquids horizontally heated, with two interfaces, when the motion is due to the combined effect of surface tension on the interface between the liquids and on the upper interface. A simplified linear analytical solution as well as a Galerkin finite element solution via FIDAP [11] is used to obtain steady flows. In the computations, gravitational effect are assumed negligible; this assumption is a reasonable first approximation in the study of the crystal growth processes in a microgravity environment. Preliminary results were reported earlier [12] and other numerical experiments on a two-layer system of liquids in conjunction with a flight experiment are in progress [13]. The analytical solution leads to some interesting results, for example, a significant reduction of the magnitude of the flow in the lower liquid layer by a proper choice of parameters, as shown
two-layer liquid system
V 1
2
4
u 1 Fig. 1. Geometry of the two-layer model: axis and velocity components.
by Doi and Koster [14]. This result is particularly relevant for crystal growth purposes and it is corroborated by numerical calculations [12] in which end effects are included, the basic differences being some structural changes in the overall flow. For example, the appearance of secondary flows near the vertical walls which could lead to a possible origin of instabilities in confined cavities. These corner flows can also be reduced by a proper choice of the Marangoni numbers for the two layers. Under these circumstances, it seems reasonable that a relatively large surface tension force on the upper surface of the encapsulant could be desirable for the processes of crystal growth from melts.
2. Physical modeL and numerical method We consider a system of two viscous, immiscible liquids that fills a rectangular cavity (see fig. 1). The upper liquid fills the region (0
E. Crespo delArco et a!.
/ Steady thermocapilla,y flows in two-layer liquid system
tension that decreases linearly with temperature, on the interface between the liquids,
337
The density is assumed to obey the following equation of state pi=p?{lai(Ti_T0)J.
UlOOlT(TT),
and on the upper surface,
The horizontal velocity is determined by the following boundary and overall mass balances: at
(72o~2T(TT), where ~ ~ assumed constant, are the surface tension temperature coefficients and a~,o20 are the values of the surface tension at some reference temperature T°. Let J’ (U1, l’), the velocity field and its components in the x and z directions. The equations of motion (continuity, momentum and energy) are =
(1) 2J’ç+ge~,
VP, +
(V. V)J7,
p
=
+
— —
(2)
U~ 0,
(8)
=
a “return flow” condition in each layer,
f~U~(z)dz
=
0,
H
dz
=
0,
(9)
0
continuity of velocity components at the interface, z = 0, U~—U~, (10) the shear stress condition on the interface,
v~V aT
aT ~+V.~VT.=x~V2T (3)
1.
0t
f’~2U2(z)
The vertical boundaries of the cavity are maintained at different constant difference temperatures sponding to a temperature ~T.correThe lower rigid wall and the upper surface are assumed insulated,
3. AnalyticaL solution
az
(11)
a~1T~~
and on the upper surface, 02T aT/ax. (12) aU2/az The solution of eqs. (4—7) with the boundary conditions (8—12) is easily found as a function of the dimensionless parameters: Marangoni numbers, ~2
=
Ma 1
In the case of two superposed infinite layers of liquids, the analytical solution can be found. Consider two layers of liquids of height H1 and H2. The continuity, Navier—Stokes and energy equations for the liquids are: au~/ax 0, (4) =
=
o-lT(z1T/L)H~/~lXl,
Ma2 and the ratios of the coefficients of thermal expansions, the viscosities and the thermal diffusivi=
ties, Qa
=
2)2 (p°aH 0 (p aH2)
2LT~/öz2,
1’
(5)
8p~/ax ~ ö
(/L/H)
=
2
ap~/az=—p~(T)g, 0
=
—
~
aT. ax
+x
(
2T.
a 1 ~ ax
Q,L=
(7)
(X/H)~’ will surface also usedefined anotherasMarangoni number theWe upper Ma2 =Q~Q~ Ma2.in
a2T \
+
~).
(~/H)2
(6)
338
E. Crespo del Arco et al.
/ Steady thermocapillary flows in two-layer liquid system
The expression for the velocity U(Z) is given in the appendix. When buoyancy forces are not considered, i.e. Ra1 Ra2 0, the velocity is: =
(J1(Z)
=
1 2(Q~+ x (Z
+
4)
(2 Ma1
—
Ma2Q~Q~)
4) (Z + 1),
4. Numerical method (13)
1 6(Q~+
Numerical computations have been performed with a Galerkin finite element method (FIDAP [11]), using 9-node quadrilateral elements with a graded mesh of 61 X 61 nodes for the velocity and temperature and the corresponding 3-node discontinuous pressure elements. The interfacial
4)
thermal boundary conditions of continuity of temperature and flux are satisfied in this case.
x{[3 Ma2Q~Q~ + 6 Ma2Q~+ 3 Ma1]Z~ —
=
=
(12(Z)
=
at Z 0. This assumption simplifies the analytical solution and is discussed in refs. [9,101.
[4Ma2Q~+ 6 Mai] Z
+
2 Ma1 (14)
where Z is the dimensionless coordinate, Z z/H1, for —H1 z 0 and Z z/H2 for 0 Z H2 (similar for X) and the dimension of the velocity is Xi/Hi. For the same case, i.e. Ra1 Ra2 0, the expression for the dimensionless temperature is easily found: =
=
=
=
Here, surface deformations are not considered, the validity of the assumption of a flat interface in thermocapillary convection is discussed elsewhere [3,61. In the computations the aspect ratio of the cavity was H1 H2 L/4. Two sets of physical parameters, (i) and (ii), have been considered for a range of the Marangoni numbers: (i) Q1~ 1, Q~ 1, corresponding to two liquids with equal thermal conductivity K2 ~1 the bot=
=
=
=
=
tom layer is a low Prandtl number melt, Pr1 0.01, with larger density, Pi 100 P2’ and the top layer a larger Prandtl number encapsulant, Pr2 1 (Ma1 80, Ma2 120, where Ma2 Q~0~Ma2). (ii) Q~ 1, Q~ 100, corresponding to two liquids with 1(2 =i~, Pr 1 = Pr2 = 0.01 and p1 = 100 p2(Ma160, Ma2 120). =
for
—
1
Z
0,
=
1 2 Ma1
T1(x,Z) =X—
-~-~
4
+
8Z3
—
Q +
Ma2Q~Q~ +
=
4
6Z2),
=
(15)
x(3Z for 0Z T2(X,Z)
1, =
x
1 —
___________
QX(Q~
x {[Mai
+
+
4)
Ma2Q~(Q~ + 2)] Z 24
=
=
The spatial accuracy has been tested by mesh refinement studies. For example, the results of using a coarser mesh with 37 x 41 nodes differ by less than 1% in velocity and about 3% in the maximum value of the streamfunction from those of a 61 X 61 nodes mesh (see table 1). (All the
—(2 Ma2Q~+ 3 Ma1)— 18 +(2 Ma1 Ma2Q~Q~)-j-~-j.(16) The general thermal boundary condition on the interface of continuity of flux is not satisfied because the derivative of the temperature is zero —
Table 1 Influence of finite-element grid on the solution for Ma1 = Ma2 =60; the 37x41 and 61x61 grids are graded 1~m~n
_________________________________________ Mesh Uimax U~m~x ~‘ma~ ~ 37 >< 41 61x61
3.41
12.4
0.180
—
1.469
3.42 12.3 0.186 —1.480 _______________________________________________
E. Crespo del Arco et al.
/ Steady thermocapillary flows in two-layer liquid system
_i’ [~jjjj~
I
~iiiii~
339
[~ I~--~ I
U
2
U
1i + Ma1
_______
2 Ma1>M3~2~ 11
2 Ma2
Q14
Ma1<2Ma2
.-
Ma
+_1_\
1
Q}i)
________
—
2
=
2 Ma1
Fig. 2. Schematic U velocity profiles along the vertical direction, Z, and associated flow patterns. Characterization of the regimes I to V.
results reported herein were obtained using the finer mesh 61 X 61.)
only one layer of liquid. In regime (II) for which Ma 1 2Ma2(1 + 1/Q1) the velocity on the upper interface is zero, U2(Z 1) 0 according to (14). When =
In order to compare the solutions obtained with FIDAP with other computations, we performed some calculations of the flow in a single layer of liquid heated horizontally for some cases considered in [5].Thus, using a graded mesh with 33 x 59 nodes, the computed values of the streamfunction and of the horizontal velocity at a reference point, are for Pr 0.015, Ma 5: U(X 2, Z 0) 1.0004 and ~1’max 0.18484 and for Ma 20: U(X= 2, Z = 0) = 2.768 and lPmax = 0.56176. These values are to be compared to the solutions in ref. [51 (31 x 91 graded mesh) for the and Ma = 5: U(X = 2, Z = 0) = 1, ~~~max = 0.2, and same value of the Prandtl number, Pr = 0.015, for Ma=20: U(X=2, Z=0)=2.8, 1I~max=0.6. =
=
=
=
=
4~i2
40
=
A U
+
Ulmax(QX=lOO) U2max (Qx = 100) Ulmax (Qx = 1)
~
U2,max
X
=
30
1/Q~),
the situation corresponds to regime (I) in fig. 2 and the velocity profile is similar to the case with
1)
U1Z 0) -
/ *
0
> 2Ma2(1 +
(Qx
20
5. Results and discussion
Ma1
+
the velocity distribution (III) corresponds to a
=
The types of flows by the analytical solution (13)—(14) arepredicted represented in fig. 2 for positive Ma1 and Ma2, which corresponds to surface tension decreasing with the temperature on both interfaces. When
=
-•-
-
I II
iv
~
~
0
30
60
90
120
150
Fig. 3. Variation versus Ma2 of the U velocity component at the interface between the liquids (U1(Z-0)) and at the upper interface (U2(Z = 1)). Comparison with the computed values of the U at the locations of the extremes for Q~ = 1 and = 1 and 100. Identification of the regimes following Ma2 = Ma2QXQ~.
E. Crespo delArco et a!.
340
/
Steady thermocapil!ary flows in two-layer liquid system
U(Xm~,1)are also displayed for the two sets of parameters (i) and (ii).
flow with two rolls in the upper layer of liquid, 2, and one roll in the lower layer, 1. The flow obtained when 2Ma1 = Ma2 is denoted by (Iv)
In fig. 4, the analytical profile, U(Z), is compared with the numerically calculated U(3,Z) and U(Xmax,Z). For = 100, i.e. the set of conditions (ii), the numerical velocity in the core region is very similar to the analytical result and it is only presented for X = 3. For = 1 there isa slight difference in the profiles when Ma1 = Ma2 60, which becomes more prominent for Ma1 = = 60. The numerically computed horizontal velocities U(X,1) and U(X,0) are displayed in fig. 5. The analytical solution which is independent of X cannot be directly compared with these computed profiles at Z = 0 and Z = 1. In the confined box, the profiles U(X,1) and U(X,0) are not constant and the recirculating
and the horizontal velocity at the liquid—liquid interface is zero, U1(Z = 0) 0, thus suppressing the liquid motion in the lower liquid. Finally, for Ma1 < 4~ì~2, the flow denoted by (V) has two counter rotating rolls in the upper and lower layers of liquids, The numerical results for the finite cavity exhibit similar profiles as shown in fig. 3 for Ma1 = 60. The straight lines represent the analytical velocities at the interface between the liquids U1(Z = 0) and at the upper surface U2(Z = 1). The maximum value of the numerically computed velocity U1(X,Z) at the interface, i.e. Ui,m~= U(Xm~,0) and at the upper surface U2m~ =
=
=
20
40 Qx=1
U
(a)
10
30
Qx=1
(b)
20
10J
~
Qx
~T~OY1
u
= 100
Qx
(c)
= 100
(d)
~____
Fig. 4. Profiles of the analytical velocity U(Z) along the vertical direction and computed solutions for = 1, Ma =60 and four sets of values of and Ma2: Q~= 1, Ma2 = 60 (a); Q~= 1, Ma2 = 120 (b); Q~= 100 Ma2 = 60 (c); Q5 = 100, Ma2 = 120 (d). Computed U solutions at X= 3, U1(3, Z) (x), and at the locations of the extremes, U2(Xmax, Z) (~): Xmax = 3.67 in (a) and Xmax = 3.87 in (b).
E. Crespo delArco et al.
/ Steady thermocapillary flows
~_I62I43~2X~
in two-layer liquid system
341
__
27.0
(c)
U::
Fig. 5. Profiles of the horizontal velocity component U
1
interface U2
for
=
U(X,0) and U2 = U(X,1) along the interface, U1, and along the upper 60, Ma2 = 60 (b); Ma1 = 60, Ma2 = 120 (c).
=
Q~= I and for three values of MaL,.~ndMa2: Ma1 = 60, Ma2 = 15 (a); Ma1 =
flow near the ends, allows the velocity to be non-zero at Z = 1 for regime (II) and Z = 0, for regime (IV) as is illustrated in fig. 8d and fig. 8a, respectively. It is noteworthy that while the agreement of the analytical velocities on the free surfaces, U1 and U2, with the magnitude of the maximal value of the numerical velocities, U(Xm~,0) and U(Xmax,1), is good (see fig. 3), the location of the
maxima, Xmax, varies with the Marangoni number. For Ma1 = Ma2, it moves closer to the cold end of the layer with increasing Marangoni number (see tables 2 and 3). The location of the maximum and the shape of the surface-velocity profile in the vicinity of the maximum (see also fig. 5) are related to the appearance of secondary recirculation zones which result in significant changes in the overall structure of the flow as is
342
E. Crespo delArco et al.
/ Steady thermocapillaryflows in two-layer liquid system
Table 2 Location and value of the maximum horizontal velocity at the middle and upper interfaces for Ma
1 = Ma2 and set (ii)
Ma1 = Ma2
Middle interface
10 20 30 40 60 80
Upper interface
Xi,max
~
U(Z = 0)
X2max
U2max
U(Z = 1)
3.44 3.44 3.50 3.50 3.56 3.62
0.76 1.43 2.02 2.54 3.46 4.26
0.71 1.43 2.14 2.86 4.29 5.71
0.21 2.41 2.63 2.78 2.96 3.08
2.16 4.38 6.64 8.92 13.55 18.21
2.14 4.29 6.43 8.57 12.86 17.14
evident in the streamline contours discussed below. There is a slight difference between the flumerical solutions computed for the two different values of due to the continuity of flux thermal boundary condition at the interface. In the analytical solution this coupling has only been satisfied for the special case of zero flux, The temperature is expected to be very sensitive to from eqs. (15) and (16). The Z dependence of the analytical temperature distribution is presented in figs. 6 and 7. In figs. 6a—6c, the set of physical properties (i) has been used for the numerical calculations. The two sets of conditions (i) and (ii) illustrate the deformation of the isotherms due to the parameter Q~for Ma1 = Ma2 = 60. This parameter only slightly influences the velocity distribution but has a strong effect on the temperature field. The streamlines for the different ratios of Marangoni numbers and the physical properties defined by (i), are presented in fig. 8 for Ma1 = 60
and for both free and rigid upper surface at Z = 1. In fig. 8 the boundary condition at the upper surface 2 corresponds to Ma2 = 120 (a); 60 (b), 30 (c), 15 (d) and Ma2 = 0 (e) (free surface). In fig. 8f, the boundary condition at the upper boundary is a rigid wall. When the surface tension temperature coefficient at the upper interface, Ma2, is large, see figs. 8a and 8b, there is an almost total suppression of the flow in the lower layer as suggested by the infinite layer results and the situation corresponds to (IV) in fig. 2. For Ma1 = 4Ma2, in fig. 8d, the analysis predicts that the horizontal velocity is zero at the top interface, i.e. U2 = 0, situation (II) in fig. 2. When the surface tension force is less important in the upper interface (figs. 8c—8f), the flow is qualitatively similar to the case in which only one layer of liquid is considered and the streamlines of the flow in the lower layer, 1, are comparable to the flows shown in refs. [5,6]. That is, there is a vortex “flywheel” being formed near the cold end, low speeds elsewhere away from interface and a sig-
Table 3
Location
and value of the maximum horizontal velocity at the middle and upper interfaces for set (i)
Ma1
Ma2
Middle interface
Upper interface
Ximax
Uim=
U(Z =0)
X2m=
60 60 60 60 60
120 60 30 15 0
0.68 3.37 3.44 3.50 3.50
3.33 3.42 4.38 4.85 5.25
0.0 4.29 6.43 7.5 8.57
3.87 3.67 3.16
60
Rigid wall
3.37
5.32
0.92
U2m=~
26.27 12.34 5.49 2.395
2.78
—2.22
2.70
—2.29
U(Z
=
30.0 12.86 4.29 0.0
—4.29
1)
E. Crespo delArco et a!.
/ Steady thermocapillary flows in two-layer liquid system
nificant separation zone near the bottom surface. The maximum values of the streamfunction and the velocity in the interface in fig. 8e are IP/Ma = 0.015 and U/Ma = 0.0875 at Ma 1 = 60 (thus, Ma/Pr = 6000) and the corresponding maximal values of the streamfunction and the velocity for Ma/Pr = 6670 are ~P/Ma = 0.016 and U/Ma = 0.095 in ref. [5]. The distribution of streamlines in figs. 8a and 8b shows that the vorticity is maximum in the cold (right) corner near the upper layer, the maximum value is 900 in fig. 8a and
~Ma2=0
343
1
z
QXIOO
0
-1
T(Z)
______________ -
1
0
1
0
1
Ma2=30
Z 0
Ma2 =120
(a) Q~=1OO
-1
~
(7 /•1~ ( j~ -
—
—
(a)Ma1=60.
Ma2
/ / / / / / ( /
—
(b)
I
—
2. 195 the in fig. 8b in dimensionless units ~/H is When surface tension force of the interface more important as in figs. 8d and 8e, the zones of large vorticity occur at the interface between the
Ma 1= 60,
\\
Ma2
=
30
~—~-y/ ( ( ( ) / /
\ ~
I (c)
Fig. 7. Analytical temperature profiles along the vertical and computed isotherm patterns for Ma1 = 60, Ma2 = 60, Q = I, Q~=100(a) and Q~=1(b).
Ma1= 60,
Ma2
=
I
120
Fig. 6. Analytical temperature profile along the vertical direction and computed isotherm patterns for = = 1, for Ma1 = 60 and for three values of Ma2 = 0 (a), 30 (b) and 120 (c).
liquids and at the lower liquid in the proximity of the wall (with maximal values of 108 and 120 in figs. 8d and 8e, respectively), and the vorticity is also important around the flywheel roll. (The streamlines obtained for the set of parameters (ii) are very similar to those obtained for the set (i).) The extension of the recirculation zones near the corners and of the core region can be seen in fig. 8. It is noteworthy that the flows displayed in fig. 8 are similar to the flows obtained in refs. [15,16] with one layer of liquid of the same aspect ratio L/H = 4 and one free surface without surface
344
E. Crespo del Arco et a!.
/ Steady thermocapi!lary flows in two-layer liquid system
tension effects. The latter flow is buoyancy driven and its instability is caused by the successive growth and stretching of the recirculation zones. Thus, in the present case, it might be expected
~
that the recirculation zones would be the cause of instabilities. In summary, the analytical solution gives an excellent approximation to the convective flows
_
cI~ ____________________________________________________ (d) Fig. 8. Streamline patterns for
________________________________________ (f)
= 1 and for various values of Ma 1 and rvla2 with extreme values of !P: (a) Ma1 = 60, 0.1708); (b) Ma Ma2 = 120 (~min= —2.7553, ~ 1 = 60, Ma2 = 60 (~~‘mjn = — 1.4804, ~‘max = 0.1860); (c) Ma1 = 60, Ma2 = 30 (‘t’mix = 07370, ~~‘max= 0.3753); (d) Ma1 = 60, ~2 = 15 (“mi,, = —0.8003, ~~‘max= 0.5723); (e) Ma1 = 60, ~2 = 0(~!’min= —0.8609, ~1’max.= 0.8719); (f) Ma1 = 60, with rigid upper wall (~‘min= — 0.8325, 1(’mxx = 0.6872). =
1’Max =
E. Crespo delArco et al.
/ Steady thermocapil!ary flows in two-layer liquid system
which are characterized by the Marangoni numbers, Ma1 and Ma2, as shown in figs. 2 and 8. The magnitude of the horizontal velocity of the flow numerically computed agrees with the analytical solution as shown in figs. 3, and 4, and the variation of the analytical solution is shown in fig. 5. Instabilities in cavities are expected to be connected with the generation of recirculating regions in the turning region near the end of the
345
and the temperature is T1(x,Z)
=~
1
AU*H~
+
4
Xi Q,~+ 5 Z4 x)—(Q~+fl+ ~[2+2Q~+Qa (Z ~, 20 +4(2K 1
—
QaK2)]
3
cavity (fig.8). The temperature field is strongly dependent on Ma 1 and Ma2, and also on the ratio of thermal diffusivities, Q~(see figs. 6 and 7). crystal growth thesignificantly convective flowFor in the lower liquid purposes, layer can be
z +
—
+
—
reduced using an appropriate choice of the parameters and this result is corroborated by the numerical calculations.
+
12
2Qa
2 z [3Qa 2 —
+
8(2K1
—
QaK2)]
12(2K
+
1
—
QaK2)]
24 (A.2) In the lower layer (2) the velocity is
Qa
Acknowledgements
(12(Z) We acknowledge Dr. P. Bontoux, J.P. Fontaine, E.C.A. acknowledges support from the MEC in J.N. Koster and T. Doi for fruitful discussions. Spain funding a visit toat the Center for GravityforFluid Mechanics University of Low Colorado at Boulder. Also E.C.A. and R.L.S. would like to acknowledge support from the Microgravity Science and Applications Division of NASA. Support from the CNRS (GDR, MFN), the CNES and from the Conseil Regional of PACA (Intern. Exchange Program) in France is also acknowledged.
=
U”
x
1
Q~1
{(
1
+
+
4/3Q~
I
+
—
4 \ 3QM.)
~2~
—6
rL
—
—
—
1 Qa
+121
1 —
Q
2K2 1 —)I 1]
+K2+
Z 1 1
Qp~
2
/ 3K1
2K2
+ +
-~- [~-
-
3
4~ + 1--————K ~ Qa 2)]}~ /2K1
~
—
—~-~--
—~-~--
“1
)j
Qa 2 —+12
(A.3)
The solution of the system (4)—(12) is U*
and the temperature is
L1 =
10 Q~
/ K
Appendix
1(Z)
—
Q~+4
T2(x,Z)=~+ AU*QaH~
X{(Q~+4)Z~+~[2+2Q,~+Qa
( Z5
2
+4(2K1 QaK2)]Z +4[Q~+2Qa+8(2Ki~QaK
/
1
x2Q~4
1+4/3Q~ \ z4 1
/ K1
2K2
—
2)JZ +
~[3Qa —2 + 12(2K1
—
QaK2)l},
(A.1)
10
\Qa
\1
346
E. Crespo delArco et al.
z3 +
+
1
—
—
12
2 +
3
3K
—
—
Q~
Qa
2 +
—
/ Steady ther,nocapil!ary flows in two-layer liquid system
12
4
1
2K2 +
—
Qa
2K1 —
K2
Q~a ,
(A.4) with K1
=
Ra1
—
Ma1/Ra1, K2 = gp~a1(z.IT/L) H~
—
Ma2/Ra2,
=
/-‘~iXi
gp~a2(~iT/L)H~ =
‘
l.L2X2
[2] A.A. Nepomnyashchy and lB. Simanovsky, Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza 3 (1984) 175. [3] A.K. Sen and S.H. Davis, J. Fluid Mech. 121 (1982) 163. [4] A. Zebib, G.M. Homsy and E. Meiburg, Phys. Fluids 28 (1985) 3467. [5] H. Ben Hadid and B. Roux, J. Fluid Mech. 221 (1990) 77. [6] D. Rivas, Phys. Fluids A 3 (1991) 280. [7] J. Metzger and D. Schwabe, Physicochem. Hydrodyn. 10 (1988) 263. [8] D. Villers and J.K. Flatten, Appl. Sci. Res. 45 (1988) 145. [9] D. Villers, PhD Thesis, Université de Mons (1989). [101 D. Villers and J.K. Platten, AppI. Sci. Res. 47(1990)177. [11] M.S. Engelman, FIDAP Users Manual (FDI, Evanston, IL, 1986). [12] E. Crespo del Arco, G.P. Extremet and R.L. Sani, Advan. Space Res. 11(7) (1991) 29. [13] J.P. Fontaine, R.L. Sani and J.N. Koster, in: Proc. AIAA, paper 920689, 30th Aerospace Science Meeting, Reno, NV, 1992.
U * = gp~H~a1A/6,i1.
[14] T. Doi and J.N. Koster, private communication, 1989. [15] J.P. Pulicani, E. Crespo del Arco, A. Randriamampianina, P. Bontoux and R. Feyret, Intern. J. Numer. Methods Fluids 10 (1990) 481.
References
[161 E. Crespo del Arco, A. Randriamampianina and P. Bon-
[1] S. Rasenat, F.H. Busse and I. Rehberg, J. Fluid Mech. 199 (1989) 519.
toux, in: Proc. Conf. on Synergetics, Order and Chaos,
Ed. M.G. Velarde (World Scientific, Singapore, 1988) p. 244.