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An exact solution for the shape of a crystal growing in a forced flow
Journal of Crystal Growth 92 (1988) 97—100 North-Holland, Amsterdam
97
AN EXACT SOLUTION FOR THE SHAPE OF A CRYSTAL GROWING IN A FORCED FLOW M. BENAMAR and Ph. BOUISSOU Groupe de Physique des Solides de l’ENS, 24 Rue Lhomond, F-75231 Paris Cedex 05, France
and P. PELCE Laboratoire de Recherche en Combustion, Université de Provence, Centre de Saint-Jérôme, Rue Henri Poincaré, F-13397 Marseille Cedex 13, France Received 30 January 1987; manuscript received in final form 30 March 1988
We determine an exact solution for the shape of an isothermal needle crystal growing in a forced flow. The shape is a parabola (respectively a paraboloid) in a two-dimensional (respectively axisymmetric) model. For a given undercooling, the Peclet number related to the velocity of the needle crystal is a function of the Peclet number associated to the flow.
1. Introduction The problem of the shape of an isothermal needle crystal growing at constant velocity in an undercooled melt at rest was solved by Ivantsov [1]. We are interested now in the case of a crystal growing in a forced flow. Still, in this case an exact solution can be found when the flow is potential and thus when viscous effects are neglected. One investigates first the case of a two-dimensional needle crystal and then the case of an axisymmetric needle crystal. The model is the following. The crystal is growing along the z axis at constant velocity v. Far at infinity upstream, the flow is uniform in velocity U, in the opposite direction of the crystal motion. The liquid is undercooled at temperature T~.The interface is assumed at constant temperature 1~ the crystallisation temperature of the planar interface, so that surface tension effects are neglected. Melt and crystal densities are assumed equal. When the liquid is at rest (U = 0), a one-parameter family can be found either in the two-dimensional or in the a.xisymmetric case. The shape is a parabola or a paraboloid and when the undercool-
ing is fixed, only the Peclet number P~= pv/2D is determined. Here p is the tip radius of the crystal and D the thermal coefficient of diffusion. We show that when forced flow is present, and when viscous effects are neglected, the parabola or the paraboloid is still a solution if the relation between the undercooling, the Peclet number P~and the Peclet number associated to the flow P~ = pU/2D is satisfied.
2. The free boundary problem One writes the stationary free boundary problem in the frame where the crystal is at rest. In the model considered, the flow satisfies the Navier—Stokes equations:
(~.v ) ~ =
1 —
v
w = 0.
Vp
+
v Llw,
(1) (2)
Here, p is the density and v the kinematic viscosity of the fluid. The temperature satisfies the diffusion equation: w~VT= D LIT.
0022-0248/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
(3)
98
M. Benamar et a!.
/ Exact solution for shape of crystal growing inforcedflow
On the crystal—melt interface are satisfied the following boundary conditions (see, for instance, Glicksman et al. [2]): —
For the velocity field,
w~n
=
—
Pt
v cos
(4) ‘
and the no-slip condition, ~, sin 0. =
—
(4’)
—
For the temperature field:
T= 1’
(5)
and Qv. n
=
Dc~[(vT)~ (vT)~] n.
(6)
—
Here, p~and p~ are the crystal and melt densities, Q the latent heat released per unit volume of solid and c the specific heat per unit volume of solid; p n and t are respectively the unit normal and tangent vectors to the interface, and 0 the angle between the normal to the interface and the direction of propagation. The subscripts s and e’ refer to the solid and the liquid sides. Far at infinity in the liquid, .
w= —(U+v)z
and
whole free boundary problem. Nevertheless, in the limit of large Reynolds number Re pU/v and small Prandtl number Pr v/D, viscous effects can be neglected outside a very thin viscous boundary layer, whose thickness can be neglected =
=
in a first approximation. Then, the flow can be assumed potential and the boundary condition (4’) can be relaxed. Furthermore, we assume crystal and melt densities to be equal, which is in practice a good approximation, i.e. we assume that no secondary flow is generated by the crystal. Then, the boundary condition (4) is replaced by the condition wn= —v cos 0. One investigates first the two-dimensional case.
3. Two-dimensional case One works in parabolic coordinates:
Here p is the tip radius of the parabola s~i 1, z the growth direction and x the transverse direction. The equation for temperature in the liquid is: 1 ~T 1 ~T =
T=Tc,~.
In the absence of forced flow (U=0) and when crystal and melt densities are assumed to be equal, the fluid is at rest in the laboratory frame and the heat transfer problem is decoupled from the Navier—Stokes equation. In this case, Ivantsov [1] has shown that the problem admits a one-parameter family of exact solutions. The shape is a paraboloid and the Peclet number related to the velocity of the needle crystal P~is determined by the undercooling. It was shown recently by McFadden and Coriell [3] that still in absence of forced flow, but when crystal and melt densities are assumed different, the same family of shapes is a solution of the whole boundary problem, butInthe relation P~ and thefree undercooling is modified. this case, the flow is generated by the difference between crystal and melt densities and is potential. When forced flow is present (U 0), we did not find in general analytical solutions to the —
—
/
~7)
x=p~I~q z=~p(q—~).
1 D ~ (h2
—
—
~
\ + -~-(h1~
8
~) ~
where
~ i~ h1
+
~
—i--—
i~/
=
, ,
i~
+~
=
.
(9)
Here uE and u~are the components of the velocity field. In order to find a simple solution, one looks to the case where T depends on ~ only, so that the shape of the crystal is still the parabola i~ 1 (T(1) 7). This implies that =
=
8~j =
P
V~~:i::TV
______
~
.~-~-?1 + ~2
1 8T 2~/~ ~ (10)
and thus u~must be chosen as U,~=
—f(1~)/2%/~~,
(11)
M.Benamar et a!.
/ Exact solution for shape of crystal growing in forcedflow
where f is for the moment an unknown function of i~. Mass conservation (2) leads to: ~df
~
=
(12)
dlJ
This flow derives from a potential if ~(h1u~)
—
~(h2u~)
=0,
(13)
99
cooling. When forced flow is present, this Peclet number is a function of the undercooling and of the Peclet number Pf. The velocity of the crystal is still undetermined. It is clear from formula (19) that at the same undercooling, the Peclet number of the crystal is smaller when the fluid is at rest compared to the case where a forced flow is present. Of course this is because the advective transports help to remove the latent heat from the crystal, this leading to a larger crystal velocity. At large Peclet number Pf, the integral in relation (19) is dominated by the neighbourhood of i~= 1
i.e., if d /
df
\
(14)
and one obtains P~= g(L~)P~, exp(_x_g-~~)dx. 2~ where g satisfies (19’)
~
x
+c~
Far upstream (i~= + cc) u,~= —(U+ v), and on the parabola (~ = 1), one applies relation (4). This leads to
This corresponds to a law pv2/D
f( ~i)
4. The axisyminetric case
=
2( U + v )~
—
2U.
(15)
0
Then, the temperature satisfies d2T
dT/ 1
=
g(~)U.
The rotational parabolic coordinates are now:
P~
(16)
x=p~cos~,
y=p/~sin~, (20)
in the liquid and T = T~in the solid. From relation (6) it follows that
dr~
(17)
—P~-~—,
=
c~,
where 4~i is the polar angle. The equation for an axisymmetric temperature field is now: 1 u~8T + 1 u~3T -~
P~)—~-— =
—
P~- exp( P~
=
F a
1h2h3
~
xexp[—P~~+Pf(2%/~—fl)]. By integrating this relation between = + cc, one obtains =
i~=
1 and
+
I
~
aT\ (21)
hr~/2ij-~~, h2=~~-t_~, h3=p~. (22) Thus, if one looks at a temperature field depend-
—
~
a
8T\
(18)
P~exp(P~ Pf) +
J1
h
‘
where
b,,—
—
~—
pD
and thus, dT
~—
Pf2%/~ ~ —
d~, (19)
~
where 4 = (7 T~) c~/Qis the dimensionless undercooling. For Pf = 0 one finds again the Ivantsov formula, i.e. the Peclet number related to the velocity of the needle crystal is determined by the under-
ing only on Un
~,
one must have (23)
=
—
and thus by mass conservation u~=
\/‘~d(v’~f)/ dfl +~
(24)
100
M.Benamar et a!.
/ Exact solution for shape of crystal growing inforced flow
One has a potential flow if ~(h
—
Thus, in both the two-dimensional and the axisyrnmetric case, at given undercooling, only the
=0,
2U~)
(25)
or 0.
=
(26)
By using far at infinity upstream and relation (4), one deduces:
f(~)
(v
=
+
u)1/7
—
Un
=
—(v
U/h.
+
U)
(27)
Peclet number of the crystal is given and not its velocity. In the absence of forced flow, it has been shown that the anisotropic surface tension breaks the continuum of solutions into a discrete set of possible states (Meiron et al. Benamar and Pomeau [6] [4], andCaroli Barbieri et [5], al. [7]). Only the fastest one was found stable with respect to tip splitting modes (Kessler and Levine [8], and Bensimon et al. [9]). A similar study remains to be done in the presence of forced flow.
The equation for the temperature field is now: dT
2D d I dT \
from which one deduces:
(28)
Note added in proof After this work was accepted by the referee, we learned and Gillthat [10].a similar result was obtained by Dash
2:~
~:~i~T= d~ —Pa-- exp(PC+Pf)
xexp(—P~i~+Pf(logfl_I])),
(29)
by using relation (6). By integrating this relation between i~= 1 and ~ = + cc, one obtains: =
P~exp(P~+ Pf) +oc exp[—P~fl + P~(log~
—
~‘
(1986)507.
(30) This relation is similar to the two-dimensional case qualitatively and the same remarks hold. At large Peclet number P~,the integral is dominated by the neighbourhood of ~U,=where 1 andhone still has 2/D = h(4) satisfies the relation p v ±00
exp
(
x
—h
dx.
[1] G.P. Ivantsov, Doki. Akad. Nauk SSSR 58 (1947) 567. [2] M.E. Glicksman, S.R. Coriel and G.B. McFadden, Ann. Rev. Fluid Mech. 18 (1986) 307. [3] G.B. McFadden and SR. Coriell, J. Crystal Growth 74
di 1
x(
References
(30’)
[4] D. Meiron, Phys. Rev. A33 (1986) 2704. [5] B. Caroli, Caroli, B. Roulet and J.S. Langer, Phys. Rev. A33 (1986)C.442. [6] M. Benamar and Y. Pomeau, Europhys. Letters 2 (1986) 307. [7] A. Barbieri, D.C. Hong and J.S. Langer, Phys. Rev. A35 (1986) 1802. [8] D. Kessler and H. Levine, Phys. Rev. Letters 57 (1986) 3069. [9] D. Bensimon, P. Pelce and B.I. Shraiman, J. Physique 48 (1987) 2081. [10] S.K. Dash and W.N. Gill, J. Heat Mass Transfer (1984) 1345.
97
AN EXACT SOLUTION FOR THE SHAPE OF A CRYSTAL GROWING IN A FORCED FLOW M. BENAMAR and Ph. BOUISSOU Groupe de Physique des Solides de l’ENS, 24 Rue Lhomond, F-75231 Paris Cedex 05, France
and P. PELCE Laboratoire de Recherche en Combustion, Université de Provence, Centre de Saint-Jérôme, Rue Henri Poincaré, F-13397 Marseille Cedex 13, France Received 30 January 1987; manuscript received in final form 30 March 1988
We determine an exact solution for the shape of an isothermal needle crystal growing in a forced flow. The shape is a parabola (respectively a paraboloid) in a two-dimensional (respectively axisymmetric) model. For a given undercooling, the Peclet number related to the velocity of the needle crystal is a function of the Peclet number associated to the flow.
1. Introduction The problem of the shape of an isothermal needle crystal growing at constant velocity in an undercooled melt at rest was solved by Ivantsov [1]. We are interested now in the case of a crystal growing in a forced flow. Still, in this case an exact solution can be found when the flow is potential and thus when viscous effects are neglected. One investigates first the case of a two-dimensional needle crystal and then the case of an axisymmetric needle crystal. The model is the following. The crystal is growing along the z axis at constant velocity v. Far at infinity upstream, the flow is uniform in velocity U, in the opposite direction of the crystal motion. The liquid is undercooled at temperature T~.The interface is assumed at constant temperature 1~ the crystallisation temperature of the planar interface, so that surface tension effects are neglected. Melt and crystal densities are assumed equal. When the liquid is at rest (U = 0), a one-parameter family can be found either in the two-dimensional or in the a.xisymmetric case. The shape is a parabola or a paraboloid and when the undercool-
ing is fixed, only the Peclet number P~= pv/2D is determined. Here p is the tip radius of the crystal and D the thermal coefficient of diffusion. We show that when forced flow is present, and when viscous effects are neglected, the parabola or the paraboloid is still a solution if the relation between the undercooling, the Peclet number P~and the Peclet number associated to the flow P~ = pU/2D is satisfied.
2. The free boundary problem One writes the stationary free boundary problem in the frame where the crystal is at rest. In the model considered, the flow satisfies the Navier—Stokes equations:
(~.v ) ~ =
1 —
v
w = 0.
Vp
+
v Llw,
(1) (2)
Here, p is the density and v the kinematic viscosity of the fluid. The temperature satisfies the diffusion equation: w~VT= D LIT.
0022-0248/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
(3)
98
M. Benamar et a!.
/ Exact solution for shape of crystal growing inforcedflow
On the crystal—melt interface are satisfied the following boundary conditions (see, for instance, Glicksman et al. [2]): —
For the velocity field,
w~n
=
—
Pt
v cos
(4) ‘
and the no-slip condition, ~, sin 0. =
—
(4’)
—
For the temperature field:
T= 1’
(5)
and Qv. n
=
Dc~[(vT)~ (vT)~] n.
(6)
—
Here, p~and p~ are the crystal and melt densities, Q the latent heat released per unit volume of solid and c the specific heat per unit volume of solid; p n and t are respectively the unit normal and tangent vectors to the interface, and 0 the angle between the normal to the interface and the direction of propagation. The subscripts s and e’ refer to the solid and the liquid sides. Far at infinity in the liquid, .
w= —(U+v)z
and
whole free boundary problem. Nevertheless, in the limit of large Reynolds number Re pU/v and small Prandtl number Pr v/D, viscous effects can be neglected outside a very thin viscous boundary layer, whose thickness can be neglected =
=
in a first approximation. Then, the flow can be assumed potential and the boundary condition (4’) can be relaxed. Furthermore, we assume crystal and melt densities to be equal, which is in practice a good approximation, i.e. we assume that no secondary flow is generated by the crystal. Then, the boundary condition (4) is replaced by the condition wn= —v cos 0. One investigates first the two-dimensional case.
3. Two-dimensional case One works in parabolic coordinates:
Here p is the tip radius of the parabola s~i 1, z the growth direction and x the transverse direction. The equation for temperature in the liquid is: 1 ~T 1 ~T =
T=Tc,~.
In the absence of forced flow (U=0) and when crystal and melt densities are assumed to be equal, the fluid is at rest in the laboratory frame and the heat transfer problem is decoupled from the Navier—Stokes equation. In this case, Ivantsov [1] has shown that the problem admits a one-parameter family of exact solutions. The shape is a paraboloid and the Peclet number related to the velocity of the needle crystal P~is determined by the undercooling. It was shown recently by McFadden and Coriell [3] that still in absence of forced flow, but when crystal and melt densities are assumed different, the same family of shapes is a solution of the whole boundary problem, butInthe relation P~ and thefree undercooling is modified. this case, the flow is generated by the difference between crystal and melt densities and is potential. When forced flow is present (U 0), we did not find in general analytical solutions to the —
—
/
~7)
x=p~I~q z=~p(q—~).
1 D ~ (h2
—
—
~
\ + -~-(h1~
8
~) ~
where
~ i~ h1
+
~
—i--—
i~/
=
, ,
i~
+~
=
.
(9)
Here uE and u~are the components of the velocity field. In order to find a simple solution, one looks to the case where T depends on ~ only, so that the shape of the crystal is still the parabola i~ 1 (T(1) 7). This implies that =
=
8~j =
P
V~~:i::TV
______
~
.~-~-?1 + ~2
1 8T 2~/~ ~ (10)
and thus u~must be chosen as U,~=
—f(1~)/2%/~~,
(11)
M.Benamar et a!.
/ Exact solution for shape of crystal growing in forcedflow
where f is for the moment an unknown function of i~. Mass conservation (2) leads to: ~df
~
=
(12)
dlJ
This flow derives from a potential if ~(h1u~)
—
~(h2u~)
=0,
(13)
99
cooling. When forced flow is present, this Peclet number is a function of the undercooling and of the Peclet number Pf. The velocity of the crystal is still undetermined. It is clear from formula (19) that at the same undercooling, the Peclet number of the crystal is smaller when the fluid is at rest compared to the case where a forced flow is present. Of course this is because the advective transports help to remove the latent heat from the crystal, this leading to a larger crystal velocity. At large Peclet number Pf, the integral in relation (19) is dominated by the neighbourhood of i~= 1
i.e., if d /
df
\
(14)
and one obtains P~= g(L~)P~, exp(_x_g-~~)dx. 2~ where g satisfies (19’)
~
x
+c~
Far upstream (i~= + cc) u,~= —(U+ v), and on the parabola (~ = 1), one applies relation (4). This leads to
This corresponds to a law pv2/D
f( ~i)
4. The axisyminetric case
=
2( U + v )~
—
2U.
(15)
0
Then, the temperature satisfies d2T
dT/ 1
=
g(~)U.
The rotational parabolic coordinates are now:
P~
(16)
x=p~cos~,
y=p/~sin~, (20)
in the liquid and T = T~in the solid. From relation (6) it follows that
dr~
(17)
—P~-~—,
=
c~,
where 4~i is the polar angle. The equation for an axisymmetric temperature field is now: 1 u~8T + 1 u~3T -~
P~)—~-— =
—
P~- exp( P~
=
F a
1h2h3
~
xexp[—P~~+Pf(2%/~—fl)]. By integrating this relation between = + cc, one obtains =
i~=
1 and
+
I
~
aT\ (21)
hr~/2ij-~~, h2=~~-t_~, h3=p~. (22) Thus, if one looks at a temperature field depend-
—
~
a
8T\
(18)
P~exp(P~ Pf) +
J1
h
‘
where
b,,—
—
~—
pD
and thus, dT
~—
Pf2%/~ ~ —
d~, (19)
~
where 4 = (7 T~) c~/Qis the dimensionless undercooling. For Pf = 0 one finds again the Ivantsov formula, i.e. the Peclet number related to the velocity of the needle crystal is determined by the under-
ing only on Un
~,
one must have (23)
=
—
and thus by mass conservation u~=
\/‘~d(v’~f)/ dfl +~
(24)
100
M.Benamar et a!.
/ Exact solution for shape of crystal growing inforced flow
One has a potential flow if ~(h
—
Thus, in both the two-dimensional and the axisyrnmetric case, at given undercooling, only the
=0,
2U~)
(25)
or 0.
=
(26)
By using far at infinity upstream and relation (4), one deduces:
f(~)
(v
=
+
u)1/7
—
Un
=
—(v
U/h.
+
U)
(27)
Peclet number of the crystal is given and not its velocity. In the absence of forced flow, it has been shown that the anisotropic surface tension breaks the continuum of solutions into a discrete set of possible states (Meiron et al. Benamar and Pomeau [6] [4], andCaroli Barbieri et [5], al. [7]). Only the fastest one was found stable with respect to tip splitting modes (Kessler and Levine [8], and Bensimon et al. [9]). A similar study remains to be done in the presence of forced flow.
The equation for the temperature field is now: dT
2D d I dT \
from which one deduces:
(28)
Note added in proof After this work was accepted by the referee, we learned and Gillthat [10].a similar result was obtained by Dash
2:~
~:~i~T= d~ —Pa-- exp(PC+Pf)
xexp(—P~i~+Pf(logfl_I])),
(29)
by using relation (6). By integrating this relation between i~= 1 and ~ = + cc, one obtains: =
P~exp(P~+ Pf) +oc exp[—P~fl + P~(log~
—
~‘
(1986)507.
(30) This relation is similar to the two-dimensional case qualitatively and the same remarks hold. At large Peclet number P~,the integral is dominated by the neighbourhood of ~U,=where 1 andhone still has 2/D = h(4) satisfies the relation p v ±00
exp
(
x
—h
dx.
[1] G.P. Ivantsov, Doki. Akad. Nauk SSSR 58 (1947) 567. [2] M.E. Glicksman, S.R. Coriel and G.B. McFadden, Ann. Rev. Fluid Mech. 18 (1986) 307. [3] G.B. McFadden and SR. Coriell, J. Crystal Growth 74
di 1
x(
References
(30’)
[4] D. Meiron, Phys. Rev. A33 (1986) 2704. [5] B. Caroli, Caroli, B. Roulet and J.S. Langer, Phys. Rev. A33 (1986)C.442. [6] M. Benamar and Y. Pomeau, Europhys. Letters 2 (1986) 307. [7] A. Barbieri, D.C. Hong and J.S. Langer, Phys. Rev. A35 (1986) 1802. [8] D. Kessler and H. Levine, Phys. Rev. Letters 57 (1986) 3069. [9] D. Bensimon, P. Pelce and B.I. Shraiman, J. Physique 48 (1987) 2081. [10] S.K. Dash and W.N. Gill, J. Heat Mass Transfer (1984) 1345.