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Phase transition picture of the soret-driven convective instability in a two-component liquid layer heated from below
Volume 72A, number 2
PHYSICS LETTERS
25 June 1979
PHASE TRANSITION PICTURE OF THE SORET-DRIVEN CONVECTIVE INSTABILITY IN A TWO-COMPONENT LIQUID LAYER HEATED FROM BELOW M.G. VELARDE Departamento de F(sica de Ruidor, Universidad Autónoma de Madrid, Cantoblanco (Madrid), Spain
and J.C. ANTORANZ Departamento de Fisica Fundamental, UNED, Ciudad Universitaria, Madrid-3, Spain Received 24 April 1979
Using a five-mode truncation of the nonlinear equations and the Landau picture, we give predictions concerning firstorder transitions, tricritical point and critical slowing down at continuous (second-order) transitions in the two-component Bénard problem.
There exists a large body of literature on the role of the Soret effect in the stability of a horizontal twocomponent fluid layer heated from below (see, for instance, refs. [1—3]). According to the known facts and theoretical predictions, for dynamically (stress) free, conducting and permeable boundaries the minimal scheme for a relevant description of the problem in a two-dimensional geometry reduces to the following five-mode approximation [4,5]: = A1 sin irlcx sin irz (la) 2kT = A ~ 3 cos irkx sin irz + A2 sin 2irz, (ib) 2JçN = A ~ 5 cos nkx sin irz + A4 sin 2irz, (ic) ,
where ~,1i, T and N1, respectively, denote the stream function (velocity potential), the temperature and the mass fraction of the heavier component. k is the wavenumber at the onset of convective instability [1—3]. x and z denote horizontal and vertical coordinates, respectively. The amplitudes A~(i = 1, 5) are unknowns to be determined. Their time evolution is governed by the qontinuity, Navier—Stokes, Fourier and Fick equations which here lead to the following ordinary differential equations: .,
—A
1
2)2]P,
3A =
[Rk(A3 +SA5) + ir
1(1 + k
(2a)
A2
2A =
~jA1A3 4ir —
2
(2b)
,
2)A + +k 3 +A1A2, 2rD(A A4 = A1A512 + 41r 2 — A4),
(2c) (2d)
=
2)rD(A ~2(1 + k 3 A5) A1A4 (2e) where the dot over the A1 denotes time derivation. The following parameters have been introduced: P is the Prandtl number, R is the Rayleigh number, which is a dimensionless measure of the temperature gradient. S is a dimensionless measure of the Soret separation. r~is the ratio of mass to heat diffusivity (Lewis num-
A5
=
—
—
—
ber). Further details about all these parameters are not needed here and may be found in ref. [3]. It is to be noted that eqs. (1) and (2) extend the Lorenz model [5] to the two-component Bénard problem [3,4]. The steady solutions of eqs. (2) are obtained by setting time derivatives to zero. Besides the trivial solution, A1 = 0 (motionless state), we have the solutions of the quartic equation 4/4—R] 16/9 A~++A~[27(1 + r~)ir R(1 + S + S/rD)]!3 = 0. (3) 64~4r~ [27ir4/4 —
These five A 1-solutions may be considered as the extrema of the following Landau potential [5]: 123
Volume 72A, number 2
PHYSICS LETTERS
25 June 1979
R R(1)
/
R>O
S
680
/
/
/
/ /
670
/ / / 660
R2
——
T LogiSi 650
-~
~
-~
-
-1
Fig. 1. Stability diagram showing the line of subcritical bifurcations, R(1). This line emerges from the tricritical point T. To the left of T, the heavy line corresponds to second-order, R(2) (normal bifurcation), transitions, as predicted by linear theory [3]. To the right of T, the broken line also belongs to the neutral stability curve of continuous transitions. The dotted line corresponds to overstabiity, as predicted by linear theory [3]. Notice that we refer to the quadrant of positive R (heating from below), and negative S (denser component migrating to the warmer plate). This is the only quadrant where overstability is predicted by linear analysis [3] and the only quadrant where our five-mode approximation gives subcritical (first-order) bifurcations. ~1max
S =1.
R
R 100
200
Fig. 2a
124
Volume 72A, number 2
PHYSICS LETTERS
25 June 1979
—15
-10
S:-1O4
-5
6625
665
-5
-10
-15
(b)
/
S= -iO-3 mover
R~1~
p
~“—_-_.
(C)
Fig. 2. Behavior of the “order parameter”, maximum value ofA 1 plotted, for various values of the Soret separation, S. (a) Continuous transition along R(2) for a value of S chosen here in the continuation of R(2) at positive values of the Soret separation (denser component migrating to the colder boundary). (b) Inverted bifurcation atS = —1O~,a value located on R(1) in the Immediate neighborhood of the point where linear theory predicts overstability. (C) Inverted bifurcation atS = io-~,on R(1). For a detailed explanation see main text.
125
Volume 72A, number 2
PHYSICS LETTERS
+ A4 [27(1 + r2)ir4/4 R] 4/9 4) = A6/6 1 1 D + 32A2ir4r2 [277r4/4 R(l + S + Sir )] /3 (4) 1 D D From eqs. (3) or (4) we predict the existence of a line of first-order transitions in the region where overstabiity is predicted by linear analysis [3]. As this firstorder (subcritical instability) line lies below the over• stability branch (see fig. 1) the fluid layer should depart from the motionless state either via a hard excitation of finite amplitude or via a transient oscillatory mode whose amplitude exponentially grows until the finite amplitude steady convection is reached. The transition should exhibit metastability and hysteretic phenomena. These predictions, already conjectured in ref. [1], agree fairly well with the experimental findings reported in ref. [2] and do not necessarily disagree with the observations given in ref. [6]. It seems difficult to accept, however, that linear oscillations can be steadily maintained in a Bénard experiment as they ought to belong to a metastable branch. On the other hand, the kind of (inverted) bifurcation [5] just described has been observed in a rather similar convective experiment on homeotropic nematic liquids heated from below. The anisotropy (or director) of the nematic liquid plays the role of “impurity” (or extra component) in the usual Bénard—Rayleigh problem [7]. The first-order line meets the second-order one at a tricritical point, where, by analogy with metamagnets and mixtures [8] and the laser [9] three “coexisting” phases become identical. Fig. 2 provides the behavior of the “order parameter”, A 1, for various values of the Soret separation. Fig. 2a corresponds to the usual continuous (second-order) transition, where, according to linear theory there is exchange of stabilities. Fig. 2b corresponds to an inverted bifurcation, or first-order transition just below the value of the Soret separation for which overstability is predicted by linear theory. Fig. 2c also corresponds to an inverted bifurcation at S = —10—3, and thus to a finite amplitude steady convection where according to linear theory overstabiity would be the dominant mode. Notice that the critical Rayleigh number for overstabiity, R”, is smaller than the cntical Rayleigh number thetheory onset of steady convection, R~,according to for linear [3]. On the other hand the onset of subcritical instability, Rth is below R°”. —
—
.
25 June 1979
Lastly, and for the record, a straightforward extension to the two-component Bénard problem of the analysis given in ref. [5] yields for rigid, heat conductrng and impermeable boundanes, the corresponding critical slowing down along the branch of continuous transitions, i.e., where exchange of stabilities takes place (fig. 3). Fairly nice agreement is found with the results reported in ref. [101. On the other hand, the five-mode approximation gives qualitatively the same picture. In conclusion, careful experiments are to be encouraged on the two-component Bénard problem where a rather varied phenomenology is predicted. If, for instance, the heavier component is magnetically tagged, a visualization of the surfaces of coexisting phases should be possible. Further details of this work wifi be reported elsewhere.
Log ~
11
‘N
,
-
126
\~
~
-~
6
-
3
2 -7
-6
-5
-4
-3
-2
-t
Log C
Fig. 3. Critical slowing down at continuous transitions in a two-component Bénard problem, 3fors; S a layer thickness To = 15of 0.6 cm, and a heat diffusional time of 320 s.= r10~ = r0(S)e~.Values: 5~ = ~, ~s. The = 21 Prandtl 5; S = i0~,~ i0 = ~ e (R — Rc)/Rc, number =isP where Rc is the critical value for the onset of steady convection, and lies on R(2).
Volume 72A, number 2
PHYSICS LETTERS
References [1] M.G. Velarde and R.S. Schechter, Phys. Fluids 15 (1972) 1707. [2) D.T.J. Hurle and E. Jakeman, J. Fluid. Mech. 47 (1971) 667; Adv. Chem. Phys. 32 (1975) 277. [3] R.S. Schechter, M.G. Velarde and J.K. Platten, Adv. Chem. Phys. 26 (1974) 265. [4] G. Veronis, 3. Marine Res. 23 (1965) 1. [5] C. Normand, Y. Pomeau and M.G. Velarde, Rev. Mod. Phys. 49(1977)581;
[6] [7] [8] [9] [10]
25 June 1979
See also: E. Wesfreid, Y. Pomeau, M. Dubois, C. Normand and P. Berg~,J. de Phys. 39 (1978) 725. J.K. Platten and G. Chavepeyer, Phys. Lett. 40A (1972) 287; J. Fluid Mech. 60 (1973) 305. E. Guyon, P. Pleranski and J. Salan, J. Fluid Mech., to be published. See, for Instance, J.M. Kincaid and E.G.D. Cohen, Phys. Rep. 22C (1975) 57. J.F. Scott et aL, Opt. Commun. 15 (1975) 13, 343. M. Giglio and A. Vendramini, Phys. Rev. Lett. 39 (1977) 1014.
127
PHYSICS LETTERS
25 June 1979
PHASE TRANSITION PICTURE OF THE SORET-DRIVEN CONVECTIVE INSTABILITY IN A TWO-COMPONENT LIQUID LAYER HEATED FROM BELOW M.G. VELARDE Departamento de F(sica de Ruidor, Universidad Autónoma de Madrid, Cantoblanco (Madrid), Spain
and J.C. ANTORANZ Departamento de Fisica Fundamental, UNED, Ciudad Universitaria, Madrid-3, Spain Received 24 April 1979
Using a five-mode truncation of the nonlinear equations and the Landau picture, we give predictions concerning firstorder transitions, tricritical point and critical slowing down at continuous (second-order) transitions in the two-component Bénard problem.
There exists a large body of literature on the role of the Soret effect in the stability of a horizontal twocomponent fluid layer heated from below (see, for instance, refs. [1—3]). According to the known facts and theoretical predictions, for dynamically (stress) free, conducting and permeable boundaries the minimal scheme for a relevant description of the problem in a two-dimensional geometry reduces to the following five-mode approximation [4,5]: = A1 sin irlcx sin irz (la) 2kT = A ~ 3 cos irkx sin irz + A2 sin 2irz, (ib) 2JçN = A ~ 5 cos nkx sin irz + A4 sin 2irz, (ic) ,
where ~,1i, T and N1, respectively, denote the stream function (velocity potential), the temperature and the mass fraction of the heavier component. k is the wavenumber at the onset of convective instability [1—3]. x and z denote horizontal and vertical coordinates, respectively. The amplitudes A~(i = 1, 5) are unknowns to be determined. Their time evolution is governed by the qontinuity, Navier—Stokes, Fourier and Fick equations which here lead to the following ordinary differential equations: .,
—A
1
2)2]P,
3A =
[Rk(A3 +SA5) + ir
1(1 + k
(2a)
A2
2A =
~jA1A3 4ir —
2
(2b)
,
2)A + +k 3 +A1A2, 2rD(A A4 = A1A512 + 41r 2 — A4),
(2c) (2d)
=
2)rD(A ~2(1 + k 3 A5) A1A4 (2e) where the dot over the A1 denotes time derivation. The following parameters have been introduced: P is the Prandtl number, R is the Rayleigh number, which is a dimensionless measure of the temperature gradient. S is a dimensionless measure of the Soret separation. r~is the ratio of mass to heat diffusivity (Lewis num-
A5
=
—
—
—
ber). Further details about all these parameters are not needed here and may be found in ref. [3]. It is to be noted that eqs. (1) and (2) extend the Lorenz model [5] to the two-component Bénard problem [3,4]. The steady solutions of eqs. (2) are obtained by setting time derivatives to zero. Besides the trivial solution, A1 = 0 (motionless state), we have the solutions of the quartic equation 4/4—R] 16/9 A~++A~[27(1 + r~)ir R(1 + S + S/rD)]!3 = 0. (3) 64~4r~ [27ir4/4 —
These five A 1-solutions may be considered as the extrema of the following Landau potential [5]: 123
Volume 72A, number 2
PHYSICS LETTERS
25 June 1979
R R(1)
/
R>O
S
680
/
/
/
/ /
670
/ / / 660
R2
——
T LogiSi 650
-~
~
-~
-
-1
Fig. 1. Stability diagram showing the line of subcritical bifurcations, R(1). This line emerges from the tricritical point T. To the left of T, the heavy line corresponds to second-order, R(2) (normal bifurcation), transitions, as predicted by linear theory [3]. To the right of T, the broken line also belongs to the neutral stability curve of continuous transitions. The dotted line corresponds to overstabiity, as predicted by linear theory [3]. Notice that we refer to the quadrant of positive R (heating from below), and negative S (denser component migrating to the warmer plate). This is the only quadrant where overstability is predicted by linear analysis [3] and the only quadrant where our five-mode approximation gives subcritical (first-order) bifurcations. ~1max
S =1.
R
R 100
200
Fig. 2a
124
Volume 72A, number 2
PHYSICS LETTERS
25 June 1979
—15
-10
S:-1O4
-5
6625
665
-5
-10
-15
(b)
/
S= -iO-3 mover
R~1~
p
~“—_-_.
(C)
Fig. 2. Behavior of the “order parameter”, maximum value ofA 1 plotted, for various values of the Soret separation, S. (a) Continuous transition along R(2) for a value of S chosen here in the continuation of R(2) at positive values of the Soret separation (denser component migrating to the colder boundary). (b) Inverted bifurcation atS = —1O~,a value located on R(1) in the Immediate neighborhood of the point where linear theory predicts overstability. (C) Inverted bifurcation atS = io-~,on R(1). For a detailed explanation see main text.
125
Volume 72A, number 2
PHYSICS LETTERS
+ A4 [27(1 + r2)ir4/4 R] 4/9 4) = A6/6 1 1 D + 32A2ir4r2 [277r4/4 R(l + S + Sir )] /3 (4) 1 D D From eqs. (3) or (4) we predict the existence of a line of first-order transitions in the region where overstabiity is predicted by linear analysis [3]. As this firstorder (subcritical instability) line lies below the over• stability branch (see fig. 1) the fluid layer should depart from the motionless state either via a hard excitation of finite amplitude or via a transient oscillatory mode whose amplitude exponentially grows until the finite amplitude steady convection is reached. The transition should exhibit metastability and hysteretic phenomena. These predictions, already conjectured in ref. [1], agree fairly well with the experimental findings reported in ref. [2] and do not necessarily disagree with the observations given in ref. [6]. It seems difficult to accept, however, that linear oscillations can be steadily maintained in a Bénard experiment as they ought to belong to a metastable branch. On the other hand, the kind of (inverted) bifurcation [5] just described has been observed in a rather similar convective experiment on homeotropic nematic liquids heated from below. The anisotropy (or director) of the nematic liquid plays the role of “impurity” (or extra component) in the usual Bénard—Rayleigh problem [7]. The first-order line meets the second-order one at a tricritical point, where, by analogy with metamagnets and mixtures [8] and the laser [9] three “coexisting” phases become identical. Fig. 2 provides the behavior of the “order parameter”, A 1, for various values of the Soret separation. Fig. 2a corresponds to the usual continuous (second-order) transition, where, according to linear theory there is exchange of stabilities. Fig. 2b corresponds to an inverted bifurcation, or first-order transition just below the value of the Soret separation for which overstability is predicted by linear theory. Fig. 2c also corresponds to an inverted bifurcation at S = —10—3, and thus to a finite amplitude steady convection where according to linear theory overstabiity would be the dominant mode. Notice that the critical Rayleigh number for overstabiity, R”, is smaller than the cntical Rayleigh number thetheory onset of steady convection, R~,according to for linear [3]. On the other hand the onset of subcritical instability, Rth is below R°”. —
—
.
25 June 1979
Lastly, and for the record, a straightforward extension to the two-component Bénard problem of the analysis given in ref. [5] yields for rigid, heat conductrng and impermeable boundanes, the corresponding critical slowing down along the branch of continuous transitions, i.e., where exchange of stabilities takes place (fig. 3). Fairly nice agreement is found with the results reported in ref. [101. On the other hand, the five-mode approximation gives qualitatively the same picture. In conclusion, careful experiments are to be encouraged on the two-component Bénard problem where a rather varied phenomenology is predicted. If, for instance, the heavier component is magnetically tagged, a visualization of the surfaces of coexisting phases should be possible. Further details of this work wifi be reported elsewhere.
Log ~
11
‘N
,
-
126
\~
~
-~
6
-
3
2 -7
-6
-5
-4
-3
-2
-t
Log C
Fig. 3. Critical slowing down at continuous transitions in a two-component Bénard problem, 3fors; S a layer thickness To = 15of 0.6 cm, and a heat diffusional time of 320 s.= r10~ = r0(S)e~.Values: 5~ = ~, ~s. The = 21 Prandtl 5; S = i0~,~ i0 = ~ e (R — Rc)/Rc, number =isP where Rc is the critical value for the onset of steady convection, and lies on R(2).
Volume 72A, number 2
PHYSICS LETTERS
References [1] M.G. Velarde and R.S. Schechter, Phys. Fluids 15 (1972) 1707. [2) D.T.J. Hurle and E. Jakeman, J. Fluid. Mech. 47 (1971) 667; Adv. Chem. Phys. 32 (1975) 277. [3] R.S. Schechter, M.G. Velarde and J.K. Platten, Adv. Chem. Phys. 26 (1974) 265. [4] G. Veronis, 3. Marine Res. 23 (1965) 1. [5] C. Normand, Y. Pomeau and M.G. Velarde, Rev. Mod. Phys. 49(1977)581;
[6] [7] [8] [9] [10]
25 June 1979
See also: E. Wesfreid, Y. Pomeau, M. Dubois, C. Normand and P. Berg~,J. de Phys. 39 (1978) 725. J.K. Platten and G. Chavepeyer, Phys. Lett. 40A (1972) 287; J. Fluid Mech. 60 (1973) 305. E. Guyon, P. Pleranski and J. Salan, J. Fluid Mech., to be published. See, for Instance, J.M. Kincaid and E.G.D. Cohen, Phys. Rep. 22C (1975) 57. J.F. Scott et aL, Opt. Commun. 15 (1975) 13, 343. M. Giglio and A. Vendramini, Phys. Rev. Lett. 39 (1977) 1014.
127