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Analogy between the convective instability in a two-component liquid layer heated from below and the laser with saturable absorber
Volume 77A, number 2,3
PHYSICS LETFERS
12 May 1980
ANALOGY BETWEEN THE CONVECTiVE INSTABILITY IN A TWO-COMPONENT LIQUID LAYER HEATED FROM BELOW AND THE LASER WITH SATURABLE ABSORBER Vittorio DEGIORGIO1 CISE, 20100 Milan, Italy
and Luigi A. LUGIATO Istituto di Fisica dell’Università di Milano, 20133Milan, Italy Received 8 February 1980 Revised manuscript received 18 March 1980
The convective instability in a two-component liquid layer is shown to be analogous to the laser with saturable absorber by applying a suitable transformation of variables to the hydrodynamic equations. The analogy is exploited for the discussion of the convective-instability steady state.
In the spirit of synergetics [1], it is tempting to look for analogies among different cooperative phenomena in physics and also outside physics. Apart from its interest per se, such an approach allows a mutual transfer of theoretical and experimental knowledge from one field to another. Typical examples of this kind of operation are the analogies between the Rayleigh—Benard instability and the laser, formulated in ref. [2], and between the Soret-driven instability and the laser, formulated in ref. [3]. In this paper we want to extend the results of refs. [2] and [3] showing the analogy between the convective instability in a two-component liquid layer heated from below [4] and the laser with saturable absorber [5]. The analogy fol. lows from a suitable transformation of variables in the Navier—Stokesequations and from the application of the single-mode approximation. The convective instability considered here arises in a two-component horizontal liquid layer heated from below when the sign of the Soret coefficient k~is such that the concentration gradient gives a stabilizing con-
tribution to the vertical density gradient, i.e. the molecules of the heavier component migrate downward (kT <0, following the usual sign convention). As a consequence, the threshold for convection becomes larger than one would predict for kT = 0, the instability shows hysteresis effects, and oscillations may occur during the evolution of the system [6,7]. The starting point of the computation is represented by the equations giving the mass, momentum and energy balance for the two-component liquid [3,7]. It is interesting to note that the cross term due to the Soret effect which appears in the Fick equation can be eliminated formally by a linear transformation of variables. By introducing the variable~’ c [SrD/(1 rD)] T, where the dimensionless variables c and T represent, respectively, the concentration of the heavier component and the temperature, S is a dimensionless measure of the Soret effect [7], and rD is the Lewis number, the conservation equations become —
aC
+
v,,
aC ~—
=
rDV 2” c
,
—
(1)
Researcher of the Italian National Research Council (CNR). 167
Volume 77A, number 2,3
PHYSICS LETTERS
~ =~ ‘3x1 x2 ~ a~T~ [(i 1_rD)(
+ -
}
~
T)
-
-
~
12 May 1980
t=B~T T, ~~T=b(r—~~T)—BT,
(6b) (6c)
C=B~~ rDC,
(6d)
—
-
= —bra IrS/(1 rD) + ‘~~]BC, (6e) where C = C + [STD/(rD 1)] Tand similarly for ~ and b = 41(1 + k2). Note that the two variables ~T and L~have a simple and direct physical meaning. —
—
—
—
+
a~+ PV2~. ~
(2)
~r V2T
(3)
=
/ ax
1
They represent the horizontally averaged temperature and concentration gradients, respectively, evaluated at the midplane z = 0.
(4)
Obviously we could have derived eqs. (6) by applying a linear transformation of variables to the equa-
where the dimensionless variables v1 and p are the ith component of the velocity and the pressure, d is the gap between the two horizontal plates confining the liquid layer, g is the acceleration of gravity, ~ the critical temperature difference between the plates, x the thermal diffusivity, P the Prandtl number, a the thermal expansion coefficent, T and ë are averaged quantities over the layer, t and x are normalized time and space coordinates, and A, = for i = 1, 2 (horizon, tal coordinates) and A1 = 1 for i = 3 (vertical axis). Clearly, the elimination procedure for the cross terms can be easily extended to the case in which the term due to the Dufour effects must be inserted into the energy conservation equation. According to ref. [4], the single-mode description for stress-free conducting and permeable boundaries leads to the following positions: (Sa)
tions derived by Velarde and Antoranz [4]. The present approach is, however, more general because the symmetrized equations (I) and (2) may be used for a broader spectrum of problems. Eqs. (6) are identical in form to the equations that rule the dynamics of a laser with saturable absorber (LSA) operating in a single-mode regime with perfect tuning between the atomic transition frequencies and the laser cavity. In fact, let us consider a resonant cavity containing N identical two-level atoms pumped to a positive inversion per atom a (amplifying or “active” atoms) and N identical two-level atoms which are not pumped enough, so that their inversion per atom ü is negative (absorbing or “passive” atoms). We indicate by ~ the square roott’D of the number inand thethe (rb)photon the polarization cavity, byof ~P(9) and active (passive) atom, respectiinversion the single vely. We denote by 7±(7~) and (~i)the transverse and longitudinal atomic relaxation time, by ,~the
(Sb)
damping constant which characterizes the cavity losses, and by G (G) the coupling constant between the active (passive) atoms and the electric field. Now eqs. (6) can
=
0,
ax~
6
~
v~(x,z, t) = B(t) cos irz cos irkx
,
T(x, z, t) = T— rz +Ar(t) sin 2irz + T(t) cos irz cos irkx
,
c(x, z, t) = ~ + rSz +A~(t)sin 2irz + C(t) cos irz cos irkx,
(Sc)
be read as the equations for a LSA [5] such that G = G, ~j/’Y.j. = ~/‘y~, provided one makes the following identifications: B 2G~/’y 1,T-+ 2~ft,~ -~
where r = L~T/i~TC, ~T is the temperature difference between the plates, and k is a dimensionless horizontal wavenumber. By substitution of eqs. (5) general equations (l)—(4), and by defining theinto newthe variables
~ —2 ~7’ft,&~,4~ ~ P K!’)’, 1 + rD S/(rD l) ~‘N/N, rD -÷ ‘y1/~’~, b -÷ 7i~/’/j’ rA/(rD 2N, andr therift, time is measured -÷ —&ft, where ~ = icy1/G in Uflits
= r 2lrAr and ~ = rS 21TA~,the following set of ordinary differential equations is obtained:
Now we can use the analysis performed in the case of the LSA [5] to rephrase the picture of the steady-
—
=
168
—
P { [1 SrD/(l —
—
TD)] T — C— B}
,
(6a)
—
.‘
—
—~
J-.
in ref. We recall that in our case Sstate < 0.presented Introducing the [4]. quantities
-
Volume 77A, number 2,3
PHYSICS LEUERS
ing the control parameter r one runs along the straight linem =nrD(l TD + ISIrD)/ISI ~nrD/!SI. This is different from the LSA, in which by varying the atomic density of the amplifyer or of the absorber one can control m and n independently. For a TD/ IS I one finds a first-order phase-transition-like picture with bistability and hysteresis. In this situation, the line m = nrD(l rD + 1 SI always crosses the curve m = a~{a+ n IS 1 rD)! + 2 [(a l)n] l/2}, whereas it crosses the line m n+ lonlyfor ISI
m 6
5
a b
/
/
~
III I
/
I
/
II
—
/
~ :
a
/
I
2
/
—
/
•
— /
•
/
————
/
/
C
— —
I
/
— - - —
- —
I~,
12 May 1980
—
——
(l—ISI)L~T—~C/rD=l,
-.
Fig. 1. Phase diagram for the convective instability in a twocomponent liquid layer heated from below [seeeqs. (7), (8)1.
The dotted line shows the overstability domain of the linear theory for P ~ 1. The curves are drawn for the unrealistic value r~= 0.5 in order to show the details for small n. The straight lines a, b and c show the curve m = nrD(1 rD + I SIrD)/ISI for different values of SI. —
a
=
l/r~ 1
n
= rS/rD(rD
~
m
,
=
r[l + rDS/(rD
—
1), (7)
—
1)
!=B2/b
one finds from eqs. (6) the following steady-state equation which is of course equivalent to the stationary equation given in ref. [4]: !h/2 [1 — m/(1 +!) + n/(1 + a)’)]
=
0
.
(8)
The case n = 0 (i.e. S = 0) corresponds to a pure Rayleigh—Benard experiment. The distribution of the steady-state solutions in the phase plane (m, n) is shown in fig. 1. In region I, which lies below the curyes m = a’4a + n I + 2 [(a 1)n] 112} and m = fl + 1, one has only the trivial motionless solution I = 0. In region II one has three distinct stationary solutions! = 0,! = I_, I = 4 with 0< I_ <4. The solution !_ is always unstable. In region II one has two stationary solutions! = I~and! = 0, the latter being unstable as we shall see later. From eqs. (7) one sees that by vary—
(9)
which represents the extension of the result L~T=
2
1
—
derived in ref. [31for the Rayleigh—Bénard case. The stability of the stationary solutions is as follows [5]: (i) The motionless solution is unstable for m > n + 1 and for m(1 + F—1)> (1 + rDP1)[rD(n + 1) + 1 + P—1(l + rD)] (dotted line in fig. 1). Hence, as ohserved in ref. [4], one can reach the convective branch only via a hard-mode excitation (using Haken’s nomenclature). (ii) The states !_ are always unstable. (iii) The stability conditions for the statesI~are involved and have been numerically analyzed only in particular ranges of the parameters [81. Let us finally recall the experiments on LSA which are relevant for the present discussion. Hysteresis effects were observed already a decade ago [9,10] However, the bistable region was carefully avoided at that time, so that experimental investigations of the bistable behavior have been performed much more recently (ref. [11] and especially ref. [12], in which the hysteresis cycle is nicely observed, together with the enhancement of fluctuations in the critical situation). The experiments reported so far on the convective instability in a two-component liquid layer heated from below [6,13,14] ,show qualitatively some of the features predicted from eqs. (6). A systematic experimental investigation is still lacking. .
169
Volume 77A, number 2,3
PHYSICS LETTERS
We thank R. Bonifacio and M. Giglio for stimulating discussion and helpful suggestions. This work was partially supported by CNR-CISE Contract 79.00764.02 and CNR Contract 78.0091.302.
12 May 1980
[71 R.S. Schechter, M.G. Velarde and J.K. Platten, Adv. Chem. Phys. 26 (1974) 265. [8] F. Mrugala and P. Peplowski, to be published.
[9] P.H. P.B. Schaefer and W.B. Barker, Appl. Phys. Lett.Lee, 36(1976)1135. [10] V.N. Lisitsyn and V.P. Chebotaev, JETP Lett. 7 (1968)
References [1] H. Haken, Synergetics (Springer, Berlin, 1977). [2] H. Haken, Phys. Lett. 53A (1975) 77. [3] V. Degiorgio, Phys. Rev. Lett. 41(1978)1293; Phys. Rev. A20 (1979) 2193. [4] M.G. Velarde and J.C. Antoranz, Phys. Lett. 72A (1979) 123. [5] L.A. Lugiato, P. Mandel, S.T. Dembinski and A. Kossakowski, Phys. Rev. A18 (1978) 238. [6] D.T.J. Hurle and E. Jakeman, I. Fluid. Mech. 47 (1971) 667.
170
i. [11] R. Salomaa and S. Stenhoim, Helsinki Univ. report TKKF-A305 (1977). [121 5. Ruscin and S.H. Bauer, Chem. Phys. Lett. 66 (1979) 100. [13] J.K. Platten and G. Chavepeyer, J. Fluid. Mech. 60 (1973) 305; D.R. Caldwell, I. Fluid. Mech. 64 (1974) 347. [14] M. Giglio, S. Musazzi, U. Perini and A, Vendramini, in: Light scattering in liquids and macromolecular solutions, eds. V. Degiorgio, M. Corti and M. Giglio (Plenum, New York, 1980), to be published.
PHYSICS LETFERS
12 May 1980
ANALOGY BETWEEN THE CONVECTiVE INSTABILITY IN A TWO-COMPONENT LIQUID LAYER HEATED FROM BELOW AND THE LASER WITH SATURABLE ABSORBER Vittorio DEGIORGIO1 CISE, 20100 Milan, Italy
and Luigi A. LUGIATO Istituto di Fisica dell’Università di Milano, 20133Milan, Italy Received 8 February 1980 Revised manuscript received 18 March 1980
The convective instability in a two-component liquid layer is shown to be analogous to the laser with saturable absorber by applying a suitable transformation of variables to the hydrodynamic equations. The analogy is exploited for the discussion of the convective-instability steady state.
In the spirit of synergetics [1], it is tempting to look for analogies among different cooperative phenomena in physics and also outside physics. Apart from its interest per se, such an approach allows a mutual transfer of theoretical and experimental knowledge from one field to another. Typical examples of this kind of operation are the analogies between the Rayleigh—Benard instability and the laser, formulated in ref. [2], and between the Soret-driven instability and the laser, formulated in ref. [3]. In this paper we want to extend the results of refs. [2] and [3] showing the analogy between the convective instability in a two-component liquid layer heated from below [4] and the laser with saturable absorber [5]. The analogy fol. lows from a suitable transformation of variables in the Navier—Stokesequations and from the application of the single-mode approximation. The convective instability considered here arises in a two-component horizontal liquid layer heated from below when the sign of the Soret coefficient k~is such that the concentration gradient gives a stabilizing con-
tribution to the vertical density gradient, i.e. the molecules of the heavier component migrate downward (kT <0, following the usual sign convention). As a consequence, the threshold for convection becomes larger than one would predict for kT = 0, the instability shows hysteresis effects, and oscillations may occur during the evolution of the system [6,7]. The starting point of the computation is represented by the equations giving the mass, momentum and energy balance for the two-component liquid [3,7]. It is interesting to note that the cross term due to the Soret effect which appears in the Fick equation can be eliminated formally by a linear transformation of variables. By introducing the variable~’ c [SrD/(1 rD)] T, where the dimensionless variables c and T represent, respectively, the concentration of the heavier component and the temperature, S is a dimensionless measure of the Soret effect [7], and rD is the Lewis number, the conservation equations become —
aC
+
v,,
aC ~—
=
rDV 2” c
,
—
(1)
Researcher of the Italian National Research Council (CNR). 167
Volume 77A, number 2,3
PHYSICS LETTERS
~ =~ ‘3x1 x2 ~ a~T~ [(i 1_rD)(
+ -
}
~
T)
-
-
~
12 May 1980
t=B~T T, ~~T=b(r—~~T)—BT,
(6b) (6c)
C=B~~ rDC,
(6d)
—
-
= —bra IrS/(1 rD) + ‘~~]BC, (6e) where C = C + [STD/(rD 1)] Tand similarly for ~ and b = 41(1 + k2). Note that the two variables ~T and L~have a simple and direct physical meaning. —
—
—
—
+
a~+ PV2~. ~
(2)
~r V2T
(3)
=
/ ax
1
They represent the horizontally averaged temperature and concentration gradients, respectively, evaluated at the midplane z = 0.
(4)
Obviously we could have derived eqs. (6) by applying a linear transformation of variables to the equa-
where the dimensionless variables v1 and p are the ith component of the velocity and the pressure, d is the gap between the two horizontal plates confining the liquid layer, g is the acceleration of gravity, ~ the critical temperature difference between the plates, x the thermal diffusivity, P the Prandtl number, a the thermal expansion coefficent, T and ë are averaged quantities over the layer, t and x are normalized time and space coordinates, and A, = for i = 1, 2 (horizon, tal coordinates) and A1 = 1 for i = 3 (vertical axis). Clearly, the elimination procedure for the cross terms can be easily extended to the case in which the term due to the Dufour effects must be inserted into the energy conservation equation. According to ref. [4], the single-mode description for stress-free conducting and permeable boundaries leads to the following positions: (Sa)
tions derived by Velarde and Antoranz [4]. The present approach is, however, more general because the symmetrized equations (I) and (2) may be used for a broader spectrum of problems. Eqs. (6) are identical in form to the equations that rule the dynamics of a laser with saturable absorber (LSA) operating in a single-mode regime with perfect tuning between the atomic transition frequencies and the laser cavity. In fact, let us consider a resonant cavity containing N identical two-level atoms pumped to a positive inversion per atom a (amplifying or “active” atoms) and N identical two-level atoms which are not pumped enough, so that their inversion per atom ü is negative (absorbing or “passive” atoms). We indicate by ~ the square roott’D of the number inand thethe (rb)photon the polarization cavity, byof ~P(9) and active (passive) atom, respectiinversion the single vely. We denote by 7±(7~) and (~i)the transverse and longitudinal atomic relaxation time, by ,~the
(Sb)
damping constant which characterizes the cavity losses, and by G (G) the coupling constant between the active (passive) atoms and the electric field. Now eqs. (6) can
=
0,
ax~
6
~
v~(x,z, t) = B(t) cos irz cos irkx
,
T(x, z, t) = T— rz +Ar(t) sin 2irz + T(t) cos irz cos irkx
,
c(x, z, t) = ~ + rSz +A~(t)sin 2irz + C(t) cos irz cos irkx,
(Sc)
be read as the equations for a LSA [5] such that G = G, ~j/’Y.j. = ~/‘y~, provided one makes the following identifications: B 2G~/’y 1,T-+ 2~ft,~ -~
where r = L~T/i~TC, ~T is the temperature difference between the plates, and k is a dimensionless horizontal wavenumber. By substitution of eqs. (5) general equations (l)—(4), and by defining theinto newthe variables
~ —2 ~7’ft,&~,4~ ~ P K!’)’, 1 + rD S/(rD l) ~‘N/N, rD -÷ ‘y1/~’~, b -÷ 7i~/’/j’ rA/(rD 2N, andr therift, time is measured -÷ —&ft, where ~ = icy1/G in Uflits
= r 2lrAr and ~ = rS 21TA~,the following set of ordinary differential equations is obtained:
Now we can use the analysis performed in the case of the LSA [5] to rephrase the picture of the steady-
—
=
168
—
P { [1 SrD/(l —
—
TD)] T — C— B}
,
(6a)
—
.‘
—
—~
J-.
in ref. We recall that in our case Sstate < 0.presented Introducing the [4]. quantities
-
Volume 77A, number 2,3
PHYSICS LEUERS
ing the control parameter r one runs along the straight linem =nrD(l TD + ISIrD)/ISI ~nrD/!SI. This is different from the LSA, in which by varying the atomic density of the amplifyer or of the absorber one can control m and n independently. For a
m 6
5
a b
/
/
~
III I
/
I
/
II
—
/
~ :
a
/
I
2
/
—
/
•
— /
•
/
————
/
/
C
— —
I
/
— - - —
- —
I~,
12 May 1980
—
——
(l—ISI)L~T—~C/rD=l,
-.
Fig. 1. Phase diagram for the convective instability in a twocomponent liquid layer heated from below [seeeqs. (7), (8)1.
The dotted line shows the overstability domain of the linear theory for P ~ 1. The curves are drawn for the unrealistic value r~= 0.5 in order to show the details for small n. The straight lines a, b and c show the curve m = nrD(1 rD + I SIrD)/ISI for different values of SI. —
a
=
l/r~ 1
n
= rS/rD(rD
~
m
,
=
r[l + rDS/(rD
—
1), (7)
—
1)
!=B2/b
one finds from eqs. (6) the following steady-state equation which is of course equivalent to the stationary equation given in ref. [4]: !h/2 [1 — m/(1 +!) + n/(1 + a)’)]
=
0
.
(8)
The case n = 0 (i.e. S = 0) corresponds to a pure Rayleigh—Benard experiment. The distribution of the steady-state solutions in the phase plane (m, n) is shown in fig. 1. In region I, which lies below the curyes m = a’4a + n I + 2 [(a 1)n] 112} and m = fl + 1, one has only the trivial motionless solution I = 0. In region II one has three distinct stationary solutions! = 0,! = I_, I = 4 with 0< I_ <4. The solution !_ is always unstable. In region II one has two stationary solutions! = I~and! = 0, the latter being unstable as we shall see later. From eqs. (7) one sees that by vary—
(9)
which represents the extension of the result L~T=
2
1
—
derived in ref. [31for the Rayleigh—Bénard case. The stability of the stationary solutions is as follows [5]: (i) The motionless solution is unstable for m > n + 1 and for m(1 + F—1)> (1 + rDP1)[rD(n + 1) + 1 + P—1(l + rD)] (dotted line in fig. 1). Hence, as ohserved in ref. [4], one can reach the convective branch only via a hard-mode excitation (using Haken’s nomenclature). (ii) The states !_ are always unstable. (iii) The stability conditions for the statesI~are involved and have been numerically analyzed only in particular ranges of the parameters [81. Let us finally recall the experiments on LSA which are relevant for the present discussion. Hysteresis effects were observed already a decade ago [9,10] However, the bistable region was carefully avoided at that time, so that experimental investigations of the bistable behavior have been performed much more recently (ref. [11] and especially ref. [12], in which the hysteresis cycle is nicely observed, together with the enhancement of fluctuations in the critical situation). The experiments reported so far on the convective instability in a two-component liquid layer heated from below [6,13,14] ,show qualitatively some of the features predicted from eqs. (6). A systematic experimental investigation is still lacking. .
169
Volume 77A, number 2,3
PHYSICS LETTERS
We thank R. Bonifacio and M. Giglio for stimulating discussion and helpful suggestions. This work was partially supported by CNR-CISE Contract 79.00764.02 and CNR Contract 78.0091.302.
12 May 1980
[71 R.S. Schechter, M.G. Velarde and J.K. Platten, Adv. Chem. Phys. 26 (1974) 265. [8] F. Mrugala and P. Peplowski, to be published.
[9] P.H. P.B. Schaefer and W.B. Barker, Appl. Phys. Lett.Lee, 36(1976)1135. [10] V.N. Lisitsyn and V.P. Chebotaev, JETP Lett. 7 (1968)
References [1] H. Haken, Synergetics (Springer, Berlin, 1977). [2] H. Haken, Phys. Lett. 53A (1975) 77. [3] V. Degiorgio, Phys. Rev. Lett. 41(1978)1293; Phys. Rev. A20 (1979) 2193. [4] M.G. Velarde and J.C. Antoranz, Phys. Lett. 72A (1979) 123. [5] L.A. Lugiato, P. Mandel, S.T. Dembinski and A. Kossakowski, Phys. Rev. A18 (1978) 238. [6] D.T.J. Hurle and E. Jakeman, I. Fluid. Mech. 47 (1971) 667.
170
i. [11] R. Salomaa and S. Stenhoim, Helsinki Univ. report TKKF-A305 (1977). [121 5. Ruscin and S.H. Bauer, Chem. Phys. Lett. 66 (1979) 100. [13] J.K. Platten and G. Chavepeyer, J. Fluid. Mech. 60 (1973) 305; D.R. Caldwell, I. Fluid. Mech. 64 (1974) 347. [14] M. Giglio, S. Musazzi, U. Perini and A, Vendramini, in: Light scattering in liquids and macromolecular solutions, eds. V. Degiorgio, M. Corti and M. Giglio (Plenum, New York, 1980), to be published.