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Asymptotic Study of Rayleigh–Bénard
Convection under Time Periodic Heating in Hele–Shaw Cell
Article in Physica Scripta · January 2006
DOI: 10.1238/Physica.Regular.071a00395
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Asymptotic Study of Rayleigh–Bénard Convection under Time Periodic Heating in Hele–Shaw Cell
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Physica Scripta. Vol. 71, 395–401, 2005
Asymptotic Study of Rayleigh-B ´
enard Convection under Time Periodic Heating in Hele-Shaw Cell
S. Aniss1, M. Belhaq1, M. Souhar2and M. G. Velarde3
1Faculty of Sciences A¨ın chock, Laboratory of Mechanics, BP 5366 Maˆarif, Casablanca, Maroc
2Lemta-Ensem, UMR 7563, 2 avenue de la Forˆet de Haye, BP 160, Vandoeuvre 54504, France
3Instituto Pluridisciplinar, UCM, Paseo Juan XXIII, 28040 Madrid, Spain
Received April 24, 2003; accepted in revised form July 15, 2004
pacsnumber: 47.27.Te Abstract
The convective instability of a horizontal Hele-Shaw liquid layer subject to a time- varying gradient of temperature is investigated. The stationary component of the temperature gradient is considered either different or equal to zero. A dimensional analysis and an asymptotic study lead to two flow regimes depending on the order of magnitude of the Prandtl number:Pr=O(1) or Pr1 andPr=O(2).
The aspect ratio of the cell,, is considered smaller than unity. This analysis closely follows a previous work by Anisset al. [Phys. Fluids12, 262 (2000)]
in which a periodic gravitational modulation was considered in terms of the two flow regimes. The aim of the present paper is to examine the effects of temperature oscillations on the onset of convective instability for these two asymptotic cases.
We have shown that for the first regime, modulation of temperature has no effect on the convective threshold. In contrast, the second regime presents a competition between the harmonic and subharmonic modes at the onset of convection.
1. Introduction
Several publications have been devoted to describe the role of the modulation of the temperature imposed on the horizontal planes of a fluid layer. The classical problem of a B´enard fluid layer was discussed by Normandet al. [2] and Koschmieder [3].
Venezian [4] considered a steady vertical temperature gradient augmented with a harmonically time-dependent small amplitude component. For free-free horizontal boundaries, it was shown that according to linear stability theory, the modulation of the temperature at the boundary may stabilize or destabilize the initially motionless state. Roppoet al. [5] completed Venezian’s linear study by performing a weakly non-linear analysis. A similar problem was earlier considered by Gershuni and Zhukhovitskii [6] letting temperature obey a periodically varying rectangular pulse. Subsequently, Rosenblat and Herbert [7] provided an asymptotic solution in the case of low modulation frequency with arbitrary modulation amplitude. Yih and Li [8] used a Galerkin method to investigate the stability when the gradient of temperature is symmetrical relative to the middle plane. Recently, Bhadauria and Bathia [9] investigated the effects of different temperature profiles imposed at the upper and lower boundaries.
After expressing these profiles in Fourier series, they examined stabilizing and destabilizing effects in the case of a modulation in phase and that out of phase. For a review, see Gershuni and Zhukhovitskii [10].
Another type of modulation, also analyzed, is that of a gravitational field which can be realized by a vertically oscillating horizontal liquid layer. Gresho and Sani [11] studied the influence of gravitational modulation on the convective threshold of a stable and an unstable motionless state. By means of a first order Galerkin method, they reduced the governing linear system to the Mathieu equation. Biringen and Peltier [12] extended to
1E-mail: saniss@hotmail.com
three dimensions the linear two-dimensional problem of Gresho and Sani [11]. Clever et al. [13] investigated the effects of gravitational modulation on a two-dimensional steady, monotonic convective threshold and constructed non-linear solutions using a Galerkin method. They also treated the case of three-dimensional oscillatory convection under gravity modulation [14].
Gershuniet al. [15] investigated the surface tension gradient (Marangoni) driven convective instability that results from par- ametric excitation of a horizontal liquid layer from the air side.
They first considered a harmonically driven heat flux imposed normal to the undeformable level open surface of the liquid in the absence of buoyancy. It was shown that for high Prandtl number liquids, the most dangerous modes are harmonics, while for low Prandtl number liquids, there is a competition between harmonic and subharmonic modes. These authors also considered the role of the thickness of the layer, the effect of Newton thermal condition on the free surface and the superposition of buoyancy and surface tension gradients [16]. Later on, Gershuniet al. [17]
took into account the deformability of the free surface and found that two different types of disturbances are dangerous for the instability of the fluid equilibrium: convective disturbances and surface disturbances. For certain values of the Galileo number, the “surface” mode of instability is more dangerous than the
“convective” one.
Recently, Aniss et al. [1] studied the influence of the gravitational modulation on the instability threshold of a horizontal Hele-Shaw layer submitted to a temperature gradient acting along or against buoyancy. It was shown that when the Prandtl number is of the order of unity or much higher, the oscillations of the cell have no effect on stability. It is only for low Prandtl number liquids that parametric instability can be induced. In the present work, we study the case of modulation of the temperature imposed on both horizontal plates. From physical point of view, our motivation is based on the fact that the effect of gravitational modulation is uniform over the entire volume of the liquid layer. However, when the temperature at the boundaries is oscillating, the equilibrium temperature gradient depends on time and on coordinates; the effects of temperature modulation is concentrated mainly in a boundary layer whose thickness decreases with increasing frequency (temperature skin effect). At low modulation frequencies, this distinction disappears and both types of modulation are equivalent. In this work, two cases are considered. In the first case, the stationary component of the applied temperature gradient is different from zero and corresponds to an unstable equilibrium configuration. In the second one, the stationary component of the temperature gradient is set to zero and corresponds to a stable configuration.
Furthermore, using the same analysis followed by Aniss et al.
C Physica Scripta 2005 Physica Scripta 71
396 S. Aniss, M. Belhaq, M. Souhar, M. G. Velarde
[1], we examine the two different linear stability problems. The first corresponds toPr=O(1) orPr1 while the second one corresponds toPr=O(2), where designates the aspect ratio of the cell. It turns out that for the first problem, modulation of temperature cannot generate convective parametric instabilities.
In the second problem, the governing system of equations is reduced to the damped Mathieu equation. Here a competition between harmonic and subharmonic modes occurs.
2. Non vanishing stationary component of the applied temperature gradient
2.1. Formulation of the problem
Consider a Newtonian liquid confined in a horizontal Hele-Shaw cell of infinite extent along thex∗direction. The thermal insulating vertical walls are located aty∗ = ±e2. In dimensional form, the temperatures at the horizontal boundaries are
T∗=
Th+Tscos(∗t∗), z∗=0 Tc+Tscos(∗t∗), z∗=d
where Th> Tc. The variables {z∗,t∗,T∗} are scaled by {d,d2/,Th−Tc}, whereis the thermal diffusivity of the liquid.
The base state corresponds to a motionless state in which the dimensionless temperature, T =(T∗−Tc)/(Th−Tc), satisfies the diffusion equation
*T
*t = *2T
*z2 (1)
subject to the conditions T =
1+acos(t), z=0 acos(t), z=1
. (2)
The quantities a=Ts/(Th−Tc) and=∗d2/ are respect- ively the amplitude and dimensionless frequency of modulation.
The solution of equations (1) and (2) is
T =1−z+B(z,t) (3)
with
B(z,t)=aRe
exp(it)sinh(z)+sinh((1−z)) sinh()
,
(4) =(i)12 =(1+i)
2
12 .
Let us consider the cases= −1 and=0. The first case corresponds to a temperature modulation applied at the lower and upper boundaries of the fluid layer, while in the other case the temperature modulation is applied only at the bottom of the liquid layer.
Under these assumptions, the linear system describing the evolution of disturbances upon the base state in the Boussinesq approximation is
div(V)=0, (5)
*V
*t∗ = −∇p∗+V+gT∗k, (6)
*T∗
*t∗ +w∗*T∗
*z∗ =T∗. (7)
Here, the dimensional quantitiesV,p∗andT∗are, respectively, the velocity, pressure, and temperature fields,k the unit vector upward, the density, the coefficient of thermal expansion
andthe dynamic viscosity. Moreover, on the vertical walls, the boundary conditions are: V =0 and **yT∗ =0 aty∗= ±e2. The boundary conditions on the horizontal walls will be discussed in the next section.
To introduce the aspect ratio, =e/d, as a smallness perturbation parameter, we follow the dimensional analysis and scales introduced by Aniss et al. [18] in which time is scaled by d2/, the coordinates (x∗,y∗,z∗) by (d,e,d), the velocity field V(u∗,v∗,w∗) by (/d,e/d2,/d) and the pressure and temperature are scaled, respectively, by /e2 and (Th−Tc).
Consequently, the system of equations (5)–(7) becomes
*u
*x +*v
*y +*w
*z =0, (8)
2Pr−1*u
*t = −*p
*x +22u+*2u
*y2, (9)
4Pr−1*v
*t = −*p
*y +42 +2*2v
*y2, (10)
2Pr−1*w
*t = −*p
*z +22w+*2w
*y2 +Ra T, (11) 2*T
*t +2w*T
*z =22T+*2T
*y2 (12)
where2 =**x22+**z22,Ra=g(T2−T1)de2the Rayleigh number of the cell andPr= /the Prandtl number,
2.2. Asymptotic study and stability
In the Hele-Shaw geometry, the order of magnitude of the Prandtl numberPrin the system (8)–(12) must be estimated to be able to drop terms of order 2. Therefore, depending on the Prandtl number we have two different problems.
2.2.1.Case Pr=O(1) or Pr 1. In this case, a first appro- ximation is obtained from the system (8)–(12) by setting 2=0. Note that the term **Vt disappears. Let us denote by uo,vo,wo,po andTo the solution of such approximation. From equation (10), the pressure is independent of y. Also, from equation (12) and using the adiabatic condition, *T*yo =0 at y= ±12, the temperatureTois independent ofy. Thus, the system (8)–(12) becomes
*uo
*x +*vo
*y +*wo
*z =0, (13)
*2uo
*y2 =*po
*x , (14)
*2wo
*y2 = *po
*z −Ra To. (15)
Integrating equations (14) and (15) with respect toy, we obtainuo
andwo. Substitutinguoandwointo the continuity equation (13) and integrating,vois determined. The solutionvowhich satisfies the boundary conditionsvo=0 aty= ±12isvo =0.
Thus, to the first order (2=0), equations (13)–(15) cannot be coupled to the energy equation (12). This case corresponds to a pseudo-singular perturbation. Therefore, the energy equation (12) is exploited at the order2by using the expansion
w=wo+2w1, T =To+2T1. (16a,b)
Asymptotic Study of Rayleigh-B´enard Convection under Time Periodic Heating in Hele-Shaw Cell 397 Inserting expressions (16) into equation (12) and keeping only
terms to order2, one obtains
*To
*t +wo*T
*z =2To+*2T1
*y2 (17)
whereT1must also satisfy the adiabatic condition on the vertical walls:*T*y1 =0 aty= ±12. In contrast to the original system (8)–
(12) which needs six boundary conditions at the horizontal walls, the first order system (13)–(15) and (17), corresponding to the Heel-Shaw geometry, requires only the relevant four boundary conditions corresponding to the free-free case: wo=To atz= 0, 1.
To perform a stability analysis, we seek the solution of the system (13)–(15) and (17) by means of a first order Galerkin method. We set
uo=g(t)(y2−14) cos(z) exp(iqx), (18a) wo= −iqg(t)(y2−14) sin(z) exp(iqx), (18b)
To =iqf(t) sin(z) exp(iqx). (18c)
We denote byf(t) and g(t) the amplitudes of the temperature and velocity field, respectively. Using the above assumptions, the system (13)–(15) and (17) is reduced to the differential equations:
df dt − h
12[Ra−Ro+4aRa(Acos(t)−Bsin(t))]f =0, (19) g(t)= r
1+r Ra
2 f(t)
with h=1+mm, m= q22 and Ro=122 (m+1)m 2 is the Rayleigh number corresponding to the marginal stability of the unmodu- lated case, [19]. The coefficientsAandBare given by
A= 43(1−)
164+2, B= (−1)
164+2. (20)
The solution of equation (19) is f(t)=foexp
h 12
(Ra−Ro)t+4aRa
(Asin(t) +Bcos(t))
(21) wherefo is an arbitrary constant depending on the initial con- ditions. We see from equation (21) that the onset of instability is given by Ra≥Ro. This criterion is the same as that of the unmodulated case given by Frick and Clever [19] whereRac= 482 andqc=. Nevertheless, at the onset of convection, the amplitudes of temperature and velocity are periodic and have the same frequency as the parametric excitation. These amplitudes are given by
f(t)=foexp 83a
(Asin(t)+Bcos(t))
.
Hence, we can conclude that if the liquid has a Prandtl number Pr=O(1) or Pr1, the modulation of the temperatures imposed on the horizontal plates cannot generate convective parametric instability. Indeed, the large friction at the vertical walls suppresses the inertial effects. This is also the case for a porous medium where the Darcy law defines the resistance force.
2.2.2.CasePr=O(2).In this case, the inertial term *V*t remains in equations (9) and (11) after performing the Hele-Shaw
approximation (2=0). Hence, settingPr=2Pr∗ withPr∗ = O(1), the system (8)–(12) becomes
*uo
*x +*vo
*y +*wo
*z =0, (22)
Pr∗−1*uo
*t = −*po
*x +*2uo
*y2 , (23)
Pr∗−1*wo
*t = −*po
*z +*2wo
*y2 +RaTo, (24)
*To
*t +wo*T
*z =2To+*2T1
*y2 (25)
wherepo(x,z,t) andTo(x,z,t). Again the energy equation (25) is obtained at the order2, in which the termT1satisfies the adiabatic condition on the vertical walls. The solution of system (22)–(25), can be sought in the form (18a, b, c). Using equation (22), we find thatvo =0 and then, after substituting expressions (18a, b, c) into equations (22)–(25) and averaging with respect toy, we obtain the differential equation
d2g dt2+2pdg
dt+hPr∗[Ro−Ra−4aRa(Acos(t)−Bsin(t))]g
=0. (26)
Here 2p=2[1+m+12Pr2∗] and the coefficientsAandBare given by equations (20). Now we shall analyze the stability of the motionless state using the amplitude equation for the velocity (26). The change of variablesg(t)=G(t) exp(−pt) and 2=t reduces equation (26) to a Mathieu equation in the form
d2G d2 + 4
2[C−4aRa(Acos(2)−Bsin(2))]G()=0 (27) withC= −hPr∗[Ra−RN], RN = −Ro[1−Pr∗M(m)]2
4Pr∗M(m) andM(m)
=2(1+m)12 . Using the Floquet theory, the general solution of the Mathieu equation (27) is
G()=P() exp() (28)
where is the Floquet exponent and P() is a -periodic or 2-periodic function. Then, the solutions of equation (26) are g(t)=P(t) exp
2 −p
t. (29)
We can see from equation (29) that the criterion for the onset of instability is
2 ≥p. (30)
Hereafter, we focus our attention on the marginal stability condition corresponding to periodic solutions of periodor 2 given by
2 =p. (31)
The solutions of equation (27) can be expressed in the form G=exp()+∞
−∞
anexp(2ni), (32a)
G2=exp()+∞
−∞
bnexp(2n+1)i. (32b)
C Physica Scripta 2005 Physica Scripta 71
398 S. Aniss, M. Belhaq, M. Souhar, M. G. Velarde
Expression (32a) corresponds to the harmonic solutions having the same frequency as the parametric excitation, while expression (32b) corresponds to the sub-harmonic ones. First, consider the harmonic solutions of equation (27). Substitution of expression (32a) into equation (27) leads to a homogeneous algebraic system.
A necessary and sufficient condition for the existence of nontrivial solutions is the vanishing of its characteristic determinant
(i)=
· · · ·
· · · ·
· 2 1 2 0 0 0 0 ·
· 0 1 1 1 0 0 0 ·
· 0 0 o 1 o 0 0 ·
· 0 0 0 −1 1 −1 0 ·
· 0 0 0 0 −2 1. −2 ·
· · · ·
· · · ·
=0 (33)
wheren= −4C+8aRa(A+iB)2(+2in)2 andnis the complex conjugate of n. As in [20], the condition (33) may be written in the following form
Ch()=1−2(0) sin2 √
C
(34) where(0) is defined from equation (33) for=0. The functions n() are actuallyn(0) and satisfyn(0)=−n(0).
Similarly, for subharmonic solutions, one obtains the corr- esponding characteristic equation
Ch()= −1+2(0) sin2 √
C
. (35)
Equations (34) and (35) can be formally written as R(,,Pr∗,a,,Ra,q)=0.
For the numerical analysis, we have used an alternative method to that considered by Gresho and Sani [11]. In order to investigate the marginal stability curvesRa(q), the characteristic equations (34) and (35) are solved for prescribed values of the Prandtl numbers Pr∗, the dimensionless frequency, and the modulation amplitude, a. For a fixed wave number qi, we calculate the Floquet exponent i using equation (31) and then we determine the corresponding value (Ra)i obeying the harmonic characteristic equation (34) or the subharmonic one (35). To visualize our results, we plot the critical Raleigh number, Rac, and wave number, qc, versus the modulation amplitude, a, or versus the dimensionless frequency, . Each of these critical curves represents the minimum of the two harmonic and subharmonic solution modes in terms of the critical Rayleigh number. Instability occurs at the lowest value of the critical Rayleigh number.
Figure 1 shows the variation of the critical Rayleigh number Rac with the modulation amplitudea, for Pr∗ =1 and=5.
Fora=0, the critical Rayleigh and wave numbers are those of the unmodulated case: Rac=482, qc=. Furthermore, one can see that in the first harmonic region, the critical Rayleigh numberRacincreases with the modulation amplitude,a, until a certain amplitude is reached at which the onset is in the form of sub-harmonic solutions. In the second harmonic region, the critical Rayleigh numberRacdecreases with increasingato reach values less than those of the unmodulated case. In contrast to the first harmonic region where we are always in the presence of a
Fig. 1.CaseTh=Tc,Pr∗=1 and=5.Racversus amplitudeafor= −1 (dashed line) and for=0 (solid line). H: harmonic solutions. SH: sub-harmonic solutions.
stabilizing effect, the second harmonic region gives rise to either a stabilizing or destabilizing effect. Note that the first harmonic region and the sub-harmonic one of the case = −1 are very small comparing to the case =0. Also, the second harmonic region in the first case is larger than that of the second case,i.e., in the second harmonic region of the case= −1, destabilization occurs for amplitude values lower than those of the corresponding region of the case=0.
The crossover amplitudes between the harmonic and the sub- harmonic solutions depend also on the dimensionless frequency . In Fig. 2, we present the variation of the critical Rayleigh number Rac versus the modulation amplitude, a, for = −1, Pr∗=1, =5 and =10. Note that the second harmonic region, for=10, is smaller than that corresponding to=5.
Accordingly, the amplitude values leading to destabilization in the case=5 are lower than those corresponding to=10.
Fig. 2. CaseTo=T1,Pr∗=1 and= −1.Racversus amplitudeafor=10 (solid line) and=5 (dashed line). H: harmonic solutions. SH: sub-harmonic solutions.
Asymptotic Study of Rayleigh-B´enard Convection under Time Periodic Heating in Hele-Shaw Cell 399
Fig. 3. CaseTh=Tc,Pr∗=1,a=1 and= −1.Racversus dimensionless frequency. H: harmonic solutions. SH: sub-harmonic solutions.
In Fig. 3, the evolution of the critical Rayleigh number versus is illustrated for= −1 anda=1. In the first region, the critical Rayleigh number, corresponding to the onset of the subharmonic solutions, increases with increasing frequency until a certain value is reached and then the onset is in the form of harmonic solutions.
In the harmonic region the critical Rayleigh numberRacdecreases with increasing to reach, as expected, an asymptotic value corresponding to the unmodulated caseRac=482. Therefore, at high frequencies, the modulation effect disappears. Indeed, the corresponding coefficientsAandBin equation (26) tend to zero whendiverges (equation (20)). This result agrees with previous works related to the case of a modulated liquid layer (Out of phase oscillations) of infinite extent in horizontal directions; see Gershuni & Zhukhovitskii [10] and Bhadauria & Bathia [9].
In Fig. 4, it is worth noting that for the parameter valuea=0.5, only harmonic solutions exist for the dependence of the critical
Fig. 4.CaseTh=Tc,a=0.5 and= −1.Racversus dimensionless frequency for different effective Prandtl numbersPr∗(harmonic solutions).
Fig. 5. CaseTh=Tc,RacversusPr∗fora=0.5,= −1 and=9 (Harmonic solutions).
Rayleigh number on the dimensionless frequency and effective Prandtl numbersPr∗. Note that the same asymptotic behaviour for largecan be observed. In Fig. 5, the evolution ofRacvesus the effective Prandtl numberPr∗ shows that beyond a certain value of Pr∗, the critical Rayleigh number tends, as expected, to the critical one of the unmodulated caseRac=482. Therefore, this result verifies that of the first asymptotic formulation,Pr=O(1) orPr1, and shows a transition between the two asymptotic problems. Similar result has been shown by Anisset al. [1] in the case of gravitational modulation.
3. Vanishing stationary component of the applied temperature gradient (Th=Tc)
In this case, the basic temperature T is still governed by the diffusion equation (1), except that Ts is now used as the temperature scale instead of (Th–Tc). Thus, we have
T =(T∗−Th)/Ts (36)
with the boundary temperatures maintained as in the previous case (Th=Tc)
T =
cos(t), z=0 cos(t), z=1
. (37)
The solution of equations (1) and (37) is
T =B(z,t) (38)
whereB(z,t) is given by equation (4).
The dimensionless system of equations governing stability is still (8)–(12) and the Rayleigh number of the cell,Ra, is now defined by
Ra= gTse2d
. (39)
We again discuss the two asymptotic cases depending on the order of magnitude of the Prandtl number.
C Physica Scripta 2005 Physica Scripta 71
400 S. Aniss, M. Belhaq, M. Souhar, M. G. Velarde 3.1. CasePr=O(1)orPr1
The governing system of equations resulting from the Hele-Shaw approximation is reduced to the differential equation
df dt − h
12[−Ro+4Ra(Acos(t)−Bsin(t))]f =0. (40) The coefficients of this equation are given in 2.2.2. From equation (40), it follows that the base state is always stable.
3.2. CasePr=O(2).
In this case, we obtain d2g
dt2 +2pdg
dt +hPr∗[Ro−4Ra(Acos(t)−Bsin(t))]g=0.
(41) The analysis of this equation follows that given in 2.2.2. for the case (Th =Tc). In the present configuration, there is one parameter less to consider,i.e., the parameteradoes not appear in equation (41). For fixed Prandtl number, it is possible to obtain a relationship between the critical Rayleigh numberRacand the corresponding wave numberqcwith the dimensionless frequency of the oscillating primary temperature field.
Figures 6 and 7 show the evolution of Rac and qc versus for the case = −1. In the first region, the onset is in the form of harmonic solutions and in the second region, the onset corresponds to the sub-harmonic solutions. Beyond =60, there are no solutions. Indeed at this value the corresponding coefficientsAandBin equation (41) tend to zero (see equation (20)). In Fig. 7, the critical wave number qc increases with for both the harmonic and subharmonic solutions, and this evolution gives rise to a jump related to the fact that we do not continue the harmonic curve and the subharmonic one beyond their intersection. In Fig. 8, we present the evolution of Rac
versus for the case =0. In this case, there is no solution after =40. We can conclude that an isothermal liquid layer
Fig. 6.CaseTh=Tc,= −1 andPr∗=1.Racversus dimensionless frequency . H: harmonic solutions. SH: sub-harmonic solutions.
Fig. 7.CaseTh=Tc,= −1 andPr∗=1.Racversus dimensionless frequency . H: harmonic solutions. SH: sub-harmonic solutions.
Fig. 8.CaseTh=Tc,=0 andPr∗ =1.Racversus. H: harmonic solutions.
SH: sub-harmonic solutions.
confined in a Hele-Shaw cell, which is generally stable, may be destabilized in the case Pr=O(2) by adding time-dependent components to the temperatures of the horizontal surfaces. In the case of a modulation applied at both the lower and upper plates (= −1), the destabilization occurs for values of the critical Rayleigh number lower than those of the case where modulation of temperature is applied only at the bottom boundary.
4. Conclusion
We have provided an asymptotic study of the influence of temperature modulation on the instability threshold of a B´enard layer in Hele-Shaw geometry. An appropriate choice of characteristic magnitudes related to the convection problem in the Hele-Shaw cell together with an asymptotic analysis lead to two different linear formulations. Each formulation corresponds to different order of magnitude of the Prandtl number. In the configuration where the stationary component of the temperature
Asymptotic Study of Rayleigh-B´enard Convection under Time Periodic Heating in Hele-Shaw Cell 401 gradient is different from zero and forPr=O(1) orPr1, the
temperature modulation has no appreciable role on the stability of the layer. Thus, we have the same criterion of stability, in terms of the critical Rayleigh numberRacand wave numberqc, as for the unmodulated case. Nevertheless, at the onset of convection, the amplitudes of the velocity field and temperature, obtained analytically, are periodic and have the same frequency as for parametric oscillations, i.e., only harmonic solutions exist and they are the most dangerous modes. IfPr=O(2), the governing system of equations can be reduced to the Mathieu equation with damping term. Floquet theory was applied to determine a simple instability criterion when competition between the harmonic and the sub-harmonic solutions exists. More precisely, the damping effect of the vertical walls can be weakened in the second asymptotic case and convective parametric instability can occur in liquids with low Prandtl numbers such as liquid metals.
In the configuration where the stationary component of the temperature gradient is set to zero and forPr=O(1) orPr1, the base state cannot be linearly destabilized. In the second asymptotic case, Pr=O(2), there is parametric convective instability and destabilization can be obtained with a modulation applied at both the upper and lower plates or with modulation applied at the lower plate only. In the former case the critical Rayleigh numbers are lowest.
Finally, due to the analogy between convection in Hele-Shaw cell and flow in a porous medium, for weak enough Rayleigh numbers, [21], [22], [23], [24], [25], our results are expected to be useful to assess the influence of temperature modulation on the onset of convective instability in a porous medium.
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Nomenclature
a=Ts/(Th−Tc): modulation amplitude of temperature d: height of the cell
e: distance between the vertical walls g: gravitational acceleration
k: unit vector upward p: dimensionless pressure p∗: dimensionlal pressure Pr: Prandtl number t: dimensionless time t∗: dimensional time
T: dimensionless temperature T∗: dimensional temperature T: dimensionless basic temperature V(u,v,w): velocity field
q: wave number
Ra= g(Th−vkTc)de2: Rayleigh number of the cell (x∗,y∗,z∗): dimensional coordinates
(x,y,z): dimensionless coordinates : coefficient of thermal expansion = de 1 the aspect-ratio of the cell ∗: dimensional frequency
=∗d2/: dimensionless frequency : density
: dynamic viscosity : kinematic viscosity : thermal diffusivity 2= *2
*x2 + *2
*z2
C Physica Scripta 2005 Physica Scripta 71
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