Effect of phase thermal modulation without stationary temperature gradient on the threshold of convection

K. Souhar

¹

Laboratory of Energy Engineering, Materials and Systems, ENSA, Ibn Zohr University, Agadir 80000, Morocco e-mail: k.souhar@uiz.ac.ma

S. Aniss

Laboratory of Mechanics, Faculty of Sciences, University Hassan II Casablanca, 20100, Morocco e-mail: s.aniss@etude.univcasa.ma

Effect of Phase Thermal

Modulation Without Stationary Temperature Gradient on the Threshold of Convection

The convective instability of a horizontal fluid layer subject to a time varying gradient of temperature is investigated. The stationary component of the temperature gradient is considered equal to zero and the oscillating components imposed on the horizontal boundaries are in phase and with the same amplitude. The aim of the present paper is to examine the effect of this type of modulation on the onset of convective instability. We show that unlike the case where the equilibrium configuration is stable in the absence of modulation, we have instability when the temperature at the horizontal boundaries is modulated in phase. Also, we observe that in the limit of low and high dimensionless fre- quency of modulation,

x<

0.5 and

x>

140, the basic state tends to a stable equilibrium configuration and for an intermediate dimensionless frequency, the system is potentially unstable. The results obtained from analytical asymptotic study for low and high dimen- sionless frequency are in good agreement with the numerical ones.

[DOI: 10.1115/1.4033644]

Keywords: stability, convection, temperature modulation, asymptotic analysis

1 Introduction

The stability of equilibrium states, which vary periodically in time, is a thematic which was analyzed in several studies since the last century. This situation occurs, for example, when the bound- ary conditions are modulated periodically in time. This modula- tion has an important role in controlling the onset of instability, for instance, control of natural convection which was found to influence the compositional uniformity in the grown crystal. The first works on the modulation of boundary conditions have started with Donnelly [1] who has investigated experimentally the time- periodic flow between concentric cylinders when the outer cylin- der is at rest and the inner has a time periodic angular velocity.

Donnelly [1] has found that at low frequencies, the onset of insta- bility of the pulsed flow generated by modulation is delayed. The study of Donnelly [1] has been extended, by Venezian [2], to the problem of Rayleigh–B enard convection to show that the thresh- old of convection can be controlled when, in addition to a station- ary gradient of temperature, an additional perturbation is applied to the surface temperatures, varying sinusoidally in time. He has performed a linear stability analysis in the free–free case and for small amplitude temperature modulation to show that the stabiliz- ing effect has a maximum in the range of low frequencies. The case of low-frequency modulation and arbitrary modulation amplitude was studied by Rosenblat and Herbert [3] using an asymptotic solution. They have investigated the linear stability problem for the free–free surfaces and have used periodicity and amplitude criteria to calculate the critical Rayleigh number. How- ever, thermal modulation with rigid–rigid boundary conditions was studied by Rosenblat and Tanaka [4] using Galerkin method.

Yih and Li [5] have also used Galerkin method to investigate the stability when the modulated gradient of temperature is

symmetrical relative to the middle plane. They have shown that the disturbances (or convection cells) oscillate either synchro- nously or with half frequency and have given numerical results for the case of zero mean gradient of temperature. Finucane and Kelly [6] have conducted an experimental study on the alteration of the onset of convection due to the temperature modulation and have also developed an approximate nonlinear analysis for the free–free boundary conditions to estimate the two-dimensional convection amplitude. Note that a weakly nonlinear analysis of Venezian’s problem has been performed by Roppo et al. [7].

Later, Or and Kelly [8] have studied the onset of temporally modulated Rayleigh–B enard convection with zero mean gradient for cases of antisymmetric and asymmetric boundary tempera- tures. They have found numerically asymptotic results giving the critical Rayleigh and wave numbers in the case of small and high frequencies. Bhadauria and Bhatia [9] have investigated the effect of different temperature profiles, imposed at the upper and lower boundaries, on the onset of convection. They have examined sta- bilizing and destabilizing effects in the case of modulation in phase and that out of phase. Aniss et al. [10] have carried out an asymptotic study of Rayleigh–B enard convection under time peri- odic heating in Hele–Shaw cell and have shown that modulation has an effect on the threshold of convection only for small Prandtl number and in this case there is a competition between the har- monic and subharmonic. Malashetty and Basavaraja [11] have an- alyzed the effect of thermal modulation on the onset of double diffusive convection in a horizontal fluid layer. They have found that low frequency symmetric modulation is destabilizing whereas the asymmetric modulation and lower wall temperature modula- tion are stabilizing. After that, Oukada et al. [12] have studied the effect of sinusoidal temperature modulation, applied at the hori- zontal boundaries of a Maxwellian liquid layer, on the onset of convection. In this study, authors have used the Floquet theory and a technique of converting a boundary value problem to an ini- tial value problem to solve the linear system of equations corre- sponding to the onset of convection. It was observed that the viscoelastic nature of the liquid has a destabilizing effect

1Corresponding author.

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received December 24, 2015; final manuscript received May 12, 2016; published online June 7, 2016. Assoc. Editor:

Andrey Kuznetsov.

compared to the Newtonian nature and a destabilizing effect of the temperature modulation. Malashetty and Swamy [13] have examined the effect of thermal modulation on the onset of convec- tion in rotating fluid layer. They have established that the instabil- ity can be enhanced by the rotation at low frequency symmetric modulation and with moderate to high frequency lower wall tem- perature modulation, whereas the stability can be enhanced by the rotation in case of asymmetric modulation. Recently, Bhadauria [14] have studied the effect of temperature modulation on the onset of thermal convection in an electrically conducting fluid sat- urated porous medium. Using the linear stability analysis, he has found that the magnetic field has a stabilizing effect and the onset of convection can be advanced or delayed by proper tuning of the frequency of modulation. Bhadauria et al. [15] have carried out a weakly nonlinear analysis by considering the temperature modula- tion of both the boundaries. They have considered a temperature profile, which is similar to the variation of the atmospheric tem- perature near the earth’s surface during one complete day–night cycle. It turns out that the modulation produces a range of stable hexagons near the critical Rayleigh number. More recently, Bhadauria et al. [16] have studied the effect of temperature and gravity modulation on convection in a rotating viscous fluid layer by performing a weakly nonlinear stability analysis resulting in the Ginzburg–Landau amplitude equation. They have shown that the temperature modulation can be used as an external means to augment/diminish heat transport in a rotating system. Bhadauria and Kiran [17] have used a sinusoidal profile to modulate the tem- perature of the boundaries. They have employed the complex non- autonomous Ginzburg–Landau equation to investigate the effect of temperature modulation on a double diffusive oscillatory mode of convection in a horizontal viscoelastic fluid layer.

The aim of the present paper is to examine the effect of tem- perature modulation on the onset of convection in an incompres- sible Newtonian fluid layer submitted to the time-periodic temperature, T

m

þ T

s

cosðx

t

Þ at the horizontal boundaries.

Note that this modulation is in phase and has zero mean of tem- perature gradient. We recall that the work of Or and Kelly [8]

has focused on the out in phase modulation and that their as- ymptotic results have been obtained from the numerical ones.

The linear problem which is periodic in time can be solved in time using one of the three approaches, Floquet theory, averaged equations, frozen or profile. A comparison between the second and the third approaches was done by Nield and Kuznetsov in the study of the effect of pulsating throughflow on the onset of convection in a horizontal porous layer [18]. The two approaches lead to the same results. The first approach is to use the Fourier series or the exponent of Floquet [19,22]. In the present work, we use Fourier series and numerically solve in space the resulting system using a Runge–Kutta shooting method. After that, we determine analytically the asymptotic critical Rayleigh and wave numbers in the case of small and high dimensionless frequencies. The numerical results are com- pared to the asymptotical ones for these two limiting cases.

Also, the deviations between asymptotical and numerical results, in terms of critical Rayleigh and wave numbers, are given for the dimensionless frequencies

x<

0.07 and

x>

140.

2 Linear Formulation

Consider a Newtonian horizontal layer of an incompressible fluid confined between two horizontal surfaces of infinite extent in the x* and y* directions, (Fig. 1). The vertical boundaries of the fluid layer are located at z

¼ 0; d. The layer is isothermal at a temperature T

m

in the absence of modulation. Then, the modulated temperature applied at the horizontal surfaces is given in the dimensional form as T

m

þ T

s

cosðx

t

Þ. The variables {z

;

t

;

T

} are scaled by {d; d

²=j;

T

s

}, where

j

is the thermal diffusivity of the fluid. The basic state corresponds to a rest state in which the dimensionless temperature, T

b

¼ T

T

m=T_s

, satisfies the diffusion equation

@T_b

@t

¼

@²

T

b

@z²

and T

_b

¼ cos ð

xt

Þ at z ¼ 0; 1 (1) where

x

¼

x

d

²=j

is a dimensionless frequency. The long-term solution of Eq. (1) is

T

_b

¼ T

1

ðzÞcosðxtÞ þ T

2

ðzÞsinðxtÞ (2) with

T

1

ð Þ ¼ z cos ð

r

ð 1 z Þ Þcos h ð Þ þ

rz

cos h ð

r

ð 1 z Þ Þcos ð Þ

rz

cos h ð Þ þ

r

cos ð Þ

r

T

2

ð Þ ¼ z sin ð

r

ð 1 z Þ Þsin h ð Þ þ

rz

sin h ð

r

ð 1 z Þ Þsin ð Þ

rz

cos h ð Þ þ

r

cos ð Þ

r ;

where

r

¼ ffiffiffiffiffiffiffiffiffi

p

x=2

Remark that this equilibrium temperature does not have a station- ary component and is similar to the basic pulsed flow between two coaxial cylinders oscillating in phase with the same frequency around their axis [20]. According to the values of

x, we present,

in Fig. 2, the temperature evolution of the basic state, T

b

, given by Eq. (2) over half a period T ¼ 2

p=x. In Fig.

2, three different regimes can be observed:

At low frequencies, for example,

x

¼ 0.1, the temperature is entirely diffused throughout the thickness of the fluid layer.

In this situation, the equilibrium state is stable.

In the case of intermediate frequencies, for example,

x

¼ 18, the temperature profile is parabolic and presents two gradients, one is positive (destabilizing) and the other is neg- ative (stabilizing).

At high frequencies, we observe two temperature gradients which are developed near the horizontal surfaces. These gra- dients allow the stabilization or the destabilization near the boundaries but are less significant than the case of intermedi- ate frequencies and instability is difficult to develop.

These qualitative observations will be confirmed by the numeri- cal and asymptotic results in Secs. 3 and 4. Note that the choice of

x

¼ 0.1,

x¼

18, and

x

¼ 140 as representative values for low, in- termediate, and high frequencies, respectively, is inspired from numerical and asymptotic results in Secs. 3 and 4. The values

x

¼ 0.1 and

x

¼ 140 correspond to the frequencies when the devi- ation between the asymptotic and numerical results is of the order of 4%. The intermediate frequency corresponds to the most unsta- ble case and it is obtained numerically for both rigid–rigid and free–free cases.

The linear system corresponding to the perturbation of the basic state is given by the following Navier–Stokes equations in the Boussinesq approximation:

Fig. 1 Horizontal fluid layer of infinite extension in thex* and y* directions with a phase modulation of the temperature at the upper and lower surfaces

r:V ¼ 0 (3)

Pr

¹@V

@t

¼ rp þ

DV

þ RaTe

z

(4)

@T

@t

þ w

@T_b

@z

¼

DT

(5)

where

Vðu;

v; wÞ and

D

¼ ð@

²=@x²

Þ þ ð@

²=@y²

Þ þ ð@

²=@z²

Þ. The Prandtl number, Pr, is given by Pr ¼

=j. The Rayleigh number

is defined by Ra ¼

bgT_s

d

³=j, whereb

is the coefficient of ther- mal dilatation and is the kinematic viscosity. We note that the Rayleigh number is not defined by a temperature difference but by means of the amplitude modulation, T

s

, as in the study of Yih and Li [5] and that of Or and Kelly [8].

To perform a stability analysis, we seek solution of the system (3)–(5) in normal modes

w ¼ wðz;

~

tÞexpðiðq

_x

x þ q

_y

yÞÞ; T ¼ T

~

ðz; tÞexpðiðq

_x

x þ q

_y

yÞÞ We denote by q

x

and q

y

the wave numbers in the x and y direc- tions, respectively. After these assumptions, the system (3)–(5) becomes

Pr

¹@

@t

M

M w

~

¼ R

_aD2

T

~

(6)

@

@t

M

T

~

þ

@T_b

@z

w

~

¼ 0 (7)

Equation (6) represents the vertical component of the vorticity and M ¼ ð@

²=@z²

Þ q

²

with q

²

¼ q

²_x

þ q

²_y

. The system (6) and (7) is associated to the free–free or to the rigid–rigid boundary conditions at z ¼ 0, 1

T

~

¼ w

~

¼

@²

w

~

@z²

¼ 0 (8)

T

~

¼ w

~

¼

@

w

~

@z

¼ 0 (9)

3 Numerical Resolution

The partial differential Eqs (6) and (7) are solved by using the Floquet theory. According to this approach, the perturbed quanti- ties are expanded in the form

ð w;

~

T

~

Þ ¼ expðl x tÞ

^p¼þ1

X

p¼1

fw

p

ðzÞ; T

_p

ðzÞg expðipxtÞ

(10)

Here, we designate by

l

¼

l_r

þ il

_i

ði

²

¼ 1Þ the Floquet expo-

nent. In this study, we are interested only in the marginal stability

Fig. 2 Time evolution of the equilibrium temperature profile with respect to the vertical dimensionless coordinatez, for different values of the dimensionless frequency,x, over half a period 0£t£T=25p=x

curves (l

_r

¼ 0) corresponding to harmonic solutions (l

_i

¼ 0).

Hereafter, we write the basic temperature in complex form T

_b

¼ GðzÞexpðixtÞ þ G

ðzÞexpðixtÞ

with GðzÞ ¼ T

1

ðzÞ iT

2

ðzÞ, and G

^*

is the complex conjugate of G. Introducing expression (10) into the system (6) and (7), an infi- nite set of equations for the eigenfunctions w

p

and T

p

is obtained.

ixp

Pr D

²

q

²

D

²

q

²

w

_p

¼ q

²

RaT

_p

(11)

ðD

²

q

²

ixpÞT

_p

¼ DGw

p1

þ DG

w

pþ1

(12) where D denotes d=dz. Now the free–free and the rigid–rigid boundary conditions (8) and (9) are, respectively,

T

_p

¼ w

_p

¼ d

²

w

_p

dz

²

¼ 0 at z ¼ 0; 1 (13) T

_p

¼ w

_p

¼ dw

_p

dz ¼ 0 at z ¼ 0;1 (14) The total modes that are retained depend on the value of the dimensionless frequency number. The system (11) and (12) is transformed into a set of first-order ordinary differential equations for the quantities w

p

, Dw

p

, ðD

²

q

²

Þw

_p;

DðD

²

q

²

Þw

_p

, T

p,

and DT

p

ðP p PÞ. The solution of the obtained boundary value problem is sought as a superposition of linearly independent solu- tions following a method often used in stability problems [21].

Each independent solution which satisfies the boundary conditions at z ¼ 0 is constructed by a Runge–Kutta numerical schema of fourth order. A linear combination of these solutions satisfying the boundary conditions at the other extreme z ¼ 1 leads to a homogeneous algebraic system for the coefficients of the combi- nation. A necessary condition for the existence of nontrivial solu- tion is the vanishing of the determinant which defines a characteristic equation relating the dimensionless frequency

x,

the Prandtl number Pr, the wave number q, and the Rayleigh number Ra. The details of the numerical algorithm are given in the appendix for the rigid–rigid case.

In practice, convergence of the numerical results was assumed when the critical Rayleigh number Ra

c

, corresponding to P in the Fourier expansion was within 2% of the one corre- sponding to P þ 1. The convergence of the numerical solutions depends greatly on the truncated temporal Fourier series. For low frequencies, the Fourier mode P is taken up to 20 to get the prescribed accuracy. However, for high frequencies, a few

Fourier modes are needed and the value P ¼ 6 was sufficient to obtain the results.

An example of marginal stability curves corresponding to the evolution of the Rayleigh number, Ra, versus the wave number, q, for

x

¼ 112.5 and Pr¼ 7 is presented in Fig. 3, for the two cases of free–free (Fig. 3(a)) and rigid–rigid (Fig. 3(b)) boundary condi- tions. The oscillation of temperature gives rise to several discon- nected branches that represent stability boundary. The different branches correspond to the parametric resonance and similar behavior is found in the marginal stability curves of Mathieu equation. Figure 4 illustrates the evolution of the critical Rayleigh number, Ra

c

, versus 1/x, for both types of boundary conditions. It turns out, from this figure, that in both cases of boundaries, the equilibrium state is less unstable in the limit of low and high dimensionless frequencies, ði:e:;

x<

0:5 and

x>

50Þ, while destabilization is maximum for an intermediate dimensionless frequency,

x

¼ 18. In the free–free case, the minimal value of the critical Rayleigh number is Ra

c

¼ 3447.02 and corresponds to the critical wave number, q

c

¼ 3.03. However, in the rigid–rigid case, the critical values are: Ra

c

¼ 7474.49 and q

c

¼ 4.01. The evolution of the critical wave number, q

c

, versus 1/x is reported in Fig. 5.

We note that for free–free boundaries, the critical wave number, q

c

, has a minimum value for an intermediate dimensionless fre- quencies,

x

¼ 5, and it tends to the value q

c

¼ 3, for low dimen- sionless frequencies. For rigid–rigid boundaries the critical wave number, q

c

, has a minimum for

x

¼ 9, and it tends to the value, q

c

¼ 4, for low dimensionless frequencies.

Fig. 3 Marginal stability curves for x5112.5 and Pr57. (a) Free–free boundaries and (b) rigid–rigid boundaries.

Fig. 4 Numerical evolution and asymptotic behavior of the criti- cal Rayleigh number, Rac, versus 1/x for free–free and rigid–

rigid boundaries. ^ð1ÞRa_c58501:77r²²;^ð2ÞRa_c518670:99 r²²,

ð3ÞRac537776:99 r²⁴;^ð4ÞRac581542:86 r²⁴.

4 Asymptotic Analysis

4.1 Low Frequencies.

In the limit

r

1, the temperature of the base state T

b

can be simplified to the following asymptotic expansion:

T

_b

¼ 1 z

6 ð 1 2z

²

þ z

³

Þr

⁴

þ o ð Þ

r⁸

cos ð

xt

Þ þ ð 1 z Þzr

²

þ o ð Þ

r⁶

sin ð

xt

Þ (15)

Furthermore, we introduce the new time variable

s

¼

xt. The sys-

tem of Eqs. (6) and (7) becomes

x

Pr

@

@s

D

²

q

²

D

²

q

²

~

w ¼ q

²

Ra T

~

(16)

D

²

q

²

x@

@s

T

~

¼

r²

ð 1 2z Þsin ð Þ þ

s r⁴

6 ð 1 6z

²

þ 4z

³

Þ cos ð Þ

s

w

~

(17) Moreover, rather than treating the time-dependent problem, we shall use a quasistatic approximation and neglect the time deriva- tive

@=@s

in Eqs. (16) and (17). The time

s

appears as a parameter.

Provided that sinðsÞ 6¼ 0, terms of order equal or higher than

r⁴

can be neglected and the partial differential equations (16) and (17) are reduced to the ordinary differential system

ðD

²

q

²

Þ

²

w

^

¼ q

²

Ra

^

T

~

(18)

ðD

²

q

²

Þ T

~

¼ ð1 2zÞ w

^

(19) where Ra

^

¼

r²

Ra sinðsÞ and w

^

¼

r²

sinðsÞ w. The numerical

~

resolution of Eqs. (18) and (19) is classical [21] and gives the critical values:

ä

free–free case: Ra

^ _c

¼ 8501:77 and q

c

¼ 3.1.

ä

rigid–rigid case: Ra

^_c

¼ 18670:99 and q

c

¼ 4.05.

Thus, the more unstable critical Rayleigh number corresponds to sinðsÞ ¼ 1 (line (1) and line (3) in Fig. 4 for free–free and rigid–rigid boundaries, respectively).

free–free case

:

Ra

_c

¼ 8501:77

r²

(20)

rigid–rigid case

:

Ra

_c

¼ 18670:99

r²

(21)

The case where sinðsÞ ¼ 0, terms of order

r⁴

arise and the system to solve is

ðD

²

q

²

Þ

²

w ¼ q

²

Ra T

~

(22) D

²

q

²

T

~

¼ 1

6 ð 1 2z Þ ð 1 þ 2z 2z

²

Þ w (23) where w ¼

r⁴

w

~

and Ra ¼

r⁴

Ra. The resolution of this system leads to the critical values

ä

free–free case: Ra

_c

¼ 37776:99 and q

c

¼ 3

ä

rigid–rigid case: Ra

_c

¼ 81542:86 and q

_c

¼ 3.95

Thus, the critical Rayleigh number is given by (line (2) and line (4) in Fig. 4 for free–free and rigid–rigid boundaries, respectively) free–free case: Ra

_c

¼ 37776:99

r⁴

(24) rigid–rigid case

:

Ra

_c

¼ 81542:86

r⁴

(25) We illustrate in Fig. 4 for both boundary conditions, the numerical and asymptotical results for low dimensionless frequencies. These curves show that the critical Rayleigh number, Ra

c

, increases rather with a good agreement with the law Ra

_c

8501:77

r²

¼ 17003:54

x¹

(Fig. 4, line (1)) obtained from Eq. (20) in the free–free case. For the rigid–rigid case, Ra

c

increases with the law Ra

_c

18670:99

r²

¼ 37341:98

x¹

(Fig. 4, line (3)) obtained from Eq. (21). We note that, for

x<

0:07, the deviation between numerical and asymptotical critical Rayleigh numbers is less than 3.5% and it is less than 3.22% for the critical wave num- ber, see Fig. 5.

4.2 High Frequencies.

When

x

1, the choice of d as the length scale is not appropriate because the instability is expected to occur in the two boundary layers of size

d

¼ ffiffiffiffiffiffiffiffiffiffiffiffiffi

2j=x

p [23,24].

Therefore, it is convenient to make the change of variables

~

z

¼

x¹⁼²

z; q

~

¼

x¹⁼²

q and

s

¼

xt. A balance of the various terms in

Eq. (7) gives the relation T

x¹⁼²

w which is then reported in Eq. (6) where the right and left-hand sides have the same magni- tude if Ra

_c

¼ Ra

⁰x³⁼²

and q

_c

¼ q

⁰x¹⁼²

. The numerical results obtained in Figs. 3 and 4 prove this behavior and allow to deter- mine Ra

⁰

and q

⁰

in the both cases of boundaries. Then, we have

free–free case: Ra

c

15:6x

³⁼²

and q

c

0:373

x¹⁼²

(26)

rigid–rigid case

:

Ra

c

27:6x

³⁼²

and q

c

0:44

x¹⁼²

(27)

Fig. 5 Numerical evolution and asymptotic behavior of the critical wave number,qc, versus

1/xfor high frequencies. (a) Free–free boundaries,q_c50:373 ffiffiffiffi px

and (b) rigid–rigid bounda- ries,qc50:44pffiffiffiffix

.

In Fig. 6, we present the numerical and asymptotical results for high dimensionless frequencies. We show for the free–free boundary condition that the critical Rayleigh num- ber increases with the frequency,

x, according to the law

Ra

_c

¼ 15:6

x³⁼²

(Fig. 6) and the critical wave number increases with the frequency as q

_c

¼ 0:373

x¹⁼²

(Fig. 5(a)).

For the rigid–rigid case, the law is given by Ra

_c

¼ 27:6

x³⁼²

(Fig. 6) and the critical wave number increases with the fre- quency as q

_c

¼ 0:44

x¹⁼²

(Fig. 5(b)). For

x>

140, the devia- tion between numerical and asymptotical critical Rayleigh numbers is less than 4.2% and it is less than 1.4% for the critical wave number. Note that the laws that we have deter- mined analytically for symmetric modulation (modulation in phase) and high frequencies were found numerically [8] with the same asymptotic behavior for antisymmetric modulation (out in phase modulation) but with other constants appearing in the asymptotic laws.

5 Conclusion

In this paper, we have studied the effect of temperature modula- tion on the threshold of convection in a horizontal fluid layer of infinite extension. This modulation is in phase and the stationary component of the resulting temperature gradient is set to zero.

The Floquet theory approach predicts that a stable equilibrium configuration corresponding to a horizontal fluid layer with con- stant temperature, on its frontiers is susceptible to become unsta- ble if this temperature oscillates in time with the same frequency.

We have shown that for high frequencies the convective instabil- ity appears with very large values of Rayleigh number in two layers of skin temperature along the horizontal walls, while for low frequencies, the instability occurs in the entire volume of fluid and also with very large Rayleigh numbers. However, for an inter- mediate frequency,

x¼

18, corresponding to Ra

c

¼ 7474.49 for the rigid–rigid case and Ra

c

¼ 3447.02 for the free–free case, the fluid layer is potentially unstable. An asymptotic study was per- formed for the two limiting cases corresponding to low and high dimensionless frequencies. We have shown that when

x>

140 and

x<

0.07, asymptotic behaviors of the critical parameters (Rayleigh and wave numbers) are in good agreement with those obtained numerically.

Finally, it is seen from this work that the type of modulation studied here causes a parametric resonance and the onset of con- vection is hastened if the frequency of modulation corresponds to the intermediate value determined numerically. However, the

onset of convection tends to be delayed in the low- and high- frequency regime.

Nomenclature

d ¼ fluid thickness D ¼

@=@z

M ¼ ð@

²=@z²

Þ q

²

p ¼ pressure Pr ¼ Prandtl number

q ¼ wave number

q

_x

, q

_y

¼ wave number in the x and y directions Ra ¼ Rayleigh number (Ra ¼

bgT_s

d

³=j)

t ¼ dimensionless time T ¼ dimensionless temperature

T

_m

¼ temperature in the absence of modulation T

_s

¼ modulation amplitude

t* ¼ time T

^*

¼ temperature

u, v, w ¼ dimensionless velocity components

V

¼ velocity field

x, y, z ¼ dimensionless Cartesian coordinates x*, y*, z* ¼ Cartesian coordinates

Greek Symbols

b

¼ thermal coefficient of volume expansion

j

¼ thermal conductivity

l

¼ Floquet exponent ¼ kinematic viscosity

q

¼ density

r

¼ frequency parameter (r ¼ ffiffiffiffiffiffiffiffiffi p

x=2

)

x

¼ dimensionless frequency of modulation

x*

¼ frequency of modulation

Subscripts

b ¼ basic state c ¼ critical Appendix

The system of Eqs. (11) and (12) is transformed into a set of first-order ordinary differential equations for the quantities, w

_p;

Dw

_p;

ðD

²

q

²

Þw

_p;

ðD

²

q

²

ÞDw

_p;

T

_p

, and DT

_p

such as

U

1þp

¼ w

_p

U

2þp

¼ Dw

_p

U

3þp

¼ ðD

²

q

²

Þw

_p

U

4þp

¼ DðD

²

q

²

Þw

_p

U

5þp

¼ T

p

U

6þp

¼ DT

_p

8 >

> >

> >

> >

> >

> <

> >

> >

> >

> >

> >

:

(A1)

with p ¼ 6ðN þ kÞ where N is the truncation order or the Fourier series and N k N. Thus, the system (10) and (11) becomes

DU

1þp

¼ U

2þp

DU

2þp

¼ q

²

U

1þp

þ U

3þp

DU

3þp

¼ U

4þp

DU

_4þp

¼ q

²

R

_a

U

_5þp

þ q

²

þ ixp Pr

U

_3þp

DU

5þp

¼ U

6þp

DU

6þp

¼ DG U

5þp

þ DG

U

7þp

þ q

²

þ ixp U

5þp

8 >

> >

> >

> >

> >

> >

> <

> >

> >

> >

> >

> >

> >

:

(A2)

The associated rigid–rigid boundary conditions are

U

1þp

¼ U

2þp

¼ U

5þp

¼ 0 at z ¼ 0; 1 (A3)

Fig. 6 Numerical evolution and asymptotic behavior of the

critical Rayleigh number, Rac, versus 1/xin the limit of high fre- quencies in the both cases of boundaries, free–free and rigid–rigid

The solution of the obtained boundary value problem (A2) is sought as a superposition of 6ð2N þ 1Þ linearly independent solu- tions U

_iþp^j

, where i ¼ 1; 2; …6. Thus

U

_iþp

¼

^{j¼6ð2Nþ1Þ}

X

j¼1

C

^iþp_j

U

^j_iþp

(A4)

Imposing the following initials values

U

^j_iþp

ð0Þ ¼

d_ðiþpÞj

¼ 0 i þ p 6¼ j 1 i þ p ¼ j

(A5)

The boundary conditions (A3) with (A5) give

C

^1þp_1þp

¼ C

^2þp_2þp

¼ C

^5þp_5þp

¼ 0 (A6) Finally, the solution of the system (A2) is

U

iþp

¼

^l¼2N

X

l¼1

ðC

^iþp_3þ6l

U

^3þ6l_iþp

þ C

^iþp_4þ6l

U

^4þ6l_iþp

þ C

^iþp_6þ6l

U

_iþp^6þ6l

Þ (A7)

A linear combination of these solutions satisfying the boundary conditions at the other extreme z ¼ 1

U

1þp

ð1Þ ¼

^l¼2N

X

l¼1

ðC

^1þp_3þ6l

U

^3þ6l_1þp

ð1Þ þ C

^1þp_4þ6l

U

^4þ6l_1þp

ð1Þ þ C

^1þp_6þ6l

U

^6þ6l_1þp

ð1ÞÞ ¼ 0

U

_2þp

ð1Þ ¼

^l¼2N

X

l¼1

ðC

^2þp_3þ6l

U

^3þ6l_2þp

ð1Þ þ C

^2þp_4þ6l

U

^4þ6l_2þp

ð1Þ þ C

^2þp_6þ6l

U

^6þ6l_2þp

ð1ÞÞ ¼ 0

U

5þp

ð1Þ ¼

^l¼2N

X

l¼1

ðC

^5þp_3þ6l

U

^3þ6l_5þp

ð1Þ þ C

^5þp_4þ6l

U

^4þ6l_5þp

ð1Þ þ C

^5þp_6þ6l

U

^6þ6l_5þp

ð1ÞÞ ¼ 0 8 >

> >

> >

> >

> >

> >

> <

> >

> >

> >

> >

> >

> >

:

(A8)

leads to an homogeneous algebraic system for the coefficients of the combination. A necessary condition for the existence of nontrivial solution is the vanishing of the determinant

U

³₁

U

₁⁴

U

₁⁶ : : :

U

^3þ12P₁

U

^4þ12P₁

U

₁^6þ12P

U

³₂

U

₂⁴

U

₂⁶ : : :

U

^3þ12P₂

U

^4þ12P₂

U

₂^6þ12P

U

³₅

U

₅⁴

U

₅⁶ : : :

U

^3þ12P₅

U

^4þ12P₅

U

₅^6þ12P

: : : : : : : : :

: : : : : : : : :

: : : : : : : : :

U

³_1þ12P

U

_1þ12P⁴

U

_1þ12P⁶ : : :

U

^3þ12P_1þ12P

U

^4þ12P_1þ12P

U

_1þ12P^6þ12P

U

³_2þ12P

U

_2þ12P⁴

U

_2þ12P⁶ : : :

U

^3þ12P_2þ12P

U

^4þ12P_2þ12P

U

_2þ12P^6þ12P

U

³_5þ12P

U

_5þ12P⁴

U

_5þ12P⁶ : : :

U

^3þ12P_5þ12P

U

^4þ12P_5þ12P

U

_5þ12P^6þ12P

¼ 0 (A9)

which can be formally written as

<ðx; q; R

_a

Þ ¼ 0

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