Weakly Nonlinear Stability Analysis of Temperature/Gravity-Modulated Stationary Rayleigh Bénard Convection in a Rotating Porous Medium

Transp Porous Med (01) 9:633 647 DOI 10.1007/s114-011-995-4 Weakly Nonlinear Stability Analysis of Temperature/Gravity-Modulated Stationary Rayleigh Bénard Convection in a Rotating Porous Medium B. S. Bhadauria P. G. Siddheshwar Jogendra Kumar Om P. Suthar Received: September 011 / Accepted: 10 December 011 / Published online: 8 December 011 Springer Science+Business Media B.V. 011 Abstract The effect of time-periodic temperature/gravity modulation on thermal instability in a fluid-saturated rotating porous layer has been investigated by performing a weakly nonlinear stability analysis. The disturbances are expanded in terms of power series of amplitude of convection. The Ginzburg Landau equation for the stationary mode of convection is obtained and consequently the individual effect of temperature/gravity modulation on heat transport has been investigated. Further, the effect of various parameters on heat transport has been analyzed and depicted graphically. Keywords Ginzburg Landau equation Porous medium Temperature modulation Gravity modulation Rotation List of Symbols Latin Symbols A Amplitude of streamline perturbation d Height of the fluid layer g Acceleration due to gravity B. S. Bhadauria (B) Department of Applied Mathematics and Statistics, School for Physical Sciences, Babasaheb Bhimrao Ambedkar University, Lucknow 605, UP, India e-mail: mathsbsb@yahoo.com P. G. Siddheshwar O. P. Suthar Department of Mathematics, Bangalore University, Bangalore, Karnataka, India e-mail: pgsmath@gmail.com O. P. Suthar e-mail: ompsuthar@gmail.com B. S. Bhadauria J. Kumar Department of Mathematics, Faculty of Science, DST Centre for Interdisciplinary Mathematical Sciences, Banaras Hindu University, Varanasi 1005, UP, India e-mail: jogendra.bhu@gmail.com

634 B. S. Bhadauria et al. k c Wavenumber η kc + π sselt number p Pressure ( ) Pr Prandtl number, δν ( κ T ) Da Darcy number, K d Va Vadasz number, ( ) Pr Da ( ) Ra Rayleigh Darcy number, βt g TdK z ( ) δνκ T Ta Taylor Darcy number, d δν T Temperature T Temperature difference across the fluid layer t Time (x, y, z) Space co-ordinates Greek Symbols δ Porosity α Coefficient of thermal expansion κ Thermal diffusivity μ Dynamic viscosity ( ) ν Kinematic viscosity, μ ρ0 ρ ω δ 1 δ φ ɛ τ Fluid density Modulation frequency Amplitude of temperature modulation Amplitude of gravity modulation Phase angle Perturbation parameter τ = ɛ t (small time scale) Other Symbol 1 + Subscripts b Basic state c Critical value 0 Value of the un-modulated case Superscripts Perturbed quantity Dimensionless quantity

Weakly Nonlinear Stability Analysis 635 1 Introduction The classical Rayleigh Bénard convection due to bottom heating is widely known and is a highly explored phenomenon. It has been extensively studied in a porous domain also (see Nield and Bejan 006). It has numerous applications in many practical problems. A comprehensive exposition on its applications in various fields is given in Nield and Bejan (006), Ingham and Pop (005)andVafai (005). The study of fluid convection in a rotating porous medium is also of great practical importance in many branches of modern science, such as centrifugal filtration processes, petroleum industry, food engineering, chemical engineering, geophysics, and biomechanics. Several studies on the onset of convection in a rotating porousmedium havebeenreported (Friedrich 1983; Patil and Vaidyanathan 1983; Palm and Tyvand 1984; Jou and Liaw 1987a,b; Qin and Kaloni 1995; Vadasz 1996a,b, 1998; Vadasz and Govender 001; Straughan 001; Desaive et al. 00; Govender 003). In most studies related to thermal instability in a rotating porous medium saturated with a Newtonian fluid, steady temperature gradient is considered. However it is not so in many practical problems. There are many interesting situations of practical importance in which the temperature gradient is a function of both space and time. This non-uniform temperature gradient (temperature modulation) can be determined by solving the energy equation with suitable time-dependent thermal boundary conditions and can be used as an effective mechanism to control the convective flow. Innumerable studies are available which explain how a time-periodic boundary temperature affects the onset of Rayleigh-Bénard convection. An excellent review related to this problem is given by Davis (1976). The study of Venezian (1969) or the Floquet theory has been extensively followed in the thermal convection problem in porous media when the boundary temperatures are timeperiodic (Caltagirone 1976; Chhuon and Caltagirone 1979; Antohe and Lage 1996; Malashetty and Wadi 1999; Malashetty and Basavaraja 00, 003; Malashetty et al. 006; Malashetty and Swamy 007; Bhadauria 007a,b; Bhadauria and Sherani 008; Bhadauria and Srivastava 010). However, fewer studies are available on convection in a rotating porous media with temperature modulation of the boundaries [see Bhadauria 007c,d; Bhadauria and Suthar 009 and references therein]. Another problem that leads to variable coefficients in the governing equations of thermal instability in porous media is the one involving vertical time-periodic vibration of the system. This leads to the appearance of a modified gravity, collinear with actual gravity, in the form of a time-periodic gravity field perturbation and is known as gravity modulation or g-jitter in the literature. Malashetty and Padmavathi (1997), Rees and Pop (000, 001, 003), Govender (005a,b), Kuznetsov (006a, b), Siddhavaram and Homsy (006); Strong (008a,b); Razi et al. (009); Saravanan and Purusothaman (009), Saravanan and Arunkumar (010), Malashetty and Swamy (011), and Saravanan and Sivakumar (010, 011) document some aspects of the problem. The studies so far reviewed concern linear stability of the thermal system in a nonrotating/rotating porous media in the absence/presence of temperature/gravity modulation, and hence address only questions on onset of convection. If one were to consider heat and mass transports in porous media in the presence of temperature/gravity modulation, then the linear stability analysis is inadequate, and the nonlinear stability analysis becomes inevitable. In the light of the above, we make a weakly nonlinear analysis of the problem using the Ginzburg Landau equation and in the process quantify the heat and mass transports in terms of the amplitude governed by the Ginzburg Landau equation. The Ginzburg Landau equation has been solved numerically, and consequently the effect of modulation on the sselt number is studied.

636 B. S. Bhadauria et al. Mathematical Formulation We consider a horizontal porous layer saturated with an incompressible viscous Boussinesq fluid confined between two parallel horizontal planes z = 0andz = d. The porous layer is heated from below and cooled from above. A Cartesian frame of reference is chosen in such a way that the origin lies on the lower plane and the z axis as vertically upward. The system is rotating about z axis with a uniform angular velocity. The governing equations for the system are given by q = 0, (1) 1 q = 1 p + ρ g δ t ρ 0 ρ 0 δ q ν q, K () γ T + 1 t δ ( q )T = κ T T, (3) ρ = ρ 0 [1 β T (T T 0 )], (4) where q is the velocity, ρ is the density of the fluid, p is hydrodynamic pressure, = (0, 0, )is the angular velocity, T is the temperature, and κ T is the thermal conductivity of the fluid. We assume the externally imposed boundary temperature to oscillate with time, according to the relations (Venezian 1969): T = T 0 + T = T 0 T [ 1 + ɛ δ 1 cos(ωt) ] at z = 0, [ 1 ɛ δ 1 cos(ωt + φ) ] at z = d, (5) where T 0 is some reference temperature, and ω is the modulation frequency. The quantity ɛ δ 1 is the amplitude of modulation. We consider small amplitude modulation such that both ɛ and δ 1 are small. We use the following scalings to non-dimensionalize the governing equations 1 3 (x, y, z)=d(x, y, z ), T = ( T ) T. (U, V, W )= δκ T d (U, V, W ), t = d κ T t, p = δμκ T K p, (6) The dimensionless form of the governing equations is given by 1 q Va t + [ ] 1 Ta ˆk q + q = p RaDa β T T (T T 0) ˆk, (7) γ T + (q ) T = T. t (8) At the basic state, the fluid is assumed to be quiescent, and thus the basic state quantities are given by ρ = ρ b (z, t); p = p b (z, t); T = T b (z, t); q b = (0, 0, 0). (9)

Weakly Nonlinear Stability Analysis 637 The basic state equations from () (4)are d p b dz = ρ bg, (10) γ T b t = T b z. (11) Solution of the aboveeq. 11 subject to the non-dimensionalized thermal boundary conditions (5) isgivenby T b (z, t) = (1 z) + ɛ δ 1 F(z, t), (1) where [ {A(λ)e F(z, t) = Re λz + A( λ)e λz} e iωt], A(λ) = 1 ( e iφ e λ) ω ( e λ e λ) ; λ = (1 i). (13) We now impose infinitesimal perturbations to the basic state in the form: q = q b + Q, T = T b + T, ρ = ρ b + ρ, (14) where primes denote the perturbation quantities. Now from Eq. 7, wehave 1 U TaV + U = p Va t, (15) 1 Va W t + W = p + RaT. (16) From Eqs. 15 and 16, we eliminate pressure term to obtain an equation in the form: [ ( 1 U Va t W )] = Ra T, (17) Also we have: 1 V + TaU + V = 0, (18) Va t Introducing stream function as U = / and W = / in the Eqs. 8, 17, and18 and using T b in the reduced system, we get the equations as ( 1 ) + Ta V T + Ra = 0, (19) Va t ( ) 1 Va t + 1 V + Ta = 0, (0) (γ t ) 1 T = (, T ) (x, z) + [ ] 1 + ɛ F δ 1. (1) 3 Stability Analysis 3.1 Linear Stability Analysis for Marginal Stationary Convection in Unmodulated System To make this article self-contained, we give a brief account of important linear stability analysis results. The linear stability results can be obtained by neglecting the nonlinear terms in

638 B. S. Bhadauria et al. the above equations. The perturbed quantities are expanded by using normal mode technique as (x, z) = A sin(k c x) sin(πz) T (x, z) = B cos(k c x) sin(πz). () V (x, z) = C sin(k c x) cos πz) In the above expressions, k c is the wave number. The Rayleigh number for stationary is given by ( Ra st = η η + π Ta ), (3) where η = π + kc, and the solution of, T and V is obtained as (x, z, t) = A sin(k c x) sin(πz) T (x, z, t) = k c η A cos(k cx) sin(πz). (4) V (x, z, t) = π Ta η A sin(k c x) cos(πz) We note at this point that the oscillatory convection is possible (see Vadasz 1998), but we focus our attention on only stationary convection in this article. 3. Nonlinear Stability Analysis We use the time variations only at the slow time scale by considering τ = ɛ t. ɛ Va τ 1 1 Ra Ta ( ) 1 + ɛ F δ 1 γɛ τ 1 0 ɛ Ta 0 Va τ + 1 0 T = (, T ) V (x, z). (5) 0 Now we use the following asymptotic expansion in Eqs. 19 1: Ra = Ra 0 + ɛ Ra + = ɛ 1 + ɛ + T = ɛt 1 + ɛ, (6) T + V = ɛv 1 + ɛ V + where Ra 0 is the critical Rayleigh number in unmodulated case. Substituting Eq. 6 in Eq. 5 and comparing like powers of ɛ on both side, we get the solutions at different orders. At first order: 1 Ra 0 Ta 0 Ta 1 0 0 1 k c 1 T 1 V 1 = 0 0. (7)

Weakly Nonlinear Stability Analysis 639 The above equations correspond to linear stability equations for stationary mode of convection. The solutions of the above equations can be given by [using Eq. 4] 1 = A(τ) sin(k c x) sin(πz) T 1 = k c η A(τ) cos(k cx) sin(πz) V 1 = π TaA(τ) sin(k c x) cos(πz), (8) Thesystem (7) gives the critical value of Rayleigh number and corresponding wave number as obtained in the previous section. At the second order, we have where and 1 Ta Ra 0 Ta 1 0 0 1 R = ( 1, T 1 ) (x, z) T V One can obtain second-order solutions as = 0 T = k c 8πη [A(τ)] sin(πz) V = 0 [ kc π = R 1 R R 3, (9) R 1 = 0, (30) = πk c η [A(τ)] sin(πz), (31) R 3 = 0. (3). (33) The horizontally averaged sselt number (τ) for the stationary mode of convection (the preferred mode in this problem) is given by [ kc ] π/kc π x=0 (1 z + T ) z dx z=0 (τ) = ]. (34) π/kc x=0 (1 z) z dx Substituting Eq. 33 in Eq. 34 and simplifying, we get At the third order, we have 1 Ra 0 Ta 3 1 0 T 3 = V 3 Ta 0 1 z=0 (τ) = 1 + k c [A(τ)] 4η. (35) R 31 R 3 R 33, (36)

640 B. S. Bhadauria et al. where T 1 R 31 = Ra 1 Va T R 3 = 1 ( 1 ), τ (37) F γ T 1 τ, (38) + δ 1 R 33 = 1 V 1 Va τ. (39) Substituting 1, T 1, T, V 1, and V from Eqs. 8 and 33 in Eqs. 37 39,weget [ η da R 31 = Va dτ R k ] c η A sin(k c x) sin(πz), (40) [ R 3 = γ k ] c da η dτ + δ F 1 k ca k3 c A3 4η cos(πz) cos(k c x) sin(πz), (41) R 33 = π Ta da Vaη dτ sin(k cx) cos(πz). (4) The solvability condition for the existence of the third-order solution is given by 1 1 z=0 π/k c x=0 [ ˆ 1 R 31 + Ra 0 ˆT 1 R 3 ˆV 1 R 33 ]dxdz = 0, (43) where ˆ 1, ˆT 1,and ˆV 1 are the solutions of the adjoint system of the first order, given by ˆ 1 = A(τ) sin(k c x) sin(πz) ˆT 1 = k c η A(τ) cos(k cx) sin(πz). (44) ˆV 1 = π Ta η A(τ) sin(k c x) cos(πz) Using Eqs. 40 4 and 44 in the Eq. 43 and simplifying, we get the Ginzburg Landau equation for stationary instability with a time-periodic coefficient in the form: ( 1 Va + γ ) [ ] da(τ) kc η F(τ)A(τ) + dτ 8η + π 4 kc Ta Va ( η + π Ta ) A 3 (τ) = 0, (45) where and I(τ) = F(τ) = Ra Ra 0 δ I(τ), (46) 1 0 [ F(z, τ) ] sin (πz) dz. (47) The solution of Eq. 45, subject to the initial condition A(0) = a 0 where a 0 is a chosen initial amplitude of convection, can be obtained by means of a numerical method. In calculations, we may assume Ra = Ra 0 to keep the parameters to the minimum.

Weakly Nonlinear Stability Analysis 641 4 Time-Periodic Gravity field Now we study the effect of a periodically varying gravitational field. For this, we assume g = g 0 (1 + ɛ δ cos(ωt)), (48) where g 0 is some reference value of the gravitational force, ω is the frequency of modulation, τ is the slow time scale defined in the previous section, and the gravitational force is given by g = (0, 0, g). It should be noted here that the basic temperature in the present case is considered to be steady, and is given by T b = 1 z. (49) Using above expression in the Eqs. 19 1, we get the equation corresponding to gravity modulation. The linear stability analysis can be performed exactly in the same way as described in the above section, and the same results (Eqs. 3 and 35) can be readily drawn. Further the system (5) reduces to the form: ɛ Va τ 1 1 Ra[1 + ɛ δ cos(ωt)] Ta γɛ τ 1 0 ɛ Ta 0 Va τ + 1 0 T = (, ) V (x, z) 0. (50) Also, we obtained here the same result at the first and second orders as reported in Eqs. 9 and 34. In particular, the sselt number for stationary mode of convection can be obtained as (τ) = 1 + k c [A(τ)] 4η. (51) At the third order, the expression in Eq. 30, in this case, are obtained as R 31 = [Ra + Ra 0 δ cos(ωτ)] T 1 1 Va T R 3 = 1 R 33 = 1 Va ( 1 1 ), (5) τ γ T 1 τ, (53) V 1 τ. (54) For the above given values of R 31, R 3,andR 33, the solvability condition (Eq. 44) produces the Ginzburg Landau amplitude equation given by ( 1 Va + γ ) [ ] da(τ) kc η F(τ)A(τ) + dτ 8η + π 4 kc Ta Va ( η + π Ta ) A 3 (τ) = 0, (55) where F(τ) = Ra + δ cos(ωτ). (56) Ra 0

64 B. S. Bhadauria et al. 5 Results and Discussions In this article, we have studied the effect of temperature and gravity modulations on thermal instability in a rotating fluid saturated porous layer. A weakly nonlinear stability analysis has been performed to investigate the effect of temperature/gravity modulation on heat transport. The temperature modulation has been considered in the following three cases: 1. In-phase modulation (IPM) (φ = 0),. Out-phase modulation (OPM) (φ = π)and 3. Lower-boundary modulated only (LBMO) (φ = i ), The effect of temperature modulation has been depicted in Figs. 1,, 3 while that of gravity modulation in Fig. 4. From the Figs. 1,, 3, we observe the following general result for sselt number : IPM < LBMO < OPM. The parameters that arise in the problem are Ta, Va,φ,δ 1,andδ and these parameters influence the convective heat transports. The first two parameters relate to the fluid and the structure of the porous medium, and the last three concern the two external mechanisms of controlling convection. Due to the assumption of low porosity medium, the values considered for Vaare 100, 150 and 00. Because small amplitude modulations are considered, the values of δ 1 and δ lie between 0 and 0.5. Further, the modulation of the boundary temperature and the vertical time-periodic fluctuation of gravity are assumed to be of low frequency. At low range of frequencies, the effect of frequencies on onset of convection as well as on heat transport is minimal. This assumption is required to ensure that the system does not pick up oscillatory (a) (b) 0.0, 0.05, 0.08 1,, 5 Va 100, Ta 60, 0.00 0.05 0.10 0.15 0.0 0.05, Ta 60, Va 100 0.00 0.05 0.10 0.15 0.0 (c) (d) 3.0 60 100 Ta 30 00 150 Va 100 0.05, Va 100, 0.00 0.05 0.10 0.15 0.0 0.05, Ta 60, 0.00 0.05 0.10 0.15 0.0 Fig. 1 Variation of sselt number with time τ for different values of (a) δ 1,(b) ω,(c) Ta,(d) Va

Weakly Nonlinear Stability Analysis 643 (a) (c).7.6.4.3. 0.0 0.05 0.08.1 Va 100, Ta 60, (b) (d).6.4.3 1 5..1 0.05, Ta 60, Va 100 3.0 Ta 30 60 100 Va 100 150 00 0.05, Va 100, 0.05, Ta 60, Fig. Variation of sselt number with time τ for different values of (a) δ,(b) ω, (c) Ta,(d) Va (a).6 (b).60 5 0.4 0.0 0.05 0.08.45.40 1 5.3 Va 100, Ta 60,..35.30 0.05, Ta 60, Va 100.5 (c) (d) 3.0 Ta 30 60 100 Va 100 150 00 0.05, Va 100, 0.05, Ta 60, Fig. 3 Variation of sselt number with time τ for different values of (a) δ 1,(b) ω,(c) Ta,(d) Va convective mode at the onset due to modulation in a situation that is conducive otherwise to stationary mode. In Fig. 1a d, we have plotted the sselt number with respect to time τ for the case of IPM. From the figures, we find that for small time τ, thevalueof does not alter and

644 B. S. Bhadauria et al. (a).7.6.4.3..1 0.0 0.05 0.08 Va 100, Ta 5, (b).6.4.3..1 1 5 0.05, Ta 5, Va 100 (c) 3.0 (d) 3.0 Ta 30 60 100 Va 100 150 00 0.05, Va 100, 0.05, Ta 60, Fig. 4 Variation of sselt number with time τ for different values of (a) δ 1,(b) ω,(c) Ta,(d) Va remains almost constant, then it increases on increasing τ, and finally becomes steady on further increasing the time τ. From the Fig. 1a, b, we observed that on increasing amplitude of modulation δ 1 and frequency of modulation ω, the value of sselt number does not alter. Thus, the effects of increasing δ 1 and ω have negligible effect on rate of heat transfer in case of IPM. From Fig. 1c, it has been examined that on increasing Darcy Taylor number Ta, the value of sselt number decreases, thus decreasing the rate of heat transport and hence stabilizing the system. The effect of Vadasz number Va on is observed in Fig. 1d. From the figure, we find that increases with increasing Va, thus advancing the convection. Further, we observed that the results obtained in this case are qualitatively similar to those of unmodulated case. Figure a d shows the plots of with time τ forthecaseofopm.fromfig.a, we find that the effect of increasing modulation amplitude δ 1 on is to increase the magnitude of, i.e., rate of heat transport increases. Figure b shows that, on increasing the frequency of modulation ω, the magnitude of does not change, but wavelength of oscillations become shorter with increasing ω. From the Fig. c, we observe that on increasing Ta,thevalueof decreases, thus the effect of increasing Ta is to decrease the rate of heat transfer, and hence to suppress the convection. Figure d displays the effect of Vaon. From the figure, we examine that on increasing Va, increases, therefore, the effect of increasing Va is to increase the rate of heat transfer across the porous layer. It is obvious from the figure that / δ1 =0.0 < / δ1 =0.05 < / δ1 =0.08. In Fig. 3a d, we depict the variation of sselt number with time τ for lowerboundary temperature modulation only. From the figure, we find qualitatively similar results to that in Fig. a d.

Weakly Nonlinear Stability Analysis 645 In Fig. 4a d, we depict the variation of with respect to slow time τ under gravity modulation. From Fig. 4a, we find that an increment in amplitude of modulation δ, increases ; however, the wavelength of oscillations remains unaltered. From Fig. 4b, which shows the effect of frequency of modulation ω on, we observe that on increasing ω, thevalue of remains unchanged wherein the wavelength of oscillations decreases. The effect of rotation has been shown in Fig. 4c, and is found to have stabilizing effect. However, from Fig.4d, we observe that on increasing Vadasz number Va, the value of increases, therefore advancing the convection. Further, the results in Fig. 4a d are found qualitatively similar to that of Figs. a d and 3a d. 6Conclusion In the present study, the effects of temperature/gravity modulation in a rotating fluid saturated porous layer have been analyzed by performing nonlinear stability analysis using Ginzburg Landau equation. On the basis of the above discussion following conclusions are drawn: (1) IPM < LBMO < OPM for temperature modulation. () The effect of increasing Taylor number Ta on is to decrease the heat transport. (3) The effect of increasing Vadasz number Va is to advance the convection for in-phase and out-phase as well as for the case when only lower plate is modulated. (4) On increasing the magnitude of modulation and frequency of modulation for the case of in-phase temperature modulation, we find negligible effect on, while there is significant effect in the cases of OPM and LBMO. (5) The results of gravity modulation are found qualitatively similar to the cases of OPM and LBMO. Acknowledgements Part of this study was done during the lien period sanctioned by Banaras Hindu University, Varanasi, India to the author BSB to work as Professor of Mathematics at Babasaheb Bhimrao Ambedkar University, Lucknow. The authors are grateful to the referees for their most useful comments. References Antohe, B.V., Lage, J.L.: Amplitude effect on convection induced by time-periodic horizontal heating. Int. J. Heat Mass Transf. 39, 111 1133 (1996) Bhadauria, B.S.: Thermal modulation of Rayleigh-Bénard convection in a sparsely packed porous medium. J. Porous Media 10(), 175 188 (007a) Bhadauria, B.S.: Double diffusive convection in a porous medium with modulated temperature on the boundaries. Transp. Porous Media 70, 191 11 (007b) Bhadauria, B.S.: Fluid convection in a rotating porous layer under modulated temperature on the boundaries. Transp. Porous Media 67, 97 315 (007c) Bhadauria, B.S.: Double diffusive convection in a rotating porous layer with temperature modulation on the boundaries. J. Porous Media 10, 569 583 (007d) Bhadauria, B.S., Sherani, A.: Onset of Darcy-convection in a magnetic fluid-saturated porous medium subject to temperature modulation of the boundaries. Transp. Porous Media 73, 349 368 (008) Bhadauria, B.S., Srivastava, A.K.: Magneto-Double Diffusive Convection in an electrically conductingfluid-saturated Porous Medium with Temperature Modulation of the Boundaries. Int. J. Heat Mass Transf. 53, 530 538 (010) Bhadauria, B.S., Suthar, O.P.: Effect of thermal modulation on the onset of centrifugally driven convection in a rotating vertical porous layer placed far away from the axis of rotation. J. Porous Media 1, 1 37 (009)

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