Transp Porous Med (2011) 90:889–909 DOI 10.1007/s11242-011-9822-x

Effect of Thermal/Gravity Modulation on the Onset of Convection in a Maxwell Fluid Saturated Porous Layer M. S. Malashetty · Irfana Begum

Received: 20 May 2011 / Accepted: 2 August 2011 / Published online: 14 August 2011 © Springer Science+Business Media B.V. 2011

Abstract The effect of thermal/gravity modulation on the onset of convection in a Maxwell fluid saturated porous layer is investigated by a linear stability analysis. Modified Darcy– Maxwell model is used to describe the fluid motion. The regular perturbation method based on the small amplitude of modulation is employed to compute the critical Rayleigh number and the corresponding wavenumber. The stability of the system characterized by a correction Rayleigh number is calculated as a function of the viscoelastic parameter, Darcy–Prandtl number, normalized porosity, and the frequency of modulation. It is found that the low frequency symmetric thermal modulation is destabilizing while moderate and high frequency symmetric modulation is always stabilizing. The asymmetric modulation and lower wall temperature modulations are, in general, stabilizing while the system becomes unstable for large values of Darcy–Prandtl number and for small frequencies. It is shown that in general the gravity modulation produces a stabilizing effect on the onset of convection for moderate and high frequency. The small frequency gravity modulation is found to have destabilizing effect on the stability of the system. Keywords

Thermal/gravity modulation · Stability · Viscoelastic fluid

List of Symbols a Wavenumber c Specific heat of solid cp Specific heat of fluid d Height of the porous layer Da Darcy number, k/d 2 g Gravitational acceleration (0, 0, −g) k Permeability of the porous layer l, m Horizontal wavenumbers M. S. Malashetty (B) · I. Begum Department of Mathematics, Gulbarga University, Gulbarga 585106, India e-mail: [email protected]

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Pr Pr D p q Ra t T T x, y, z

Prandtl number, ν/κ Darcy–Prandtl number, Pr D = φ 2 Pr /Da Pressure Velocity vector (u, v, w) Rayleigh number, βgT dk/νκ Time Temperature Temperature difference between the walls Space coordinates

Greek symbols β Thermal expansion coefficient φ Porosity ε Amplitude of modulation  Frequency of modulation ϕ Phase angle γ Ratio of specific heat (ρc)m /(ρc p )f κ Thermal diffusivity λ¯ Stress relaxation parameter

Deborah number, λ¯ κ/φd 2 χ Normalized porosity, φ/γ μ Dynamic viscosity ν Kinematic viscosity ρ Density ω Dimensionless frequency of modulation, d 2 γ /κ Other symbols ∇12 ∇2

2 ∂2 + ∂∂y 2 ∂x2 ∂2 ∇12 + ∂z 2

Subscripts b Basic state c Critical f Fluid m Porous medium 0 Reference value s Solid

Superscripts ∗ Dimensionless quantity  Perturbed quantity

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1 Introduction Thermal convection induced by oscillating forces resulting from either oscillating or modulated gravitational forces or a combination of the two has received much attention in the fluid dynamics research community in recent time. There have been many investigations concerning how a time-dependent boundary temperature affects the onset of Rayleigh– Benard convection. Most of the findings relevant to these problems have been reviewed by Davis (1976). In case of small amplitude temperature modulation, a linear stability was performed by Venezian (1969), Rosenblat and Herbert (1970), Rosenblat and Tannaka (1971), and Roppo et al. (1984) have studied the effect of temperature modulation on the onset of thermal convection in a fluid layer. The studies on the effect of thermal modulation on the onset of thermal convection in a fluid-saturated porous medium have received marginal attention. The effect of time dependent wall temperature on the onset of convection in a porous medium has been studied by Caltagirone (1976), Rudraiah and Malashetty (1990), Malashetty and Wadi (1999). Further, Malashetty and Basavaraja (2002, 2003, 2004) have examined the single and double diffusive convections in a fluid saturated anisotropic porous layer subject to time-dependent wall temperature. Recently Bhadauria (2007) studied the effect of thermal modulation on the onset of convection in a layer of sparsely packed porous medium bounded by rigid boundaries. A physically important class of problems involves convection in a fluid layer in the presence of complex body forces. Such forces can arise in a number of ways. For example, when a system with density gradient is subjected to vibrations, the resulting buoyancy forces, which are produced by the interaction of the density gradient with the gravitational field, have a complex spatio-temporal structures. The time dependent gravitational field is of interest in space laboratory experiments, in areas of crystal growth and other applications. It is also of importance in the large-scale convection of atmosphere. Many theoretical and experimental studies dealing with materials processing or physics of fluids under the micro-gravity conditions aboard an orbiting spacecraft have been carried out in recent years by Nelson (1991). Owing to several unavoidable sources of residual acceleration experienced by a spacecraft, the gravity field in an orbiting laboratory is not constant in a micro-gravity environment, but is a randomly fluctuating field. This fluctuating gravity is referred to as g-jitter. It is reported in the literature Wadih et al. (1988, 1990) that vibrations can either substantially enhance or retard heat transfer and thus drastically affect the convection. The effect of gravity modulation on a convectively stable configuration can significantly influence the stability of a system by increasing or decreasing its susceptibility to convection. In general, a distribution of stratifying agency that is convectively stable under constant gravity field is introduced. Certain combinations of thermal gradients, physical properties, and modulation parameters may lead to parametric resonance and, hence, to the instability of the system. Gershuni et al. (1970) and Gresho and Sani (1970) were the first to study the effect of gravity modulation on the stability of a heated fluid layer. Their results show that the stability of the layer being heated from below is enhanced by gravity modulation; but in case of heating from above, the layer is destabilized. Wadih and Roux (1988) presented a study on the instability of the convection in an infinitely long cylinder with gravity modulation oscillating along the vertical axis. Their analyses established that the onset of convection is altered under the modulation of constraints. Experiments on the response of Rayleigh–Benard convection to gravity modulation were carried out by Rogrers et al. (2005). The study of the effect of gravity modulation on the onset of convection in porous medium is comparatively of recent origin. Malashetty and Padmavathi (1997) studied the effect of small amplitude gravity modulation on the onset of convection in fluid and fluid saturated

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porous layer. They found that low frequency oscillations have significant effect on the stability of the system. Bardan and Mojtabi (2000) made an analytical and numerical study of convection in a porous cavity in the presence of vertical vibrations. They found that the vibrations stabilize the quiescent state. Govender (2004) made stability analysis to investigate the effect of low amplitude gravity modulation on convection in a porous layer heated from below. It was shown that increasing the frequency of vibration stabilizes the convection. Recently, Saravanan and Purusothaman (2009) carried out an investigation to find out the effect of non-Darcian effects in an anisotropic porous medium and found that non-Darcian effects significantly affect the synchronous mode of instability. More recently, Saravanan and Sivakumara (2010) studied the effect vibration on the onset of convection in a horizontal fluid saturated porous layer considering arbitrary amplitude and frequency. It is demonstrated that vibrations can produce a stabilizing or a destabilizing effect depending on their amplitude and frequency for a porous layer heated from below. Quite recently non-Newtonian fluids housed in fluid-based systems, with and without porous matrix, have been extensively used in application situations and hence warrant the attention they have been duly getting. In the asthenosphere and the deeper mantle it is an important rheological process (see Lowire 1997). Rudraiah et al. (1990) have investigated the effect of thermal modulation on the onset of convection in a viscoelastic fluid-saturated porous medium using Oldroyd model, and the effect of anisotropy on the problem has been analyzed by Malashetty et al. (2006a,b). More Recently, Shivakumara et al. (2011) investigated effect of thermal modulation on the onset of convection in Walters B viscoelastic fluid-saturated porous medium. The other application areas of viscoelastic fluid saturated porous media are flow through composites, timber wood, snow systems, and rheology of food transport. Regulation of convection in these application situations is important and the study of this is the motive for this article. We investigate in this article the effect of thermal/gravity modulation on the onset of convection in a Maxwell fluid saturated porous layer. The amplitude and frequency of the modulation are externally controlled parameters and hence the onset of convection can be delayed or advanced by the proper tuning of these parameters. The problem has potential application in achieving major enhancement of mass; momentum and heat transfer in the geothermal context and related areas.

2 Basic Equations We consider an infinite horizontal porous layer saturated with a Maxwell fluid, confined between the planes z = 0 and z = d, with the vertically downward gravity force g acting on it. A Cartesian frame of reference is chosen with the origin in the lower boundary and the z-axis vertically upwards. The porous medium is assumed to be isotropic and is in local thermal equilibrium with fluid phase. The Boussinesq approximation, which states that the variation in density is negligible everywhere in the conservations except in the buoyancy term, is assumed to hold. We use the modified Darcy–Maxwell model to describe the flow in the porous media. Under these assumptions, the basic governing equations are ∇.q = 0,    ∂ ρ0 ∂q μ 1 + λ¯ + ∇ p − ρg + q = 0, ∂t φ ∂t k

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(2.1) (2.2)

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∂T + q.∇T = κ∇ 2 T, ∂t ρ = ρ0 [1 − β(T − T0 )] , γ

(2.3) (2.4)

where q = (u, v, w) is the velocity, p the pressure, φ represents the porosity, λ¯ stress relaxation parameter, μ the viscosity, k the permeability, ρ the density, κ the thermal diffusivity, β is the coefficient of volume expansion. Further, γ = (ρc)m /(ρc p )f , (ρc)m = (1 − φ)(ρc)s + φ(ρc p )f , c p is the specific heat of the fluid, at constant pressure, c is the specific heat of the solid, the subscripts f, s, and m denote fluid, solid, and porous medium values, respectively. In this article we investigate how two different types of modulation alter the stability conditions. 2.1 Thermal Modulation In this section we present the thermal modulation problem. The external driving force is modulated harmonically in time by varying the temperatures of lower and upper horizontal boundary. Accordingly, we take T [1 + ε cos t] at z = 0, 2 T T (z, t) = T0 + [1 − ε cos(t + ϕ)] at z = d, 2 T (z, t) = T0 +

(2.5) (2.6)

where ε represents small amplitude of modulation (which is used as a perturbation parameter to solve the problem),  the frequency, ϕ the phase angle. We consider three types of thermal modulation, viz., Case (a) : symmetric (in phase, ϕ = 0), Case (b) : asymmetric (out of phase, ϕ = π ), and Case (c) : only lower wall temperature is modulated while the upper one is held at constant temperature (ϕ = −i∞). 2.2 Basic State The basic state of the fluid is quiescent and is given by qb = (0, 0, 0), T = Tb (z, t),

p = pb (z, t), ρ = ρb (z, t).

(2.7)

The temperature T = Tb (z, t) is a solution of γ

∂ Tb ∂ 2 Tb , =κ ∂t ∂z 2

(2.8)

and pressure pb (z, t) balances the buoyancy force. The solution of Eq. 2.8 subject to the boundary conditions (2.5) and (2.6) is  

  2z T 1− + εRe a(λ)eλz/d + a(−λ)e−λz/d e−it , (2.9) Tb = T0 + 2 d where  λ = (1 − i)

γ d 2 2κ

1/2

 , a(λ) =

 e−iϕ − e−λ . eλ − e−λ

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and Re stands for the real part. We do not record the expressions of pb and ρb as these are not explicitly required in the remaining part of the paper. We now superimpose infinitesimal perturbations on the quiescent basic state and study the stability of the system. 2.3 Linear Stability Analysis Let the basic state be disturbed by an infinitesimal perturbation. We now have q = qb + q  ,

T = Tb + T  ,

p = pb + p  ,

ρ = ρb + ρ  ,

(2.10)

where prime indicates that the quantities are infinitesimal perturbations. Introducing Eq. 2.10 into Eqs. 2.1 to 2.4 and using basic state solutions, and neglecting the nonlinear terms in perturbations, we obtain the linearized equations governing the perturbations in the form,    ρ0 ∂q ∂ μ   ¯ 1+λ (2.11) + ∇ p − ρ g + q = 0, ∂t φ ∂t k   ∂ Tb ∂T  (2.12) γ + w = κ∇ 2 T  , ∂t ∂z (2.13) ρ  = −βρ0 T  . We eliminate ρ  between Eqs. 2.11 and 2.13 and then by operating curl twice on Eq. 2.11 we eliminate p  from it, and then render the resulting equation and Eq. 2.12

2 dimensionless using the following transformations (x  , y  , z  ) = d(x ∗ , y ∗ , z ∗ ), t  = d κφ t ∗ , w  =   ∗  κ  ∗   d 2 γ to obtain (after dropping the asterisks for d w , T , Tb = T T , Tb , ω = κ simplicity)      1 ∂ 2 ∂ ∂ 2 (2.14) ∇ w + ∇ w = Ra 1 + ∇12 T, 1+ ∂t Pr D ∂t ∂t   ∂T 1 ∂ − ∇ 2 T = −w b , (2.15) χ ∂t ∂z dk ∂2 . The dimensionless groups that appear are Ra = βgT , the νκ ∂ y2 ¯ φ 2 Pr λκ Rayleigh number, = φd 2 , the Deborah number, Pr D = Da , the Darcy–Prandtl number (Pr = κν is the Prandtl number, Da = dk2 is the Darcy number), and χ = γφ , the normalized

where ∇12 =

∂2 ∂x2

+

porosity. The buoyancy due to thermal gradient is characterized by the Rayleigh number, the ratio between viscous and thermal diffusivities is given by the Prandtl number, the ratio between porosity, Prandtl number and Darcy number is given by Darcy–Prandtl number, the viscoelastic character of the fluid appears in the relaxation parameter (which is also known as the Deborah number). The Deborah number is a dimensionless number used in rheology to characterize how fluid and material is. The smaller the Deborah number, the more fluid the material appears. The parameter that relates to the relaxation time to the thermal diffusion time is of order one for most viscoelastic fluids. The normalized porosity χ, is expressed in terms of the porosity of the porous medium, φ, and the solid to fluid heat capacity ratio, γ . Since 0 < φ < 1, it is clear that 0 < χ < 1. The Prandtl number affects the stability of the porous system through the combined dimensionless group known as Darcy–Prandtl number (which is also known as the Vadasz number). The influence and significance of Darcy–Prandtl number in porous medium convection has been discussed by Vadasz (1998) in his excellent work. Readers may refer to this paper for details. The range of values for

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Darcy–Prandtl number for the Maxwell fluid in porous media is not available. Therefore, we consider wide range of values for the Darcy–Prandtl number. Equations 2.14 and 2.15 are to be solved subject to the impermeable and isothermal boundary conditions w = T = 0, at z = 0, 1.

(2.16)

We now combine Eqs. 2.14 and 2.15 to obtain a single differential equation for the vertical component of velocity w as          1 ∂ 1 ∂ 2 ∂ ∂ 2 2 2 ∂ Tb 1+ w = −Ra 1 + ∇ +∇ −∇ ∇1 w ∂t Pr D ∂t χ ∂t ∂t ∂z (2.17) The boundary conditions (2.16) can also be expressed in terms of w in the form ∂ 2w = 0, at z = 0, 1. (2.18) ∂z 2 Using Eq. 2.9, the dimensionless temperature gradient appearing in Eq. 2.17 may be written as ∂ Tb = −1 + ε f, (2.19) ∂z where

  f = Re A(λ)eλz + A(−λ)e−λz e−iωt ,   ω 1/2 λ e−iϕ − e−λ A(λ) = and λ = (1 − i) . 2 eλ − e−λ 2 w=

3 Method of Solution We now seek the eigenfunctions w and eigenvalues Ra of Eq. 2.17 for the  basic temperature gradient given by Eq. 2.19 that departs from the linear profile ∂ Tb /∂z = −1 by quantities of order ε. We therefore assume the solution of Eq. 2.17 in the form: (w, Ra) = (w0 , Ra0 ) + ε (w1 , Ra1 ) + ε 2 (w2 , Ra2 ) + · · ·

(3.1)

Substituting Eq. 3.1 into Eq. 2.17 and equating the coefficients of various powers of ε on either side of the resulting equation, we obtain the following system of equations up to the order of ε 2 : (3.2) Lw0 = 0,    ∂ ∂ 2 2 (3.3) Lw1 = Ra1 1 + ∇1 w0 − Ra0 1 + f ∇1 w0 , ∂t ∂t       ∂ ∂ ∂ Lw2 = Ra2 1 + ∇12 w0 − Ra1 1 + f ∇12 w0 + Ra1 1 + ∇12 w1 ∂t ∂t ∂t   ∂ f ∇12 w1 , −Ra0 1 + (3.4) ∂t 

where L=

 1+

∂ ∂t



     1 ∂ 2 1 ∂ ∂ ∇ + ∇2 − ∇ 2 − Ra0 1 + ∇12 Pr D ∂t χ ∂t ∂t

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and w0 , w1 , w2 are required to satisfy the boundary conditions of Eq. 2.18. We now assume the solutions for Eq. 3.2 in the form w0 = W0 (z) exp [i(lx + my)], where W0 (z) = W0n (z) = sin(nπ z), n = 1, 2, 3. . . and l, m are the wave numbers in the x y-plane (n) such that l 2 + m 2 = a 2 . The corresponding eigenvalues Ra0 = Ra0 are given by (n)

Ra0 =

2 1  2 2 n a + π2 . 2 a

(3.5)

For a fixed value of the wave number a the least eigenvalue occurs for n = 1, and is given by Ra0 =

2 1  2 a + π2 . 2 a

(3.6)

Ra0 assumes the minimum value Ra0c = 4π 2 and ac = π.

(3.7)

These are the values reported by Horton and Rogers (1945) and Lapwood (1948). Equation 3.3 is inhomogeneous and poses a problem because of the presence of the resonance terms. The solvability condition requires that the time independent part of the right-hand side must be orthogonal to w0 . The term independent of time on the right-hand side is Ra1 ∇12 w0 so that Ra1 must be zero. It follows that all the odd coefficients, i.e., Ra1 , Ra3 , … in Eq. 3.1 must vanish. The equation for w1 then takes the form Lw1 = Ra0 a 2 Re {1 − iω } f sin(π z).

(3.8)

We solve Eq. 3.8 for w1 by expanding the right-hand side in a Fourier series and inverting the operator L term by term. Thus, we obtain   ∞  Bn (λ) −iωt 2 w1 = Ra0 a Re (1 − iω ) sin(nπ z) . (3.9) e M(ω, n) n=1

The detail of the algebra is presented in Appendix through Eqs. A1–A9. The equation for w2 becomes   ∂ 2 2 Lw2 = −Ra2 a w0 + Ra0 a 1 + (3.10) f w1 . ∂t We do not require the solution of this equation but need to merely use it to determine Ra2 , the first non-zero correction to Ra0 . The solvability condition requires that the time-independent part of the right-hand side of Eq. 3.10 be orthogonal to sin(π z) and this results in the following equation: Ra2 =

∞ Ra02 a 2   ( 2 Bn λ) Cn . 2

(3.11)

n=1

The detail is presented in Appendix through Eqs. A10–A17. In the above equation the summation extends over even values of n for case (a), odd values of n for case (b), and all integer values of n for case (c). Equation 3.10 could now be solved for w2 if desired, and the procedure may be continued to obtain further corrections to w and Ra. However, we shall stop at this step. The value of Rayleigh number Ra obtained by this procedure is the eigenvalue corresponding to the

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eigenfunctions w, which, though oscillating, remains bounded in time. Ra is a function of horizontal wavenumber a and the amplitude of modulation ε, accordingly we expand Ra (a, ε) = Ra0 (a) + ε 2 Ra2 (a) + ε 4 Ra4 (a) + · · · ,

(3.12)

a = a0 + ε a2 + · · · ,

(3.13)

2

where Ra0 and a0 are the Rayleigh number and wavenumber, respectively, for the unmodulated system. The Rayleigh number Ra as a function of wavenumber a has a least value Ra Rac which occurs at a = ac and the critical wavenumber occurs when ∂∂a = 0. In view of Eq. 3.12 we have ∂ Ra0 ∂ Ra0 + ε2 + · · · = 0. ∂a ∂a

(3.14)

Equation 3.14 in view of Eq. 3.13 takes the form         3 ∂ 2 Ra0 ∂ 2 Ra0 ∂ Ra2 ∂ Ra0 2 1 ∂ Ra0 2 +ε a1 + ε a1 + a2 + + · · · = 0. ∂a0 2 ∂a0 ∂a02 ∂a03 ∂a02 (3.15) Equating the coefficients of like powers of ε on both sides of Eq. 3.15 we get   ∂ Ra0 = 0, a1 = 0, a2 = − (∂ Ra2 /∂a0 ) / ∂ 2 Ra0 /∂a02 . ∂a0

(3.16)

The critical Rayleigh number is then given by Rac (a, ε) = Ra0c + ε 2 Ra2c + ε 4 Ra4c + · · · = Ra0 (a0 ) +    1 2 ∂ Ra0 /∂a02 a12 + (∂ Ra0 /∂a0 ) a2 + Ra2 (a0 ) + · · · ε (∂ Ra0 /∂a0 ) a1 + ε 2 2 (3.17) In view of Eq. 3.16, the above equation reduces to Rac (a, ε) = Ra0 (a0 ) + ε 2 Ra2 (a0 ) + · · ·

(3.18)  2 The critical value of the Rayleigh number Ra is thus computed up to O ε by evaluating Ra0 and Ra2 at a = a0 . It is only when one wishes to evaluate Ra4 , a2 must be taken into account (see also Venezian 1969). If Ra2c is positive, the effect of modulation is to stabilize the system as compared to the unmodulated system. When Ra2c is negative, the effect of modulation is to destabilize   the system as compared to the unmodulated system. To the order of O ε 2 , Ra2c is obtained for the three cases (a) in-phase temperature modulation, (b) out-phase temperature modulation, and (c) only lower wall temperature modulation.

4 Gravity Modulation In this section we study the effect of gravity modulation for the Maxwell fluid-saturated porous layer confined between the planes z = 0 and z = dunder the influence of a time-periodically varying gravity force g ≡ (0, 0, −g(t)) acting on it, where g(t) = g0 (1 + ε cos t) with g0 the constant gravity in an otherwise unmodulated system, ε the small amplitude of modulation,  the frequency and t the time. A uniform adverse temperature gradient

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T = (Tl − Tu ) where Tl > Tu is maintained between the lower and upper boundary. Within the Boussinesq approximation, the governing equations are essentially same as in Sect. 2 except Eq. 2.2 in place of which we take the following equation    ρ0 ∂q ∂ μ ¯ 1+λ + ∇ p − ρg (t) + q = 0, (4.1) ∂t φ ∂t k where all the quantities have their usual meaning as defined earlier. The basic state is quiescent and is described by qb = (0, 0, 0) ,

p = pb (z, t), T = Tb (z), ρ = ρb (z).

(4.2)

Using these into Eqs. 2.3, 2.4, and 4.1 one can obtain ∂ pb = −ρb g (t) , ∂z

   d 2 Tb = 0, ρb = ρ0 1 − β Tb − T0 . 2 dz

(4.3)

Thus Eq. 4.3 together with boundary conditions possess the following solutions: 1 Tb (z) = Tl − (Tl − Tu ) z, d   1 2 pb (z, t) = ρb βg (t) Tl z − (Tl − Tu ) z . 2d

(4.4) (4.5)

Following the analysis of Sect. 2.3, the non-dimensional linearized perturbation equation, are given by      1 ∂ 2 ∂ ∂ ∇ w + ∇ 2 w = Ra (1 + ε cos t) 1 + ∇12 T, (4.6) 1+ ∂t Pr D ∂t ∂t   1 ∂ (4.7) − ∇ 2 T = w, χ ∂t where the parameters Ra, Pr D , , and χ are as defined earlier. Following the analysis of Sect. 3, we obtain Ra0 same as the one given by Eq. 3.6 and Ra2 in the form, Ra2 =

∞ Ra02 a 2  Cn . 2

(4.8)

n=1

The expression for Cn is essentially the same as in Sect. 3. The variation of Ra2c with ω for different values of , Pr D , and χ are depicted in Figs. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 and the results are discussed in the next section.

5 Result and Discussion In this article we make an analytical study of the effect of time-periodically varying temperature and gravity field on the onset of thermal convection in a Maxwell fluid-saturated porous layer using linear stability theory. The regular perturbation method based on small amplitude of modulation is employed to compute the critical value of Rayleigh number and wavenumber. The expression for the critical correction Rayleigh number Ra2c is computed as a function of the frequency of modulation, Deborah number, Darcy–Prandtl number, and normalized porosity and the effect of these parameters on the stability of the system is

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100 Symmetric temperature modulation Γ Γ Γ Γ

= 0.1 = 0.3 = 0.5 = 0.9 PrD = 5, χ = 0.1

80

ω

60

40

20

0 -0.01

0.00

0.03

0.06

0.09

Ra2c/Ra0c Fig. 1 Variation of Ra2c /Ra0c with ω for different values of Deborah number 150 Symmetric temperature modulation PrD = 5 PrD = 10 PrD = 50 PrD = 100 Γ = 0.1, χ = 0.1

ω

100

50

0 -0.05

0.0

0.1

0.2

0.3

0.4

0.5

Ra2c/Ra0c Fig. 2 Variation of Ra2c /Ra0c with ω for different values of Darcy–Prandtl number Pr D

discussed. The sign of Ra2c characterizes the stabilizing or destabilizing effect of modulation. A positive Ra2c indicates that the modulation effect is stabilizing while a negative Ra2c indicates that the modulation effect is destabilizing, compared to the system in which the modulation is absent. We present below the results for three different wall temperature oscillating mechanisms, where case a and b correspond to symmetric and asymmetric temperature modulation, respectively, and case c is for the lower wall temperature modulation only. Clearly, we observe that, the results of case a is different; but case b and c are similar. In Figs. 1, 2, and 3, the variation of critical correction Rayleigh number Ra2c with frequency ω for different governing parameters is revealed for the case of symmetric temperature modulation. We find from these figures that for small frequencies the critical correction

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M. S. Malashetty, I. Begum 200 Symmetric temperature modulation χ = 0.1 χ = 0.3 χ = 0.5 χ = 0.9

150

ω

PrD = 5, Γ = 0.1

100

50

0 -0.015 -0.01

0.00

0.01

0.02

0.03

0.04

0.05

Ra2c/Ra0c Fig. 3 Variation of Ra2c /Ra0c with ω for different values of normalized porosity χ

Rayleigh number Ra2c is negative indicating that the effect of symmetric modulation is destabilizing. On the other hand, for moderate and high frequencies the critical correction Rayleigh number Ra2c is positive indicating that the effect of symmetric modulation is to stabilize the system. There are two peak values of critical correction Rayleigh number Ra2c one positive and another negative. Let ω∗ represent the frequency at which Ra2c changes its sign from negative to positive, then the modulated system may be classified as destabilized or stabilized, compared with the unmodulated system, according as ω < ω∗ or ω > ω∗ . First Ra2c decreases to its maximum destabilizing value and then increases to its maximum stabilizing value to finally decrease to zero as the frequency increases from zero to infinity. The maximum stabilization or destabilization can be achieved at critical frequency ωc (positive/negative) depending on the value of the other parameters. Further, at some particular value of the frequency ω = ω0 , the effect of modulation ceases, i.e., Ra2c = 0. These critical frequencies depend on the parameters governing the system. Figure 1 depicts the effect of Deborah number on critical correction Rayleigh number Ra2c , for the fixed value of Darcy–Prandtl number and normalized porosity in respect of symmetric modulation. We observe from this figure that an increase in the value of increases the magnitude of Ra2c . At small frequencies Ra2c increases negatively, while Ra2c increases positively with the Deborah number at moderate and high frequencies indicating that the effect of Deborah number is to destabilize the system for small frequencies while its effect is to stabilize the system for moderate and high values of ω. It is apparent that the Deborah number reinforces the effect of the symmetric modulation. In Fig. 2 the variation of Ra2c with ω is displayed for different values of Darcy–Prandtl number with the value of Deborah number and normalized porosity kept fixed. This figure indicates that the peak negative value of Ra2c increases with increasing Pr D , and the range of frequencies in which Ra2c becomes positive increases with Pr D . Further, with an increase in ω, the critical correction Rayleigh number decreases much faster for smaller Darcy–Prandtl number. Thus, the Darcy–Prandtl number enhances the destabilizing effect at small frequencies while at moderate and high frequencies it enhances the stabilizing effect of symmetric modulation.

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20 Asymmetric temperature modulation Γ = 0.1 Γ = 0.3 Γ = 0.5 Γ = 0.9

15

ω

PrD = 5, χ = 0.1

10

5

0

0

1

2

3

4

5

5.5

Ra2c/Ra0c Fig. 4 Variation of Ra2c /Ra0c with ω for different values of Deborah number 100 Asymmetric temperature modulation PrD = 5 PrD = 10 PrD = 50 PrD = 100

80

Γ = 0.1, χ = 0.1

ω

60

40

20

0 -0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Ra2c/Raoc Fig. 5 Variation of Ra2c /Ra0c with ω for different values of Darcy–Prandtl number Pr D

We present in Fig. 3 the effect of normalized porosity χ on the critical correction Rayleigh number when other parameters are fixed in respect of symmetric modulation. From the figure it is evident that an increase in the value of χ increases the magnitude of the Ra2c . At small modulation frequencies, Ra2c is negative indicating that the normalized porosity destabilizes the system while at moderate and high frequency Ra2c is positive and there by indicating that it stabilizes the system. The normalized porosity χ has a strong influence on the oscillatory behavior of the convective flows and in the present case also χ has a stabilizing effect on the thermal modulation. The effect of asymmetric modulation of the wall temperatures on the onset of convection in a horizontal, porous layer saturated by a Maxwell fluid is, in general, to inhibit the onset of

123

902

M. S. Malashetty, I. Begum 100 Asymmetric temperature modulation χ = 0.1 χ = 0.3 χ = 0.5 χ = 0.9

80

PrD = 5, Γ = 0.1

ω

60

40

20

0 0.0

0.1

0.2

0.3

0.4

0.5

Ra2c/Ra0c Fig. 6 Variation of Ra2c /Ra0c with ω for different values of normalized porosity χ

convection over the whole range of frequencies (see Figs. 4, 5, 6) except for small frequencies and high Darcy–Prandtl number in which case the asymmetric modulation destabilizes the system (see Fig. 5). Figure 4 shows the variation of Ra2c with ω and the Deborah number

for fixed value of Darcy–Prandtl number and normalized porosity. We find from the figure that an increase in the value of increases the critical correction Rayleigh number indicating the stabilizing effect of Deborah number. The effect of Darcy–Prandtl number on the critical correction Rayleigh number when other parameters are fixed is shown in Fig. 5. We observe from this figure that as Pr D increases Ra2c increases indicating the stabilizing effect of Darcy–Prandtl number. It is important to note from this figure that for Pr D > 10 there is a small range of values of the frequencies for which Ra2c becomes negative indicating that the effect is destabilizing. The effect of normalized porosity χ on the stability of the system for the case of asymmetric temperature modulation for the fixed value of Pr D and is shown in Fig. 6. We observe from this figure that an increase in the value of χ increases the critical correction Rayleigh number Ra2c indicating that the normalized porosity χ enhances the stabilizing effect of modulation. The effect of lower wall temperature modulation is found to be qualitatively similar to the case of asymmetric modulation and we therefore omit the discussion of the same (see Figs. 7, 8, and 9). The effect of thermal modulation disappears for high frequencies irrespective of the type of thermal modulation. So far we discussed the effect of thermal modulation on the onset of convection. We shall now discuss the results of gravity modulation. The variation of critical correction Rayleigh number Ra2c with frequency ω for different parameters is shown in Figs. 10, 11, and 12. We find from these figures that Ra2c is positive over entire range of values of ω indicating that the effect of gravity modulation is to stabilize the system. However, there exists a small range of frequency over which gravity modulation has destabilizing effect for certain range of values of governing parameters. Figure 10 shows the variation of critical correction Rayleigh number Ra2c with frequency ω for different values of Deborah number for gravity modulation. We observe from this figure that as increases the value of the critical correction Rayleigh number Ra2c increases,

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Effect of Thermal/Gravity Modulation

903

30 Lower wall temperature modulation Γ Γ Γ Γ

= 0.1 = 0.3 = 0.5 = 0.9 PrD = 5, χ = 0.1

25

ω

20

15

10

5

0 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Ra2c/Ra0c Fig. 7 Variation of Ra2c /Ra0c with ω for different values of Deborah number 150 Lower wall temperature modulation PrD = 5 PrD = 10 PrD = 50 PrD = 100 Γ = 0.1, χ = 0.1

ω

100

50

0 -0.05

0.00

0.05

0.10

0.15

0.20

0.25

0.30

Ra2c/Ra0c Fig. 8 Variation of Ra2c /Ra0c with ω for different values of Darcy–Prandtl number Pr D

indicating the stabilizing effect of Deborah number on the gravity modulated fluid layer saturated porous medium. It is reported from this figure that for ≤ 0.3 the critical correction Rayleigh number Ra2c becomes negative over a small range of values frequencies; indicating that the gravity modulation is destabilizing for this range of frequencies. In Fig. 11 the effect of Darcy–Prandtl number Pr D on the stability of the system is displayed. It is found that the magnitude of critical correction Rayleigh number Ra2c increases negatively with increasing Pr D when frequency is small. However, the trend reverses for moderate and high frequency. Therefore, increase of Darcy–Prandtl number enhances the destabilizing effect of gravity modulation when frequency is small where as it enhances the stabilizing effect of modulation for moderate and high frequency. In Fig. 12 we display

123

904

M. S. Malashetty, I. Begum 100 Lower wall temperature modulation χ = 0.1 χ = 0.3 χ = 0.5 χ = 0.9

80

PrD = 5, Γ = 0.1

ω

60

40

20

0 0.00

0.02

0.04

0.06

0.08

0.10

0.12

Ra2c/Ra0c Fig. 9 Variation of Ra2c /Ra0c with ω for different values of normalized porosity χ 30 Gravity modulation 30

= 0.1 = 0.3 = 0.5 = 0.9 PrD = 5, χ = 0.1

20

25 20

ω

Γ Γ Γ Γ

25

15

Γ = 0.1

10 5 0 -0.05

0.00

0.05

0.10

0.15

0.20

ω

Ra2c/Ra0c

15

10

5

0

0

2

4

6

Ra2c/Ra0c Fig. 10 Variation of Ra2c /Ra0c with ω for different values of Deborah number

the effect of normalized porosity on the critical correction Rayleigh number when other parameters are fixed. We observe from this figure that as χ increases the value of Ra2c increases, indicating the stabilizing effect of normalized porosity on the gravity modulated fluid layer saturated porous medium. However, for a very small frequency and χ = 0.1 the critical correction Rayleigh number becomes negative indicating the destabilizing effect of normalized porosity. The analysis presented in this article is based on the assumptions that the amplitude of the modulation is very small and that the convective currents are weak so that nonlinear effects may be neglected. The violation of these assumptions would alter the results significantly only when the modulating frequency ω is small. This is due to the fact that the perturbation

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Effect of Thermal/Gravity Modulation

905

150 Gravity modulation PrD = 5 PrD = 10 PrD = 50 PrD = 100

100

ω

Γ = 0.1, χ = 0.1

50

0 -0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Ra2c/Ra0c Fig. 11 Variation of Ra2c /Ra0c with ω for different values of Darcy–Prandtl number Pr D 150 Gravity modulation χ = 0.1 χ = 0.3 χ = 0.5 χ = 0.9 Γ = 0.1, PrD = 5

ω

100

50

0 -0.05

0.0

0.1

0.2

0.3

0.4

0.5

0.55

Ra2c/Ra0c Fig. 12 Variation of Ra2c /Ra0c with ω for different values of normalized porosity χ

method imposes the condition that the amplitude of εw1 should not exceed that of w0 which in tern gives the condition ω > ε. Thus validity of the results obtained here depends on the value of frequency ω of modulation. When ω is sufficiently small (i.e., the period of modulation is large) the modulation affects the entire volume of the fluid and hence the disturbances grow large. However, for large values of ω the effect of modulation is confined only to narrow boundary layer near the boundary. Therefore, outside this thickness the temperature and gravity field takes a mean value leading to the equilibrium state of the unmodulated case. Thus the effect of modulation is significant only for the small and moderate values of ω. Since the amplitude of the modulation is an externally controlled parameter, the finite amplitude instabilities can be avoided by not allowing the amplitude to become very large.

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The amplitude of the convection currents, however, cannot be controlled, but rather is determined by the nonlinear interactions. It is important that the flow fields considered remain of small amplitude throughout a cycle of modulation, otherwise the assumption that the nonlinear terms are small is violated.

6 Conclusion The effect of thermal/gravity modulation on the onset of convection in a Maxwell fluid-saturated porous layer is studied using a linear stability analysis and the following conclusions are drawn: 1. The small frequency symmetric thermal modulation is destabilizing while moderate and high frequency symmetric modulation is always stabilizing. The asymmetric modulation and lower wall temperature modulations are, in general, stabilizing for all frequencies. However, in these two cases there is a range of small ω for which the system becomes unstable when Pr D > 10. Thus, the thermal modulation can destabilize a mode that is stable in the unmodulated case, or stabilize an unstable mode, with the stability characteristics depending on the parameters governing the system. 2. Gravity modulation has both stabilizing and destabilizing effect in a Maxwell fluid-saturated porous layer; stabilization or destabilization depends on the frequency and other governing parameters. 3. The effect of increase in Deborah number , Darcy–Prandtl number Pr D and normalized porosity χ is to reinforce the effect of thermal/gravity modulation. 4. High Darcy–Prandtl number has a strong influence on the stability of the system. 5. The effect of thermal/gravity modulation disappears for large frequency in all the cases. Acknowledgments This study is supported by University Grants Commission, New Delhi, under the Special Assistance Programme DRS Phase II. The authors thank the reviewers for their useful suggestions.

Appendix We solve Eq. 3.8 for w1 by expanding the right-hand side in a Fourier series expansion and inverting the operator L term by term. So we take eλz sin(mπ z) =

∞ 

gnm (λ)sin(nπ z).

(A1)

n=1

where 1 gnm (λ) = 2

eλz sin(nπ z) sin(mπ z)dz

0

  4nmπ 2 λ 1 + (−1)n+m+1 eλ  . = − (n − m)2 π 2 + λ2 (n + m)2 π 2 + λ2

(A2)

We now define M(ω, n) = B1 + iωB2 ,

123

(A3)

Effect of Thermal/Gravity Modulation

907

where   2  2 ω2  2 2 ω2  2 a + n2π 2 − a2 + n2π 2 + a2 + π 2 + a + n2π 2 , χ Pr D Pr D 2   2   2  2 2 2 2 2 2 2 2 2  a +n π a +n π

ω a + n π B2 = + − a2 + π 2 . − Pr D χ χ Pr D B1 =

(A4) (A5)

It is easily seen that L[sin(nπ z)e−iωt ] = M(ω, n)sin(nπ z)e−iωt . From Eq. 3.8, we obtain  Lw1 = Ra0 a Re (1 − iω ) 2

∞ 

(A6) 

[A(λ)gn1 (λ) + A(−λ)gn1 (−λ)] sin(nπ z)e

−iωt

, (A7)

n=1

so that



 ∞  Bn (λ) −iωt sin(nπ z)e , w1 = Ra0 a Re (1 − iω ) M(ω, n) 2

(A8)

n=1

where Bn (λ) = A(λ)gn1 (λ) + A(−λ)gn1 (−λ).

(A9)

The equation for w2 can then be written as

  ∂ f w1 . Lw2 = −Ra2 a 2 w0 + Ra0 a 2 1 + ∂t

To simplify Eq. A10 for w2 , we need   ∂ 1+ f w1 = Re {1 − 2iω } f w1 . ∂t

(A10)

(A11)

Equation A10 then reads as Lw2 = −Ra2 a 2 w0 + Ra0 a 2 Re {1 − 2iω } f w1 .

(A12)

We do not require the solution of Eq. A12 but shall use it to determine the correction Rayleigh number Ra2 . The solvability condition requires that the time-independent part of right-hand side of Eq. A12 be orthogonal to w0 = sin(π z). To that end we multiply the right-hand side of Eq. A12 by sin(π z) and integrate between the limits 0 and 1 to obtain 

1 Ra2 = 2Ra0 Re 1 − 2iω f w1 sin(π z)dz,

(A13)

0

where the over bar indicates time average. Now from Eq. 3.8 we obtain,   ∞  |Bn (λ)|2 Ra0 a 2 2 f w1 sin(π z) = Re (1 + iω ) sin (nπ z) 2 M(ω, n) n=1

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so that 1

  ∞  |Bn (λ)|2 Ra0 a 2 f w1 sin(π z)dz = Re (1 + iω ) . 4 M(ω, n)

(A14)

n=1

0

Substituting Eq. A14 into Eq. A13, we get Ra2 =

∞   Ra02 a 2  |Bn (λ)|2 Re (1 + iω )(1 − 2iω )M ∗ (ω, n) . 2 2 |M(ω, n)| n=1

(A15)

Now,   Re (1 + iω )(1 − 2iω )M ∗ (ω, n) = B1 B3 − ωB2 B4 , and |M(ω, n)|2 = B12 + (ωB2 )2 , where B3 = 1 + 2ω2 2 and B4 = ω . Thus, we may write Ra2 =

∞ Ra02 a 2  |Bn (λ)|2 Cn , 2

(A16)

n=1

where 16π 4 n 2 ω2 B1 B3 − ωB2 B4   , Cn = |Bn (λ)|2 =  2 . (A17) ω + (n + 1)4 π 4 ω2 + (n − 1)4 π 4 B12 + (ωB2 )2 References Bardan, G., Mojtabi, A.: On the Horton–Rogers–Lapwood convcetive instability with vertical vibration: onset of convection. Phys. Fluids 12, 2723 (2000) Bhadauria, B.S.: Thermal modulation of Rayleigh–Benard convection in a sparsely packed porous medium. J. Porous Media 10(2), 175–188 (2007) Caltagirone, J.P.: Stabilite d’une couche poreuse horizontale soumise a des conditions aux limite periodiques. Int. J. Heat Mass Transf. 19, 815–820 (1976) Davis, S.H.: The stability of time periodic flows. Ann. Rev. Fluid Mech. 8, 57–74 (1976) Gershuni, G.Z., Zhukhovitskii, E.M., Iurkov, I.S.: On convcetive stability in the prescence of periodically varying parameter. J. Appl. Math. Mech. 34, 442–452 (1970) Govender, S.: Stability of convection in a gravity modulated porous layer heated from below. Transp. Porous Media 57, 113 (2004) Gresho, P.M., Sani, R.L.: The effect of gravity modulation on the stability of a heated fluid layer. J. Fluid Mech. 40, 783–806 (1970) Horton, C.W., Rogers, F.T.: Convection currents in a porous medium. J. Appl. Phys. 16, 367–370 (1945) Lapwood, E.R.: Convection of a fluid in a porous medium. Proc. Cambridge Phil. Soc. 44, 508–521 (1948) Lowire W.: Fundamentals of Geophysics University Press, Cambridge (1997) Malashetty, M.S., Padmavathi, V.: Effect of gravity modulation on the onset of convection in porous layer. J. Porous Media 1(3), 219–226 (1997) Malashetty, M.S., Wadi, V.S.: Rayleigh Benard convection subjected to time dependent wall temperature in a fluid saturated porous layer. Fluid Dyn. Res. 24, 293–308 (1999) Malashetty, M.S., Basavaraja, D.: Effect of thermal/gravity modulation on the onset of convection in a horizontal anisotropic porous layer. Int. J. Appl. Mech. Eng. 8(3), 425–439 (2003) Malashetty, M.S., Basavaraja, D.: Effect of time-periodic boundary temperatures on the onset of double diffusive convection in a horizontal anisotropic porous layer. Int. J. Heat Mass Transf. 47, 2317–2327 (2004) Malashetty, M.S., Basavaraja, D.: Rayleigh–Benard convection subject to time dependent wall temperature/gravity in a fluid saturated anisotropic porous medium. Heat Mass Transf. 38, 551–563 (2002) Malashetty, M.S., Shivakumara, I.S., Kulkarni, S., Swamy, M.: Convective instability of Oldroyd-B fluid saturated porous layer heated from below using a thermal non-equilibrium model. Transp. Porous Media 64, 123–139 (2006)

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Malashetty, M.S., Siddheshwar, P.G., Swamy, M.: Effect of thermal modulation on the onset of convection in a viscoelastic fluid saturated porous layer. Transp. Porous Media 62, 55–79 (2006) Nelson, E.S.: An examination of anticipation of g-jitter on space station and its effects on materials processes, NASA TM, 103775 (1991) Rogrers, J.R., Pesch, W., Brausch, O., Schatz, M.F.: Complex-ordered patterns in shaken convection. Phys. Rev. E. 71, 066214-1-18 (2005) Roppo, M.H., Davis, S.H., Rosenblat, S.: Benard convection with time periodic heating. Phys. Fluids 27, 796– 803 (1984) Rosenblat, S., Herbert, D.M.: Low frequency modulation of thermal instability. J. Fluid Mech. 43, 385– 389 (1970) Rosenblat, S., Tannaka, G.A.: Modulation of thermal convection instability. Phys. Fluids 14, 1319–1322 (1971) Rudraiah, N., Malashetty, M.S.: Effect of modulation on the onset of convection in a sparsely packed porous layer. ASME J. Heat Transf. 122, 685–689 (1990) Rudraiah, N., Radhadevi, P.V., Kaloni, P.N.: Effect of modulation on the onset of thermal convection in a viscoelastic fluid-saturated sparsely packed porous layer. Can. J. Phys. 68, 214–221 (1990) Saravanan, S., Purusothaman, A.: Floquet instability of a gravity modulated Rayleigh–Benard problem in an anisotropic porous medium. Int. J. Therm. Sci. 48, 2085 (2009) Saravanan, S., Sivakumara, T.: Onset of filtration convection in a vibrating medium: the Brinkman model. Phys. Fluids 22, 034104(1–15) (2010) Shivakumara, I.S., Jinho, Lee., Malashetty, M.S., Sureshkumar, S.: Effect of thermal modulation on the onset of convection in Walters B viscoelastic fluid-saturated porous medium. Transp. Porous Media 87, 291– 307 (2011) Vadasz, P.: Coriolis effect on gravity-driven convection in a rotating porous layer heated from below. J. Fluid Mech. 376, 351–375 (1998) Venezian, G.: Effect of modulation on the onset of thermal convection. J. Fluid Mech. 35, 243–254 (1969) Wadih, H., Zahibo, N., Roux, B.: Effect of gravity jitter on natural convection in a vertical cylinder. In: Koster, J.N., Sani, R.L. (eds.) Low Gravity Fluid Dynamics and Transport Phenomena, pp. 309–354. AIAA, New York (1990) Wadih, M., Roux, B.: Natural convection in a long vertical cylinder under gravity modulation. J. Fluid Mech. 193, 391–415 (1988)

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