Transp Porous Med (2013) 98:59–79 DOI 10.1007/s11242-013-0133-2
Effect of Thermal Modulation on the Onset of Convection in a Porous Medium Layer Saturated by a Nanofluid J. C. Umavathi
Received: 3 December 2012 / Accepted: 29 January 2013 / Published online: 13 February 2013 © Springer Science+Business Media Dordrecht 2013
Abstract The effect of time-periodic temperature modulation at the onset of convection in a Boussinesq porous medium saturated by a nanofluid is studied analytically. The model used for the nanofluid incorporates the effects of Brownian motion. Three types of boundary temperature modulations are considered namely, symmetric, asymmetric, and only the lower wall temperature is modulated while the upper wall is held at constant temperature. The perturbation method is applied for computing the critical Rayleigh and wave numbers for small amplitude temperature modulation. The shift in the critical Rayleigh number is calculated as a function of frequency of modulation, concentration Rayleigh number, porosity, Lewis number, and thermal capacity ratio. It has been shown that it is possible to advance or delay the onset of convection by time-periodic modulation of the wall temperature. The nanofluid is found to have more stabilizing effect when compared to regular fluid. Low frequency is destabilizing, while high frequency is always stabilizing for symmetric modulation. Asymmetric modulation and only lower wall temperature modulation is stabilizing for all frequencies when concentration Rayleigh number is greater than one. Keywords
Thermal modulation · Nanofluid · Onset of convection · Porous medium
List of Symbols c cp (ρc)m dp g DB DT
Nanofluid specific heat at constant pressure Specific heat of the nanoparticle material Effective heat capacity of the porous medium Nanoparticle diameter Gravitational acceleration Brownian diffusion coefficient (m2 /s) Thermophoretic diffusion coefficient (m2 /s)
J. C. Umavathi (B) Department of Mathematics, Gulbarga University, Gulbarga 585 106, Karnataka, India e-mail:
[email protected]
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hp H jp jp,T k kB km kp Le NA NB p∗ p q Ra Rm Rn t∗ t T∗ T Tc∗ Th∗ TR (u, v, w) v vD ∗ vD VT (x, y, z) (x ∗ , y ∗ , z ∗ )
J. C. Umavathi
Specific enthalpy of the nanoparticle material Dimensional layer depth (m) Diffusion mass flux for the nanoparticles Thermophoretic diffusion Thermal conductivity of the nanofluid Boltzman’s constant Effective thermal conductivity of the porous medium Thermal conductivity of the particle material Lewis number Modified diffusivity ratio Modified particle-density increment Pressure Dimensionless pressure, p ∗ K /μαm Energy flux relative to a frame moving with the nanofluid velocity v Thermal Rayleigh–Darcy number Basic-density Rayleigh number Concentration Rayleigh number Time Dimensionless time, t ∗ αm /σ H 2 Nanofluid temperature T ∗ −T ∗ Dimensionless temperature, T ∗ −Tc∗ c h Temperature at the upper wall Temperature at the lower wall Reference temperature Dimensionless Darcy velocity components (u ∗ , v ∗ , w ∗ ) H/αm Nanofluid velocity Darcy velocity εv Dimensionless Darcy velocity (u ∗ , v ∗ , w ∗ ) Thermophoretic velocity Dimensionless Cartesian coordinate (x ∗ , y ∗ , z ∗ )/H ; z is the vertically upward coordinate Cartesian coordinates
Greek symbols αm
Thermal diffusivity of the porous medium,
β˜ ε εt μ μ˜ ρ ρp σ φ∗ φ Ω
Proportionality factor Porosity of the medium Amplitude of the modulation Viscosity of the fluid Effective viscosity of the porous medium Fluid density Nanoparticle mass density Heat capacity ratio Nanoparticle volume fraction φ ∗ −φ ∗ Relative nanoparticle volume fraction, φ ∗ −φc∗ c h Dimensional frequency
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km
(ρc p )f
Effect of Thermal Modulation on the Onset of Convection
ω ψ
61
Dimensionless frequency =Ω H 2 /k Phase angle: ψ = 0, symmetric modulation; ψ = π, antisymmetric modulation; ψ = −i∞, only lower wall temperature modulation
1 Introduction Natural convection, or buoyancy driven convection, is the heat removal strategy adopted in a wide variety of industries ranging from transportation, HVAC, and energy production and supply to electronics, textiles and paper production, geophysical problems, nuclear reactors to name a few (Choi 1999). Conventional heat transfer liquids have low thermal conductivity. Nanofluids are mixtures of base fluid such as water or ethylene glycol with a very small amount of nanoparticles such as metallic or metallic oxide particles (Cu, Cuo, Al2 O3 , SiO, TiO), having dimensions from 1 to 100 mm, with very high thermal conductivities. It was Choi (1995) who christened the term “nanofluid”. A significant feature of nanofluids is thermal conductivity enhancement, a phenomenon which was first reported by Masuda et al. (1993). The phenomenon suggest the possibility of using nanofluids in advanced nuclear systems (Buongiorno and Hu 2005). Another recent application of nanofluid is nano-drug delivery Kleinstreuer et al. (2008). Eastman et al. (2004) conducted a comprehensive review on thermal transport in nanfluids to conclude that a satisfactory explanation for the abnormal enhancement in thermal conductivity and viscosity of nanofluids needs further studies. Buongiorno (2006) conducted a comprehensive study to account for the unusual behavior of nanofluids based on inertia, Brownian diffusion thermophoresis, diffusiophoresis, Magnus effects, fluid drainage and gravity settling, and proposed a model incorporating the effects of Brownian diffusion and the thermophoresis. With the help of these equations, studies were conducted by Tzou (2008a,b), Kim et al. (2004, 2006, 2007), and more recently by Nield and Kuznetsov (2009, 2010). Natural convection in nanofluids is still poorly understood compared to forced convection (Abu-Nada 2009). For example, Khanafer et al. (2003) reported an increase in heat transfer in Cu-water nanofluids in a two-dimensional rectangular enclosures with the increase in concentration of suspended nanoparticles, while Putra et al. (2003) reported that in natural convection, using Al2 O3 and CuO nanofluids the heat transfer coefficient was smaller than that in a clear fluid. Wen and Ding (2006) reported a reduction in heat transfer after changing a clear fluid to a nanofluid while Abu-Nada et al. (2008) showed the enhancement of heat transfer in natural convection in nanofluids at higher values of the Rayleigh number. All this suggests that more research in this area is needed, especially due to the prospects of application of nanofluids in various electronic cooling devices (Ghasemi and Aminossadati 2009). Because of their unique properties as heat transfer fluids, nanofluids are being looked up on as great coolants of the future. Thus studies need to be conducted involving nanofluids in porous media and without it. Kuznetsov and Nield (2010c) studied the onset of thermal instability in a porous medium saturated by a nanofluid using Brinkman model and incorporating the effects of Brownian motion and thermophoresis of nanoparticles. They found that the critical thermal Rayleigh number can be reduced or increases by a substantial amount, depending on whether the basic nanoplarticle distribution is top-heavy or bottom-heavy by the presence of the nanoparticles. The same Horton–Rogers–Lapwood problem was investigated by Nield and Kuznetsov (2009) for the Darcy model. The effect of local thermal non-equilibrium among the particle, fluid, and solid-matrix phases was investigated using a three-temperature model by Kuznetsov and Nield (2010a). They came to the conclusion that in some circum-
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stances the effect of LTNE can be significant, but for a typical dilute nanofluid (with lagre Lewis number and with small particle–two-fluid heat capacity ratio) the effect was small. The onset of double-diffusive nanofluid convection in a layer of a saturated porous medium was analysed by Kuznetsov and Nield (2010b). They concluded that for the case when Soret and Dufour parameters are negligible, the non-oscillatory mode was expected when nanoparticle Rayleigh number was positive, a situation which physically corresponds to the fact that for oscillations to occur two of the buoyancy forces have to be in opposite directions. The effect of local thermal non-equilibrium on the onset of convection in a porous medium layer saturated by a nanofluid using Brinkman model was also studied by Kuznetsov and Nield (2011). Nield and Kuznetsov (2011) found analytically the effect of vertical throughflow on the onset of convection in a horizontal layer of a porous medium saturated by a nanofluid. Thermal instability in a porous medium layer saturated with a viscoelastic nanofluid was analyzed by Sheu (2011). His results indicated that there was a computation among the processes of thermophoresis, Browinian diffusion, and viscoelasticity that causes the convection to set in through oscillatory rather than stationary modes. Agarwal et al. (2011) and Agarwal and Bhaduria (2011) studied thermal instability in a rotating porous layer saturated by a nanofluid for top-heavy and bottom-heavy suspension considering Darcy model. Bhaduria and Agarwal (2011a,b) studied the effect of local thermal non-equilibrium on linear and nonlinear thermal instability in a horizontal porous medium saturated by a nanofluid. Agarwal et al. (2012) analyzed double-diffusive convection in a horizontal porous medium saturated by a nanofluid, for the case when the base fluid of the nanofluid is itself a binary fluid such as salty water. Recently, Chand and Rana (2012) studied the onset of thermal convection in rotating nanofluid layer saturated by a Darcy–Brinkman porous medium. Boundary and internal source effects on the onset of Darcy–Brinkman convection in a porous layer saturated by nanofluid was studied by Yadav et al. (2012). Nield and Kuznetsov (2012) studied the linear stability theory for the Horton–Rogers–Lapwood problem for the porous medium saturated by a nanofluid with thermal conductivity and viscosity dependent on the nanoparticle volume fraction. They found these parameters increase the critical value the Rayleigh number when compared to constant viscosity and thermal conductivity. One of the effective mechanism to control convection is by maintaining a non-uniform temperature gradient across the boundaries. The non-uniform temperature gradient may be generated by (i) appropriate heating or cooling at the boundaries, (ii) through flow, (iii) appropriate distribution of heat sources, and (iv) radiative heat transfer (see e.g., Rudraiah and Malashetty 1990). These are concerned only with space-dependent temperature gradient. However, in many practical problems, the non-uniform temperature gradient is a function of both space and time. This is to be determined by solving energy equation with suitable time-dependent temperature boundary conditions. There are many studies available in the literature concerning how a time-periodic boundary temperature affects the onset of Rayleigh–Benard convection. Most of the findings related to these problems have reviewed by Davis (1976). In case of small amplitude temperature modulations, a linear stability analysis was performed by Venezian (1969). He has established that the onset of convection can be delayed or advanced by the out of or in phase modulation of the boundary temperatures, respectively as compared to the unmodulated system. Rosenblat and Herbert (1970), found the asymptotic solution of the low frequency and arbitrary amplitude thermal modulation problem. The solution is discussed from the viewpoint of the stability or otherwise of the basic state, and possible stability criteria are analyzed. They have also made some comparison with known experimental results. Rosenblat and Tanaka (1971) have also studied the effect of thermal modulation on the onset of Rayleigh–Benard convection when the temperature gradient has both a steady and time periodic component. It has
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Effect of Thermal Modulation on the Onset of Convection
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been found that, in general, there is enhancement of the critical value for a suitably defined Rayleigh number. Finucane and Kelly (1976) performed both theoretical and experimental investigation of the thermal modulation in a horizontal fluid layer. A numerical analysis of the linear stability equations indicated that the linear assumption is valid at the low frequencies of modulation. A nonlinear analysis employing the shape assumption and free boundary conditions was developed and examined numerically. They found both experimentally and numerically that at low frequencies the modulation is destabilizing, whereas at high frequencies some stabilization is apparent. The above-mentioned studies have reported that the effect of thermal modulation is to alter the critical value of Rayleigh number by comparison with the unmodulated, steady case. Malashetty and Basavaraja (2002, 2004, 2003) and recently Shivakumara et al. (2011) have studied the effect of thermal modulation on the onset of convection in a horizontal fluid layer. All these investigations are restricted to a viscous fluid. To our knowledge the studies on the effect of temperature modulation on the convection in a horizontal porous layer saturated by a nanofluid are not available in the literature. The main objective of this work is to study the effect of time-periodic boundary temperature on the onset of convection in a horizontal porous layer saturated by a nanofluid. 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. Therefore, temperature modulation can be used as a mechanism to delay convection to achieve higher efficiencies in case of material processing applications and advance it for achieving major enhancement of mass, momentum and heat transfer for nanofluids. These results can also be applicable in industrial sectors, transportation, electronics, medical, energy, and the environment.
2 Mathematical Formulation We consider a nano-fluid saturated porous layer, confined between two infinite horizontal plates situated at z ∗ = 0 and z ∗ = H . We select a coordinate frame in which the z-axis is aligned vertically upward. Further, in addition to a fixed temperature difference between the walls, an additional perturbation is applied to the wall temperatures, varying sinusoidally in time. Thus, the wall temperatures are 1 Tt + T [1 + εt cos (Ωt)] at z ∗ = 0 2 1 T1 − T [1 − εt cos (Ωt + φ)] at z ∗ = H, 2
(1) (2)
where εt represents a small amplitude, Ω the frequency of modulation and φ the phase angle. The Oberbeck–Boussinesq approximation is employed. The Buongiorono model treats the nanofluid as a two-component mixture (base fluid plus nanoparticles) with the assumptions that the flow is incompressible, there is no chemical reactions with negligible external forces, viscous dissipation, radiative heat transfer for dilute mixture. It is also assumed that the nanoparticle and base fluid are in local thermal equilibrium. The conservation equations take the form ∇ ∗ · v∗D = 0.
(3)
∗ is the nanofluid Darcy velocity. We write v∗ = (u ∗ , v ∗ , w ∗ ). Here vD D
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The conservation equation for the nanoparticles, in the absence of thermophoresis diffusion is given by (Nield and Kuznetsov 2009) 1 ∗ ∂φ ∗ + vD · ∇φ ∗ = ∇ ∗ · DB ∇ ∗ φ ∗ , ∗ ∂t ε
(4)
where φ ∗ is the nanoparticle volume fraction, ε is the porosity, T ∗ is the temperature, and DB is the Brownian diffusion coefficient. If one introduces a buoyancy force and adopts the Boussinesq approximation, and uses the Darcy model for a porous medium, then the momentum equation can be written as 0 = −∇ ∗ p ∗ −
μeff ∗ v + ρg. K D
(5)
Here ρ is the overall density of the nanofluid, which we now assume to be given by ρ = φ ∗ ρp + 1 − φ ∗ ρ0 1 − βT T ∗ − T0∗ ,
(6)
where ρp is the particle density, ρ0 is a reference density for the fluid, and βT is the thermal volumetric expansion. The thermal energy equation for a nanofluid can be written as (ρc)m
∂T ∗ ∗ + (ρc)f vD · ∇ ∗ T ∗ = km ∇ ∗2 T ∗ + ε (ρc)p DB ∇ ∗ φ ∗ · ∇T ∗ . ∂t ∗
(7)
The conservation of nanoparticle mass requires that 1 ∗ ∂φ ∗ + vD · ∇ ∗ φ ∗ = DB ∇ ∗2 φ ∗ . ∗ ∂t ε
(8)
Here c is the fluid specific heat (at constant pressure), km is the overall thermal conductivity of the porous medium saturated by the nanofluid, and c p is the nanoparticle specific heat of the material constituting the nanoparticles. ∗ = (u ∗ , v ∗ , w ∗ ). We write vD In the context of modeling transport in porous media, Eqs. (3) and (4) are standard. Equation (8) involves just intrinsic quantities in the sense that the average is being taken over the nanofluid only and the solid matrix is not involved. The question thus reduces to whether the terms within the square brackets on the right-hand side of Eq. (7) need modification. We recall that in nanofluids the particles are so small that for practical purpose they remain in suspension in a uniform manner. We emphasize our assumption that the nanoparticles are suspended in nanofluid using either surfactant or surface charge technology, something that prevents particles from agglomeration and deposition on the porous matrix. We suggest that then it is reasonable to assume as a first approximation that no modification to Eq. (7) is necessary (following Kuznetsov and Nield 2010a,b; Nield and Kuznetsov 2012). We assume that the volumetric fractions of the nanoparticles are constant on the boundaries. Thus, the boundary conditions are w ∗ = 0, φ ∗ = φ0∗ at z ∗ = 0 ∗
∗
w = 0, φ =
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φ1∗
∗
at z = H.
(9) (10)
Effect of Thermal Modulation on the Onset of Convection
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2.1 Basic State The basic state is quiescent and the temperature Tb , density ρb , and the pressure pb satisfy → ρb − g + ∇ pb = 0 ∂T ∗ (ρc)m ∗ = km ∇ 2 T ∗ ∂t d2 φb∗ = 0. dz 2
(11) (12) (13)
Following Nield and Kuznetsov (2009), Eqs. (12) and (13) are considered from (5) and (6). The solutions of Eq. (12) satisfying the thermal conditions given by (1) and (2) is Tb = T1 (z) + εt T2 (z, t) T 2z where T1 (z) = 1− 2 H
T2 (z, t) = Re b (λ) eλz/H + b (−λ) e−λz/H e−iωt 1/2 (ρc)m ωH 2 with λ = (1 − i) 2km −iφ − e−λ T e b (λ) = 2 eλ − e−λ
(14) (15) (16) (17)
and Re stands for real part. The expression for pb and ρb is not given as they are not explicitly required in the subsequent analysis. 2.2 Perturbation Solution We now superimpose perturbations on the basic solution. We write v = v ,
p = pb + p , T = Tb + T , φ = φb + φ ,
(18)
where v , p , T and φ represents the perturbed quantities. We introduce dimensionless variables as follows. We define (x, y, z) = x ∗ , y ∗ , z ∗ /H, t = t ∗ αm /σ H 2 , (u, v, w) = u ∗ , v ∗ , w ∗ H /αm , p = p ∗ K /μαm , φ ∗ − φ0∗ T ∗ − Tc∗ σΩ H2 , T = ∗ , ω= , φ= ∗ ∗ ∗ φ1 − φ0 Th − Tc αm
(19)
where αm =
ρc p m km , σ = . ρc p f ρc p f
(20)
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Substituting Eqs. (19) and (20) in (3)–(8), and linearised by neglecting products of primed quantities. The following equations are obtained 0 = −∇ p − v + Ra T eˆ z − Rn φ eˆ z ∂T ∂2T ∂T ∂φ ∂ Tb NB ∂ Tb ∂ Tb +w = + + + ∂t ∂z ∂z 2 Le ∂z ∂z ∂z ∂z 1 ∂φ 1 1 + w = ∇ 2 φ σ ∂t ε Le w = 0, T = 0, φ = 0 at z = 0, 1.
(21) (22) (23) (24)
where the parameter Le is a Lewis number, and Ra is the familiar thermal Rayleigh–Darcy number. The new parameter Rn may be regarded as a concentration Rayleigh number. NB is a modified particle density increment. In deriving Eq. (21), Oberbeck–Boussinesq approximation is used (neglecting a term proportional to the product of φ and T ). This assumption is likely to be valid in the case of small temperature gradients in a dilute suspension of nanoparticles. For the case of regular fluid (not a nanofluid), the parameters Rn , and NB are zero. The remaining equations are reduced to the familiar equations for the Horton–Roger– Lapwood problem. According to Buongiorno (2006), for most nanofluids investigated so far Le/ φ1∗ − φ0∗ is large, of order 105 –106 , and since the nanoparticle fraction decrement ∗ φ1 − φ0∗ is typically no smaller than 10−3 this means that Le is large, of order 102 –103 . Also Bhaduria and Agarwal (2011b) have taken the value of Le = 10. The six unknowns u , v , w , p , T , φ can be reduced to three by operating on (21) with eˆ z .curl curl and using the identity curl curl ≡ grad div − ∇ 2 together with (3). The result is ∇ 2 w = −Ra∇H2 T + Rn∇H2 φ .
(25)
Here ∇H2 is the two-dimensional Laplacian operator on the horizontal plane. Combining equations (25), (22), and (23), we obtain equation for the vertical component of velocity w in the form (dropping prime)
∂ 1 ∂ Rn ∂ ∇2 − ∇2 − ∇2w + − ∇ 2 ∇12 w ∂t σ ∂t Le ε ∂t
∇2 ∂ Tb 1 ∂ (26) − ∇12 w = 0. −Ra ∂z σ ∂t Le The dimensional basic temperature gradient is given by ∂ Tb = −1 + ε f. ∂z Here, f is the modulation temperature gradient and is given by f = Re A (λ) eλz + A (−λ) eλz e−iωt , where
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λ 2
(28)
e−iϕ − e−λ eλ − e−λ σ ω 1/2 . λ = (1 − i) 2
A (λ) =
(27)
(29)
Effect of Thermal Modulation on the Onset of Convection
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3 Method of Solution We obtain the perturbation technique to obtain eigen functions w and eigen values Ra of Eq. (26) for the basic temperature distribution, which departs from the linear profile (∂ Tb /∂z = −1) by quantities of order εt . Thus, the eigen values of present problem differ from those of porous media saturated with nanofluid analogue of Nield and Kuznetsov (2009) by quantities of order εt . Since the adopted technique is based on small amplitudes, εt has to be less than unity. We therefore assume the solution of (26) in the form w = w0 + εt w1 + εt2 w2 + · · · Ra =
Ra0 + εt2
(30)
Ra2 + · · · ,
(31)
where Ra0 is the critical Rayleigh number for the unmodulated convection in an porous medium saturated with nanofluid. Substituting Eqs. (30) and (31) into (26) and equating coefficients of like powers of εt , we obtain the following system of equations (32) Lw0 = 0 Ra1 2 2 Ra0 ωG 2 Ra0 f Lw1 = (33) ∇ ∇1 − ∇1 + ∇12 w0 Le σ Le f f ωG ωG Ra1 2 Ra2 2 ∇ − Ra0 + ∇ 2 ∇12 w1 + Ra1 + ∇2 + ∇ ∇12 w0 , Lw2 = Le σ Le σ Le Le (34) where
L=
∂ − ∇2 ∂t
1 ∂ ∇2 − σ ∂t Le
Rn ∇ + ε 2
∂ Ra0 2 2 2 − ∇ ∇12 − ∇ ∇1 ∂t Le
We now assume the marginally stable solutions for Eq. (32) in the form w0 = sin (nπ z) exp [i (lx + my)] , n = 1, 2, 3, . . .
(35)
and y-directions, respectively, such that l 2 +m 2
= where l and m are the wave numbers in the x α 2 . The corresponding eigen values are given by 2 2 2 n π + α2 Rn Le Ra0 = . (36) − α2 ε For a fixed value of the wave number α, the least eigen value occurs at n = 1 and is given by 2 2 π + α2 Rn Le − Ra0 = (37) α2 ε We note that Ra0 attains its critical value, Ra0c at α = αc = π, where Rn Le . (38) ε We note that, as far as the steady state is concerned, there is no distinction between the Rayleigh number obtained by Nield and Kuznetsov (2009) for non-oscillatory convection in the absence of thermophoresis diffusion (NA = 0) and the expression obtained as in Eq. (38). Equation (33) is inhomogeneous and its solution poses a problem due to the presence of resonance terms. The solvability condition requires that time-independent part of right-hand side of (33) should be orthogonal to w0 . The term independent of time on the right-hand is Ra0c = 4π 2 −
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J. C. Umavathi
Ra1 2 2 Le ∇ ∇1 w0
so that Ra1 = 0. It follows that all the odd coefficients, i.e., Ra1 , Ra3 , . . . in Eq. (31), must vanish. If we expand the right-hand side of Eq. (33) in a Fourier series of the form eλz sin (mπ z) =
∞
gnm (λ) sin (nπ z)
(39)
n=1
Then 1 gnm (λ) = 2 0
−4nmπ 2 λ 1 + (−1)n+m+1 e z eλz sin (mπ z) sin (nπ z) dz = λ2 + (n + m)2 π 2 λ2 + (n − m)2 π 2 (40)
We thus obtain
L sin (nπ z) e−iωt = L (ω, n) sin (nπ z) e−iωt ,
(41)
where 2 2 3 n π + α2 ω2 2 2 Rn 2 2 2 2 n π +α − L (ω, n) = − α n π + α2 σ Le ε Rn Le α2 2 2 n π + α 2 4π 2 + + Le ε 2 2 2 2 2 2 2 n π +α n π + α2 Rn α 2 +iω + + . Le σ ε From Eq. (33), we have Ra0 α 2 ω A (λ) gn1 (λ) + A (−λ) gn1 (−λ) e−iωt I.P. Lw1 = σ n Ra0 n 2 π 2 + α 2 α 2 −iωt − A (λ) gn1 (λ) + A (−λ) gn1 (−λ) e R.P. Le n (42) We obtain w1 , by inverting the operator L term by term, in the form Bn (λ) Ra0 α 2 ω w1 = I.P. σ L (ω, n) 2 2 2 2 Bn (λ) Ra0 n π + α α − R.P. sin (nπ z) e−iωt , Le L (ω, n)
(43)
where Bn (λ) = A (λ) gn1 (λ) + A (−λ) gn1 (−λ) The solution of the homogeneous equation corresponding to (42) involves a term proportional to sin (π z). However, addition of such term to the complete solution of Eq. (42) merely amounts to a renormalization of w0 because all the terms proportional to sin (π z) can then
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be grouped to define a new w0 with corresponding definition for w1 , w2 , . . .. Hence, we can assume that w0 is orthogonal to all other wn ’s. From Eq. (34), we get ωG ∇2 ∇2 f (44) Lw2 = −Ra0 + ∇12 w1 + R2 · ∇12 w0 σ Le Le We shall not require the solution of this equation but merely use it to determine R2 . The solvability condition requires that the time-independent part of the right-hand side of (44) must be orthogonal to sin (π z). Multiplying Eq. (47) by sin (π z) and integrating between 0 and 1, we obtain 2Le Ra0 R2 = ∇2
1 0
ωG ∇2 f + w1 sin (π z) dz, Le σ
(45)
where the upper bar denotes the time average. From Eq. (33), we have f w1 sin (π z) =
1 w1 Lw1 Ra0 a 2
(46)
Using Eqs. (42) and (43) and finding the time average, we obtain w1 Lw1 , which yields from Eqs. (45) and (46). 2 2 n π + α 2 |Bn |2 Ra02 α 2 ω2 R2 = 2 2 + L (ω, n) + L ∗ (ω, n) , σ2 Le2 4 n π + α2 |L (ω, n)|2 (47) L ∗ (ω, n)
is the complex conjugate of L (ω, n). The critical value of R2 , denoted where by R2c , is obtained at the wave number given by equation αc = π for the following three different cases. Case (i) Oscillatory wall temperature field is symmetric (φ = 0). Case (ii) Oscillatory wall temperature field is asymmetric (φ = π ). Case (iii) Only lower wall temperature is modulated and upper one is held at constant temperature (φ = −i∞). Case (i): Oscillatory wall temperature field is symmetric (φ = 0). The oscillatory temperature field is symmetric when φ = 0 and it is found that 16n 2 π 2 ω2 |bn |2 (say), if n is even |Bn (λ)|2 = (48) = 2 4 0, if n is odd ω2 + (n + 1) π 4 ω2 + (n − 1) π 4 The R2c =
Roc αc2 A |bn |2 2 , 2 A + B2 n
(49)
where A = Re [L (ω, n)]
(50)
and B = Im [L (ω, n)]
(51)
The summation in Eq. (49) extends over even values of n. Case (ii) Oscillatory wall temperature field is asymmetric (φ = π). This case is corresponding to out of phase temperature modulation with φ = π and we obtain |bn |2 (say), if n is odd |Bn (λ)|2 = 0, if n is even
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J. C. Umavathi 300
ω
200
In-phase temperature modulation Rn = 0 Rn = 0.1 Rn = 1 Rn = 2 Rn = 3 Le = 10 ε = 0.9 σ = 10
100
0 -0.4
-0.3
-0.2
-0.1
0.0
0.1
R2C Fig. 1 Variations of R2c with ω for different values of Rn
200 In-phase temperature modulation
ω
150
Rn = 0 Rn = 1 Le = 10 σ = 10
ε = 0.5
100
50
10
5
0 -0.8
-0.6
-0.4
-0.2
0.0
0.2
R2C Fig. 2 Variations of R2c with ω for different values of Rn and ε
The R2c is same as in Eq. (49) with the summation extending over odd values of n only. Case (iii) Only lower wall temperature is modulated while the upper one is held at constant temperature (φ = −i∞). 2 This is the case corresponds to φ = −i∞ and we have |Bn (λ)|2 = |b4n | . Again R2c is same as in Eq. (49) but the summation extends over all values of n. The variation of R2c with ω for different physical parameter is shown in Figs. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12, and the results are discussed in the next section.
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250 In-phase temperature modulation Rn = 0 200 Rn = 1 ε = 0.9 σ = 10
ω
150
100 Le = 10 50
20
30
20
Le = 10
30 0 -0.4
-0.3
-0.2
-0.1
0.0
0.1
0.2
R2C Fig. 3 Variations of R2c with ω for different values of Rn and Le 300
ω
200
In-phase temperature modulation Rn = 0 Rn = 0 Le = 10 ε = 0.9
15 100
10
σ=5
0 -0.5
-0.4
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
R2C Fig. 4 Variations of R2c with ω for different values of Rn and σ
4 Results and Discussion The effect of time-periodic temperature modulation on the onset of convection in a horizontal layer of a porous medium saturated by a nanofluid is investigated. A perturbation technique with amplitude of the modulating temperature as a perturbation parameter is used to find the critical thermal Rayleigh number as a function of frequency of the modulation, concentration Rayleigh number (Rn ), porosity (ε), Lewis number (Le), and heat capacity ratio (σ ). The stability of the system is characterized by the sign of correction critical Rayleigh number (R2c ). A positive and negative R2c , respectively represents a stabilizing and destabilizing effect of thermal modulation on the system as compared to unmodulation system.
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J. C. Umavathi 300 Out-of-phase temperature modulation Rn = 0 Rn = 0.1 Rn = 1 Rn = 2 Rn = 3 Le = 100 ε = 0.9 σ = 10
ω
200
100
0 -0.2
0.0
0.2
0.4
0.6
R2C Fig. 5 Variations of R2c with ω for different values of Rn
The analytical expression defined for R2c is computed for various values of physical parameters for the following three cases. (i) Oscillating wall temperature field is symmetric, i.e., wall temperatures are modulated in phase (φ = 0) (ii) Oscillating wall temperature filed is asymmetric, i.e., wall temperature are modulated out of phase modulation (φ = π) and (iii) Only the temperature of the bottom wall is modulated, the upper wall being held at a fixed constant temperature, (φ = −i∞) The results obtained for the above are shown graphically in Figs. 1, 2, 3, and 4 for φ = 0, Figs. 5, 6, 7, and 8 for φ = π and Figs. 9, 10, 11, and 12 for φ = −i∞ for the case of regular fluid (Rn = 0) and for nanofluid (Rn = 1). Figure 1 shows the variation of R2c with ω, for different values of concentration Rayleigh number Rn . Following Nield and Kuznetsov (2009), Rn > 0 indicates top heavy nanoparticles and Rn < 0 indicates bottom heavy nanoparticles. Therefore the same analysis is considered in this work. For both regular and nanofluid, for small frequencies R2c is negative indicating that the symmetric modulation has destabilizing effect while for moderate and large values of frequency its effect is stabilizing. The peak value of R2c occurs around ω = 41.616 for destabilizing effect and at ω = 101.69 for stabilizing effect for both regular and nanofluid for Rn = 0.1. It is also noted from Fig. 1 that as the concentration Rayleigh number Rn increases, the magnitude of correction Rayleigh number R2c decreases indicating that the effect of Rn is to delay the onset of convection. Besides the curves for different values of Rn are close to zero when the modulation frequency is small. Hence, the modulation has the little effect on the stability of the system when ω approaches zero value. As ω increases, |R2c | increases to its maximum value initially and then starts decreasing with further increase in ω. When ω is very large, all the values for different Rn coalesce and |R2c | approaches to zero. This means that the modulation with large frequency will have no substantial effect on the stability characteristics of the system. Figure 1 also shows that the peak negative and positive value of R2c decreases with an increase in the value of Rn . The peak value of R2c is
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150
Out-of-phase temperature modulation Rn = 0 Rn = 1 Le = 10 ε = 0.9 100 σ = 10
ω
ε= 1
50
5
10
0 -5
-4
-3
-2
-1
0
1
R2C Fig. 6 Variations of R2c with ω for different values of Rn and ε 300
ω
200
Out-of-phase temperature modulation Rn = 0 Rn = 1 ε = 0.9 σ = 10
100
0
Le = 100, 150, 200
Le = 100, 150, 200
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
0.20
R2C Fig. 7 Variations of R2c with ω for different values of Rn and Le
observed for regular fluid when compared to nanofluid. Therefore the onset of convection is delayed for nanofluid when compared to regular fluid. Therefore nanofluid is having more stabilizing effect when compared to regular fluid. The effect of porosity (ε) on the stability of the system in the presence of symmetric modulation is shown in Fig. 2. We observe that as ε increases the value of |R2c | becomes large indicating that the large value of ε increases the effect of modulation. Here also the peak value of |R2c | is large for regular fluid when compared to nanofluid. That is the effect of porosity shows more stabilizing nature for nanofluid when compared with regular fluid and its influence is more for values of ε < 1 compared with the values of ε > 1. Malashetty and Basavaraja (2002) considered the value for porous parameter (k2 ) in the range of 1 to
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ω
200
Out-of-phase temperature modulation Rn = 0 Rn = 1 Le = 100 ε = 0.9
100 15 10
σ=5 σ = 5, 10, 15
0 -0.6
-0.4
-0.2
0.0
0.2
R2C Fig. 8 Variations of R2c with ω for different values of Rn and σ
10. Following Malashetty and Basavaraja (2002), in this paper also the porosity ε is chosen from 1 to 10. It is also observed that as ω increases |R2c | increases to its maximum value initially and then starts decreasing with further increase in ω. When ω is very large, all the curves for different porosity ε coincide and |R2c | approaches to zero which is the similar nature observed as in Fig. 1. Figure 3 depicts the variation R2c with frequency ω, for different values of Lewis number Le for the case of symmetric modulation. It can be seen that an increase in the value of Lewis number decreases the value of |R2c | indicating that, the effect of increasing Le is to reduce the effect of thermal modulation for regular and nanofluid. It is also observed that the peak value of |R2c | for regular fluid is more than that for nanofluid. As ω increases |R2c | increases to its maximum value initially and then starts decreasing with further increase in ω. When ω is large, all the curves for different Lewis number coincide and |R2c | approaches to zero for both regular and nanofluids. The effect of thermal capacity ratio σ and ω on R2c is similar to the effect of Lewis number as seen in Fig. 4 for symmetric modulation of the wall temperature. That is as σ increases, |R2c | decreases for both regular and nanofluid. The values of σ are chosen followed by Sheu (2011). The peak value of R2c is observed for regular fluid compared to nanofluid. The results obtained for the case of asymmetric modulation are presented in Figs. 5, 6, 7, and 8. It is seen that, the effect is stabilizing over the whole range of frequencies for nanofluid with Rn > 1 as seen in Fig. 5. For regular fluid and for Rn = 0.1, for small frequencies R2c is negative indicating that the asymmetric modulation has destabilizing effect while for moderate and large value of frequency its effect is stabilizing. It is observed that as Rn increases, the value of R2c also increases, indicating that the effect of increasing Rn is to make the system more stable. Thus nanofluid is to delay the onset of convection compared to the regular fluid. The effect of ω on R2c is similar as in Fig. 1 for regular fluid and for Rn < 1. The effect of porosity ε on R2c for asymmetric modulation is shown in Fig. 6. Its effect is similar to symmetric modulation as seen in Fig. 2. That is, as ε increases, the value of |R2c | also increases for both regular and nanofluid. The peak value of R2c is for regular
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300 Gravity modulation
ω
200
Rn = 0 Rn = 0.1 Rn = 1 Rn = 2 Rn = 3 Le = 50 ε = 0.9 σ = 10
100
0 -0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
R2C Fig. 9 Variations of R2c with ω for different values of Rn 150
ω
100
Gravity modulation Rn = 0 Rn = 1 Le = 10 σ = 10
50
ε = 10 1
5 0 -6
-5
-4
-3
-2
-1
0
1
R2C Fig. 10 Variations of R2c with ω for different values of Rn and ε
fluid when compared to nanofluid. The effect of ω on R2c is similar as that for symmetric modulation (Fig. 2) as seen in Fig. 6. The effect of Lewis number on |R2c | is shown in Fig. 7 for asymmetric modulation. It is observed that the value of R2c increases with increase in Lewis number. The effect of increasing Lewis number is to delay the onset of convection for both regular and nanofluid. It is also observed from the figure that the effect of Lewis number is stabilized over the whole range of the frequencies. For regular fluid, for small frequencies R2c is negative indicating that the asymmetric modulation has destabilizing effect, while for moderate and large values of frequency its effect is stabilizing.
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J. C. Umavathi 300 Gravity modulation Rn = 0 Rn = 1 ε = 0.9 σ = 10
ω
200
100 Le = 10 100 Le = 10
50
100
50
0 -1.2
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
R2C Fig. 11 Variations of R2c with ω for different values of Rn and Le 300
ω
200
Gravity modulation Rn = 0 Rn = 1 ε = 0.9 Le = 100
σ = 10, 20, 30 100
σ = 10, 20, 30 0 -0.4
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
R2c Fig. 12 Variations of R2c with ω for different values of Rn and σ
The effect of thermal capacity ratio σ and ω on R2c for asymmetric modulation are shown in Fig. 8. Its effect is similar to symmetric modulation (Fig. 4). That is as σ increases. |R2c | decreases for both regular and nanofluid. The peak value of R2c is observed for regular fluid compared to nanofluid. The variation of R2c as a function of ω for different values of concentration Rayleigh number Rn and porosity ε is shown in Figs. 9 and 10, respectively, for only lower temperature modulation. From these figures, it is evident that the effect of increase of Rn and ε has quantitatively similar effect as that for a symmetric temperature modulation (Figs. 5, 6). That is the effect of increasing Rn decreases the magnitude of the correction Rayleigh number for values of Rn < 1, whereas for values of Rn > 1, the effect is stabilizing over the whole range
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of frequencies. An increase in the value of Rn increases the value of R2c . As the porosity ε increases, |R2c | also increases for both regular and nanofluid. The effect of Lewis number on |R2c | for only lower wall temperature modulation is seen in Fig. 11. As Lewis number increases |R2c | decreases for both regular and nanofluid which is the similar result observed for symmetric modulation (Fig. 3). The effect of thermal capacity ratio σ on R2c for only lower wall temperature modulation is the similar nature observed for asymmetric modulation (Fig. 8). That is, for regular fluid, as σ increases |R2c | decreases, indicating that the effect of increasing σ is to make the system more stable. For nanofluid, the effect is stabilizing over the whole range of frequencies. For nanofluid also, as σ increases the value of R2c increases.
5 Conclusion The effect of thermal modulation on onset of convection in a horizontal layer of porous medium saturated by a nanofluid is studied using a linear stability analysis and the following conclusions are drawn. • Low frequency symmetric modulation is destabilizing while high frequency symmetric modulation is always stabilizing in all types of modulations considered. • The large values concentration Rayleigh number is found to be stabilizing in the presence of thermal modulation. That is the nanofluid is found to be more stabilizing compared to regular fluid in all types of modulation considered. • The effect of increasing the porosity is to destabilize the system for low frequencies in all types of modulations. • The effect of Lewis number is found to be stabilizing and the large Lewis number fluid systems are more stable in the presence of thermal modulation. • The large values of thermal capacity ratio is to stabilize the system in all types of modulations considered for both regular and nanofluids. • Asymmetric modulation and only lower wall temperature modulation is stabilizing for all frequencies for values of concentration Rayleigh number greater than 1 and for all values of thermal capacity ratio for nanofluid. For regular fluid and the values of Rn < 1 the effect is similar to symmetric modulation. The results of this study indicate that the onset of convection is delayed for nanofluid when compared to regular fluid. The porosity advances the onset of convection, whereas Lewis number and thermal capacity ratio delay the onset of convection for low frequencies and is always stabilizing for high frequencies in all types of modulations considered.
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