Transp Porous Med (2011) 87:585–602 DOI 10.1007/s11242-010-9702-9

Natural Convection in a Nanofluid Saturated Rotating Porous Layer: A Nonlinear Study B. S. Bhadauria · Shilpi Agarwal

Received: 14 November 2010 / Accepted: 17 December 2010 / Published online: 12 January 2011 © Springer Science+Business Media B.V. 2011

Abstract The present paper deals with linear and nonlinear analysis of thermal instability in a rotating porous layer saturated by a nanofluid. Momentum equation with Brinkman term, involving the Coriolis term and incorporating the effect of Brownian motion along with thermophoresis has been considered. Linear stability analysis is done using normal mode technique, while for nonlinear analysis, a minimal representation of the truncated Fourier series, involving only two terms, has been used. Stationary and oscillatory modes of convection have been studied. A weak nonlinear analysis is used to obtain the concentration and thermal Nusselt numbers. The behavior of the concentration and thermal Nusselt numbers is investigated by solving the finite amplitude equations using a numerical method. Obtained results have been presented graphically and discussed in details. Keywords Nanofluid · Porous medium · Natural convection · Rotation · Horton–Roger–Lapwood problem · Brinkman model List of Symbols Variables DB Brownian diffusion coefficient DT Thermophoretic diffusion coefficient Da Darcy number Pr Pradtl number d Dimensional layer depth kT Effective thermal conductivity of porous medium km Thermal diffusivity of porous medium

B. S. Bhadauria (B) · S. Agarwal Department of Mathematics, Faculty of Science, DST-Centre for Interdisciplinary Mathematical Sciences, Banaras Hindu University, Varanasi 221005, India e-mail: [email protected] S. Agarwal e-mail: [email protected]

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Le NA NB p g Ra Rm Rn t T Tc Th v vD (x ∗ , y ∗ , z ∗ ) Ta Greeks α β ε μ μ¯ ρf ρp (ρc)f (ρc)m (ρc) p γ φ ν ψ α ω

Lewis number Modified diffusivity ratio Modified particle-density increment Pressure Gravitational acceleration Thermal Rayleigh–Darcy number Basic density Rayleigh number Concentration Rayleigh number Time Temperature Temperature at the upper wall Temperature t the lower wall Nanofluid velocity Darcy velocity εv Cartesian coordinates Taylor number

Horizontal wave number Proportionality factor Porosity Viscosity of the fluid Effective viscosity of the porous medium Fluid density Nanoparticle mass density Heat capacity of the fluid Effective heat capacity of the porous medium Effective heat capacity of the nanoparticle material m Parameter defined as (ρc) (ρc)f Nanoparticle volume fraction Kinematic viscosity μ/ρf Stream function Wave number Frequency of oscillation

Subscript b Basic solution Superscripts * Dimensional variable  Perturbation variable Operators ∇2 ∇12

∂2 ∂x2 ∂2 ∂x2

+ +

123

∂2 ∂ y2 ∂2 ∂z 2

+

∂2 ∂z 2

Natural Convection in a Nanofluid Saturated Rotating Porous Layer

587

1 Introduction Effective cooling and heating techniques are required in most of the industries such as power manufacturing, transportation, electronics, etc. Cooling techniques are greatly needed for cooling any sort of high energy device. It was found that common fluids have limited heat transfer capabilities because of their low heat transfer capacity. However, some metals are found to have very high thermal conductivity, may be three to four times higher than the common fluids. Therefore, it is required to make a substance by combining these two, which will behave like a fluid and have thermal conductivity of a metal. Nanofluids are such substances, made by suspending the nanoparticles (having diameter 1–100 nm) in the common fluids, called base fluids. Presence of small amount of these nanoparticles in the base fluids increases the thermal conductivity of the fluids by 15–40%. Prior to the use of nano-sized particles of metals and metal oxides to make nanofluids, some research was done on the effects of putting millimeter- or micrometer-sized particles inside fluid. Although these particles improved the thermal conductivity of the fluid, they created other problems such as settling, producing drastic pressure drops, clogging channels, and premature wear on channels and components. However, nano-sized particles have an advantage over milli and micro-sized particles because they approach the size of the molecules in the fluid. This helps the nanoparticles not settling down due to gravity and thus avoid clogging and wearing of channels. 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 ), having dimensions from 1 to 100 nm. It was Choi (1995) who first proposed this term “nanofluid”. Studies pertaining to thermal conductivity enhancement by nanofluids have been conducted by Masuda et al. (1993), Eastman et al. (2001), Das et al. (2003), and others. They claimed a 10–30% increase in thermal conductivity by using very low concentrations of nanofluids. These reports led Buongiorno and Hu (2005) to suggest the possibility of using nanofluids in advanced nuclear systems. Another recent application of the nanofluid flow is in the delivery of nano-drug as suggested by Kleinstreuer et al. (2008). Eastman et al. (2004) made a comprehensive review on thermal transport in nanofluids and concluded that despite several attempts, a satisfactory explanation for the abnormal enhancement in thermal conductivity and viscosity in nanofluids is yet to be found. Various other studies have thus been conducted to determine the governing mechanisms in nanoscale, including a modified Maxwell model accounting for the ordered nanolayer near the particle–fluid interface by Yu and Choi (2003), Brownian motion of nanoparticles in fluids by Jang and Choi (2004) and Kumar and Murthy (2005), ballistic nature of heat transport within nanoparticles by Keblinski and Cahill (2005) and Chen (2001), and thermal lagging in nanoparticles with a large surface-area-to-volume ratio by Vadasz (2006). Buongiorno (2006) conducted an extensive study of convective transport in nanofluids, but focused on explaining the further heat transfer enhancements observed during convective situations. He finally suggested a new model based on the mechanics of nanoparticles/base-fluid relative velocity, and derived the conservation equations based on Brownian diffusion and thermophoresis. Based on this model, studies have been conducted by Kim et al. (2004, 2006, 2007), and Tzou (2008a,b). Convection in porous media is of practical applications in modern science and engineering, including food and chemical processes, rotating machineries like nuclear reactors, petroleum industry, biomechanics and geophysical problems. Convection in porous medium has been studied by many authors including Horton and Rogers (1945), Lapwood (1948), Nield (1968), Rudraiah and Malashetty (1986), Murray and Chen (1989), Malashetty (1993),

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Bhadauria (2007a), Vafai (2005), Nield and Bejan (2006). Since nanofluids are being looked upon as great coolants of the future, studies need to be conducted involving nanofluids in porous media and without it. There are several studies available in which phenomena related to the onset of convection in a rotating porous medium have been investigated. Few of them are: Pearlstein (1981), Chakrabarti and Gupta (1981), Patil and Vaidyanathan (1983) Patil et al. (1990), Qin and Kaloni (1995), Vadasz (1998), Desaive et al. (2002), Govender (2006, 2008), Maharaj and Saneshan (2005), Riahi (2005). Bhadauria (2007b,c, 2008a), Bhadauria and Suthar (2008b), and Vanishree and Siddheshwar (2010). The Rayleigh B´enard problem of thermal instability in a horizontal nanofluid with local thermal non-equilibrium was investigated by Nield and Kuznetsov (2010). The corresponding problem for porous medium (Horton–Rogers–Lapwood problem) was investigated by Nield and Kuznetsov (2009) using Darcy model, and extended by Kuznetsov and Nield (2010a) for local thermal non-equilibrium. Further Kuznetsov and Nield (2010b) studied thermal instability in a horizontal porous medium saturated by nanofluid, considering Brinkman model and using both free–free and rigid–rigid boundaries. They also studied double diffusive convection in a porous medium layer (Kuznetsov and Nield 2010c). Agarwal and Bhadauria (2010) studied the same problem of thermal instability in a rotating porous layer saturated by a nanofluid for top heavy and bottom heavy suspension for the Darcy Model. Assuming that the nanoparticles being suspended in the nanofluid using either surfactant or surface charge technology, preventing the agglomeration and deposition of these on the porous matrix, in the present article, we study the linear and nonlinear thermal instability in a rotating porous medium saturated by nanofluid, using Horton–Roger–Lapwood problem based on the Brinkman’s Model.

2 Governing Equations We consider a porous layer saturated by a nanofluid, confined between two horizontal boundaries at z = 0 and z = d, heated from below and cooled from above. The boundaries are impermeable and perfectly thermally conducting. The porous layer is extended infinitely in x and y-directions, and z-axis is taken vertically upward with the origin at the lower boundary. The porous layer is rotating uniformly about z-axis with uniform angular velocity . The Coriolis effect has been taken into account by including the Coriolis force term in the momentum equation, whereas, the centrifugal force term has been considered to be absorbed into the pressure term. In addition, the local thermal equilibrium between the fluid and solid has been considered, thus the heat flow has been described using one equation model. Th and Tc are the temperatures at the lower and upper walls respectively such that Th > Tc . Employing the Oberbeck–Boussinesq approximation, the governing equations to study the thermal instability in a nanofluid saturated rotating porous medium are Buongiorno (2006), Kuznetsov and Nield (2010a,b); ∇ · vD = 0 ρf ∂vD μ 2 = −∇ p + μ∇ ¯ 2 vD − vD + (vD × ) ε ∂t K δ   + φρp + (1 − φ) {ρ(1 − β(T − Tc ))} g   DT ∂T + (ρc)f vD · ∇T = km ∇ 2 T + ε(ρc) p DB ∇φ · ∇T + ∇T · ∇T (ρc)m ∂t Tc

123

(1)

(2) (3)

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589

∂φ 1 DT 2 + vD · ∇φ = DB ∇ 2 φ + ∇ T ∂t ε Tc

(4)

where vD = (u, v, w) is the Darcy velocity. Assuming temperature and volumetric fraction of the nanoparticles to be constant at the stress-free boundaries, we may assume the boundary conditions on T and φ to be: v = 0, T = Th , φ = φ1 at z = 0,

(5)

v = 0, T = Tc , φ = φ0 at z = d.

(6)

where φ1 is greater than φ0 . The dimensionless variables are considered as given below: (x ∗ , y ∗ , z ∗ ) = (x, y, z)/d, t ∗ = tkT /γ d 2 (u ∗ , v ∗ , w ∗ ) = (u, v, w)d/kT , p ∗ = pK /μkT φ − φ0 T − Tc φ∗ = T∗ = φ1 − φ0 Th − Tc km where kT = (ρc) ,γ = f dropping the asterisk)

(ρc p )m (ρc p )f .

The non dimensionalized governing equations are (after

∇ ·v = 0 (7) √ Da ∂v ˆ − Rm eˆz + RaT eˆz − Rnφ eˆz (8) = −∇ p + Da∇ 2 v − v + T a(v × k) Pr ∂t NA NB NB ∂T + v · ∇T = ∇ 2 T + + ∇φ · ∇T + ∇T · ∇T (9) γ ∂t Le Le 1 1 2 ∂φ NA 2 + v · ∇φ = ∇ φ+ ∇ T (10) ∂t ε Le Le v = 0, T = 1, φ = 1 at z = 0, (11) v = 0, T = 0, φ = 0 at z = 1

(12)

The dimensionless parameters in the above equations are  2 K 2 , is the Taylor’s number νδ μ , is the Prandtl number, ρf kT μK ¯ , is the Darcy’s number, μd 2 kT , is the Lewis number, DB ρgβ K d(Th − Tc ) , is the Rayleigh−Darcy number, μk T   ρp φ0 + ρ(1 − φ0 ) gK d , is the basic density Rayleigh number, μk T (ρp − ρ)(φ1 − φ0 )gK d , is the concentration Rayleigh number, μk T ε(ρc) p (φ1 − φ0 ) , is the modified particle density increment, (ρc)f

 Ta = Pr = Da = Le = Ra = Rm = Rn = NB =

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and NA =

DT (Th − Tc ) , is the modified diffusivity ratio which is similar DB Tc (φ1 − φ0 ) to the Soret parameter in cross diffusion thermal instability.

3 Basic Solution Basic solutions of the above problem is given by v = 0, T = Tb (z), φ = φb (z).

(13)

Substituting Eq. 13 in Eqs. 9 and 10, we get d2 Tb NA NB N B dφb dTb + + dz 2 Le dz dz Le



dTb dz

2 =0

(14)

Using an order of magnitude analysis, Kuznetsov and Nield (2010a) showed that the second and third terms in Eq. 14 are small, therefore we have: d2 φb d2 Tb = 0, = 0. 2 dz dz 2 The dimensionless boundary conditions are

(15)

Tb = 1, φb = 1, at z = 0,

(16)

Tb = 0, φb = 0, at z = 1.

(17)

Solving Eq. 15, subject to conditions (16) and (17), we obtain: Tb = 1 − z

(18)

φb = 1 − z.

(19)

4 Stability Analysis We now superimpose perturbations on the basic state as given below: v = v , p = pb + p  , T = Tb + T  , φ = φb + φ  .

(20)

For the sake of simplicity of calculations, we consider the case of two-dimensional rolls, assuming all physical quantities to be independent of y. Substituting the expression (20) in Eqs. 8–10, and using the expressions (18) and (19), eliminating the pressure term, and introducing the stream function, we obtain the system of nonlinear perturbed equations as  2      Da ∂ Da ∂ ∂ψ − Da∇ 2 + 1 ∇ 2 − = − Da∇ 2 + 1 Ra∇12 T − Rn∇12 φ Pr ∂t ∂x Pr ∂t   ∂ 2 ∂ψ (21) +T a 2 ∂z ∂x ∂(ψ, T ) ∂T ∂ψ γ + = ∇2T + (22) ∂t ∂x ∂(x, z)   1 ∂ψ 1 2 NA 2 ∂(ψ, φ) ∂φ + = ∇ φ+ ∇ T+ (23) ∂t ε ∂x Le Le ∂(x, z)

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Equations 21–23 are solved subject to stress-free, isothermal, iso-nanoconcentration boundary conditions given by ∂ 2ψ = T = φ = 0 at z = 0, 1. (24) ∂z 2 The linear stability analysis is well studied and reported by Kuznetsov and Nield (2010b). The critical Rayleigh numbers for stationary and oscillatory onset of convection and the frequency of oscillations are given by 2    1 + Daδ 2 δ 4 + T aπ 2 δ 2 Le   (25) Ra st = Rn − NA + ε α 2 1 + Daδ 2     Le  2 δ2  Rn osc 4 4 2 2 2 Da   ω γ Le + δ − N A δ + 2 1 + Daδ δ − ω γ Ra = ε α Pr ω2 Le2 + δ 4   2   T aπ 2 Da 2 2 (26) +  2  ωDa 2  ω γ Pr + 1 + Daδ δ α 2 1 + Daδ 2 + ψ=

ω2 =

−X 2 +



Pr

X 22 − 4X 1 X 3

(27)

2X 1

where

   Da 2 δ + γ 1 + Daδ 2 Pr    2  γ − Le  Da 2 Da X 2 = Rnα 2 Leδ 2 2 + N A + T aπ 2 Le2 γ 1 + Daδ 2 − δ Pr ε Pr    2   Da Da 2 δ2 δ 4 + Le2 1 + Daδ 2 +δ 2 γ 1 + Daδ 2 + Pr Pr 2      Da 2  2 2 γ − Le 2 2 2 4 2 X 3 = Rn Leα δ + T aπ δ γ 1 + Daδ − + N A 1 + Daδ δ ε Pr    2 Da 2  δ γ 1 + Daδ 2 + +δ 6 1 + Daδ 2 Pr X 1 = δ2

Da 2 Le2 Pr 2

and δ2 = π 2 + α2 It is clear from Eq. 27 that oscillatory convection is possible only when X 22 − 4X 1 X 3 > 0

(28)

To perform a nonlinear stability analysis we take the following Fourier expressions: ψ = T = φ=

∞ ∞   n=1 m=1 ∞ ∞   n=1 m=1 ∞  ∞ 

Amn sin(mαx) sin(nπ z)

(29)

Bmn (t) cos(mαx) sin(nπ z)

(30)

Cmn (t) cos(mαx) sin(nπ z)

(31)

n=1 m=1

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We take the modes (1,1) for stream function, and (0,2) and (1,1) for temperature and nanoparticle concentration as ψ = A11 (t) sin(αx) sin(π z)

(32)

T = B11 (t) cos(αx) sin(π z) + B02 (t) sin(2π z)

(33)

φ = C11 (t) cos(αx) sin(π z) + C02 (t) sin(2π z)

(34)

where the amplitudes A11 (t), B11 (t), B02 (t), C11 (t), and C02 (t) are functions of time and are to be determined. Substituting Eqs. 32–34 in Eqs. 21–23 taking the orthogonality condition with the eigenfunctions associated with the considered minimal model, we get dA11 (t) = D11 (t) dt

 1 2 dB11 (t) =− δ B11 (t) + α A11 (t) + πα A11 (t)B02 (t) dt γ  dB02 (t) 1  απ A11 (t)B11 (t) − 8π 2 B02 (t) = dt 2γ 

 απ dC11 (t) α NA 2 C 11 (t) =− A11 (t)C02 (t) + A11 (t) + δ + B11 (t) dt ε ε Le Le

  1 απ NA C02 (t) dC02 (t) = A11 (t) C02 (t) − 8π 2 + C11 (t) dt 2 ε Le Le

dD11 (t) 1 = T2 (RnC02 (t) − Ra B11 (t)) − T1 (t)A11 (t) dt P2 (Da/Pr )2   Da Da 2 2 δ D11 (t) + 2 D11 (t) −P2 2 Pr Pr

 Da 2 Ra  2 + α δ B11 (t) + α A11 (t) + πα A11 (t) B02 (t) Pr γ   α δ2 πα C02 (t) A11 (t) + A11 (t) + −Rn (C02 (t) + N A B11 (t)) ε ε Le

(35) (36) (37) (38) (39)

(40) The above system of simultaneous autonomous ordinary differential equations can be solved numerically using some numerical methods. The six-mode differential Eqs. 35–40 has an interesting property in phase-space: ∂ A˙11 ∂ B˙11 ∂ B˙02 ∂ C˙11 ∂ C˙02 ∂ D˙11 + + + + + ∂ A11 ∂ B11 ∂ B02 ∂C11 ∂C02 ∂ D11   2 δ 4π 2 δ2 4π 2 2P2 Da(Daδ 2 − 1) <0 =− + + + + γ γ Le Le Pr

(41)

which indicates that the system is dissipative and bounded. One may also conclude that the trajectories of Eqs. 35–40 will be confined to the finiteness of the ellipsoid. Thus the effect of the parameters Rn, Le, N A , Pr, Da, ε, and γ on the trajectories is to contract them to a point. To conclude, it can be said that the set of initial points in the phase space occupies a region V (0) at time t = 0.

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However, for steady non-linear analysis, Eqs. 35–40 can be solved exactly to get the solution in terms of A11 as −W2 + W22 − 4W1 W3 A211 (42) = 8 2W1 α A11  B11 = − 2 (43) δ + α 2 A211 /8  α 2 A211 /8  (44) B02 = −  2 π δ + α 2 A211 /8    α 2 N A A211 /8 δ2 N A ε 2 Leα A11 1  − −     C11 = 2 2 ε ε δ + α 2 Le2 A211 /8 Le δ 2 + α 2 A211 /8 ε δ 2 + α 2 A211 /8 C02 =

εLe2 α 2

π





A211 /8   ε 2 δ 2 + α 2 Le2 A211 /8

 Le



δ2 N δ2

+ α2

A



A211 /8

−

 α 2 N A A211 /8  +  2 π δ + α 2 A211 /8

1 − ε





(45) 

2 A A11 /8  2  ε δ 2 + α 2 A11 /8

α2 N

(46)

where W1 = Le2 αc4 T1  W2 = T1 α 2 δ 2 ε 2 + Le2 + T2 Rn Leεα 3 (N A + 1) − T2 Raα 3 Le2 W3 = T1 ε 2 δ 4 − T2 Rnαδ 2 ε (N A ε − Le) − T2 Raαδ 2 ε 2 and P2 = αδ 2  2 T1 = Daδ 2 + 1 P2 + T aαπ 2  T2 = Daδ 2 + 1 α 2 .

Equating the discriminant in the Eq. 42 to zero, we rearrange the same to obtain the steady state finite amplitude Rayleigh number in the form: Q 4 − Q 24 − 4Q 3 Q 5 (47) Ra f = 2Q3 where  Q 1 = T1 δ 2 α 2 ε 2 + Le2 + T2 Rn Leεα 3 (N A + 1) Q 2 = T1 δ 4 ε 2 − T2 Rnαδ 2 ε (εN A − Le) Q 3 = T22 Le4 ε 6 Q 4 = 4T1 T2 α 5 Le2 ε 2 δ 2 − 2T2 Q 1 Le2 ε 3 Q 5 = Q 21 − 4T1 Q 2 α 4 Le2 .

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5 Heat and Nanoparticle Concentration Transport The thermal Nusselt number, N u(t) is defined as Heat transport by (conduction + convection) Heat transport by conduction ⎤ ⎡ 2π/αc  ∂ T dx  ∂z  ⎦ = 1 + ⎣ 0 2π/αc ∂ Tb 0 ∂z dx

N u(t) =

(48)

z=0

Substituting Eqs. 18 and 33 in Eq. 48, we get N u f (t) = 1 − 2π B02 (t).

(49)

The nanoparticle concentration Nusselt number, N u φ (t) is defined similar to the thermal Nusselt number. Following the procedure adopted for arriving at N u(t), one can obtain the expression for N u φ (t) in the form: N u φ (t) = (1 − 2πC02 (t)) + N A (1 − 2π B02 (t)).

(50)

6 Results and Discussion First, we discuss the marginally stable state. For it, from (25) we have the expression for Ra st convection, i.e. when ω = 0, as 2    1 + Daδ 2 δ 4 + T aπ 2 δ 2 Le st   (51) − NA + Ra = Rn ε α 2 1 + Daδ 2 when we have taken a first term Galerkin approximation, i.e., N = 1 in Eqs. 29–31. When T a = 0, we will get     1 + Daδ 2 δ 4 Le − NA + Ra st = Rn (52) ε α2 This result agrees with the result obtained by Kuznetsov and Nield (2010b). When Da = 0, the minimum is attained at α = π, with   Le st − N A + 4π 2 Ra = Rn (53) ε On the other √ hand , when Da is very large in comparison with unity, the minimum is attained at α = π/ 2, and the value becomes   Le 27π 4 − NA + Da (54) Ra st = Rn ε 4 So, we can say that in the absence of nanoparticles we get an already established result that the critical Rayleigh–Darcy number is equal to 4π 2 when Da = 0, and when Da is very large, it comes out to be 27π 4 /4. Thus, we see that on applying a single term Galerkin approximation we get the same standard results without any overestimate. Since the value of Le is very large , and that of N A is quite small, so the coefficient of Rn is large and positive in Eq. 51. So, as a result of the approximations made so far, we can say that the value of critical Rayleigh Number is increased by a considerable amount

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by the presence of nanoparticles, in the case when Rn is positive, the nanoparticle distribution being a bottom heavy one. Also, the coefficient of T a would also be contributing to increase the value of critical Rayleigh number, thus bringing a stabilizing effect on the system. If we write Eq. 53 in the form   Le Ra st + Rn N A − (55) = 4π 2 ε to get the left hand side as a linear combination of the thermal Rayleigh number, Ra, and the concenteration Rayleigh number Rn, we get an analogous situation of the double

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 1 Neutral stability curves for different values of a Rn, b Le, c N A , d Da, e T a, and f γ

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diffusive problem dealt by Nield and Bejan (2006) and stated by Nield and Kuznetsov (2009). It can also be compared with the bioconvection problem dealt by Kuznetsov and Avramenko (2004). We depict the results, corresponding to the linear stability, in Fig. 1a–f for the Rayleigh numbers corresponding to both stationary and oscillatory convection. In figures, we present neutral stability curves for Ra st and Ra osc versus the wave number α for Rn = 10, Le = 10, N A = 10, T a = 10, Da = 0.1, ε = 0.4, γ = 1, Pr = 1, with variation in one of

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 2 Variation of Thermal Nusselt Number with thermal Rayleigh number Ra for a different values of Rn, b different values of Le, c different values of N A , d different values of Da, e different values of T a, f different values of ε

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these parameters. From all the figures we observe that initially when α is small, the onset of convection occurs as oscillatory convection. Then at intermediate value of α, the critical value for onset of convection is achieved through oscillatory convection. Finally when α is large, mode of convection changes to stationary convection. Therefore, it can be said that Exchange of Stabilities takes place (Chandrasekhar 1961). From Fig. 1a,b and d–f we find, respectively, that on increasing the values of Rn, Le, Da, T a and γ the value of the Rayleigh number is increasing, thus making the

(a)

(c)

(e)

(b)

(d)

(f)

Fig. 3 Variation of concentration Nusselt number with thermal Rayleigh number Ra for a different values of Rn, b different values of Le, c different values of N A , d different values of Da, e different values of T a, f different values of ε

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system more stabilized i.e. delaying the onset of convection. Further from Fig. 1c we see that the system becomes less stabilized on increasing the value of N A . Here we present the results for nonlinear steady problem and depict the variation in Nusselt numbers with respect to the Rayleigh number Ra. From Figs. 2a–e and 3a– e, we find that initially as the value of Ra increases the value of Nusselt number also increases, thus increasing the rate of heat and concentration transfer. But when the value of Ra becomes very large, the value of the Nusselt number becomes almost constant and approaches to a fixed value. Thus after a certain value of Ra change in the rate of heat and concentration transfer attains a near constant value. In Figs. 2d and 3d we observe that as Da decreases, the value of N u and N u φ increases, thus showing an increase in the rate of mass transfer of nano particles. From Fig. 3c, we find that on increasing N A , the value of Nusselt number N u φ increases, thus showing an increase in the mass transfer increases. In Figs. 4 and 5, we draw, respectively, streamlines and isotherms for Racr = 10147.4 and Ra = Racr × 10 at Rn = 4, Le = 100, N A = 5, Da = 0.01, T a = 10, and ε = 0.04. First we consider Fig. 4a–b for streamlines. From the figures we observe that the magnitude of the stream function is more in Fig. 4b than in Fig. 4a. From Fig. 5a–b, which shows the

1.0

(a)

-1.6

(b)

0.8

-4.8

6.4

-4.4E-16

-6.4

Z

0.6

0.4 4.8 3.2

0.2

-3.2

1.6

0.0 0.0

0.2

0.4

0.6

0.8

1.0

1.2

X

Fig. 4 Streamlines for Rn = 4, Le = 100, N A = 5, ε = 0.04, Da = 0.01, T a = 10, a Ra cr = 10147.4, b Ra = Ra cr × 10 1.0

1.0

(a)

0.20 0.40

0.10

0.10

(b)

0.30

0.30

0.50

0.8

0.70

0.6

0.60

Z

Z

0.6

0.40

0.4

0.50

0.4 0.50

0.60 0.40

0.40

0.40

0.2

0.2 0.60 0.70

0.80 0.90

0.0

0.50

0.70

0.60

0.0

0.0 0.2

0.20 0.40

0.8

0.4

0.6

X

0.8

1.0

1.2

0.90

0.0

0.80

0.2

0.4

0.6

0.8

1.0

1.2

X

Fig. 5 Isotherms for Rn = 4, Le = 100, N A = 5, ε = 0.04, Da = 0.01, T a = 10, a Ra cr = 10147.4, b Ra = Ra cr × 10

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isotherms, we find that isotherms are flat near the walls, while they are in the form of contour in the middle of the porous layer. Further the contour becomes more vertical in case (b) showing the effect of increased Rayleigh number. Figure 6 shows the time dependent fields for different values of ψ, T, φ, for different parameter combinations at different times. It is evident that with the passage of time, magnitudes of stream function, isotherms and isonanohalines increases. In all the figures, for ψ, the sense of motion in the subsequent cells is alternately identical with and opposite to that of the adjoining cell. The effect of the parameters Rn, Le, Pr is to expand the cell size as their values increase, all other parameters and the magnitude of the stream function remaining constant. Whereas, the effect of Da, ε, γ is to decrease the cell size, while N A has no effect on the stream cell size.

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In case of isotherms, at the starting time, convection occurs which changes to conduction after some time but approaches to convection again at a later stage, with the magnitude of isotherms increasing with time. Initially, the cells are narrower near to z = 0 and broaden as we move towards z = 1. This implies that convection is more near the top layer compared to the lower layer as heating starts. This indicates a high velocity profile towards the top layer implying a high value of kinetic energy towards the top. Continued heating makes the system approach towards a conduction stage which very soon approaches again to a convection situation, where the trend observed is opposite to the initial stage. The temperature cells broaden near z = 0 and narrower towards z = 1. The convection is more widespread near the bottom indicating a high amount of kinetic energy for the lower layers. For the isonanohalines, it is observed that the concentration is more near the walls and less in the middle of the system. The particles remain concentrated toward the walls and enhance the convection towards the walls. This is well in agreement with the trend observed for isotherms. 7 Conclusions We have investigated thermal instability in a horizontal porous layer saturated by a nanofluid, and rotating about z-axis with uniform angular velocity ω. The Brinkman model with Coriolis term in the momentum equation has been considered. The porous layer is heated from below and cooled from above. The distribution of nanoparticles is considered to be bottom heavy, i.e., the density of nanoparticles decreases as we go up in the porous layer. The linear stability analysis is made with the help of normal mode technique, however, for weakly nonlinear analysis a truncated Fourier series representation having only two terms is considered. We draw the following conclusions: 1. For linear stability, oscillatory mode of convection is found to be the critical mode of onset of convection in most of the graphs. However as the value of wavenumber increases, there is a shift from oscillatory convection to stationary convection, i.e., exchange of stabilities holds. 2. Increments in the values of Le, Rn, T a, Da, and γ stabilize the system more, however an increment in N A makes the system less stabilized. 3. The values of N u and N u φ increase on increasing the value of Ra, become constant on further increasing Ra. 4. Magnitude of stream functions increases on increasing the value of Ra. 5. The isotherms are flat near the boundaries, while they are in the form of contours in the center of the porous medium. 6. For the time dependent fields of ψ, T , and φ, the magnitude of these increase as time passes. Also, it is clear that in the starting time, we have convection, then a state of near conduction arrives, which finally approaches to a strong convection stage. Acknowledgment The author greatly acknowledges the financial assistance from Banaras Hindu University as a research fellowship.

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