cross diffusion convection in a newtonian fluid-saturated rotating porous medium

Transp Porous Med (2013) 98:683–697DOI 10.1007/s11242-013-0166-6

Cross Diffusion Convection in a NewtonianFluid-Saturated Rotating Porous Medium

B. S. Bhadauria · I. Hashim · Jogendra Kumar ·Alok Srivastava

Received: 16 June 2012 / Accepted: 17 April 2013 / Published online: 22 May 2013© Springer Science+Business Media Dordrecht 2013

Abstract Effect of rotation on linear and nonlinear instability of cross-diffusive convection inan anisotropic porous medium saturated with Newtonian fluid has been investigated. Normalmode technique has been used for linear stability analysis, however nonlinear analysis isdone using spectral method, involving only two terms. The Darcy model with Coriolis terms,has been employed in the momentum equation. Nonlinear analysis is used to find the thermaland concentration Nusselt numbers. The effects of various parameters, including Soret andDufour parameters, on stationary and oscillatory convection, have been obtained, and showngraphically.

Keywords Rayleigh number · Porous medium · Cross diffusion · Soret parameter ·Dufour parameter · Heat-mass transfer

B. S. Bhadauria (B)Department of Applied Mathematics, School for Physical Sciences,Babasaheb Bhimrao Ambedkar University, Lucknow 226025, Indiae-mail: [email protected]; [email protected]

I. HashimSchool of Mathematical Sciences, Faculty of Science and Technology,Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysiae-mail: [email protected]

J. KumarSchool of Basic and Applied Sciences, Galgotias University, Greater Noida 201306, UP, Indiae-mail: [email protected]

A. SrivastavaDepartment of Mathematics, Faculty of Science, Banaras Hindu University, Varanasi 221005, Indiae-mail: [email protected]

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684 B. S. Bhadauria et al.

List of Symbols

Latin Symbols

l,m Horizontal wave numbersa Wave numberac Critical wave numberd Depth of the porous layerDu Dufour Parameter (αT κ12/αSκ11)

K Permeability of the porous mediump Pressureq Velocity (u, v, w)RaT Thermal Rayleigh number (RaT = αT gKz(�T )d/νκ11)

RaS Concentration Rayleigh number (RaS = αSgKz(ΔS)d/νκ22)

Sr Soret parameter (αSκ21/αT κ11)

RaT c Critical Rayleigh numberg Gravitational acceleration (0, 0,−g)t TimeK 2 π2(a2 + 1)Ta Taylor number, (2�Kz/νδ)

2

T Temperature�T Temperature difference between the wallsS Concentration�S Concentration difference between the wallsH Rate of Heat transport per unit areaJ Rate of mass transport per unit areaNu Thermal Nusselt numberNuS Concentration Nusselt numberx, y, z Space co-ordinate

Greek Symbols

αT Thermal expansion coefficientαS Concentration expansion coefficientκ11 Thermal diffusivity of the fluidκ12 Cross diffusion due to S componentκ21 Cross diffusion due to T componentκ22 Concentration diffusivity of the fluid� Angular velocity vector(0, 0,�)ω Vorticity vector, ∇ × qτ Diffusivity ratio, κ22/κ11

ξ Mechanical anisotropy parameter, Kx/Kz

δ Porosityρ Densityμ Dynamic viscosityν Kinematic viscosity, μ/ρ0

σ Growth rate of fluidψ Stream function

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Newtonian Fluid-Saturated Rotating Porous Medium 685

Other symbols

b Basic statec Critical0 Reference statei Unit normal vector in x-directionj Unit normal vector in y-directionk Unit normal vector in z-direction

∇21

∂2

∂x2 + ∂2

∂y2 , Horizontal Laplacian

∇2 ∇21 + ∂2

∂z2

D d/dzi

√−1

1 Introduction

The interest in the study of two or multi-component convection has been developed as aresult of the marked difference between single- component and multicomponent systems. Incontrast to single- component systems, here convection sets are even when density decreaseswith height, that is, when the basic state is hydrostatically stable. The study of double-diffusive convection in a rotating porous media is motivated both theoretically and by itspractical applications in engineering and science. Solidification and centrifugal casting ofmetals, food and chemical process, rotating machinery, petroleum industry, biomechanicsand geophysical problems are some of the important areas of applications of this study. Indouble-diffusive convection, an occurrence of remarkable impact is the presence of “crossdiffusion.” This means that the flux of a component depends not only on its own gradient, butalso on the gradient of the other component. In a system where two diffusing properties arepresent, instabilities can occur only if one of the components are destabilizing. When heatand mass transfer occur simultaneously in a moving fluid, the relation between the fluxesand the driving potentials are of more intricate in nature. It has been found that an energyflux can be generated not only by temperature gradient but also by composition gradients aswell. The energy flux caused by a composition gradient is called the Dufour or diffusion-thermo effect. On the other hand, mass fluxes can also be created by temperature gradientsand this is the Soret or thermal-diffusion effect. If the cross-diffusion terms are included inthe species transport equations, then the situation will be quite different. Due to the cross-diffusion effects, each property gradient has a significant influence on the flux of the otherproperty.

An excellent review of most of the findings related to double diffusive convection hasbeen given by Nield and Bejan (2006). The onset of thermal instability in a horizontal porouslayer was first studied extensively by Horton and Rogers (1945) and Lapwood (1948). How-ever, Nield (1968) was the first to investigate double-diffusive generalization of the HortonRogers Lapwood problem. The linear stability analysis of the thermosolutal convection ina sparsely packed porous layer was made by Poulikakos (1986) using the Darcy–Brinkmanmodel. The double- diffusive convection in porous media in the presence of Soret and Dufourcoefficients has been analyzed by Rudraiah and Malashetty (1986). Murray and Chen (1989)have extended the linear stability theory, by taking into account the effects of temperature-dependent viscosity and volumetric expansion coefficients and nonlinear basic salinity profile.

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Bahloul et al. (2003) have carried out an analytical and numerical study of the double-diffusiveconvection in a shallow horizontal porous layer under the influence of Soret effect. Malashettyand Gaikwad (2002) studied the effect of cross diffusion for Soret and Dufour coefficients onthe double-diffusive convection in an unbounded vertically stratified two component systemwith compensating horizontal thermal and solute gradients. Malashetty et al. (2006) inves-tigated the double-diffusive convection in a two-component couple stress liquid layer withSoret effect using both linear and nonlinear stability analyses. An analytical study of linearand nonlinear double-diffusive convection with Soret and Dufour effects in couple stress fluidwas studied by Gaikwad et al. (2007). Gaikwad et al. (2009a,b) investigated the linear andnonlinear double-diffusive convection in a porous layer with Soret and cross-diffusion effect,respectively. Saravanan and Jegajothi (2010) have studied the finger-type double-diffusiveconvective instability in a fluid-saturated porous considering Darcian porous medium andinvoking a local thermal non-equilibrium condition by taking into account the energy trans-fer between the fluid and solid phases. Very recently Altawallbeh et al. (2013) have studiedlinear and nonlinear double-diffusive convection in a saturated anisotropic porous layer withSoret effect and internal heat source. Some more studies on double-diffusive convectionin a porous medium are due to Kuznetsov and Nield (2008, 2011), Nield and Kuznetsov(2011), Malashetty and Swamy (2010), Malashetty and Begum (2011); Govender (2011)and Malashetty and Biradar (2012).

The thermal instability in a rotating porous medium subject to uniform temperature gradi-ent was first investigated by Friedrich (1983). Some of the other researchers who have studiedthe thermal instability in a rotating porous medium are; Patil and Vaidyanathan (1983); Palmand Tyvand (1984); Jou and Liaw (1987a,b); Qin and Kaloni (1995); Vadasz (1996a,b, 1998);Vadasz and Govender (2001); Straughan (2001); Desaive et al. (2002); Govender (2003) andMalashetty and Swamy (2007).

However, there are only few studies available in the literature in which the double-diffusiveconvection has been investigated in a rotating porous medium. The linear and nonlineardouble-diffusive convection in a rotating porous medium was first investigated by Chakrabartiand Gupta (1981). Then Rudraiah et al. (1986) studied the effect of rotation on linear andnonlinear double-diffusive convection in a sparsely packed porous medium. Guo and Kaloni(1995) used the Lyapunov direct method and studied the nonlinear stability problem ofdouble-diffusive convection in a rotating sparsely packed porous layer. Patil et al. (1989,1990) studied double-diffusive convection in a rotating porous medium using linear stabilityanalysis. Lombardo and Mulone (2002) studied the nonlinear stability of conduction diffusionsolution of a fluid mixture, heated and salted from below and saturating a rotating porousmedium, using Lyapunov direct method. Malashetty and Heera (2008) studied the effectof rotation on the onset of double-diffusive convection in a horizontal anisotropic porouslayer.

The cross-diffusive convection is encountered in many systems in industry and nature,and in the present context, is of particular interest in the study of extraction of metals fromores where a mushy layer is formed during solidification of a metallic alloy. It is found thatthe quality and structure of the resulting solid can be controlled by influencing the transportprocess externally, which can be done by thermal modulation, gravity modulation or byrotation. However in the present study, we use rotation as an external means to influencethe transport process, thereby controlling the quality and structure of the resulting solid.Furthermore, many of the previous studies have modeled the mushy layer as isotropic porousmedium, however realistically, the permeability of the porous medium is anisotropic (Nieldand Kuznetsov 2007). It is with these motives that we have made a weak nonlinear analysis

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of hydrodynamic stability and studied the effect of rotation on cross- diffusive convection inan anisotropic porous medium saturated with Newtonian fluid.

2 Mathematical formulation

We consider an anisotropic porous medium saturated by a Newtonian fluid, confined betweentwo parallel horizontal planes at z = 0 and z = d , heated from below and cooled fromabove. The planes are infinitely extended horizontally in x and y directions and are free,also assuming the z- axis as vertical. Considering that the system is rotating about z- axiswith a constant angular velocity�. We consider temperature T0 +�T, T0 with concentrationS0 +�S, S0 at z = 0 and z = d respectively. Boussinesq approximation is applied to accountthe effect of density variations. Cross-diffusion terms are included in the temperature andconcentration equations, and under these postulates the governing equations are as given by

∇.q = 0 (1)

μK .q + 2ρ0

δ�× q = (−∇ p + ρg) (2)(

∂

∂t+ q.∇

)T = κ11∇2T + κ21∇2S (3)

(∂

∂t+ q.∇

)C = κ22∇2S + κ21∇2T (4)

ρ = ρ0 [1 − αT (T − T0)+ αS(S − S0)] (5)

where the symbols used in above equations are given in the nomenclature. The basic stateof fluid is assumed to be quiescent initially, which is then perturbed by imposing a smallperturbation. The system is then non-dimensionalized using the following transformations:

(x, y, z) = (x∗, y∗, z∗)d, t = d2

κ11t∗, q = κ11

d q∗, T = (�T )T ∗, C = (�C)C∗, p =μκ11

K p∗. After dropping the asterisks, we obtain the non-dimensionalized form of the equa-tions as:

∇.q = 0 (6)

qa + √Ta

(k × q

)= −∇ p + [RaT T − τRaS S]k (7)

∂T

∂t+ (q.∇)T − w = ∇2T + τDu

RaS

RaT∇2S (8)

∂S

∂t+ (q.∇)S − w = τ∇2S + Sr

τ

RaT

RaS∇2T (9)

The non-dimensional parameters which appear in the above equations are defined in theNomenclatures. The boundaries are considered to be free-free, isothermal and isohaline,therefore the appropriate boundary conditions are

w = ∂2w

∂z2 = T = S = 0 at z = 0 and z = 1. (10)

We now eliminate the pressure p from Eq. (7) by operating curl on it, we obtain

1

ξω − √

Ta∂q∂z

=[

RaT

(∂T

∂yi − ∂T

∂xj

)− τRaS

(∂S

∂yi − ∂S

∂xj

)](11)

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where ω = ∇ × q, denotes the vorticity vector. Once again applying curl on Eq. (11), we getthe equation

Q−√Ta∂ω

∂z=

[RaT

{i∂2T

∂x∂z+ j

∂2T

∂y∂z−∇2

1 T k

}−τRaS

{i∂2S

∂x∂z+ j

∂2S

∂y∂z−∇2

1 Sk

}]

(12)

where ∇21 =

(∂2

∂x2 + ∂2

∂y2

)is the horizontal Laplacian, and Q = (Q1, Q2, Q3), where

Q1 = 1

ξ

∂2v

∂y∂x+ ∂2w

∂x∂z−

(∂2v

∂y2 + 1

ξ

∂2u

∂z2

), Q2 = 1

ξ

∂2u

∂x∂y+ ∂2w

∂y∂z− 1

ξ

(∂2v

∂x2 + ∂2v

∂z2

)

and Q3 = −(

∇21 + 1

ξ

∂2

∂z2

)w.

3 Linear Stability Analysis

To perform the linear stability analysis, we take the z-component of the Eqs. (11) and (12)and also linear form of the Eqs. (8) and (9). A single equation form these four equations isobtained for the vertical component of velocity w as

[{(∂

∂t− ∇2

) (∂

∂t− τ∇2

)− DuSr∇4

}(∇1

2 + 1

ξ

∂2

∂z2

)+ ξTa

∂2

∂z2

}

−{(

∂

∂t− τ∇2 − Sr∇2

)RaT −

(∂

∂t− ∇2 − Du∇2

)τRaS

}∇2

1

]w = 0 (13)

For the solution of (13), we use normal mode analysis as

w = W (z) exp[i(lx + my)+ σ t)]etc., (14)

where l and m are the horizontal wave numbers, and σ is the growth rate, in general a complexquantity. Now using the boundary conditions for W

W = d2W

dz2 = d4W

dz4 = 0 at z = 0 and z = 1, (15)

we obtain the expression for the thermal Rayleigh number RaT as

RaT =[σ + (1 + Du)(π2 + a2)

][σ + (τ + Sr)(π2 + a2)

] τRaS +[(

a2 + 1

ξπ2

)+ ξπ2Ta

]

×[(σ + π2 + a2

) (σ + τ

(π2 + a2

)) − DuSr(π2 + a2

)2]

a2[σ + (τ + Sr)(π2 + a2)

] , (16)

where a = √(l2 + m2) is the horizontal wave number.

To achieve the marginal state through stationary convection we must take σ as real andσ = 0. Under this condition, the expression for Rayleigh number Rast

T for the onset ofstationary convection is given by

RastT = (1 + Du)

(τ + Sr)τRaS + (π2 + a2)(τ − DuSr)[(a2 + 1

ξπ2)+ ξπ2Ta]

a2(τ + Sr). (17)

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To obtain the critical value of Rayleigh number, we put ∂RastT /∂a = 0, and get

RastT =

π2(τ − DuSr)[1 +

√1ξ

+ ξTa]2 + τRaS(1 + Du)

(τ + Sr). (18)

The corresponding expression for critical value of wave number a = ac is

ac = π

(1

ξ+ ξTa

)1/4

. (19)

It is obvious that the oscillatory convection are possible only if some additional constraints,like rotation, magnetic field, and salinity gradient are present in the system. For oscillatoryconvection, we write σ = σr + iσi . At the marginal state σr = 0 and σi �= 0, thereforeputting σ = iσi into Eq. (16) and making the denominator as real, we obtain

RaoscT = �1 + iσi�2, (20)

where

�1 =(a2τ

(σ 2

i + (1 + Du)δ4(Sr + τ))

RaS − δ2((−1 + Sr)σ 2

i + δ4 (DuSr − τ) (Sr + τ)) (π2Taξ + δ2

1

))a2

(σ 2

i + δ4(Sr + τ)2)

�2 =(a2δ2τ(−1 − Du + Sr + τ)RaS + (

σ 2i + δ4

(τ 2 + Sr(1 + Du + τ)

)) (π2Taξ + δ2

1

))a2

(σ 2

i + δ4(Sr + τ)2) ,

δ2 = a2 + π2 and δ12 = a2 + 1

ξπ2

Since oscillatory Rayleigh number RaoscT should be real, therefore we must have �2 = 0

(since σi �= 0 in oscillatory case)

σ 2i =

(a2δ2τ(1 + Du − Sr − τ)RaS − δ4

(τ 2 + Sr(1 + Du + τ)

) (π2Taξ + δ2

1

))(π2Taξ + δ2

1)

(21)

4 Nonlinear analysis

Although we have performed the linear stability analysis to know the criteria for onset ofconvection but to extract some more informations related to convection, we need to performthe nonlinear analysis of the above problem. For nonlinear analysis, we eliminate the pressureterm from the Eq. (7) by operating curl on it, and make all physical quantities independent of y.Furthermore, introduction of the stream functionψ such that u = ∂ψ/∂z andw = −∂ψ/∂x ,in the above resulting equation and in Eqs. (8)–(9) results (taking ∂/∂t ≡ 0, for steady-state)(

∂2

∂x2 + 1

ξ

∂2

∂z2

)ψ + ξTa

∂2ψ

∂z2 = τRaS∂S

∂x− RaT

∂T

∂x(22)

(∂2

∂x2 + ∂2

∂z2

)T + τDu

RaS

RaT

(∂2

∂x2 + ∂2

∂z2

)S − ∂ψ

∂z

∂T

∂x+ ∂ψ

∂x

∂T

∂z− ∂ψ

∂x= 0

(23)

τ

(∂2

∂x2 + ∂2

∂z2

)S + Sr

τ

RaT

RaS

(∂2

∂x2 + ∂2

∂z2

)T − ∂ψ

∂z

∂S

∂x+ ∂ψ

∂x

∂S

∂z− ∂ψ

∂x= 0

(24)

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A local nonlinear stability analysis shall be performed using a spectral method. This study willhelp in understanding the physics of the problem with minimum mathematical expressions.Furthermore, the results can be used as starting point to generalize it for full nonlinearproblem. Also it is to be noted that the effect of nonlinearity is to distort the temperature andconcentration fields through the interaction of ψ and T, ψ and S, and ψ and V , respectively.As a result a component of the form sin(2π z) will be generated. Therefore, the minimalexpression which describes the finite amplitude convection is of the form

ψ = A1(t)sin(πax)sin(π z) (25)

T = B1(t)cos(πax)sin(π z)+ C1(t)sin(2π z) (26)

S = D1(t)cos(πax)sin(π z)+ E1(t)sin(2π z) (27)

V = F1(t)sin(πax)cos(π z)+ G1(t)sin(2πx) (28)

where amplitudes A1(t), B1(t),C1(t), D1(t), E1(t), F1(t), and G1(t) are functions of time,but for steady- state, we assume them to be constants, and V is zonal velocity induced byrotation. Putting these values in Eqs. (22)–(24) and equating the coefficients of like terms ofthe resulting equations, we obtain

[π2

(a2 + 1

ξ

)+ ξπ2Ta

]A1 + πaRaT B1 − τπaRaS D1 = 0 (29)

πa A1 + π2(a2 + 1)B1 + τDuRaS

RaTπ2(a2 + 1)D1 + π2a A1C1 = 0 (30)

8π2C1 + 8τDuRaS

RaTπ2 E1 − π2a A1 B1 = 0 (31)

πa A1 + Sr

τ

RaT

RaSπ2(a2 + 1)B1 + τπ2(a2 + 1)D1 + π2a A1 E1 = 0 (32)

8π2 E1 + 8Sr

τ

RaT

RaSπ2C1 − π2a A1 D1 = 0 (33)

Now eliminating all amplitudes with the help of Eqs. (30)–(33), except A1, will results in asingle equation for A1, quadratic in (A2

1/8), as

4π4a4

(τ − DuSr)

{π2

(a2 + 1

ξ

)+ ξπ2Ta

}(A2

1

8

)2

+ 2π2a2 K 2

(τ − DuSr)

(2DuSr + τ 2 + 1

) {π2

(a2 + 1

ξ

)+ ξπ2Ta

}(A2

1

8

)

− 2π4a4

(τ − DuSr){τRaS(Du − τ)+ RaT (1 − Sr)}

(A2

1

8

)

−{π2

(a2 + 1

ξ

)+ ξπ2Ta

}(τ − DuSr)K 4

+π2a2 {τRaS(1 + Du)− RaT (τ + Sr)} K 2 = 0, (34)

where K 2 = π2(a2 + 1). Once we determine the value of A1, we can find the rate ofheat and mass transfer in terms of the thermal and concentration Nusselt numbers, respec-tively. The expressions for thermal and concentration Nusselt numbers are obtained as given

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below:

Nu = (1 − 2πC1)+ τDuRaS

RaT(1 − 2πE1) (35)

NuS = (1 − 2πE1)+ Sr

τ 2

RaT

RaS(1 − 2πC1) (36)

First, we find the value of C1 and E1 in terms of A1 from Eqs. (30)–(33), and then put theminto Eqs. (35), (36), to calculate the values of thermal and concentration Nusselt numbers.

5 Results and Discussions

The parameters that arise in the problem are Du,Sr, τ,RaS, ξ and Ta, and these influencethe convective heat and mass transports. The first five parameters are related to the fluid andthe structure of the porous medium, and the last one concerns with the external mechanismof controlling the convective flow in the porous medium.

The values of Du and Sr which are very small, have been taken from the work ofMalashetty and Biradar (2012). Since κS < κT , therefore small values of τ have beentaken in the calculations. Furthermore, positive values of RaS are considered and in sucha case one gets positive values of RaT , and these signify the assumption of a situation inwhich we have cool fresh water overlying warm salty water. In cross diffusion, this situ-ation is conducive for the appearance of salt-fingers that arises in a stationary regime ofonset of convection. Also we take ξ < 1 under the situation that horizontal permeabilityis less that the vertical permeability. Since we are interested only in the moderate rotationeffect, therefore values of Ta, which is externally controlling parameter, are taken to besmall.

We have obtained the expressions for stationary and oscillatory Rayleigh numbers andcomputed their corresponding critical values using the expressions (17) and (20), respec-tively. Furthermore, we have used base e logarithm in drawing the figures for stationary andoscillatory Rayleigh numbers. The weak nonlinear theory, based on truncated Fourier series,represents the information about the rates of heat and mass transfer. In Figs. 1a–e and 2a–e,we depict respectively, variation in the critical values of stationary and oscillatory thermalRayleigh numbers with respect to concentration Rayleigh number, for fixed values of variousparameters. From both Figs. 1 and 2, we observe that initially the value of RaT is small andincreases slightly on increasing RaS . But at higher values of RaS , increase in the values ofRaT is also high, thus making the system highly stabilized. At high RaS , the values of RaT

increase with respect to RaS almost linearly.From Figs. 1a and 2a, 1b and 2b, 1c and 2c, we observe that on increasing the values

of mechanical anisotropy ξ , diffusivity ratio τ and Taylor number Ta respectively, values of(Rast

T )c and (RaoscT )c increase, thus stabilizing the system. The effect of Taylor number Ta

is same as found by Vadasz (1998). On increasing the value of Soret number Sr and Dufourparameter Du, we find respectively, from Figs. 1d and 2d, and 1e and 2e that values of (Rast

T )cincrease, however the value of (Raosc

T )c remains unaffected.Furthermore, we find from Figs. 1a–d and 2a, c–e that the effect of changing the values

of various parameters is more at low values of RaS and decreases on increasing the values ofRaS . This effect becomes negligible at high RaS . Thus these parameters plays a dominatingrole at small values of Rs, however their effects become negligible at higher Rs. Also thevalues of parameters play more significant roles at higher solute Rayleigh number RS asobserved in Figs. 1e and 2b.

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(a) (b)

(c) (d)

(e)

Fig. 1 a–e Variation of critical value of Stationary Rayleigh number RastT with solute Rayleigh number RaS

On comparing the present results with the results of Malashetty and Begum (2011), wefind that the effects of ξ,Ta,Rs and τ on RaT are similar.

Results related to nonlinear stability analysis are obtained by using truncated represen-tation of series which are shown graphically. The results, corresponding to steady-state, arepresented in Figs. 3a–f and 4a–f for thermal Nusselt number Nu and concentration Nusseltnumber NuS , respectively. From the figures, we find that initially when RaT is small, thevalues of Nu and NuS are small, increase on increasing the value of RaT . But when RaT

becomes very large, the values of Nu and NuS become constants. Thus, there is no furtherenhancement in heat and mass transfer at very high RaT .

In Fig. 3a–f, we depict the variation of thermal Nusselt number Nu with respect to thermalRayleigh number RaT , for different parameters. From Fig. 3a, we observe that on increasingthe value of mechanical anisotropic parameter ξ,Nu decreases. Thus effect of increasing ξ

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τ

ξ

τ

ξ

(a) (b)

(c)

(e)

(d)

Fig. 2 a–e Variation of critical value of oscillatory Rayleigh number RaoscT with solute Rayleigh number RaS

is to decrease the rate of heat transfer, therefore suppressing the convection. Actually whenξ increases, then either Kx increases or Kz decreases, and so in both these cases fluid flowthrough porous medium decreases in vertical direction in comparison to the flow in horizontaldirection. This delays the onset of convection, and thus decreases the heat transport in thesystem. From Fig. 3b, c we found that on increasing the value of Dufour parameter Du,Nuincreases while trend is reverse for Soret parameter Sr. Thus effect of increasing Du is toincrease the rate of heat transfer, thereby advancing the convection, while decrease in therate of heat transfer due to increase in Sr, means suppression of convection. Figure 3d showsthat for small values of RaT ,Nu takes lower values on increasing diffusivity ratio τ , however

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(a) (b)

(c) (d)

(e) (f)

Fig. 3 a–f Variation of thermal Nusselt number Nu with Rayleigh number RaT

at large values of RaT this trend is reverse and at further large values of RaT ,Nu becomesalmost constant. From Fig. 3e, we observe that on increasing the value of Taylor numberTa,Nu decreases, and so the rate of heat transfer. Therefore rotation has stabilizing effect.Figure 3f shows that on increasing the value of concentration Rayleigh number RaS,Nuincreases. The effects of ξ and Ta are found to be compatible with the result of Malashettyand Begum (2011).

In Fig. 4a–f, we depict the variation of concentration Nusselt number Nu with thermalRayleigh number RaT , for different parameters. Figure 4a, b, e shows qualitatively similarresults for concentration Nusselt number NuS as shown in Fig. 3a, b, e for thermal Nusseltnumber Nu. From Fig. 4c, we observe that on increasing the value of Soret number Sr,concentration Nusselt number also increases, thus advancing the convection. Figure 4d, fshows that on increasing the values of diffusivity ratio τ and concentration Rayleigh number

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(a) (b)

(c) (d)

(e) (f)

Fig. 4 a–f Variation of concentration Nusselt number NuS with Rayleigh number RaT

RaS,NuS decreases i.e., rate of mass transfer decreases, thus inhibiting the convection. Here,the effects of ξ and Ta on NuS are same as that in Malashetty and Begum (2011).

6 Conclusion

The effect of cross diffusion on the onset of double-diffusive convection in a rotating horizon-tal porous layer, saturated with Newtonian fluid, has been considered. The problem has beensolved analytically, and linear and nonlinear analysis have been performed. The followingconclusions are drawn:

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1. The effects of increasing mechanical anisotropy ξ , diffusivity ratio τ and Taylor numberTa are found to delay the onset of convection, thus stabilizing the system.

2. Effect of increasing the value of Soret number Sr is to decrease the value of RaT , thusadvancing the onset of convection.

3. An increment in Dufour parameter Du, first advances the convection and then suppressesthe convection as RaS increases.

4. Initially at the onset of convection the heat and mass transports are due to conductiononly, therefore the values of Nu and NuS are small. However as convection starts, Nuand NuS increase and achieve maximum values with time. Then at the steady-state theheat and mass transports become constant.

5. The values of Nu and NuS decrease on increasing ξ and Ta, however increase on increas-ing Du. This shows that the effects of increasing ξ and Ta is to decrease the heat and masstransports, thus stabilizing the system. However, in case of Du it has opposite effect.

6. On increasing τ and RaS , Nu increases, but it decreases on increasing Sr and Ta, however,the trend is totally reversed in the case of NuS .

Acknowledgment Author BSB is grateful to Banaras Hindu University, Varanasi for sanctioning the liento work as Professor of Mathematics at Department of Applied Mathematics, BB Ambedkar University,Lucknow-226025, India.

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cross diffusion convection in a newtonian fluid-saturated rotating porous medium

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