Transp Porous Med (2011) 87:229–250 DOI 10.1007/s11242-010-9677-6

Non-Linear Two Dimensional Double Diffusive Convection in a Rotating Porous Layer Saturated by a Viscoelastic Fluid Anoj Kumar · B. S. Bhadauria

Received: 13 September 2010 / Accepted: 14 October 2010 / Published online: 3 November 2010 © Springer Science+Business Media B.V. 2010

Abstract Double diffusive convection in a rotating anisotropic porous layer, saturated by a viscoelastic fluid, heated from below and cooled from above has been studied making linear and non-linear stability analyses. The fluid and solid phases are considered to be in equilibrium. In momentum equation, we have employed the Darcy equation which includes both time derivative and Coriolis terms. The linear theory based on normal mode method is considered to find the criteria for the onset of stationary and oscillatory convection. A weak non-linear analysis based on minimal representation of truncated Fourier series analysis containing only two terms has been used to find the Nusselt number and Sherwood number as functions of time. We have solved the finite amplitude equations using a numerical scheme. The results obtained, during the above analyses, have been presented graphically and the effects of various parameters on heat and mass transfer have been discussed. Finally, we have drawn the steady and unsteady streamlines, isotherms, and isohalines for various parameters. Keywords Viscoelastic fluid · Rotation · Porous medium · Double diffusive convection · Heat-mass transfer · Streamlines · Isotherms and isohalines

List of Symbols Latin Symbols l, m Horizontal wave numbers a Wave number ac Critical wave number d Depth of the porous layer ˆ + K z (kˆ k) ˆ K Permeability of the porous medium, K x (iˆiˆ + jˆ j) A. Kumar · B. S. Bhadauria (B) DST-Centre for Interdisciplinary Mathematical Sciences, Department of Mathematics, Faculty of Science, Banaras Hindu University, Varanasi 221005, India e-mail: [email protected] A. Kumar e-mail: [email protected]

123

230

A. Kumar, B. S. Bhadauria

p g q RaT Ra S RaTc t Ta T T S S H J (x, y, z)

Pressure Gravitational acceleration, (0, 0, −g) Velocity (u, v, w) Thermal Rayleigh number RaT = (αT gK z d(T )/νκT z ) Concentration Rayleigh number Ra S = (α S gK z d(S)/νκ S ) Critical Rayleigh number Time Taylor number, (2K z /νδ)2 Temperature Temperature difference between the walls Concentration Concentration difference between the walls Rate of heat transport per unit area Rate of mass transport per unit area Space co-ordinate

Greek Symbols λ1 Relaxation time λ2 Retardation time αT Thermal expansion coefficient α S Concentration expansion coefficient κT Thermal diffusivity of the fluid κ S Concentration diffusivity of the fluid  Angular velocity vector (0, 0, ) η Thermal anisotropy parameter ω Vorticity vector, ∇ × q τ Diffusivity ratio, κ S /κT z ξ Mechanical anisotropy parameter δ Porosity ρ Density μ Dynamic viscosity ν Kinematic viscosity, μ/ρ0 σ Growth rate of fluid γ Rescaled wave number, a 2 /π 2 ψ Stream function Other Symbols b Basic state c Critical 0 Reference state iˆ Unit normal vector in x-direction jˆ Unit normal vector in y-direction kˆ Unit normal vector in z-direction 2 2 ∇12 ∂∂x 2 + ∂∂y 2 ∇2 D i

∇12 + d/dz √ −1

123

∂2 ∂z 2

Non-Linear Two Dimensional Double Diffusive Convection

231

1 Introduction The study of double diffusive convection in a rotating porous medium is of great practical importance in many branches of Science and Engineering such as Centrifugal filtration process, Petroleum industry, Food engineering, Chemical engineering, Geophysics, and Biomechanics etc. An excellent review of most of the findings related to the above subject has been given by Nield and Bejan (2006). The onset of thermal instability in a horizontal porous layer was first studied extensively by Horton and Rogers (1945) and Lapwood (1948). However, Nield (1968) was first to investigate double diffusive generalization of the Horton–Rogers–Lapwood problem. Some other researchers who have worked on double diffusive convection in a porous medium are; Taunton et al. (1972), Patil and Rudraiah (1980), Patil and Vaidyanathan (1982), Griffith (1981). The onset of double diffusive convection in a horizontal porous layer has been investigated by Rudraiah et al. (1982) using a weak non-linear theory. The linear stability analysis of the thermosolutal convection in a sparsely packed porous layer was made by Poulikakos (1986) using the Darcy–Brinkman model. The double diffusive convection in porous media in the presence of Soret and Dufour coefficients 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 non-linear basic salinity profile. Malashetty (1993) made a linear stability analysis to determine the effects of anisotropic thermo convective currents on the double diffusive convection in a sparsely packed porous medium. Bahloul et al. (2003) have carried out an analytical and numerical study of the double diffusive convection in a shallow horizontal porous layer under the influence of Soret effect. Malashetty and Swamy (2005) studied the linear and non-linear double convection in a fluid saturated anisotropic porous layer using uniform temperature gradient. Very recently, Gaikwad et al. (2009) investigated double diffusive convection in a anisotropic porous medium considering cross-diffusion effects and using linear and non-linear analyses. Further the problem of thermal instability in a rotating porous medium subject to uniform temperature gradient was first investigated by Friedrich (1983). Others who have studied the thermal instability in a rotating porous medium are; Patil and Vaidyanathan (1983); Palm and 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), and Govender (2003); Malashetty and Swamy (2007). However, there are only few studies available in the literature in which the double diffusive convection has been investigated in a rotating porous medium. The study of double diffusive convection in a rotating porous media is motivated both theoretically and by its practical applications in engineering and science. Solidification and centrifugal casting of metals, food and chemical process, rotating machinery, petroleum industry, biomechanics, and geophysical problems are some of the important areas of applications of this study. Then, Rudraiah et al. (1986) studied the effect of rotation on linear and non-linear double diffusive convection in a sparsely packed porous medium. Guo and Kaloni (1995) used the Lyapunov direct method and studied the non-linear stability problem of double diffusive convection in a rotating sparsely packed porous layer. Lombardo and Mulone (2002) studied the nonlinear stability of the conduction diffusion solution of a fluid mixture, heated and salted from below and saturating a rotating porous medium, using Lyapunov direct method. Recently, Malashetty and Heera (2008) studied the effect of rotation on the onset of double diffusive convection in a horizontal anisotropic porous layer.

123

232

A. Kumar, B. S. Bhadauria

The flow of non-Newtonian fluids in a porous layer is of great interest in different areas of modern Sciences, engineering and Technology like material processing, Petroleum, Chemical and nuclear industries, Geophysics, and Bio-mechanics engineering. Some oil sands contain waxy crudes at shallow depth of the reservoirs which are considered to be viscoelastic fluid. In these situations, a viscoelastic model of a fluid serves to be more realistic than the Newtonian model. Viscoelastic fluids exhibit unique patterns of instabilities such as the overstability that is not predicted or observed in Newtonian fluid. The nature of convective motions in a thin horizontal layer of viscoelastic fluid which is heated from below, in the classical Rayleigh–Benard convection geometry, has been the subject of discussion in the literature for nearly four decades. Herbert (1963) and Green (1968) were first to analyze the problem of oscillatory convection in an ordinary viscoelastic fluid of the Oldroyd type under the condition of infinitesimal disturbances. Later on, Rudraiah et al. (1989, 1990) had studied the onset of stationary and oscillatory convection in a viscoelastic fluid of porous medium. Laroze et al. (2007) have analyzed the effect of viscoelastic fluid on bifurcations of convective instability, he analyzed that the nature of the convective solution depends largely on the viscoelastic parameters. Kim et al. (2003) studied the thermal instability of viscoelastic fluids in porous media, conducting linear and non-linear stability analyses to obtain the stability criteria. Later on, Yoon et al. (2004) has studied the onset of oscillatory convection in a horizontal porous layer saturated with viscoelastic fluid by using linear theory. Sheu et al. (2008) investigated the chaotic convection of viscoelastic fluids in porous media and deduced that the flow behavior may be stationary, periodic, or chaotic. Stability analysis of double diffusive convection of Maxwell fluid in a porous medium heated from below has been investigated by Wang and Tan (2008). Malashetty et al. (2009) studied the onset of double diffusive convection in a binary viscoelastic fluid saturated anisotropic porous medium considering both linear and non-linear stability analyses. Malashetty and Swamy (2010) studied the onset criterion for oscillatory convection is affected by the viscoelastic parameters in fluid layer. Kumar and Bhadauria (2010) studied the thermal instability in a rotating porous layer saturated by viscoelastic fluid. Although some literature on double diffusive convection in a rotating porous medium saturated by ordinary fluid is available, very little attention has been devoted to the study of double diffusive convection in a rotating porous medium saturated by viscoelastic fluid. To the best of authors knowledge, till date no study is available in which the effect of rotation on double diffusive convection has been investigated in a porous medium saturated with viscoelastic fluid. Therefore, purpose of the present work is to study the effect of rotation on the onset of double diffusive convection in a horizontal saturated porous layer of a viscoelastic fluid. In this article, we intend to find the onset criteria for steady and unsteady cases considering linear and non-linear stability analyses. Further we investigate how heat and mass transfer are affected by rotation in a viscoelastic fluid saturated porous medium for unsteady state. We also investigate the nature of flow patterns in the form of streamlines, isotherms, and isohalines.

2 Governing Equations We have considered a viscoelastic fluid saturated horizontal porous layer confined between two parallel horizontal planes at z = 0 and z = d, a distance d apart. The planes are infinitely extended horizontally in x and y directions. A Cartesian frame of reference is chosen in such a way that the origin lies on the lower plane and the z-axis as vertical upward. The system is rotating about z-axis with a constant angular velocity . Adverse temperature and

123

Non-Linear Two Dimensional Double Diffusive Convection

233

concentration gradients are applied across the porous layer and the lower and upper planes are kept at temperature T0 +T , concentration S0 +S, and T0 , S0 , respectively. Finally, the Soret and Dufour effects are assumed to be negligible. Oberbeck–Boussinesq approximation is applied to account the effect of density variations. Then, the governing equations in the non-dimensional form (Appendix A) are given by ∇.q = 0     ∂ ∂ √ 1 + λ2 T a(kˆ × q) qa + 1 + λ1 ∂t ∂t     ∂ ∂ = − 1 + λ1 ∇ p + 1 + λ1 [RaT T − τ Ra S S]kˆ ∂t ∂t

(1)

(2)

∂T ∂2T + (q.∇)T − w = η∇12 T + 2 ∂t ∂z ∂S 2 + (q.∇)S − w = τ ∇ S ∂t

(3) (4)

The appropriate boundary conditions in the non-dimensionalized form are given by w=T =S=0

at z = 0 and z = 1.

(5)

where T a = (2K z /νδ)2 , Taylor number, RaT = αT gK (T )d/νκT z , thermal Rayleigh number, Ra S = α S gK (S)d/νκ S , concentration Rayleigh number, qa = ( ξ1 u, ξ1 v, w), the anisotropic modified velocity vector, τ = κ S /κT z , diffusivity ratio, ξ =K x /K z , the mechanical anisotropy parameter, η=κT x /κT z , the thermal anisotropy parameter, and ν = μ/ρ0 , kinematic viscosity, further we have taken γ = 1 to reduce the number of non-dimensional parameters. We now eliminate the pressure p from Eq. 2 by operating curl on it and get an equation in the form 

     ∂ 1 ∂ √ ∂q ∂ 1 + λ2 Ta ω − 1 + λ1 = 1 + λ1 ∂t ξ ∂t ∂z ∂t      ∂T ˆ ∂T ˆ ∂S ˆ ∂S ˆ × RaT i− j − τ Ra S i− j ∂y ∂x ∂y ∂x

(6)

where ω = ∇ × q, denotes the vorticity vector. Now once again applying curl on Eq. 6, we get the equation 

∂ 1 + λ2 ∂t





∂ Q + 1 + λ1 ∂t



√

  ∂ ∂ω Ta = 1 + λ1 ∂z ∂t

× [RaT ∇12 T − τ Ra S ∇12 S]

(7)

where ∇12 = ∂ 2 /∂ x 2 + ∂ 2 /∂ y 2 is the horizontal Laplacian with Q= (Q 1 , Q 2, Q 3 ), where

1 ∂2v ∂2w ξ ∂ y∂ x + ∂ x∂z ∂2 −(∇12 + ξ1 ∂z 2 )w.

Q1 =

− ( ∂∂ yv2 + 2

1 ∂2u ξ ∂z 2 ),

Q2 =

1 ∂2u ξ ∂ x∂ y

+

∂2w ∂ y∂z

−

1 ξ

∂2v ∂x2

+

∂2v ∂z 2

, and Q 3 =

123

234

A. Kumar, B. S. Bhadauria

3 Linear Stability Analysis We now perform the linear stability analysis, which is also useful in the non-linear stability analysis. For this, we linearize the Eqs. 3–4 and Eqs. 6–7 and obtain   ∂ ∂2 2 − η∇1 − 2 T = w (8) ∂t ∂z   ∂ − τ ∇2 S = w (9) ∂t     ∂ 1 ∂ √ ∂w ωz = 1 + λ1 (10) Ta 1 + λ2 ∂t ξ ∂t ∂z      ∂ ∂ √ ∂ωz 1 ∂2 1 + λ2 Ta ∇1 2 + w + 1 + λ 1 ∂t ξ ∂z 2 ∂t ∂z   ∂ [RaT ∇12 T − τ Ra S ∇12 S] = 1 + λ1 (11) ∂t where ωz = ∇ × w, w is vertical component of velocity. By eliminating T, S and ωz from the above equations, we get a single equation for the vertical component of the velocity in the form        ∂ ∂ ∂ 2 ∂2 1 ∂2 2 2 2 − η∇1 − 2 − τ∇ 1 + λ2 ∇1 + ∂t ∂z ∂t ∂t ξ ∂z 2

 2     ∂2 ∂ ∂ ∂ 1 + λ2 ξ T a 2 − 1 + λ1 + 1 + λ1 ∂t ∂z ∂t ∂t      ∂ ∂ ∂2 (12) × − τ ∇ 2 RaT − − η∇1 2 − 2 τ Ra S ∇12 w = 0. ∂t ∂t ∂z The boundary conditions in terms of w are given by w=

∂ 2w ∂ 4w = = 0 at z = 0 and z = 1. ∂z 2 ∂z 4

(13)

To obtain the solution of the unknown fields, we use normal mode technique and write w = W (z) exp[i(lx + my) + σ t)]

(14)

where l and m are the horizontal wave numbers and σ is the growth rate, in general a complex quantity as σ = σr + iσi . We substitute Eq.14 into the Eq. 12, and obtain a single ordinary differential equation for W (z) in the form    1 {σ + ηa 2 − D 2 }{σ − τ (D 2 − a 2 )}{(1 + λ2 σ )2 −a 2 + D 2 ξ +(1 + λ1 σ )2 ξ T a D 2 } + (1 + λ1 σ )(1 + λ2 σ )a 2 

 × {σ − τ (D 2 − a 2 )}RaT − {σ + ηa 2 − D 2 }τ Ra S W (z) = 0.

(15)

The boundary conditions in terms of W are given by W =

123

d4 W d2 W = = 0 at z = 0 and z = 1. dz 2 dz 4

(16)

Non-Linear Two Dimensional Double Diffusive Convection

235

 where a = (l 2 + m 2 ) is the horizontal wave number and D ≡ d/dz. The above problem can be regarded as an eigen value problem in the parameters RaT , Ra S , a, λ2 , λ2 , τ , and T a. The solutions of the boundary value problem are assumed to have the form W (z) = An sin(nπ z). where An denotes the amplitude which gives the minimum Rayleigh number when n = 1, showing that W (z) = A1 sin(π z) is the eigen function for the marginal stability. Substituting W (z) = A1 sin(π z) into Eq. 15, we obtain the expression for the Rayleigh number RaT =

(σ + δ22 ) [(1 + λ2 σ )2 δ12 + (1 + λ1 σ )2 π 2 ξ T a](σ + δ22 ) τ Ra + . S (σ + τ δ 2 ) (1 + λ1 σ )(1 + λ2 σ )a 2

(17)

For a single component fluid saturating the rotating anisotropic porous layer i.e., Ra S = 0, we obtain RaT =

(σ + δ22 )[(1 + λ2 σ )2 δ12 + (1 + λ1 σ )2 ξ π 2 T a] (1 + λ1 σ )(1 + λ2 σ )a 2

where δ 2 = (π 2 + a 2 ), δ12 = (a 2 +

(18)

π2 ), δ22 = (ηa 2 + π 2 ). ξ

3.1 Stationary State For the existence of neutral stability, the real part of σ must be zero. Further for stationary convection to occur, we must have σ = 0. From Eq. 17, we get Rayleigh number for the stationary convection as   2 2 Ra S st 2 (δ1 + ξ π T a) (19) RaT = δ2 + 2 a2 δ In the case of single component system, Ra S = 0, therefore, Eq. 19 becomes RaTst =

δ22 (δ12 + ξ π 2 T a) a2

(20)

If in the above equation, we rescale the parameters in the form RT = RaT /π 2 , γ = a 2 /π 2 and put η = ξ = 1, then we recover RTst =

(1 + γ ) [1 + γ + T a] γ

(21)

the result of Vadasz (1998), for the case of a rotating isotropic porous layer. By putting ∂ RaTst /∂a = 0, we obtain the expression for critical Rayleigh number corresponding to the critical value of wave number a = ac which satisfies the equation ηξ a 8 + 2ηξ π 2 a 6 + {(η − 1)π 2 ξ Ra S + ηξ π 4 − (1 + ξ 2 T a)π 4 }a 4 −2(1 + ξ 2 T a)π 6 a 2 − (1 + ξ 2 T a)ξ 2 π 8 T a = 0.

(22)

3.2 Oscillatory State The oscillatory convections are possible only if some additional constraints such as rotation, magnetic field, and salinity gradient etc. are present in the system. For marginal oscillatory state, we have σr = 0, therefore, putting σ = iσi in Eq. 17, we obtain

123

236

A. Kumar, B. S. Bhadauria

RaTosc =

[δ22 (1 − λ1 λ2 σi2 ) + (λ1 + λ2 )σi2 ](L 1 − L 3 σi2 ) a 2 (1 + λ21 σi2 )(1 + λ22 σi2 ) −

2L 2 [(1 − λ1 λ2 σi2 ) − δ22 (λ1 + λ2 )] a 2 (1 + λ21 σi2 )(1 + λ22 σi2 )

+

(τ 2 δ 2 δ22 + σi2 )τ Ra S (τ 2 δ 4 + σi2 )

+ iσi  (23)

where =

[(1 − λ1 λ2 σi2 ) − δ22 (λ1 + λ2 )](L 1 − L 3 σi2 ) a 2 (1 + λ21 σi2 )(1 + λ22 σi2 ) +

2L 2 [δ22 (1 − λ1 λ2 σi2 ) + (λ1 + λ2 )σi2 ] a 2 (1 + λ21 σi2 )(1 + λ22 σi2 )

+

(τ δ 2 − δ22 )τ Ra S (τ 2 δ 4 + σi2 )

.

(24)

Since physical quantity RaTosc should be real, so we have either σi = 0 or  = 0 in Eq. 23. But σi  = 0, therefore, we must have  = 0. Putting  = 0, we get a cubic equation in σi2 in the form K 1 (σi2 )3 + K 2 (σi2 )2 + K 3 (σi2 ) + K 4 = 0

(25)

where K 1 = λ1 λ2 L 3 K 2 = (τ δ 2 − δ22 )λ21 λ22 τ a 2 Ra S + τ 2 δ 4 λ1 λ2 L 3 − 2L 2 [δ22 λ1 λ2 − (λ1 + λ2 )] −[1 − δ22 (λ1 + λ2 )]L 3 − λ1 λ2 L 1 K 3 = (τ δ 2 − δ22 )(λ21 + λ22 )τ a 2 Ra S + 2L 2 δ22 + [1 − δ22 (λ1 + λ2 )]L 1 −2L 2 [δ22 λ1 λ2 − (λ1 + λ2 )]τ 2 δ 4 − τ 2 δ 4 [1 − δ22 (λ1 + λ2 )]L 3 −τ 2 δ 4 λ1 λ2 L 1 K 4 = (τ δ 2 − δ22 )τ a 2 Ra S + 2L 2 τ 2 δ 4 δ22 + τ 2 δ 4 [1 − δ22 (λ1 + λ2 )]L 1 and L 1 = (δ12 + ξ π 2 T a),

L 2 = (λ2 δ12 + λ1 ξ π 2 T a), and L 3 = (λ22 δ12 + λ21 ξ π 2 T a)

Since the Eq. 25 is a cubic equation in σi2 , so it can have one or more than one positive real roots. Now if there is no positive real root then no oscillatory motion is possible. The number of positive real roots depends upon the number of change of sign of coefficients K i ’s in Eq. 25. However, we have found that Eq. 25 has only one positive real root for some values of fixed parameters λ2 , λ2 , τ , and T a. The expressions for the critical thermal Rayleigh number for stationary convection is given in the Eq. 19. It is found that the critical thermal Rayleigh number and wave number are independent of the viscoelastic parameters and, therefore, same as in the double diffusive convection in a rotating porous medium saturated with Newtonian fluid. Also, the critical Rayleigh number for the oscillatory mode is given in Eq. 23 with  = 0. It is found to be the function of the relaxation time, retardation time, solute Rayleigh number, diffusivity ratio, and the Taylor number.

4 Finite Amplitude Analysis Although in the previous section we have studied the linear stability analysis, which is sufficient to obtain the stability criteria but to obtain the additional information like rate

123

Non-Linear Two Dimensional Double Diffusive Convection

237

of heat transfer and the convection amplitudes we need to perform the non-linear stability analysis. This will be one step forward in understanding the non-linear mechanism of the double diffusive convection. For simplification, we consider the case of two dimensional rolls, and thus make all physical quantities as independent of y. We eliminate the pressure term from the Eq. 2 by operating curl on it and introduce the stream function ψ such that u = ∂ψ/∂z, w = −∂ψ/∂ x in the above resulting equation and in Eqs. 3–4, we obtain the expressions as       ∂ 2 ∂2 ∂ 2 1 ∂2 ∂ 2ψ 1 + λ1 ψ + 1 + λ + ξTa 2 2 2 2 ∂t ∂x ξ ∂z ∂t ∂z     ∂ ∂ ∂S ∂T = 1 + λ1 1 + λ2 τ Ra S − RaT (26) ∂t ∂t ∂x ∂x    2  ∂ψ ∂ T ∂T ∂2 ∂ψ ∂ T ∂ψ ∂ + − + = η 2 + 2 T (27) ∂t ∂z ∂ x ∂ x ∂z ∂x ∂x ∂z    2  ∂ψ ∂ S ∂ ∂S ∂2 ∂ψ ∂ S ∂ψ + + − + =τ S. (28) ∂t ∂z ∂ x ∂ x ∂z ∂x ∂x2 ∂z 2 A local non-linear stability analysis shall be performed and hence we will take the following Fourier expressions: ψ = T = S=

∞ ∞   n=1 m=1 ∞ ∞   n=1 m=1 ∞ ∞  

Amn sin(max)sin(nπ z)

(29)

Bmn (t)cos(max)sin(nπ z)

(30)

Cmn (t)cos(max)sin(nπ z)

(31)

n=1 m=1

In what follows we take the modes (1, 1) for stream function, and (0, 2) and (1, 1) for temperature and concentration ψ = A11 (t)sin(ax)sin(π z)

(32)

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

(33)

S = C11 (t)cos(ax)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. 26–28 and equating the coefficients of like terms of the resulting equations, then we have dA11 (t) = D11 (t) dt

dB11 (t) = −a A11 (t) − δ22 B11 (t) − πa A11 (t)B02 (t) dt πa dB02 (t) = A11 (t)B11 (t) − 4π 2 B02 (t) dt 2 dC11 (t) = −a A11 (t) − τ δ 2 B11 (t) − πa A11 (t)C02 (t) dt dC02 (t) πa = A11 (t)C11 (t) − 4π 2 τ C02 (t) dt 2

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

123

238

A. Kumar, B. S. Bhadauria

dD11 (t) = −L (δ12 + ξ π 2 T a)A11 (t) + 2(λ2 δ12 + λ1 ξ π 2 T a)D11 (t) dt  dB11 (t) + a RaT B11 (t) − τ a Ra S C11 (t) + a RaT (λ1 + λ2 ) − τ a Ra S dt  d2 B11 (t) d2 C11 (t) dC11 (t) ×(λ1 + λ2 ) − τ a Ra S λ1 λ2 + a RaT λ1 λ2 dt dt 2 dt 2 (40) where L = 1/[λ22 δ22 + λ21 ξ π 2 T a]. We know that it is difficult to solve the above non-linear system of autonomous differential equations for time-dependent variables analytically thus we have to solve it by using a numerical method. After determining the value of the amplitude functions A11 (t), B11 (t), B02 (t), C11 (t), C02 (t), and D11 (t), we will obtain the expressions for the Nusselt number and Sherwood number as function of time. For the qualitative point of view, we now perform the steady analysis by setting the left hand side of Eqs. 35–40 equal to zero with the consideration that all the amplitudes are constants and eliminating all amplitudes except A11 , yields a equation for A11 as

τ a 2 Ra S a 2 RaT 2 2 + A11 [δ1 + ξ π T a] − 2 = 0. (41) [δ + a 2 (A211 /8)] [τ 2 δ 2 + a 2 (A211 /8)] Solution A11 = 0 gives only the pure conduction solution which is possibly a solution though it is unstable when RaT is sufficiently large, therefore, the remaining solutions with x = A211 /8 are given by A1 x 2 + A2 x + A3 = 0

(42)

where A1 = a 4 (δ12 + ξ π 2 T a) A2 = a 2 (δ12 + ξ π 2 T a)(τ 2 δ 2 + δ22 ) − a 4 (RaT − τ Ra S ) A3 = τ 2 δ 2 δ22 (δ12 + ξ π 2 T a) − τ a 2 (τ δ 2 RaT − δ22 Ra S ) The required root of Eq. 42 is,    A2 1 = −A2 + A22 − 4 A1 A3 8 2 A1

(43)

Now setting the radical in Eq. 43 to vanish, we obtain the expression for finite amplitude Rayleigh number (RaTF ) which characterize finite amplitude motions and the expression for Finite amplitude Rayleigh number is defined as    1 RaTF = −B2 + B22 − 4B1 B3 (44) 2B1 where B1 = a 4 B2 = 2a 2 [(τ 2 δ 2 − δ22 )(δ12 + ξ π 2 T a) − τ a 2 Ra S ] B3 = [(τ 2 δ 2 + δ22 )(δ12 + ξ π 2 T a) + τ a 2 Ra S ]2 −4(δ12 + ξ π 2 T a)[τ 2 δ 2 δ22 (δ12 + ξ π 2 T a) + τ a 2 δ22 Ra S ].

123

Non-Linear Two Dimensional Double Diffusive Convection

239

It is evident that the viscoelastic fluid saturated porous layer of simple fluid type becomes Newtonian when the flow is steady and weak. Hence, as far as the weak finite amplitude steady convection is concerned, there is no distinction between Newtonian fluid and viscoelastic binary fluid. In steady state, the system given by Eqs. 35–39 becomes identical with Newtonian binary fluid system for steady finite case. Therefore, the steady finite amplitude motions are independent of the viscoelastic parameters and the results coincide with the Newtonian fluid convection (see Malashetty and Heera (2008)), therefore, we have not given here the details of it.

5 Results and Discussions The results which are given in the Eqs. 19, 23 with  = 0, and (44) have been presented graphically in the Figs. 1a–g and 2a–g. In Figs. 1 and 2, we draw the neutral stability curves and depict the variation of the thermal Rayleigh number RaT for oscillatory, stationary, finite amplitude, and Newtonian fluid case with respect to the wave number a, for fixed values of λ1 = 0.9, λ2 = 0.7, Ras = 103 , ξ = 0.2, η = 0.9, T a = 30 and τ = 0.3, respectively, with variation in one of the parameters. The linear stability theory expresses the criteria of stability in terms of the critical thermal Rayleigh number RaT , below which the system is stable while unstable above. In Fig.1a–g, we have drawn neutral stability curves for oscillatory Rayleigh number, corresponding to Newtonian binary fluid and viscoelastic fluid. From the Fig.1c–g, we observed that on increasing the value of parameters concentration Rayleigh number Ras , mechanical anisotropy parameter ξ , thermal anisotropy parameter η, Taylor number T a, and diffusivity ratio τ , the value of RaT increases. Further the bifurcation points where the overstable solutions for Newtonian fluid branch off the viscoelastic fluid can be seen easily for different values of the parameters. From Fig.1a, we find that the value of RaT increases at small wave number a and decreases at large a on increasing the value of relaxation parameter λ1 . Thus, the effect of increasing the value of λ1 is to delay the onset of convection when a is small and advance it when a is large. From Fig.1b, we see qualitatively similar results as shown in Fig.1a, the value of RaT decreases when wave number a is very small and increases when a is large, on increasing the value of retardation parameter λ2 , thus the effect of increasing the value of λ2 is to advance the convection initially and delay later on. From the Fig.1d,f, we find that bifurcation points shift toward right as the value of ξ and T a increases, while they remain constant in other figures. In Fig. 2a–e, we depict the variation of stationary Rayleigh number RaTst and finite amplitude Rayleigh number RaTF with respect to wave number a. From the Fig. 2a–e, it is clear that on increasing the value of parameters Ra S , ξ, T a, and τ , the value of RaT increases, however, it decreases and then increases on increasing η. Further, one can clearly identify, from the Fig. 2a–d, the bifurcation points where the stationary solutions branch off finite amplitude solutions for the parameters Ra S , ξ, η, and T a. Also, we see from these figures that convection sets in as stationary at small value of wave number a, changes to finite amplitude convection for a good range of values of a and then becomes stationary at large values of a. The results related to the weak non-linear analysis have been shown in Figs. 3a–g and 4a–g, respectively, for quantities of heat and mass transfer across the porous layer, and are computed in terms of Nusselt number N u and Sherwood number Sh. In Figs. 3 and 4, we plot, respectively, the variation in the Nusselt number N u and Sherwood number Sh with respect to time t for unsteady motion. The values of parameters are fixed at λ1 = 0.8, λ2 =

123

240

A. Kumar, B. S. Bhadauria

a

b

c

d

e

f

g

Fig. 1 a–g Neutral stability curves for oscillatory Rayleigh number with respect to wave number a

123

Non-Linear Two Dimensional Double Diffusive Convection

a

b

c

d

241

e

Fig. 2 a–e Neutral stability curves for stationary Rayleigh number with respect to wave number a

0.2, Ra S = 200, ξ = 0.6, η = 0.3, T a = 60, τ = 0.5, while one of them is varied. From the figures, we find that rate of heat and mass transfer increases for some parameters and decreases for others as the values of the parameters increased. Also, we find that initially when t is small, heat and mass transfer oscillate and finally become constant and approach steady state value, when t is increased further. In both Figs. 3 and 4, we find qualitatively similar results. In Figs. 5–7, we draw, respectively, for steady state case, the streamlines, isotherms, and isohalines for a viscoelastic fluid saturated porous layer at ξ = 0.7, η = 0.8, T a = 30, Ra S = 500, and τ = 0.5. Figures 5–7a are drawn at RaT = RaT,c = 927.8, while Figs. 5– 7b are at RaT = 10 × RaT,c . From the Figs. 5a, b, we observed that on increasing the value of Rayleigh number, the magnitude of stream function increases, which shows that

123

242

A. Kumar, B. S. Bhadauria

a

b

c

d

e

f

g

Fig. 3 a–g Variation of Nusselt number N u with respect to time t for unsteady case

123

Non-Linear Two Dimensional Double Diffusive Convection

a

b

c

d

e

f

243

g

Fig. 4 a–g Variation of Sherwood number Sh with respect to time t for unsteady case

123

244

a

A. Kumar, B. S. Bhadauria

b

Fig. 5 a, b Streamlines for fixed values of parameters at a RaT = RaT,c and b RaT = 10 × RaT,c

a

b

Fig. 6 a, b Isotherms for fixed values of parameters at a RaT = RaT,c and b RaT = 10 × RaT,c

a

b

Fig. 7 a, b Isohalines for fixed values of parameters at a RaT = RaT,c and b RaT = 10 × RaT,c

convection occurs more faster at higher RaT . From Fig. 6a, b, we observed that isotherms are flat near the walls and become contours in middle (Fig. 6a), while they become more flat on increasing the value of RaT (Fig. 6b). In Figs. 7, we found qualitatively similar results for isohalines, as shown in Figs. 6 for isotherms.

123

Non-Linear Two Dimensional Double Diffusive Convection

245

a

b

a

b

c

a

b

c

c

Fig. 8 Streamlines, isotherms, and isohalines for different values of λ1 for time-dependent case

In Figs. 8– 9, we depict the effects of viscoelastic parameters on streamlines, isotherms, and isohalines at different time for unsteady state. In both the figures, for ψ, the sense of motion in the subsequent cells is alternately identical and opposite to that of the adjoining cell. The magnitude of stream function ψ changes initially when t is small, and remains constant at intermediate value of time and again changes on increasing the time further. From Fig. 8, we see that the effect of increasing the value of λ1 is to decrease the magnitude of stream function as well as cell size. Further from Fig. 9, we find that the effect of λ2 is to increase the cell size and magnitude of stream function. For isotherms, we see that convection occurs at initial time, which changes to conduction or slow convection on increasing t, and again changes into convection on increasing t further. In the case of isohalines, we find that initially

123

246

A. Kumar, B. S. Bhadauria

a

b

c

a

b

c

a

b

c

Fig. 9 Streamlines, isotherms, and isohalines for different values of λ2 for time-dependent case

convection occurs and it slows down on increasing the time. On further increasing t again convection takes place.

6 Conclusion In this article, we have investigated the onset of double diffusive convection in a rotating horizontal porous layer saturated with viscoelastic fluid, heated from below and cooled from above. The problem has been solved analytically. Also, we have performed the steady and

123

Non-Linear Two Dimensional Double Diffusive Convection

247

unsteady non-linear analyses. The graphs for streamlines, isotherms, and isohalines have been drawn for steady case and for transient case. The following conclusions are drawn: (i) In the neutral stability curves, it is found that the effect of relaxation parameter λ1 is to delay or advance the onset of convection, depending upon the values of a. However, we found qualitatively similar results for retardation parameter λ2 . (ii) It is found that, the effect of increasing the parameters Ra S , ξ, η, T a, and τ is to stabilize the system at marginal oscillatory state for Newtonian fluid and viscoelastic fluid. (iii) Also we found that, the effects of increasing Ra S , ξ, T a, and τ is to stabilize the system at marginal stationary state as well as for finite amplitude convection, however, the effect of increasing η is to advance the convection initially and then stabilize the system. (iv) In the unsteady case, we found that the value of N u and Sh increases for some parameters and decreases for other parameters. For small t, we found the oscillatory behavior of the Nusselt number N u and Sherwood number Sh. Further at large t, both N u and Sh approach their steady state values. (v) The magnitudes of ψ, T , and S are found to increase or decrease depending on the values of parameters. Acknowledgments This work was done during the Junior Research Fellowship awarded by UGC to the author AK. The financial assistance from UGC is gratefully acknowledged by both the authors.

Appendix A The governing equations for double diffusive convection in a viscoelastic fluid saturated rotating porous medium are given by Kim et al. (2003), Sheu et al. (2008), and Malashetty et al. (2009) 

∂ 1 + λ2 ∂t



∇.q = 0     μ ∂ 2ρ0 ∂ q + 1 + λ1  × q = 1 + λ1 (−∇ p + ρg) K ∂t δ ∂t   ∂ + q.∇ T = ∇.(κT .∇T ) γ ∂t   ∂ + q.∇ S = κ S ∇ 2 S ∂t ρ = ρ0 [1 − τT (T − T0 ) + τ S (S − S0 )]

(A1) (A2) (A3) (A4)

(A5) where q is velocity (u, v, w), μ is the dynamic viscosity, K is the permeability tensor ˆ + K z (kˆ k), ˆ κT is the thermal diffusivity tensor, κTx (iˆiˆ + jˆ j) ˆ + κTz (kˆ k), ˆ T is K x (iˆiˆ + jˆ j) temperature, αT is thermal expansion coefficient, τ S is the concentration expansion coefficient, κT is thermal diffusivity of the fluid, κ S is concentration diffusivity of the fluid, γ is the ratio of heat capacities, ρ is the the density, while ρ0 is the reference density. The boundary conditions are given by, w = T = S = 0 at z = 0 and z = d.

(A6)

123

248

A. Kumar, B. S. Bhadauria

At the basic state, fluid is assumed to be quiescent. The quantities at the basic state are given by, qb = (0, 0, 0), p = pb (z), T = Tb (z), S = Sb (z), and ρ = ρb (z)

(A7)

with the conditions d pb d2 Tb d2 Sb = −ρb g, = 0, = 0. 2 dz dz dz 2

(A8)

We now imposing a small perturbation at the basic state in the form q = qb + q  , T = Tb + T  , S = Sb + S  , ρ = ρb + ρ 

(A9)

where primes denote the perturbations, substituting Eq. A9 in Eqs. A1–A5 and using basic state Eq. A8, we obtain the perturbed equations in the form ∇.q = 0   ∂ μ  ∂ 2ρ0 1 + λ2 q + 1 + λ1  × q ∂t K ∂t δ   ∂ = − 1 + λ1 [∇ p  − ρ0 (τT T  − τ S S  )g] ∂t





∂T  dTb ∂2T  + (q  .∇)T  + w  = κTx ∇12 T  + κTz ∂t dz ∂z 2 d S ∂ S b + (q  .∇)S  + w  = κS ∇ 2 S . ∂t dz

(A10)

(A11) (A12) (A13)

Equations A10–A13 are non-dimensionalized by using the following transformations: d2 ∗  κT ∗  t ,q = q , T = (T )T ∗ , S = (S)S ∗ κT d d2 ∗ d2 ∗ λ1 = λ1 , λ2 = λ κT κT 2

(x  , y  , z  ) = (x ∗ , y ∗ , z ∗ )d, t =

and p  = 1–5.

μκT K

p ∗ to obtain non-dimensional equations (dropping the asterisks for simplicity)

References Bahloul, A., Boutana, N., Vasseur, P.: Double diffusive and Soret induced convection in a shallow horizontal porous layer. J. Fluid Mech. 491, 325–352 (2003) Desaive, Th., Hennenberg, M., Lebon, G.: Thermal instability of a rotating saturated porous medium heated from below and submitted to rotation. Eur. Phys. J. B29, 641–647 (2002) Friedrich, Einflu R.: der Prandtl-Zahl auf die Zellularkonvektion in einem rotierenden mit Fluid gesattigten porosen Medium. Z. Angew. Math. Mech 63, 246–249 (1983) Gaikwad, S.N., Malashetty, M.S., Prasad, K.R.: Linear and non-linear double diffusive convection in a fluidsaturated anisotropic porous layer with cross-diffusion effects. Transp. Porous Media 80, 537–560 (2009) Govender, S.: Oscillatory convection induced by gravity and centrifugal forces in a rotating porous layer distant from the axis of rotation. Int. J. Eng. Sci 41(6), 539–545 (2003) Green, T. III.: Oscillating convection in an elasticoviscous liquid. Phys. Fluids 11, 1410–1412 (1968) Griffith, R.W.: Layered double-diffusive convection in porous media. J. Fluid Mech. 102, 221–248 (1981) Guo, J., Kaloni, P.N.: Nonlinear stability problem of a rotating doubly diffusive porous layer. J. Math. Anal. Appl. 190, 373–390 (1995)

123

Non-Linear Two Dimensional Double Diffusive Convection

249

Herbert, D.M.: On the stability of visco-elastic liquids in heated plane Couette flow. J. Fluid Mech. 17, 353–359 (1963) Horton, C.W., Rogers, F.T.: Convection currents in a porous medium. J. Appl. Phys. 16, 367–370 (1945) Jou, J.J., Liaw, J.S.: Transient thermal convection in a rotating porous media confined between two rigid boundaries. Int. Commun. Heat Mass Transf. 14, 147–153 (1987a) Jou, J.J., Liaw, J.S.: Thermal convection in a porous medium subject to transient heating and rotating. Int. J. Heat Mass Transf. 30, 208–211 (1987b) Kim, M.C., Lee, S.B., Kim, S., Chung, B.J.: Thermal instability of viscoelastic fluids in porous media. Int. J. Heat Mass Transf. 46, 5065–5072 (2003) Kumar, A., Bhadauria, B.S.: Thermal instability in a rotating anisotropic porous layer saturated by a viscoelastic fluid. Int. J. Non-linear Mech. 453–454 (2010) Lapwood, E.R.: Convection of a fluid in a porous medium. Proc. Cambridge Philos. Soc. 44, 508–521 (1948) Laroze, D., Maritnez-Mardones, J., Bragard, J.: Thermal convection in a rotating binary viscoelastic liquid mixture. Eur. Phys. J. Special Top. 146, 291–300 (2007) Lombardo, S., Mulone, G.: Necessary and sufficient conditions of global nonlinear stability for rotating double-diffusive convection in a porous medium. Continuum Mech. Thermodyn. 14, 527–540 (2002) Malashetty, M.S.: Anisotropic thermoconvective effects on the onset of double diffusive convection in a porous medium. Int. J. Heat Mass Transf. 36(9), 2397–2401 (1993) Malashetty, M.S., Heera, R.: The effect of rotation on the onset of double diffusive convection in a horizontal anisotropic porous layer. Transp. Porous Media 74, 105–127 (2008) Malashetty, M.S., Swamy, M.: Linear and non-linear double diffusive convection in a fluid saturated anisotropic porous layer. In: Proceedings of the International Conference on Advanced Applied Mathematics, pp. 253–264 (2005) Malashetty, M.S., Swamy, M.: The effect of rotation on the onset of convection in a horizontal anisotropic porous layer. Int. J. Therm. Sci. 46, 1023–1032 (2007) Malashetty, M.S., Swamy, M.: The onset of double diffusive convection in a viscoelastic fluid layer. J. NonNewtonian Fluid Mech. 465–466 (2010) Malashetty, M.S., Tan, W., Swamy, M.: The onset of double diffusive convection in a binary viscoelastic fluid saturated anisotropic porous layer. Phys. Fluids 21, 084101–084111 (2009) Murray, B.T., Chen, C.F.: Double-diffusive convection in a porous medium. J. Fluid Mech. 201, 147–166 (1989) Nield, D.A.: Onset of thermohaline convection in a porous medium. Water Resour. Res. 4(3), 553–560 (1968) Nield, D.A., Bejan, A.: Convection in Porous Media. Springer-Verlag, New York (2006) Palm, E., Tyvand, A.: Thermal convection in rotating porous layer. Z. Angew. Math. Phys. 35, 122–123 (1984) Patil, P.R., Rudraiah, N.: Linear convective stability and thermal diffusion of a horizontal quiescent layer of a two component fluid in a porous medium. Int. J. Eng. Sci. 18, 1055–1059 (1980) Patil, P.R., Vaidyanathan, G.: Effect of variable viscosity on thermohaline convection in a porous medium. J. Hydrol. 57, 147–161 (1982) Patil, P.R., Vaidyanathan, G.: On setting up of convective currents in a rotating porous medium under the influence of variable viscosity. Int. J. Eng. Sci 21, 123–130 (1983) Poulikakos, D.: Double-diffusive convection in a horizontally sparsely packed porous layer. Int. Commun. Heat Mass Transf. 13, 587–598 (1986) Qin, Y., Kaloni, P.N.: Nonlinear stability problem of a rotating porous layer. Quart. Appl. Math. 53, 129–142 (1995) Rudraiah, N., Malashetty, M.S.: The influence of coupled molecular diffusion on double diffusive convection in a porous medium. ASME J. Heat Transf. 108, 872–876 (1986) Rudraiah, N., Shrimani, P.K., Friedrich, R.: Finite amplitude convection in a two component fluid saturated porous layer. Int. J. Heat Mass Transf. 25, 715–722 (1982) Rudraiah, N., Shivakumara, I.S., Friedrich, R.: The effect of rotation on linear and non-linear double-diffusive convection in a sparsely packed porous medium. Int. J Heat Mass Transf. 29, 1301–1316 (1986) Rudraiah, N., Kaloni, P.N., Radhadevi, P.V.: Oscillatory convection in a viscoelastic fluid through a porous layer heated from below. Rheol. Acta 28, 48–53 (1989) Rudraiah, N., Radhadevi, P.V., Kaloni, P.N.: Convection in a viscoelastic fluid-saturated sparsely packed porous layer. Can. J. Phys. 68, 1446–1453 (1990) Sheu, L.J., Tam, L.M., Chen, J.H., Chen, H.K., Lin, K.T., Kang, Y.: Chaotic convection of viscoelastic fluids in porous media. Chaos Solitons Fractals 37, 113–124 (2008) Straughan, B.: A sharp nonlinear stability threshold in rotating porous convection. Proc. Roy. Soc. Lond. A 457, 87–93 (2001) Taunton, J.W., Lightfoot, E.N., Green, T.: Thermohaline instability and salt fingers in a porous medium. Phys. Fluids 15, 748–753 (1972)

123

250

A. Kumar, B. S. Bhadauria

Vadasz, P.: Stability of free convection in a rotating porous layer distant from the axis of rotation. Transp. Porous Media 23, 153–173 (1996a) Vadasz, P.: Convection and stability in a rotating porous layer with alternating direction of the centrifugal body force. Int. J. Heat Mass Transf. 39(8), 1639–1647 (1996b) Vadasz, P.: Free Convection in Rotating Porous Media, Transport Phenomena in Porous Media. pp. 285– 312. Elsevier, Amsterdam (1998) Vadasz, P., Govender, S.: Stability and stationary convection induced by gravity and centrifugal forces in a rotating porous layer distant from the axis of rotation. Int. J. Eng. Sci. 39(6), 715–732 (2001) Wang, S., Tan, W.: Stability analysis of double-diffusive convection of Maxwell fluid in a porous medium heated from below. Phys. Lett. A 372, 3046–3050 (2008) Yoon, D.Y., Kim, M.C., Choi, C.K.: The onset of oscillatory convection in a horizontal porous layer saturated with viscoelastic liquid. Transp. Porous Media 55, 275–284 (2004)

123