ANALYTICAL STUDY OF DDC IN A BINARY FLUID SATURATED ANISOTROPIC POROUS LAYER

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1 Bulletin of the Marathwada Mathematical Society Vol. 13, No. 2, December 2012, Pages ANALYTICAL STUDY OF DDC IN A BINARY FLUID SATURATED ANISOTROPIC POROUS LAYER S. N. Gaikwad 1 and S. S. Kamble 2 1, Department of Mathematics, Gulbarga University, Jnana Ganga Campus, Gulbarga , India. 2. Department of Mathematics, Government First Grade College, Chittapur , Karnataka, India. Abstract Analytical study of double diffusive convection (DDC) in a horizontal fluid saturated anisotropic porous layer is studied analytically, using a linear stability analyses. The linear theory is based on the usual nomlal mode technique. The Darcy model extended to include time derivative term with anisotropic permeability is used to describe the flow through porous media. The Rayleigh number for both stationary and oscillatory modes is obtained. The effect of anisotropy parameters, solute Rayleigh number, Lewis number and Prandtl number on the stationary and oscillatory convection is shown graphically. 1 1 INTRODUCTION Most of the studies have usually been concerned with homogeneous isotropic porous structures. In a porous medium, due to the structure of the solid material in which the fluid flows, there can be a pronounced anisotropy in such parameters as permeability or thermal diffusivity. The novelties introduced by anisotropy have only recently been studied. The geological and pedological processes rarely form isotropic medium as is usually assumed in transport studies. In geothermal system with a ground structure composed of many strata of different permeabilities, the overall horizontal permeability may be up to ten times as large as the vertical component. Processes such as sedimentation, compaction, frost action, and reorientation of the solid matrix are responsible for the creation of anisotropic natural porous media. However, in many practical situations the porous materials are anisotropic in their mechanical and thermal properties Anisotropy is generally a consequence of preferential orientation or asymmetric geometry of porous matrix or fibers, and is in fact, encountered in numerous systems in 1 Keywords: Double diffusive convection, porous layer. Marathwada Mathematical Society, Aurangabad, India, ISSN

2 38 S. N. Gaikwad and S. S. Kamble industry and nature. It is particularly important in a geological context, since sedimentary rocks generally have a layered structure; the vertical direction is often much less than in the horizontal direction. Anisotropy can also be a characteristic of artificial porous material like pelleting used in chemical engineering process, fiber materials used in insulating purposes. Thermal convection in anisotropic porous media has attracted researchers only in the last three decades, despite its great relevance in engineering applications. There are few investigations available on the thermal convection in a fluid saturated anisotropic porous layer heated from below [4, 5]. Castinel and Combarnous [4] have studied the Rayleigh-Benard convection in an anisotropic porous layer both, experimentally and theoretically. Epherre [5]extended his stability analysis to a porous medium with anisotropy in thermal diffusivity also. Tyvand and Storesletten [17] investigated the problem concerning the onset of convection in an anisotropic porous layer in which the principal axes were obliquely oriented to the gravity vector. Storesletten [14] made similar analysis for an anisotropic thermal diffusivity with axis non-orthogonal to the layer; the findings here are significantly different from the orthogonal case Quin and Kaloni [11]. The effect of anisotropy of thermal instability in a fluid saturated porous medium subjected to an inclined temperature gradient of finite magnitude was analyzed by Parthiban and Patil [10] using the, Gelerkin technique. Payne et al. [9] studied penetrative convection in a porous medium with anisotropic permeability. They reported that the onset of convection may be via Hopf bifurcation rather than stationary convection. The much study on convection in an anisotropic porous layer saturated with twocomponent fluid is not available except the work by Tyvand [18]. Therefore, the aim of the present paper is to study double diffusive convection in a binary fluid saturated anisotropic porous layer. 2 MATHEMATICAL FORMULATION We consider an infinite horizontal fluid saturated porous layer confined between the planes z = 0 and z = d, with the vertically downward gravity force g acting on it. A uniform adverse temperature gradient T = (T t T u ) and a stabilizing concentration gradient s = (S t S u ) where T t > T u and S t > S u are maintained between the lower and upper surfaces. A Cartesian frame of reference is chosen with the origin in the lower boundary and the z-axis vertically upwards. The porous medium is assumed to possess horizontal isotropy in both thermal and mechanical properties. The Darcy model that includes the time derivative term with anisotropic permeability is used to describe the flow through porous media. With these assumptions the governing equations are : ρ 0 ϵ q = 0 (2.1) q + p + µk q ρg = 0, (2.2) t

3 Analytical study of ddc in a binary fluid γ T t + (q )T = (D T ), (2.3) ϵ S t + (q )S = K s 2 S, (2.4) ρ = ρ 0 [1 β T (T T 0 ) + β S (S S 0 )], (2.5) where q = (u, v; w) is the velocity, p the pressure, ϵ represents the porosity, K = Kx 1 (ii + jj) + Kt 1 (kk) the inverse of the anisotropic permeability tensor, T the temperature, S the concentration, D = D s (ii + jj) + D t (kk) the anisotropic heat diffusion tensor. ρ, µ, β T and β S denote the density, viscosity, thermal and solute expansion coefficients respectively, and K S is the mass diffusivity.lt is hereby stated that permeability and heat diffusion are most strongly anisotropic than solute diffusivity. Therefore, we ignore the solute anisotropy. Unfortunately, we have no experimental support for this, because measurement of anisotropic diffusivity is lacking. Further, γ = (ρc)m (ρc p) f, (ρc) m = (1 ϵ)(ρc) sd + ϵ(ρc p ) f, c p is the specific heat of the fluid at constant pressure, c is the specific heat of the solid, the subscripts f, sd and m denote fluid, solid and porous medium values respectively. The justification for the inclusion of the time derivative term in the Darcy equation is discussed in detail by Vadasz [19, 20] For simplicity, we assume the ratio of specific heat of fluid and solid phase to be unity. 2.1 Basic state The basic state of the fluid is assumed to be quiescent and is given by, q b = (0, 0, 0), p = p b (z), T = T b (z), S = S b (z), ρ = ρ b (z), (2.6) Using these into Eqs. (2.1)- (2.5) one can obtain, dp b dz = ρ bg, d 2 T b dz 2 = 0, d 2 S b dz 2 = 0, ρ b = ρ 0 [1 β T (T b T 0 ) + β S (S b S 0 )]. (2.7) Then the conduction state temperature and concentration are given by T b = T d z + T t, S b = S d z + S t. (2.8) 2.2 Perturbed state On the basic state we superpose perturbations in the form q = q b + q, T = T b (z) + T, S = S b (z) + S, p = p b (z) + p, ρ = ρ b (z) + ρ, (2.9) where the primes indicate perturbations. Substituting Eq. (2.9) into Eqs. (2.1) (2.5) and using basic state Eqs. (2.6) (2.7) and the transformations (x, y, z ) =: (x, y, z )d, t = t (γd 2 /D t ), (u, v, w ) = (D t /d)(u, v, w ), T = ( T )T, S = ( S)S. (2.10) to render the resulting equations dimensionless, we obtain (after dropping the asterisks

4 40 S. N. Gaikwad and S. S. Kamble for simplicity), [ 1 γp r D t 2 + ( 2 h ξ z 2 [ t )] w Ra T 2 h T + Ra s 2 hs = 0, (2.11) ) 2h 2 (η + z 2 ] + q ] [ ϵ n t 1 Le 2 + q T w = 0 (2.12) S w = 0, (2.13) where Ra T = β T g T dk z /vd z, the Darey Rayleigh number, Ra s = β s g SdK z /vd z, solute Rayleigh number, P r D = ϵvd 2 /K z D z, the Darey Prandtl number. The Prandtl number for water at 60 o F is 7.0 (approx.), P r = 0.044, and Glycerine P r = 7250 (Bansal, [1],)Le = D z /K s, the Lewis number. For free dendritic growth of alloys. the range of the Lewis number is 1 to 200 (Ramirez and Beckermann.[12]). ξ = K x /K z, mechanical anisotropy parameter. η = D x /D z, thermal anisotropy parameter, ϵ n = ϵ/γ is the normalized porosity. To restrict the parameter space to the minimum. we set ϵ = γ = 1. Eqs. (2.11)- (2.13) are to be solved for impermeable. isothermal and isohaline boundaries. Hence the boundary conditions for the perturbation variables are given by w = T = S = 0 at z = 0, 1. (2.14) 3 LINEAR STABILITY THEORY In this section we predict the thresholds of both marginal and oscillatory convections using linear theory. The eigenvalue problem defined by Eqs. (2.11)- (2.13) subject to the boundary conditions (2.14) is solved using the time-dependent periodic disturbances in a horizontal plane. Assuming that the amplitudes of the perturbations are very small. we write w W (z) T = Θ(z) exp [i(lx + my) + σi], (3.1) S Φ(z) where l.m are the horizontal wavenumbers and σ is the growth rate. Infinitesimal perturbations of the rest state may either damp or grow, depending on the value of the parameter σ. Substituting Eq. (3.1) into the linearized version of Eqs. (2.11)- (2.13) we obtain [ ] σ (D 2 a 2 ) + ( D2 P r D ξ a2 ) W + a 2 Ra r Θ a 2 Ra s Φ = 0, (3.2) [ σ (D 2 ηa 2 ) ] Θ W = 0, (3.3) [ σ 1 ] Le (D2 a 2 ) Φ W = 0, (3.4)

5 Analytical study of ddc in a binary fluid where D = d/dz and a 2 = l 2 + m 2. The boundary conditions (2.14) now read W = Θ = Φ = 0 at z = 0, 1. (3.5) We assume the solutions of Eqs. (3.2)- (3.4) satisfying the boundary conditions (3.5) in the form W (z) W 0 Θ(z) = Θ 0 sin nπz, (n = 1, 2, 3,...) (3.6) Φ(z) Φ 0 The most unstable mode corresponds to n = 1 (fundamental mode). Therefore, substituting Eq. (3.6) with n = 1 into Eqs. (3.2)- (3.4), we obtain a matrix equation δ 2 σp r 1 D + δ2 1 a 2 Ra r a 2 Ra s W σ + δ2 2 0 Θ 0 = 0, (3.7) 1 0 σ + δ 2 Le 1 Φ 0 0 where δ 2 = π 2 + a 2, δ1 2 = π2 ξ 1 + a 2 and δ2 2 = π2 + ηa 2.!he condition of nontrivial solution of above system of homogeneous linear equations (3.7) yields the expression for thermal Rayleigh number in the form Ra T = (σ + δ2 2 ) a 2 ( σδ 2 P r D + δ ) a2 Ra s σ + δ 2 Le 1, (3.8) The growth rate σ is in general a complex quantity such that σ = σ r + Iσ 1. The system with σ r < 0 is always stable, while σ r > 0, it will become unstable. For neutral stability state σ r = Stationary state For the validity of the principle of exchange of stabilities (i.e., steady case), we have σ = 0 (i.e., σ r = σ i = 0) at the margin of stability. Then the Rayleigh number at which marginally stable steady mode exists becomes Ra St T = 1 ( ) a 2 a 2 + π2 (ηa 2 + π 2 ) + ζ ( ηa 2 + π 2 a 2 + π 2 occurs at the wavenum- The minimum value of the stationary Rayleigh number Ra St T ber a = a St cr where (a St cr ) 2 = α satisfies the equation ) LeRa S. (3.9) ηa 4 + 2π 2 ηa 3 + π 2 ( π 2 (η ξ 1 ) + LeRa S (η 1) ) α 2 2π 6 ξ 1 α π 8 ξ 1 = 0. (3.10) It is important to note that the critical wavenumber a St cr depends on the solute Rayleigh number apart from the Lewis number and anisotropic properties. This result is in contrast to the case of isotropic porous medium. For an isotropic porous media, that is when ξ = η = 1, Eq. (3.9) gives Ra St T = 1 a 2 (a2 + π 2 ) 2 + LeRa S, (3.11)

6 42 S. N. Gaikwad and S. S. Kamble which is the classical result for a double diffusive convection in an isotropic porous media (see e.g. Nield and Bejan [8]). For single component fluid,ra S = 0, the expression for stationary Rayleigh number given by Eq. (3.9) reduces to Ra St T = 1 ( ) a 2 a 2 + π2 (ηa 2 + π 2 ), (3.12) ξ which is the one obtained by Storesletten [14]for the case of single component fluid saturated anisotropic porous layer. Further, for an isotropic porous medium, ξ = η = 1, the above Eq. (3.12) reduces to the classical result Ra St T = π2 (a 2 + 1) 2 a 2, (3.13) which has the critical value Ra St T cr = 4π2 for a St cr = 1 obtained by Horton and Rogers [6] and Lapwood [7]. 3.2 Oscillatory state We now set σ = iσ i in Eq. (3.8) and clear the complex quantities from the denominator, to obtain Ra T = 1 + iσ i 2. (3.14) Here the expression for 1 and 2 are not presented for brevity. Since Ra T is a physical quantity, it must be real. Hence, from Eq. (3.14) it follows that either σ i = 0 (steady onset) or 2 = 0(σ i 0, Oscillatory onset). For oscillatory onset or 2 = 0(σ i 0), this gives an expression for frequency of oscillations in the form (on dropping the subscript i) σ 2 = a2 Ra S (δ 2 2 δ2 Le 1 ) (δ δ2 δ 2 2 P r 1 D ) ( ) δ 2 2. (3.15) Le Now Eq. (3. 14) with 2 = 0, gives Ra 0sc T = 1 ( a 2 δ1δ σ2 δ 2 ) + Ra S(σ 2 + δ 2 δ2 2Le 1 ) P r D σ 2 + (δ 2 Le 1 ) 2. (3.16) The analytical expression for oscillatory Rayleigh number given by Eq. (3.16) is minimized with respect to the wavenumber, after substituting for σ 2 (> 0) from Eq. (3.15), for various values of physical parameters in order to know their effects on the onset of oscillatory convection. 4 RESULTS AND DISCUSSION Analytical study of double diffusive convection in a two-component Boussinesq fluid saturated anisotropic porous layer is investigated analytically using the linear theory. In the linear stability theory the Rayleigh number for the stationary and oscillatory convection is obtained. The variation of stationary and oscillatory Rayleigh number with scaled wavenumber α for various parameter values are shown in Figs.1-5. From

7 Analytical study of ddc in a binary fluid these figures, we observe that for smaller values of the wavenumber of each curve is a margin of the oscillatory instability and at some fixed α depending on the other parameters the over- stability disappears and the curve forms the margin of stationary convection. The effect of mechanical anisotropy parameter ξ for the fixed values of η = 0.3, Le = 5, Ra S = 100, P r D = 10, on both the stability curves is depicted in Fig 1. It can be observed that an increase in ξ decreases the minimum of the Rayleigh number for both the states, indicating that, the effect of increasing mechanical anisotropy parameter ξ; is to advance the onset of stationary and oscillatory convection. Fig 2 indicates the effect of thermal anisotropy parameter η on stationary and oscillatory curves for fixed values of ξ = 0.5, Le = 5, Ra s = 100, P r D = 10. It is observed that critical value of Rayleigh number increases with η indicating that the effect of thermal anisotropy parameter 17 is to inhibit the onset of oscillatory.convection. Fig 3 depicts the effect solute Rayleigh number Ra s on the stationary and oscillatory curves for fixed values of η = 0.3, Le = 5, ξ = 0.5, P r D = 10. We find that the effect of increasing Ra s is to increase the critical value of the Rayleigh number and the corresponding wave number. Thus the solute Rayleigh number Ra s has a stabilizing effect on the double diffusive convection in porous medium. In Fig 4 the neutral stability curves for different values of Lewis number Le are drawn for fixed values of η = 0.3, ξ = 0.5, Ra s = 100, P r D = 10. It is observed that with the increase of Le the critical values of Rayleigh number and the corresponding wavenumber for both the modes increase. Therefore, the effect of Le is to advance the onset of oscillatory convection where as, its effect is inhibit the stationary onset. The neutral stability curves for different values of Darcy-Parndtl number P r D is presented in Fig 5 for fixed values of ξ = 0.5, η = 0.3, Ra s = 100, Le = 5. From this figure it is evident that for small and moderate values of P r D the critical value of Rayleigh number decreases with the increase of P r D, however this trend is reversed for large values of P r D. The variation of the critical Rayleigh number R r,cr with solute Rayleigh number Ra s for different values of the governing parameters is depicted in Figs The effect of the mechanical anisotropy parameter ξ on both the stationary and oscillatory convection for fixed values of η = 0.5, Le = 3.0 is shown in Fig 6. We observe from this figure that an increase in the value of ξ decreases the critical Rayleigh number for the stationary and oscillatory modes implying that the effect is destabilizing. Fig. 7 displays the effect of thermal anisotropy parameter η on the stationary and oscillatory convection for fixed values of ξ = 0.5, Le = 3.0. We find that an increase in the value of thermal anisotropy parameter increases the critical Rayleigh number for the stationary and oscillatory modes. Thus the effect of increasing the thermal anisotropy parameter is to stabilize the system. The effect of Lewis number on the stationary convection for fixed values of ξ = 0.5, η = 0.5 is shown in Fig 8. We find that increase in the value of Lewis number increases the critical Rayleigh number for stationary mode and is to stabilize the system. For oscillatory mode, an increase in the value of Lewis number decreases the critical Rayleigh number implying that the Lewis number destabilizes the system.

8 44 S. N. Gaikwad and S. S. Kamble

9 Analytical study of ddc in a binary fluid... 45

10 46 S. N. Gaikwad and S. S. Kamble 5 CONCLUSIONS Analytical study of double diffusive convection in a horizontal anisotropic poious layer saturated with a Boussinesq fluid is studied analytically using linear stability theory. The effect of increasing the mechanical anisotropy parameter ξ is to advance the onset of oscillatory convection. It is found that the thermal anisotropy parameter delay the onset of oscillatory convection. There exist critical value for solute Rayleigh number and Lewis number such that the convection mode switches from stationary to oscillatory when the values of these parameters exceeds the critical values. The Darcy-Prandtl number has a dual effect on the oscillatory mode, in the sense that there is a critical value say P rd such that a strong destabilizing effect is observed when P r D < P rd, while for P r D > P rd, this effect is reversed and hence the system is stabilized. The stabilizing effect of P r D becomes less significant when P r D >> P rd The Lewis number advances the onset of oscillatory convection while, it delays the onset of stationary convection. Acknowledgement: This work is supported by the University Grants Commission, New Delhi under the Major Research Project with F.No (SR) dated References [1] Bansal, J. L., Viscous Fluid Dynamics, 2nd edn., Oxford & IBH Publishing Co.Pvt. Ltd., New Delhi, [2] Banu, N., and Rees, D. A. S., Onset of Darcy-Benard convection using a thermal non- equilibrium model, Int. J. Heat Mass transfer, 45 ( 2002), [3] Bormann, A. S., The onset of convection in the Rayleigh -Benard problem for compressible fluids, Continuum. Mech. Themodyn, (2001), [4] Castinel, G., and Combarnous, M., Critere d apparition de la convection naturelle dans une couche poreuse anisotrope, C.R. Hebd. Seanc. Acad. Sci, Paris B, 278 (1974), [5] Epherre, J. F., Critere d apparition de la convection naturelle dans une couche poreuse anisorrope, Rev. Gen. Therm. 168 (1975), [6] Horton, C. W., and Rogers, F. T., Convection currents in a porous medium, J. Appl. Phys., 16 (19f5), [7] Lapwood, E. R., Convection of a fluid in a porous medium, Proc. Camb. Phil., Soc., 44 (1948), [8] Nield, D. A., and Bejan, A., Convection in porous media, 3rd edn. Springer- Verlag, New York, (2006).

11 Analytical study of ddc in a binary fluid [9] Payne, L. E., Rodrigues, J. F., and Straughan, B., Effect of anisotropic permeability on Darcy s law, Math.Mech.Apll.Sci., 24 (2001), [10] Parthiban, C., and Patil P. R., Effect of inclined temperature gradient on thermal instability in an anisotropic porous medium, Warme- und Stoffubertragung, 29 (1993), [11] Qin, Y., and Kaloni, P. N., Convective instabilities in anisotropic porous media, Stud.appl.Math., 9 (1994), [12] Ramirez, J. C., and Beckermann, C., Examination of binary alloy free dendritic growth theories with a phase -field model, Acta Materiala, 53 (2005), [13] Rudraiah, N., Kaloni, P. N., and Radhadevi,P. V., Oscillatory convection in a viscoelastic fluid through a porous layer heated from below, Rheol. Acta, 28 (1989), 48. [14] Storesletten, L., Natural convection in a horizontal porous layer with anisotropic thermal diffusivity, Trans. Porous Med., 12 (1999), [15] Straughan, B., Stability and Wave Motion in Porous Media, 115, Springer, Amsterdam, [16] Straughan, B., The Energy Method, Stability, and Nonlinear Convection, 2nd ed., Springer, Amsterdam [17] Tyvand, P. A., and Storesletten, L., Onset of convection in an anisotropic porous medium with oblique principal axis, J. Fluid Mech.,.226 (1991) [18] Tyvand, P.A., Thermohaline instability in anisotropic porous media, Water Resour. Res., 16 (1980), [19] Vadasz, P., Coriolis effect on gravity-driven convection in a rotating porous layer heated from below, J. Fluid Mech. 376 (1998a), [20] Vadasz, P., Free convection in rotating porous media, Transport Phenomena in Porous media (eds. D. B. ingham and I. Pop), Elsevier, Oxford, (1998b)