Analysis of Linear stability on double diffusive convection in a fluid saturated anisotropic porous layer with Soret effect

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1 Available online at Pelagia esearch Library Advances in Applied cience esearch, 0, 3 (3):6-67 IN: COEN (UA): AAFC Analysis of Linear ability on double diffusive convection in a fluid saturated anisotropic porous layer with oret effect. N. Gaikwad * and.. Kamble epartment of Mathematics, Gulbarga University, Jnana Ganga Campus, Gulbarga, Karnataka, India epartment of Mathematics, Government Fir Grade College, Chittapur, Karnataka, India _ ABAC he double diffusive convection in a horiontal anisotropic porous layer saturated with a Boussinesq fluid, which is heated and salted from below in the presence of oret coefficient is udied analytically using linear ability analysis based on the usual normal mode technique. he generalied arcy model is employed for the momentum equation. he effect of mechanical anisotropy parameter, thermal anisotropy parameter, Lewis number and oret parameter on ationary and oscillatory convection are shown graphically. Keywords: ouble diffusive convection, oret parameter, Anisotropic porous layer, Critical ayleigh number, Lewis number. _ INOUCION he problem of convection induced by temperature and concentration gradients or by concentration gradients of two species, known as double diffusive convection, has attracted considerable intere in the la several decades. If gradients of two ratifying agencies having different diffusivities are simultaneously present in a fluid layer, a variety of intereing convective phenomena can occur that are not possible in single component fluids. he double diffusive convection in porous media has also become important in recent years because of its many applications in geophysics, particularly in saline geothermal fields where hot brines remain beneath less saline, cooler ground waters. A comprehensive review of the literature concerning double diffusive convection in a binary fluid saturated porous medium may be found in the book by Nield and Bejan [7]. Excellent review articles on double diffusive convection in porous media include those by Mojtabi and Charrier-Mojtabi ([3], [4]) and Mamou []. In a syem where two diffusing properties are present, inabilities can occur only if one of the component is deabiliing. If the cross diffusion terms are included in the species transport equations, then the situation will be quite different. ue to the cross diffusion effect, each property gradient has a significant influence on the flux of the other property. A flux of salt caused by a spatial gradient of temperature is called the oret effect. here are many udies available on the onset of double diffusive convection in a porous medium with and without cross diffusion effects (see e.g. Nield and Bejan, [7]. hermal convection in a binary fluid driven by the oret and ufour effects has been inveigated by Knobloch [8]. He has shown that equations are identical to the thermosolutal problem except for a relation between the thermal and solute ayleigh numbers. he double diffusive convection in a porous medium in the presence of oret and ufour coefficients has been analyed by udraiah and Malashetty [5]. his work has been extended to weak nonlinear analysis by udraiah and iddheshwar [6]. he effect of temperature dependent viscosity on double diffusive convection in an anisotropic porous medium in the presence of oret coefficient has been udied by Patil and ubramanian [9]. traughan and Hutter [5] have inveigated the double diffusive convection with oret effect in a porous layer using arcy-brinkman model. Pelagia esearch Library 6

2 . N. Gaikwad et al Adv. Appl. ci. es., 0, 3(3):6-67 Bahloual et al. [] have carried out an analytical and numerical udy of the double diffusive convection in a shallow horiontal porous layer under the influence of oret effect. ecently, Mansour et al. [] have inveigated the multiplicity of solutions induced by thermosolutal convection in a square porous cavity heated from below and subject to horiontal solute gradient in the presence of oret effect. Mo of the udies have usually been concerned with homogeneous isotropic porous ructures. However during the la one decade, the effect of non-homogeneity and anisotropy of the porous medium have also been udied. he geological and pedagogical processes rarely form isotropic media as is usually assumed in transport udies. In geothermal syem with a ground ructure composed of many rata of different permeabilities, the overall horiontal permeability may be up to ten times as large as the vertical component. Process such as sedimentation, compaction, fro action, and reorientation of the solid matrix are responsible for the creation of anisotropic natural porous media. Anisotropy can also be a characteriic of artificial porous material like pelleting used in chemical engineering process, fiber materials used in insulating purposes. here are many inveigations available on the thermal convection in a single component fluid saturated anisotropic porous layer heated from below. A theoretical analysis of non-linear thermal convection in an anisotropic porous media is performed by Kvernvold and yvand [7]. Nilsen and toresletten [] have udied the problem of natural convection in both isotropic and anisotropic porous channels. yvand and toresletten [8] inveigated the problem concerning the onset of convection in an anisotropic porous layer in which the principal axes were obliquely oriented to the gravity vector. Natural thermal convection in horiontal anisotropic porous layers heated from below or in vertical cavities filled with an anisotropic porous layer subjected to a conant heat flux, as described in the work of egan et al. [0]. ome other udies reported the anisotropy and heterogeneous character of porous media, and a summary of these can be found in the book of Nield and Bejan [7]. ecently many authors have udied the effect of anisotropy on the onset of convection in a porous layer (see e.g., Govinder [0], []: Malashetty and wamy [3]; Malashetty and Heera [4]). Although some work on double diffusive convection in an isotropic porous medium is available (Malashetty and Heera [4], attention has not been given to the udy of double diffusive convection in an anisotropic porous medium with oret effect. he main objective of this udy is therefore to inveigate the effect of oret coefficient, mechanical and thermal anisotropy on the double diffusive convection in a fluid saturated porous layer using linear analysis. MAHEMAICAL FOMULAION A horiontal porous layer held between two walls at 0 and d saturated with a Boussinesq fluid, which is heated and salted from below, is considered. he porous medium is assumed to possess isotropy in horiontal plane in both thermal and mechanical properties. A conant gradient of temperature and salinity is maintained between the two walls. he generalied arcy model has been employed for the momentum equation. With these assumptions the basic governing equations of motion are. q 0, () ρ0 q p µ K. q + ρ g, ε t () γ + ( q. ) (. ), t (3) ε + ( q. ) 3 t + ε, (4) ρ ρ0 β ( 0 ) + β ( 0 ) (5) Where, q is the velocity vector ( u, v, w ), ρ is the density, t is time, p is pressure, µ is the dynamic viscosity, K is permeability tensor, g is gravitational acceleration, γ is specific heat ratio, is temperature, is temperature difference between the walls, is salinity difference between the walls, is thermal diffusivity, ε is the porosity, is solute concentration, is solute diffusivity, 3 is cross diffusion due to component, is thermal expansion coefficient, temperature of cold walls. β is solute expansion coefficient, β b is the temperature of hot walls, 0 is the Pelagia esearch Library 6

3 . N. Gaikwad et al Adv. Appl. ci. es., 0, 3(3):6-67. Basic ate he basic ate of the fluid is assumed to be quiescent and is given by, ( 0,0,0 ), p p ( ), ( ), ( ), ( ) q b b b b b Using equation (6), equations () to (5) yield dp b b b ρbg, d 0, d 0, d d d ρ ρ. (6) ρ ρ β ( ) + β ( ). (7) b 0 b 0 b 0. Perturbed ate On the basic ate we superpose perturbations in the form ( x y t) ( x y t) ( x y t) ( ) ρ ρ ρ ( ) q q + q,,,, ( ) +,,,, ( ) +,,,, b b b p p ( ) + p x, y,, t, ( ) + x, y,, t b b where primes indicate perturbations. We consider only two dimensional diurbances and define ream function ψ by ψ,, ( u w ) ψ x (8) (9) Introducing (8) in equations () - (5) and using basic ate equations (7) and the transformations x t ψ x,,, t, ψ,,, d d ( ) where, * * * * * d / is the effective thermal diffusivity in vertical direction. o render the resulting equations dimensionless, we obtain (after dropping the aerisks and ε and γ are set equal to unity for simplicity). + + ψ + Pr t x ξ x x ψ ( ψ, ) η + + t x x ( x, ) ψ ( ψ, ) r + t Le x ( x, ) where, Pr is the arcy Prandtl number arcy ayleigh number anisotropy parameter ream function. β g dk v x (0), (), (), (3) vd ε, ξ is the mechanical anisotropy parameter ( K x / K ), K β g dk, is the solute ayleigh number v β 3, r is the oret parameter x β, Le is the Lewis number Equations () - (3) are solved for ress-free, isothermal, isohaline boundary conditions, namely, ψ 0 at 0,. (4) is, η is thermal, ψ is the Pelagia esearch Library 63

4 . N. Gaikwad et al Adv. Appl. ci. es., 0, 3(3): LINEA ABILIY ANALYI In this section, we discuss the linear ability analysis, which is very useful in the local non-linear ability analysis discussed in the next section. o make this udy we neglect the Jacobian in equations () and (3) and assume the solutions to be periodic waves of the form ψ Ψ sinπ α x σt e Θcosπ α x sinπ Φ cosπ α x. (5) ubituting equations (5) into the linearied version of equations () (3), we get σ Pr ( σ a ) a + a Ψ π α Θ + π α Φ, (6) + Θ π α Ψ, (7) σ + a Φ + r a Θ π α Ψ Le, (8) where, σ is growth rate, α is wave number, Ψ is the dimensionless amplitude of ream function, Φ is dimensionless amplitude of concentration perturbation, Θ is dimensionless amplitude of temperature perturbation, a π ( α + ), a π α + a π η α +. ξ and ( ) For non-trivial solution of Ψ, Θ and Φ, we require 4 4 a 3 a a a a a a a σ + a + + σ + + a a + + π α ( ) σ Pr Le Pr Pr Le Le Pr a a a + + a π α π α a r 0. Le + Le (9) 3. tationary ate For the validity of principle of exchange of abilities (i.e., eady case), we have 0 ability. hen the ayleigh number at which marginally able eady mode exis becomes ( ) ( )( ) η α + π α + α + ξ + α Le α ( α + )( r Le + ) he minimum value of the ayleigh number satisfies the equation. (0) occurs at the critical wave number 4 3 Le η x + η x + η + ( η ) x x 0 ξ π ξ ξ. () σ at the margin of α α where α x It is important to note that the critical wavenumber αc depends on the solute ayleigh number apart from its dependence on Lewis number and anisotropic properties. his result is in contra to the case of thermally isotropic porous medium. In the absence of oret effect, the ationary ayleigh number given by equation (0) reduces to c c Pelagia esearch Library 64

5 . N. Gaikwad et al Adv. Appl. ci. es., 0, 3(3):6-67 ( ) ( )( ) α ( α + ) η α + π α + α + ξ + α Le. () Equation () coincides with the results of Malashetty and wamy []. In case of single component fluid saturated porous layer, that is, when 0, the ationary ayleigh number given by equation () reduces to ( η α + ) π ( α + )( α + ξ ) α ( α + ). (3) Equation (3) coincides with that of toresletten [7] for the case of single component fluid saturated anisotropic porous layer. Further for isotropic porous medium, ξ, η, the equation (3) reduces to the classical result π ( α + ), (4) α which has the critical value c 4π for α obtained by Horton and ogers [4] and Lapwood [5]. he c critical ayleigh number c for marginal ate is computed from equation (0) for different values of the parameters and the results are discussed in section Oscillatory ate We put σ iω (ω is real) in equation (9) and rearrange the terms to get the oscillatory ayleigh number at the margin of ability, in the form osc Pr a a + + a Le Pr a Pr a + a r a π α + a r a a a Pr a Pr a a a a a a a a a a a Le Le Le Pr with the non-dimensional frequency ω in the form a a + a Le Le Pr a a a ω + a π α π α a r +. Le osc 4 a a a Le a + + Le Pr Pr osc he critical ayleigh number c for oscillatory ate is computed from equation (5) for different values of the parameters and the results are discussed in section 4. EUL AN ICUION he double diffusive convection in a horiontal anisotropic porous layer saturated with Boussinesq fluid, which is heated and salted from below in the presence of oret effect is udied analytically using linear ability analyses. he effect of mechanical anisotropy parameter, thermal anisotropy parameter, Lewis number and oret parameter on ationary and oscillatory convection are shown graphically and the results are discussed in this section. he variation of the critical ationary and oscillatory ayleigh number, (5) (6) c with solute ayleigh number osc for different values of the governing parameters is depicted in Figs. 4. he effect of the mechanical anisotropy parameter ξ on the ationary and oscillatory convection is shown in Fig.. We observe from this figure that an increase in the value of ξ decreases the critical ayleigh number for both the ationary and oscillatory modes implying that the effect is deabiliing. Further we observe that the effect of anisotropy parameter is insignificant for large solute ayleigh number. Pelagia esearch Library 65

6 . N. Gaikwad et al Adv. Appl. ci. es., 0, 3(3):6-67 η0.5, r0.005, Le3. ξ0.5, r0.005, Le3. ξ(0.5, 0., 0.05, 0.0) η(0.5, 0., 0.05, 0.0) c c ationary η ationary Fig : Variation of ationary and oscillatory critical ayleigh number c with solute ayleigh number for different values of mechanical anisotropy parameter ξ. Fig. : Fig : Variation of ationary and oscillatory critical ayleigh number number c with solute ayleigh for different values of thermal anisotropy parameter η. ξ0.5, r0.005, η0.5. ξ0.5, Le, η0.5. c Le(.5,.0,,.0) c ationary.5 Le(.5,.0,,.0).5 r(0., 0.005, 0.0, -0.0, -0.).0.0 ationary r(-0., -0.0, 0.005, 0.0, 0.) Fig. 3: Variation of ationary and oscillatory critical ayleigh number c with solute ayleigh number for different values of Lewis number Le. Fig 4: Variation of ationary and oscillatory critical ayleigh number with solute ayleigh number c for different values of oret parameter r. Fig. displays the effect of thermal anisotropy parameter η on both the ationary and oscillatory convection. It is apparent that an increase in the value of thermal anisotropy parameter η increases the critical ayleigh number for Pelagia esearch Library 66

7 . N. Gaikwad et al Adv. Appl. ci. es., 0, 3(3):6-67 both the ationary and oscillatory modes. hus the effect of increasing the thermal anisotropy parameter is to abilie the syem. he effect of Lewis number Le on the ationary and oscillatory convection is shown in Fig. 3. We find that increase in the value of Lewis number Le increases the critical ayleigh number for ationary and oscillatory modes. hus the effect of Lewis number is to abilie the syem in the ationary and oscillatory modes Fig.4 depicts the effect of oret parameter r on the ationary and oscillatory convection. We observe that the negative oret parameter abilies the syem while positive oret parameter deabilies in the ationary mode. In the oscillatory mode, the negative oret coefficient has deabiliing effect where as the positive oret coefficient has a abiliing effect. CONCLUION An analytical udy of double diffusive convection in a fluid saturated anisotropic porous layer with oret effect is udied using a linear ability analysis. We observe from this udy that the value of critical ayleigh number increases asymptotically with to indicate the abiliing effect of the solute ayleigh number on the syem in ationary and oscillatory modes. he effect of anisotropic properties is felt only for small values of. In each mode the effect of the mechanical anisotropy parameter ξ is to deabilie the syem while the effect of thermal anisotropy parameter η is to abilie the syem. he effect of Le is to abilie the syem in the ationary mode while in the oscillatory mode the trend reverses. he negative oret parameter abilies the syem while positive oret parameter deabilies in case of ationary mode while its effect reverses in case of oscillatory mode. Acknowledgement his work is supported by the University Grants Commission (UGC) New elhi under the Major esearch Project F. No /009 () dated EFENCE [] Bahloul A, Boutana N, Vasseur P, J. Fluid Mech., 003, 49, [] Mansour A, Amahmid A, Hasnaoui M, Bourie M, Numerical Heat-ransfer, 006, 49(A), [3] Mojtabi A, Charrier Mojtabi MC, Hand book of Porous Media, Marcel ekker, New York, 000, pp [4] Mojtabi A, Charrier Mojtabi MC, Handbook of Porous Media, nd edn. aylor and Francis, New York, 005, pp [5] traughan B, Hutter K, Proc. oyal oc. London A., 999, 455, [6] Horton C W, ogers F, J. Applied Physics, 945, 6, [7] Nield A, Bejan A, Convection in porous media. pringer-verlag, Berlin, 006. [8] Knobloch E, Phys. Fluids, 980, 3(9), [9] Lapwood E, Proc. Camb. Phil. oc., 948, 44, [0] egan G, Vasseur P, Bilgen E, Heat Mass ransfer, 995, 38(), [] toresletten L, ransport phenomena in porous media, Oxford, 998, pp 6-83, [] Mamou M, ransport Phenomena in porous Media, Elsevier, Oxford, II, 00, pp [3] Malashetty M, wamy M, ransp. Porous Media, 007, 67, [4] Malashetty M, Heera, ranspot in Porous Media, 007, 74, [5] udraiah N, Malashetty M, AME J. Heat ransfer, 986, 08, [6] udraiah N, iddheshwar P G, Heat and Mass ransfer, 998, 33(4), [7] Kvernvold O, yvand P A, J. Fluid Mech., 979, 90, [8] yvand P A, toresletten L, J. Fluid Mech., 999, 6, [9] Patil P, ubramanian L, Fluid yn. es., 99,, [0] Govinder, ransp. Porous Media, 006, 64, [] Govinder, ransport in Porous Media, 007, 69, [] Nilsen, toresletten L, AME J. Heat ransfer, 990,, [3] Pradeep Kumar, Adv. in Appl. ci. es., 0, 3(), [4] Kishan N, hrinivas M, Adv. in Appl. ci. es., 0, 3(), [5] ana G C, Kango K, Adv. in Appl. ci. es., 0, (3), [6] reenadh, Nanda Kishore, rinivas, Hemadri reddy, Adv. in Appl. ci. es., 0, (6), 5-. Pelagia esearch Library 67