Onset of Darcy-Brinkman Convection in a Maxwell Fluid Saturated Anisotropic Porous Layer

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1 Journal of Applie Flui Mechanics Vol. 9 No. 4 pp Available online at IN EIN DOI: /acapub.jafm Onset of Darcy-Brinkman Convection in a Maxwell Flui aturate Anisotropic Porous Layer. N. Gaikwa an A. V. Javaji Department of Mathematics Gulbarga University Kalaburagi Jnana Ganga CampusGulbarga Karnatak Inia Corresponing Author sngaikwa009@yahoo.co.in (Receive January 8 015; accepte eptember 015) ABRAC In the present stuy the onset of Darcy-Brinkman ouble iffusive convection in a Maxwell flui-saturate anisotropic porous layer is stuie analytically using stability analysis. he linear stability analysis is base on normal technique. he moifie Darcy-Brinkmam Maxwell moel is use for the momentum equation. he yleigh number for stationary oscillatory an finite amplitue convection is obtaine analytically. he effect of the stress relaxation parameter solute yleigh number Darcy number Darcy-Prantl number Lewis number mechanical an thermal anisotropy parameters an normal porosity parameter on the stationary oscillatory an finite amplitue convection is shown graphically. he nonlinear theory is base on the truncate representation of the Fourier series metho an is use to fin the heat an mass transfer. he transient behavior of the Nusselt an herwoo numbers is obtaine by solving the finite amplitue equations using the Runge-Kutta metho. Keywors: Double iffusive convection; Darcy brinkman Maxwell moel; Porous layer Anisotropy; Heat an mass transfer. NOMENCLAURE a c Da g H J K Le l m Nu p Pr D q h t x yz Overall horizontal wave number specific heat Darcy number height of the porous layer gravitational acceleration rate of heat transport per unit area rate of mass transport per unit area permeability tensor Lewis number Horizontal wave numbers Nusselt number pressure Darcy Prantl number velocity vector solute yleigh number thermal yleigh number solute concentration herwoo number salinity ifference between the walls time temperature temperature ifference between the walls space coorinates thermal expansion coefficient solute expansion coefficient imensionless amplitue of concentration perturbation normalize porosity ratio of specific heats thermal anisotropy parameter thermal iffusivity solute iffusivity iffusivity stress relaxation parameter stress relaxation time porosity ynamic viscosity e effective viscosity kinematics viscosity imensionless amplitue of temperature perturbation flui ensity growth rate angular velocity of rotation mechanical anisotropy parameter stream function

2 . N. Gaikwa an A. V. Javaji / JAFM Vol. 9 No. 4 pp Other ymbols h x y h z ubscripts an uperscripts b basic state c critical f flui h horizontal m porous meium 0 reference value s soli imensionless quantity ' perturbe quantity F finite amplitue Osc oscillatory state t stationary state 1. INRODUCION Double iffusive convection in porous meium is prevalent in nature occurring such iverse areas as in polymer processing chemical separation techniques an transport in biological system. ypical examples of natural porous meia inclue sanstone woo an human tissue incluing lungs an bloo vessels. Unerstaning the nature the behavior an stability characteristics of viscoelastic flui in porous meia is also important in many engineering fiels for example in oil recovery processes paper an textile manufacturing wetting an rying processes an composite manufacturing processes see for instance Khuzhayorov et al. (000) an Khan et al. (007). Double-iffusive convection is referre to buoyancy riven flows inuce by combine temperature an concentration graients. he onset of ouble iffusive convection in a flui saturate in porous meium is regare as a classical problem ue to its wie range of applications in many engineering fiels such as evaporative cooling of high temperature systems agricultural prouct storage soil sciences enhance oil recovery packe-be catalytic reactors an the pollutant transport in unergroun. A etaile review of the literature concerning ouble iffusive convection in binary flui in a porous meium was given by Niel an Bejan (006) revisan an Bejan (1990) an Malashetty an Kollur (011). hermal convection in binary flui riven by the oret an Dufour effects was investigate by Knobloch (1980) an showe that the equations were ientical to the thermosolutal problem except relation between the thermal an solutal yleigh numbers. It is well known that many applications in engineering isciplines as well as in circumstances linke to moern porous meia involve high permeability porous layer. For instance in biomeical hyroynamic stuies a thin fibrous surface layer coating bloo vessels (enothelial surface layer) is foun to be a highly permeable high porosity porous meium is stuie by Khale an Vafai (003). In such circumstances the use of a non-darcy moel which takes care of bounary an/or inertia effects is of funamental an practical interest to obtain accurate results. Further it is believe that the results of this stuy are useful in briging the gap between a non-porous case in which Da an a ense porous meium in which Da 0. A better unerstaning of the characteristics of the Darcy Brinkman equation is therefore an important part of more practical problems an thus forms a motivation of the present report. For the low porosity meia the viscous effects near the bounary are negligible. In such situations Darcy s laws is a goo approximation for the momentum equation. he main avantage of Darcy s flow moel is that it linearizes the momentum equation an thus reuces a significant amount vali for flow through regular structures over the whole spectrum of the porosity. his moel is silent about ifficulty in solving the governing equations. Further the classical Darcy moel is the flow structure near the bouning surfaces where close packing of the porous material is not possible. Brinkmam moel is vali for a sparsely packe porous meium wherein there is more winow flui to flow so that the istortions of velocity give rise to the usual shear force. An analytical an the numerical stuy of ouble iffusive convection with parallel flow in a horizontal sparsely packe porous layer uner the influence of constant heat an mass flux was performe using a Brinkmam moel by Amahmi et al. (1999). As some new technologically significant materials are iscovere acting like non-newtonian fluis therefore mathematicians physicists an engineers are actively conucting research in rheology.maxwell fluis can be consiere as a special case of a Jeffreys-Olroy B flui which contain relaxation an retaration time coefficients. Maxwell s constitutive relation can be recovere from that corresponing to Jeffreys-Olroy B fluis by setting the retaration time to be zero. everal fluis such as glycerin crue oils or some polymeric solutions behave as Maxwell fluis. Recently interest in viscoelastic flows through porous meia has grown consierably ue to the emans of such iverse fiels as biorheology geophysics chemical an petroleum inustries. he works on convective instability threshols for viscoelastic fluis in porous meia ( ) can be foun in the literature an have not been given much attention. Recently Wang an an (008) have mae the stability analysis of ouble iffusive convection of Maxwell flui in a porous meium. It is worthwhile to point out that the first viscoelastic rate type moel which is still use wiely ue to Maxwell (1866). Maxwell i not 1710

3 . N. Gaikwa an A. V. Javaji / JAFM Vol. 9 No. 4 pp evelop this moel for polymeric liquis he recognize that such a flui has a means for storing energy characterizing its viscous nature. Malashetty et al. (009) have stuie ouble iffusive convection in a viscoelastic flui saturate porous layer using Olroy moel. More recently Awa et al. (010) use the Darcy-Brinkman-Maxwell moel to stuy linearstability analysis of a Maxwell flui with cross-iffusion an ouble-iffusive convection. hey foun that the effect of relaxation time is to ecrease the critical Darcy-yleigh number. Although many works are available on the use of non-darcy moels to stuy flow an heat transfer in porous meia in the recent past the works on ouble iffusive convection in an anisotropic sparsely packe porous layer are very sparse an it is in much-to-be esire state. In the present paper we inten to analyze of a Darcy-Brinkmam binary Maxwell flui saturate anisotropic porous layer. Our objective is to stuy how the onset criterion for oscillatory convection is affecte by Maxwell flui an the other parameters an also to know their heat an mass transfer in a more general porous meium in limiting cases some previously publishe results can be recovere as the particular cases of our results.. MAHEMAICAL FORMAION We consier an infinite horizontal flui-saturate anisotropic porous layer confine between the planes z 0 an z with vertically ownwar gravity force g acting on it. A uniform averse temperature graient l u an a stabilizing concentration graient l u ( l u an l u) are impose at the bottom an top bounaries respectively. he bounaries are impermeable an we assume that the flui an soli phases are in local thermal equilibrium. A Cartesian frame of reference is chosen with origin in the lower bounary an z -axis vertically upwar. he velocities are assume to be small so that the avective an Forchheimer inertia effects are ignore. he Boussinesq approximation which states that the variation in ensity is negligible everywhere in the conservations except in the buoyancy term is assume to hol. he Darcy- Brinkman Maxwell moel is employe to escribe the flow in the porous meia. he basic state is assume to be quiescent an we superpose perturbations on the basic state. he equations for the perturbation are. q 0 (1) 0 q 1 p0[1 ( 0) ( 0)] g t t e qkq. () ( q. ) w (3) t (. q ) ( ) (4) t where q ( u v w) is the velocity p is pressure is the relaxation time g 00 gis the acceleration ue to gravity is the flui viscosity is the flui ensity 0 is the reference ensity an are the temperature an concentration respectively an is the porosity of the porous meium K Kx iiky jj Kz kk is the inverse of the permeability tensor an κ Τ x ii y jj z kk is the thermal iffusivity tensor. ( c) = m ( c) m (1 )( c p ) c f p is the ( c p ) f specific heat of the flui at constant pressure c is the specific heat of the soli the subscripts f s an m enote flui soli an porous meium values respectively an f e an are the thermal an solute expansion coefficients flui viscosity effective viscosity an solute iffusivity respectively. It is hereby state that permeability is most strongly anisotropic than solute iffusivity. herefore we ignore the solute anisotropy. Notice that in the case = 0 the moel reuces to the Newtonian binary flui. By operating curl twice on Eq. () we eliminate p an then rener the resulting equation an Eqs. (3) an (4) imensionless using the following transformations * * * * x y z x y z t t z * ( uvw ) z ( u v w ) (5) to obtain non-imensional linear equations as (on ropping the asterisks for simplicity) 1 1 w h h t PrD t 1 4 h Da w z h q. w 0 t z 1 q. w 0 t Le (8) where z / (6) (7) relaxation parameter. he 1711

4 . N. Gaikwa an A. V. Javaji / JAFM Vol. 9 No. 4 pp stress relaxation parameter is the ratio of a relaxation time characterizing the intrinsic fluiity of a material an the characteristic timescale. he smaller the stress relaxation parameter the more flui the material appears. g Kz / z the thermal yleigh number g Kz/ z the solute yleigh number Da ek z / f the Darcy number Pr D / K zz the Darcy-Prantl number. he parameters (namely Pr D an Le) epen on the properties of the flui. It is worth mentioning here that the Darcy Prantl number Pr D inclues the Prantl number Darcy number porosity an the specific heat ratio. It epens on the properties of the flui an on the nature of porous matrix. he Prantl number affects the stability of the porous system through this combine imensionless group. Le z / s the ratio between thermal an solutal iffusivities is characterize by the Lewis number. kx / kz the mechanical anisotropy parameter x / z is the thermal anisotropy parameter / normalize porosity. he normalize porosity is expresse in terms of the porosity of the porous meium an the soli to flui heat capacity ratio. Equation (6) - (8) are solve for stress free isothermal an isosolutal bounary conitions. Hence the bounary conitions for the perturbation variables are given by w w 0 at z 01. (9) z 3. LINEAR ABILIY ANALYI In this section we preict the threshol of both stationary an oscillatory convection using linear theory. he Eigen value problem efine by Eqs. (6) - (8) subject to the bounary conitions (9) is solve using the time-epenent perioic isturbances in a horizontal plane upon assuming that amplitues are small enough an can be expresse as z z w W z exp i lx my t (10) where l an m are the wave numbers in the horizontal plane an is the growth rate. Infinitesimal perturbations of the rest state may either amp or grow epening on the value of the parameter. ubstituting Eq. (9) into Eqs. (6) - (8) we obtain 1 D a W a a PrD (11) 1 DaD a a D W 1 D a W 0 Le D a (1) W 0 (13) where D z / an a l m. In case of stressfree bounary conitions it is possible to solve analytically the system of Eqs. (11) - (13). his is a stanar Eigenvalue-Eigen function problem. Here the yleigh number is taken as the Eigenvalue an it is expresse as a function of the other parameters which govern the stability of the system. he corresponing bounary conitions are W D W 0 at Z 01 (14) he solutions of Eqs. (11) - (13) satisfying the bounary conitions (14) are assume in the from z z W z W0 0 sin nz n (15) 0 he most unstable moe corresponing to n 1 (funamental moe). herefore substituting Eq (15) with n 1into Eqs. (11) - (13) an using the solvability conition we obtain a matrix equation of the form 4 Da 1 Pr a a D 1 W Le (16) where a 1 1 a an a. he conitions of non-trivial solutions of system of homogenous linear equations (16) yiels the expression for the thermal yleigh number in the 4 1 Da Pr 1 D a form (17) 1 Le 3.1 Marginal tate For valiity of the principle of exchange of stabilities (i.e. steay case) we have 0 (i.e. r i 0 ) at the margin of stability. hen the yleigh number at which marginally stable steay moe exists becomes 171

5 . N. Gaikwa an A. V. Javaji / JAFM Vol. 9 No. 4 pp st a Da a a a a Le a (18) he above result is inepenent of the relaxation time an ientical to that of the Newtonian problem. In the absence of Da 0 i.e. for ensely packe porous meium Eq. (18) reuces to st a a Le a a a (19) which is ientical with Malashetty an wamy (010). Further for an isotropic porous meium that is when 1 Eq. (19) gives a st Le (0) a which is the one obtaine by Gaikwa an Bharati (013). 3. Oscillatory tate We now set i in Eq. (17) an clear the complex quantities from the enominator to obtain 1ii (1) where 4 1 Da a 1 a Pr D 1 1 Le 1 Le 4 1 Da 1 a 1 Le 1 Le a Pr D 1 () (3) ince is a physical quantity it must be real. Hence from Eq. (1) it follows that either i 0 (steay onset) or 0 i 0 (oscillatory onset).for oscillatory onset 0 i 0 an this gives an expression for frequency of oscillations in the form (on ropping the subscript i ) a0 a1 a 0 (4) where the coefficients are given by a0 1 a1 Le PrD PrD 1 Da PrD Da PrD 1 a Le PrD a Pr D PrD Le 1 DaPrD Le 4 1 D Le D 1 1 a Le 1 Le PrD Da Pr Le a 1 Pr a Pr D. Now Eq. (3) with 0 gives 4 1 Da a Le 1 osc a Pr D 1 1 Le (5) he expression for the oscillatory yleigh number given by Eq. (5) in the limit Da 0 for ensely packe porous meium coincies exactly with the one given by Gaikwa an Bharati (013) for isotropic case after making necessary rescaling. 4. WEAK NONLINEAR ANALYI In this section we consier the nonlinear analysis using a truncate representation of the Fourier series consiering only two terms. We consier the early stages of nonlinear convection when the basic structure of the convection is still etermine bythe behavior of the linearize solution. In the immeiate vicinity of the stability bounary we coul evelop a weakly nonlinear analysis in which the amplitues are no longer small but finite. A weak nonlinear stability analysis is performe using a truncate representation of the Fourier series metho. For simplicity of analysis we confine ourselves to the two-imensional rolls so that all the physical quantities are inepenent of y. We introuce a stream function such that u z w x into the Eqs. (3) - (4) to obtain 1 1 t PrD t x x 1 4 Da x z 0 t x z x z x 0 (6) (7) 1713

6 . N. Gaikwa an A. V. Javaji / JAFM Vol. 9 No. 4 pp t Le x z x (8) As mentione earlier we set 1 for st simplicity. For flows with linear stability analysis is not vali one has to take into account the non-linear effects. he first effect of non-linearity is to istort the temperature an concentration fiels through the interaction of an also. he istortion of these fiels will correspons to a change in the horizontal mean i.e. a component of the form sin z will be generate. hus a minimal Fourier series which escribes the finite amplitue free convection is given by A()sin( t ax)sin( z) (9) B()cos( t ax)sin( z) C()sin( t z) (30) E()cos( t ax)sin( z) F()sin( t z) (31) where the amplitues A() t B() t C() t E() t anf() t are to be etermine from the ynamics of the system. ubstituting equations (9)-(31) into the couple nonlinear system of partial ifferential equations (6)-(8) an equating the coefficients of like terms we obtain the following non-linear autonomous system of ifferential equations X D t (3) where X A B C E F with D D D D D D D an D1A D aa B aac a D3 4 C AB 1 D4 aa E aaf Le 1a 4 D5 AE F Le 4 Pr D D 11 Da A D6 Pr. D a B D a E D 4 he non-linear system of autonomous ifferential equations is not suitable to analytical treatment for the general time-epenent variable an we have to solve it using a numerical metho. However one can make qualitative preictions as iscusse below. he system of Eqs. (3) is uniformly boune in time an possesses many properties of the full problem. Like the original Eqs. ()-(4) Eqs. (3) must be issipative. hus the volume in the phase space must contract. In orer to prove volume contraction we must show that the flow fiel has a constant negative ivergence. Inee A B C E A t B t C t E t PrD 1 Da 4 F F t Le Le Le (33) which is always negative an therefore the system is boune an issipative. As a result the trajectories are attracte to a set of measure zero in the phase; in particular they may be attracte to a fixe point a limit cycle or perhaps a strange attractor. From Eq. (33) we conclue that if a set of initial points in the phase occupies a region PrD 4 1 Da Le Vt V0exp t Le Le (34) his expression inicates that the volume ecreases exponentially with time. We can also infer that the relaxation parameter an Lewis number ten to enhance contraction. 4.1 teay Finite Amplitue Motions From qualitative preictions we look into possibility of an analytical solution. In case of steay motions Eqs. (6) - (8) can be solve in close form. he steay-state solutions are useful because they preict that a finite-amplitue solution to the system is possible for subcritical values of the yleigh number an that the minimum values of for which a steay solutions is possible lie below the critical values for instability perturbation setting the left-han sie of Eqs. (3) equal to zero an writing the amplitues BCE an F in terms of A we get Ax 1 Ax A3 0 (35) A where x an A1a Le 1 Da A 1 Da Le a a Le Le A3 1 Da a Le he require root of Eq. (35) is given by 1 1 x A A 4 AA 1 3. A1 (36) When we set the raical in the above equation 1714

7 . N. Gaikwa an A. V. Javaji / JAFM Vol. 9 No. 4 pp vanish we obtain the expression for the finite amplitue yleigh number F which characterizes the onset of finite amplitue steay motions. he finite- amplitue yleigh number can be obtaine in the form 1 F 1 B B 4 B1B3 (37) B1 where 8 B1 a Le B a Le a DaLe a Le adale ale B3 a Le a DaLe a Da ale1 ada1 a adale adale ale 1 4aDaLe ale 1 adale adale 1 ale 1. he expression for the steay finite-amplitue yleigh number given by Eq. (37) is evaluate for critical values an the results are iscusse. 4. Heat an Mass ransport In the stuy of convection in fluis the quantification of the heat an the mass transport is important. his is because the onset of convection as the yleigh number is increase is more reaily etecte by its effect on the heat an mass transport. In the basic state heat an mass transport is by conuction alone. If H an Jare the rate of heat an mass transport per unit area respectively then H total z an z z0 J total z (38) z z 0 where the angular brackets correspon to a horizontal average an z total 0 x z t an z total 0 x z t (39) substituting Eq. (30) - (31) into Eqs. (39) an using the resultant equation in Eq. (38) we get z z H 1 C an J 1 F. (40) he Nusselt (Nu) an herwoo (h) numbers are respectively efine by H Nu 1 C an J h 1 F (41) writing C an F in terms of A an substituting into Eqs. (41) we obtain a x Nu 1 a x x h 1. x Le a (4) (43) he secon term on the right-han sies of Eqs. (4) an (43) represent the convective contributions to heat an mass transport respectively. Further we solve the system of autonomous Eqs. (3) numerically using the Runge Kutta metho an trace the transient curves for heat an mass transfer for various values of the parameters. 5. REUL AND DICUION he onset of ouble iffusive convection of a Maxwell flui in a saturate anisotropic Darcy- Brinkman porous layer which is heate an salte from below is investigate analytically using both linear an nonlinear theories. he linear theory is use to obtain the criterion for the onset of stationary an oscillatory convection. he weakly nonlinear theory provies the quantification of heat an mass transport. he neutral stability curves in the Osc a plane for various parameter values are as shown in figures1-5. We fixe the values for the parameters except the varying parameter. From these figures it is clear that the neutral curves are connecte in a topological sense. his connection allows the linear stability criteria to be expresse in terms of the critical yleigh number c below which the system is stable an unstable above. he stationary critical yleigh number is foun to be inepenent of the viscoelastic parameter an therefore concurs with the classical results of ouble iffusive convection in porous meium for Newtonian flui (see e.g Niel an Bejan) (006). In figure 1 the effect of Darcy number Da on the onset of the oscillatory convection when all other parameters are fixe is shown. We observe that the minimum of yleigh number for oscillatory moe increases with increasing Darcy number Da. he size of Darcy number Da is relate to the importance of viscous effects at the bounaries an reuction in Da ecreases this effect which allows the flui to move more freely thereby ecreasing the oscillatory yleigh number. he neutral stability curves for ifferent values of scale stress relaxation parameter for the 1715

8 . N. Gaikwa an A. V. Javaji / JAFM Vol. 9 No. 4 pp oscillatory moe is presente in Fig.. We observe that increasing the stress relaxation parameter results in a ecrease of the minimum of the oscillatory yleigh number up to a particular value of. An further increase of this tren reverses. hus the effect of an increase in the value of the stress relaxation parameter is to estabilize the system in oscillatory moe. Furthermore we fin from this figure that the minimum of the yleigh number shift towar the smaller values of the wave number with an increase of the stress relaxation parameter. permeability permeability K x fixe an then increase vertical K increases the critical yleigh z number. Further we fin that the minimum of the yleigh number shift towars the smaller values of the wave number with increasing mechanical anisotropy parameter. his inicates that the cell with increases with an increase of mechanical anisotropy parameter. Fig. 3. Oscillatory neutr al stability curves for ifferent values of Mechanical anisotropy parameter ξ. Fig. 1. Oscillatory neutr al stability curves for ifferent values of Darcy number Da. Fig. 4. Oscillatory neutr al stability curves for ifferent values of hermal anisotropy parameter η. Fig.. Oscillatory neutr al stability curves for ifferent values of tress relaxation parameter λ. In Fig. 3 the effect of the mechanical anisotropy parameter on the oscillatory yleigh number is epicte. We observe that the minimum of the yleigh number ecreases with increasing mechanical anisotropy parameter inicating that the mechanical anisotropy parameter avances the onset of oscillatory convection. he effect of mechanical anisotropy can be unerstoo as follows; let us keep the vertical permeability K z fixe an then an increase horizontal permeability K x reuces the critical yleigh number. his is ue to the fact that increase permeability enhances the flui mobility in the vertical irection an hence convection sets in early. On the other han keep horizontal he effect of thermal anisotropy parameter with Osc for the fixe values of other parameters is isplaye in figure 4. It is foun that Osc for the oscillatory moe ecrease. herefore the effect of is to avance the onset of oscillatory convection. Figure 5 epicts the effect of solute yleigh number on the neutral stability curves for oscillatory moe. We fin that the effect of increasing is to increase the value of the yleigh number for oscillatory moe. hus the solute yleigh number has a stabilizing effect on the ouble-iffusive convection in porous meium. 1716

9 . N. Gaikwa an A. V. Javaji / JAFM Vol. 9 No. 4 pp fin that the quantities namely the critical yleigh number for stationary an oscillatory moes is increasing function of the solute yleigh number. It is clear that for the parameters chosen for these figures the oscillatory convection sets in prior to the stationary convection. Fig. 5. Oscillatory neutr al stability curves for ifferent values of olute yleigh number s. Figure 6 isplays the variation of c with for ifferent values of Darcy number Da. It is observe that with an increase of Da the oscillatory an the stationary critical yleigh number increase implying that Da has a stabilizing effect on the system. Fig. 7 reveals the effect of stress relaxation parameter on the critical yleigh number for the stationary an oscillatory moes with the solute yleigh number. he critical yleigh number for the oscillatory moe ecreases with an increase of inicating that the effect of the stress relaxation parameter is to estabilize the system in the oscillatory moe. In Fig. 8 we show the effect of the mechanical anisotropy parameter on the critical yleigh number c is shown. We fin that increasing anisotropy parameter ecreases the critical yleigh number inicating that the mechanical anisotropy parameter estabilizes the system. Fig. 6. Variation of the critical yleigh number with solute yleigh number for ifferent values of the Darcy number Da. Fig. 8. Variation of the critical yleigh number with solute yleigh number for ifferent values of mechanical anisotropy parameter ξ Fig. 7. Variation of the critical yleigh number with solute yleigh number for ifferent values of stress relxation parameter λ. he etaile behavior of the critical yleigh number for stationary an oscillatory moes with respect to the solute yleigh number is analyze in the c plane through figures 6-9. We In figure 9 the variation c with for ifferent values of thermal anisotropy parameter is presente. It is notice that an increase of the critical yleigh number ecreases for oscillatory moe whereas it increases for the stationary moe implying that the effect of increasing thermal anisotropy parameter is to avance the onset of oscillatory convection an is to inhibit the onset of stationary convection as compare to the isotropic case. o know the transient behavior of Nusselt an 1717

10 . N. Gaikwa an A. V. Javaji / JAFM Vol. 9 No. 4 pp herwoo numbers the autonomous system of orinary ifferential equations is solve numerically using the Runge-Kutta metho with suitable initial conitions. hen Nu an h are evaluate as a function of time t. he transient behavior of Nu an h is shown graphically through the figures It is observe that both Nu an h start with a conuction state value i.e. 1 at t 0 an then oscillate perioically about their steay state value i.e. close to 3 for t 0. his perioic variation of Nu an h is very short live an ecays as time progresses. he values of Nusselt an herwoo then ten towar their steay- state value of 3. increase in the value of the mechanical anisotropy parameter suppresses both heat an mass transfer. In the Figs. 13(a) & (b) the effect of thermal anisotropy parameter is to suppress both Nu an h. From Fig. 14 (a) an (b) we observe that an increase in the value of Le Lewis number enhances both heat an mass transfer. Fig. 10 (b). Variation of the herwoo number with time for ifferent values of the Darcy number Da. Fig. 9. Variation of the critical yleigh number with solute yleigh number for ifferent values of thermal anisotropy parameter η. Fig. 11 (a). Variation of the Nusselt number with time for ifferent values of stress relaxation parameter λ. Fig. 10 (a). Variation of the Nusselt number with time for ifferent values of the Darcy number Da. he effect of the Darcy number is to increase the amplitue of the oscillations of heat an mass flux marginally is shown in figs 10(a) & (b). Figs 11(a) & (b) show the effect of the relaxation parameter on heat an mass transfer. We fin that an increase in increases both Nu an h marginally. Figures 1(a) & (b) show the effect of mechanical anisotropy parameter. We observe that an Fig. 11 (b). Variation of the herwoo number with time for ifferent values of stress relaxation parameter λ. 1718

11 . N. Gaikwa an A. V. Javaji / JAFM Vol. 9 No. 4 pp CONCLUION he onset of ouble iffusive convection of a Maxwell flui in a saturate anisotropic Darcy- Brinkman porous layer is stuie analytically using both linear an non-linear stability analyses. he linear analysis is base on the usual normal moe technique while the non-linear analysis is base on truncate representation of the Fourier series. he moifie Darcy-Brinkman Maxwell moel is use for the momentum equation. he following conclusions are rawn: 1. he oscillatory moe is most favorable for a system with moerate an high values of the Darcy number Da an olute yleigh number. However for the large values of stress relaxation parameter mechanical anisotropy parameter an thermal anisotropy parameter is avance the onset of oscillatory convection.. he effect of increasing the value of thermal anisotropy parameter in the presence of critical yleigh number is to allow the onset of convection to be stationary rather than oscillatory. he value of stress relaxation parameter an mechanical anisotropy parameter avance the onset of oscillatory convection. he effect of increasing the value of Darcy number Da in the presence of critical yleigh number elay the onset of convection whereas the stress relaxation parameter an mechanical anisotropy parameter avances. Fig. 1 (b). Variation of the herwoo number with time for ifferent values of mechanical anisotropy parameter ξ. ACKNOWLEDGMEN his work is supporte by UGC Research Fellowship in cience for Meritorious tuents (RFM) [voi UGC Ltr. No.F-4-1/006(BR)/ (BR) ate ]. One of the authors (Anuraha V J) acknowleges UGC for awaring the Research Fellowship. 3. he effect of Darcy number Da an increases both heat an mass transfer. While the mechanical anisotropy parameter suppresses an thermal anisotropy parameter avances. he transient Nusselt an herwoo numbers approach the steay state values for large time. Fig. 13 (a). Variation of the Nusselt number with time for ifferent values of mechanical anisotropy parameter η. Fig. 1 (a). Variation of the Nusselt number with time for ifferent values of mechanical anisotropy parameter ξ. Fig. 13 (b). Variation of the herwoo number with time for ifferent values of thermal anisotropy parameter η. 1719

12 . N. Gaikwa an A. V. Javaji / JAFM Vol. 9 No. 4 pp Fig. 14 (a). Variation of Nu with time for ifferent values of Le. Fig. 14 (b). Variation of h with time for ifferent values of Le. REFERENCE Amahmi A. M. Hasnaoui M. Mamou an P. Vasseur (1999). Double-iffusive parallel flow inuce in a horizontal Brinkman porous layer subjecte to constant heat an mass fluxes: analytical an numerical stuies. Heat Mass ransf Awa F. G. P. ibana an.. Motsa (010). On the linear stability analysis of a Maxwell flui with ouble-iffusive convection. Appl. Math. Moel Bertola V. an E. Cafaro (006). hermal instability of viscoelastic fluis in Horizontal porous layers as initial problem. Int. J. Heat Mass ransfer Gaikwa. N. an B.. Biraar (013).he onset of ouble iffusive convection in a Maxwell flui saturate porous layer. pecial topics an reviews in porous meium Khale A. R. A. an K. Vafai (003). he role of porous meia inmoeling flow an heat transfer in biological tissues. Int.J. Heat Mass ransf Khan M. M. aleem C. Fetecau an. Hayat (007). ransient oscillatory an constantly accelerate non-newtonian flow in a porous meium. Int. J.Non. Linear. Mech Khuzhayorov B. J. L. Auriault an P. Royer (000). Derivation of macroscopic Itration law for transient linear viscoelastic flui flow in porous meia. Int. J.Eng. ci Kim M. C.. B. Lee. Kim an Chung (003). hermal instability of viscoelastic fluis in porous meia. Int. J. Heat Mass ransfer Knobloch E. (1980). Convection in binary fluis. he Physics of Fluis 3(9) Malashetty M. an M. wamy (007). he onset of convection in a viscoelastic liqui saturate anisotropic porous layer. ransp. Porous Meia 67() 03. Malashetty M.. an M. wamy (010). he onset of convection in a binary flui saturate anisotropic porous layer. Int. J. hermal ciences. Malashetty M.. an P. Kollur (011). he onset of ouble iffusive convectionin a couple stress flui saturate anisotropic porous layer. ransport in Porous Meia 86() Malashetty M.. an. Kulkarni (009). he convective instability of Maxwell flui saturate porous layer using a thermal non-equilibrium moel. Int. J. Non-Newtonian Flui Mech16 9. Malashetty M... Wenchang an M. wamy (009). he onset of ouble convection in a binary viscoelastic flui saturate anisotropic porous layer. Phys.Fluis Malashetty M.. M. wamy an R. Heera (009). he onset of convection in a binary viscoelastic flui saturate porous layer. Z. Angew. Math. Mech 89(5) 356. Maxwell J. C. (1866). On the ynamical theory of gases Philos. rans. R. oc. Lononer. A Niel D. A. an A. Bejan (006). Convection in Porous Meia 3 r E. Berlin pringer. Niel D. A. an A. Bejan (006). Convection in Porous Meium pringer NewYork NY UA 3r eition. heu L. J. J. H. Chen H. K. Chen L. M. am an Y. C. Chao (009). A unifie system escribing ynamics of chaotic convection. Chaos olitons Fractals.41. heu L. J. L. M. J. H. am H. K. Chen K.. Chen Lin an Y. Kang (008). Chaotic convection of viscoelastic fluis in porous meia. Chaos olitons Fractals an W. an. Mausoka (007). tability analysis of a Maxwell flui in a porous meium heate from below. Phys. Lett.A revisan O. V. an A. Bejan (1990). Combine heat an mass transfer by natural convection in a porous meium. Avances in Heat ransfer Wang. an W. C. an (008). tability analysis of ouble-iffusive convection of Maxwell flui in a porous meium heate from below. Phys.Lett. A

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