Convective Transport of Nanofluid Saturated with Porous Layer

Transcription

1 erican Journal o lied Matheatics 7; 5(): -3 htt:// doi:.648/j.aja.75. ISSN: (Print); ISSN: 33-6X (Online) Convective ransort o Nanoluid Saturated with Porous Layer Jada Pratha Kuar, Jawali Channaasaa Uavathi, Channakeshava Murthy Deartent o Matheatics, Gularga University, Kalauragi, Karnataka, India Eail address: _ratha@yahoo.co (J. P. Kuar) o cite this article: Jada Pratha Kuar, Jawali Channaasaa Uavathi, Channakeshava Murthy. Convective ransort o Nanoluid Saturated with Porous Layer. erican Journal o lied Matheatics. Vol. 5, No., 7,. -3. doi:.648/j.aja.75. Received: Noveer, 6; cceted: Noveer 6, 6; Pulished: January 8, 7 stract: In the resent article, the onset o convection in a horizontal layer o orous ediu saturated y ananoluid is investigated analytically using linear and weakly nonlinear analysis. he odel used or the nanoluid incororates the eect o rownian otion and therohoresis. he eect o Raleigh-Darcy nuer, Lewis nuer, odiied diusivity ratio, on the staility o the syste is investigated. Stationary and Oscillatory odes o convections has een studied. he linear staility analysis is ased on noral ode technique, while then on linear theory is ased on the truncated reresentation o Fourier series ethod. weekly nonlinear analysis is used to otain the concentration and theral Nusselt nuer. he ehavior o the concentration and theral Nusselt nuer is investigated y solving the inite alitude equations. Otained results have een resented grahically and discussed in details. Keywords: Nanoluid, Porous Mediu, Instaility, Natural Convection. Introduction he ter nanoluid reers to a liquid containing a susension o suicronic solid articles (nanoarticles). he ter was coined y Choi []. he characteristic eature o nanoluids is theral conductivity enhanceent, a henoenon oserved y Masuda et al. []. his henoenon suggests the ossiility o using nanoluids in advanced nuclear systes (uongiorno and Hu [8]). Nanoluids are ixtures o ase luid such as water or ethylene-glycol with a very sall aount o nanoarticles such as etallic or etallic oxide articles (Cu, Cuo, l O 3 ), having diensions ro to n. uongiorno [5] conducted a corehensive study to account or the unusual ehavior o nanoluids ased on inertia, rownian diusion, therohoresis, diusiohoresis, Magnus eects, luid drainage and gravity settling, and roosed a odel incororating the eect o rownian diusion and the therohoresis. Studies ertaining to theral conductivity enhanceent y nanoluids have een conducted y Eastan et al. [3], Das et al. [4] and others. hey claied a -3% increase in theral conductivity y using very low concentrations o nanoluid. Due to alication o nanoluid and orous edia theory in drying, reezing o oods and alications in every day technology such as icrowave heating, raid heat transer ro couter chis via use o orous etal ors and their use in heat ies. One o the ost signiicant scientiic challenges in the industrial area is cooling, which alies to any diverse roductions including icroelectronics, transortation and anuacturing. echnological develoents such as icro-electronics devices oerating at high seeds, high ower engines, a righter otical devices and driving increases theral loads, requiring advances in cooling. here are several studies availale in which henoena related to the onset o convectional in a orous ediu have een investigated. Few o the are Perlstein [6], Chakraarti and Guta [7], Patil and Vidyanathan [8], Vadasz [9]. Convection in orous ediu has een studied y any authors including Horton and Roger [], Lawood [], Nield [], Rudraiah and Malshetty [3], Murray and Chen [4], hadauria [5], Vaai [6], Neild and ejan [7]. Recently, Nield and kuznetsov [8, 9] or the Darcy Model, Nield and Kuznetsov [] also studied local theral non-equiliriu and low ast vertical late or nanoluids. he sae role was studied y garwal et al [] and hadauria et al []. Uavathi [3] exained the staility o

2 Jada Pratha Kuar et al.: Convective ransort o Nanoluid Saturated with Porous Layer orous ediu saturated with a nanoluid with theral conductivity and viscosity deendent on the nanoarticle volue raction in the case o tie deendent wall teerature. linear and weakly nonlinear staility analysis on the onset o convection in a horizontal orous ediu saturated with a Maxwell nanoluid was studied y Uavathi et al [4]. Recently Uavathi and Pratha kuar [5] nalysed the onset o convection in a orous ediu layer saturated with an Oldroyd nanoluid.we study the linear and nonlinear analysis o theral instaility in a orous layer saturated y nanoluids in this resent article.. Conservation Equations or a Nanoluid First, we outline the derivation o conservation equation alicale to a nanoluid in the asence o a solid atrix. Later we odiy these equations to the case o a orous ediu saturated y the nanoluid. he uongiorno odel treats the nanoluid as a two coonents ixture (ase luid lus nanoarticles) with the ollowing assution here v is the nanoluid velocity. he conservation equation or the nanoarticles in the asence o cheical reactions is where φ is nanoarticle volue raction, ρ is the nanoarticle ass density and is the diusion ass lux or the nanoarticles, given as the su o two diusion ters (rownian diusion and therohorosis)y (herohoretic is the article equivalent o the soret eect in gaseous or liquid ixtures) here D is the rownian diusion coeicient given y the Einstein-stokes equation D () () (3) k = (4) 3πµd where k oltzann's constant, µ is the viscosity o the luid and d is the nanoarticle diaeter. Use has een ade o the exression. v = φ + v φ = j ρ j = j, + j, = ρd φ ρd µ V = ɶ β (5) ρ or the therohoretic velocity V here ρ is the luid density and the roortionality actor β ɶ is given y j ɶ k β =.6 k + k where k and k are the theral conductivities o the luid and the article aterial. Hence the therohoretic diusion lux is given y (6) j, = ρφv = ρd (7) where the therohoretic diusion coeicient is given y D µ = ɶ β φ (8) ρ Eq.() and (3) then roduce the conservation equation in the or he oentu equation or a nanoluid takes the sae or as or a ure luid, ut it should e reeered that µ is strong unction o φ. I one introduce a uoyancy orce and adots the oussinesq aroxiation, then the oentu equation can e written as where (9) v ρ + v. v = + µ v + ρg, () he nanoluid density φ + v. φ =. D φ + D ρ = φρ + ( φ) ρ () ρ can e aroxiated y the ase-luid density ρ when φ is sall. hen when the oussinesq aroxiation is adoted the uoyancy ter is aroxiated y ρg [ φρ + ( φ){ ρ( β( ))}] g () he theral energy equation or a nanoluid can e written as ρc +. =. + h. v q j (3) where c is the nanoluid seciic heat, is the nanoluid teerature, h is the seciic enthaly o the nanoarticle aterial and q is the energy lux, relative to a rae oving with the nanoluid velocityv, given y q = k + h j, (4) where k is the nanoluid theral conductivity. Sustituting Eq (4) in Eq (3) yields

3 erican Journal o lied Matheatics 7; 5(): -3 3 ρc +. =.( k ) c. v j (5) In deriving this equation use has een ade o a vector identity and the act (deriving ro assution (7)) that h = c,where c is the nanoarticle seciic heat o the aterial constituting the nanoarticles, while c is the seciic heat (at constant ressure) o the luid. hen sustitution o Eq (3) in Eq (5) gives inal or ρc + v..( k ) t = + D φ. + ρ c. D 3. Conservation Equations or a Porous Mediu Saturated y a Nanoluid (6) We consider a orous ediu whose orosity is denoted y ε and ereaility yk. suscrit s will now e used to denote roerties o the solid atrix. he Darcy velocity is denoted y v D.his is related to v y vd = εv.we now have to deal with the ollowing our ield equation (corresonding to Eq (), (), (6), (9) or total ass, oentu, theral energy and nanoarticle resectively).. v = D vd ρ + v D. vd ε t ε = + µ ɶ µ vd vd K + φρ + ( φ ){ ρ ( β ( ))} g ( ρc) ( ) D ( ) + ρc v = k +. ε ( ρc) D φ. D + φ + vd. φ =.[ D φ + D ] ε (7) (8) (9) () Here we have introduced the eect o viscosity µɶ, the eective heat caacity ( ρc) and the eective theral conductivityk o the orous ediu. In deriving Eq (7)- () we have assued that the rownian otion and therohoriesis rocesses reains coherent while volue averages over a reresentative eleentary volue are taken. his assution can e questioned. In the context o odeling transort in orous edia, Eq (7) and (8) are standard. Eq () involves just intrinsic quantities in the sense that the average is eing taken over the nanoluid only and the solid atrix is not involved. he question thus reduces to whether the ters within the square rackets on the right-hand side o Eq (9) need odiication we recall that in nanoluids the articles are so sall that or ractical uroses they reain in susension in a unior anner. We ehasize our assution that the nanoarticles are susended in nanoluid using either suractant or surace charge echnology, soething that revents articles ro aggloeration and deosition on the orous atrix. We suggest that then it is reasonale to assue as a irst aroxiation that no odiication to Eq (9) is necessary. 4. lication to the Horton Rogers Lawood Prole We select a coordinate rae in which the z-axis is aligned vertically uwards. We consider a horizontal layer o a orous ediu conined etween the lanes z = and z = H. Fro now on asterisks are used to denote diensional variales (reviously an asterisk has not een needed ecause all the variales were diensional). Each oundary wall is assued to e iereale and erectly therally conducting. he teeratures at the lower and uer wall are taken to e h and c, the orer eing the greater. For silicity, Darcy s law is assued to hold and the Oereck oussinesq aroxiation is eloyed. Hoogeneity and local theral equiliriu in the orous ediu are assued he reerence teerature is taken to e c in the linear theory eing alied here the teerature change in the luid is assued to e sall in coarison with Eq (8) () takes the or c µ = vd + K φ ρ + φ ρ β We write v = ( u, v, w ). { } ( ) ( ( C )) ( c) g ( ρc) + ( ρc) v D = ( k ) + ε ρ D. ( D φ + ). C D φ + v. D ϕ = D ϕ + ε D ( ) () () (3) We assue that the teerature and the voluetric raction o the nanoarticles are constant on the oundaries. hus the oundary conditions are w =, =, φ = φ at z = h (4)

4 4 Jada Pratha Kuar et al.: Convective ransort o Nanoluid Saturated with Porous Layer w =, =, φ = φ at z = H C (5) We recognize that our choice o oundary conditions iosed on φ is soewhat aritrary. It could e argued that zero article lux on the oundaries is ore realistic hysically, ut then one is aced with the role that it aears that no steady-state solution or the asic conduction equations is then ossile, so that in order to ake analytical rogress it is necessary to reeze the asic roile orφ, and at that stage our choice o oundary conditions is seen to e quite realistic. We introduce diensionless variales as ollows. We deine ( x, y, z) = ( x, y, z )/ H, t = t α / σh ( u, v, w) = ( u, v, w ) H /, = K / α µα φ φ φ =, =, φ C φ h C (6) k ( ρc ) where α =, σ = (7) ( ρc ) ( ρc ) then Eq(7) and ()-(5) take the or (8) = v Reˆ + Ra eˆ Rnφe ˆ (9) z z z α ρgβkh here Le =, Ra =, D µα N [ ρφ + ρ( φ)] gkh R =, µα D ε( ρc) =, N = D ( φ φ ) ( ρc), (3) (3) (3) (33) he araeter Le is a Lewis nuer and Ra is the ailiar theral Rayleigh Darcy nuer. he new. v = + v. = + φ. + NN Le. N Le φ N + v. φ = φ + σ ε Le Le w =, =, φ = at z = w =, =, φ = at z = ( ρ ρ)( ϕ ϕ ) gkh Rn = µα c ( ϕ ϕ ) araeters R and Rn ay e regarded as a asic-density Rayleigh nuer and a concentration Rayleigh nuer, resectively. he araeter N is a odiied diusivity ratio and is soewhat siilar to the Soret araeter that arises in cross-diusion henoena in solutions, while N is a odiied article-density increent. In the sirit o the Oereck oussinesq Eq (9) has een linearize y the neglect o a ter roortional to the roduct o φ and. his assution is likely to e valid in the case o sall teerature gradients in a dilute susension o nanoarticles. 4.. asic Solution We seek a tie-indeendent quiescent solution o Eq(8) (33) with teerature and nanoarticle volue raction varying in the z-direction only, that is a solution o the or v=,= (z), Eq (3) and (3) reduce to (z) (34) (35) Using the oundary conditions (3) and (33) Eq (4) ay e integrated to give nd sustitution o this into Eq (34) gives he solution o Eq (37) satisying Eq (3) and (33) is (36) (37) (38) he reainder o the asic solution is easily otained y irst sustituting in Eq (4) to otain φ and then using integration o Eq (9) to otainp. ccording to uongiorno [5], or ost nanoluids investigated so ar Le/( φ φ ) is large, o order 5 6, and since the nanoarticle raction decreent is tyically no 3 saller than this eans so that Le is large, o order 3, while N is no greater than aout. hen the exonents in Eq (37) and (38) are sall and so to a good aroxiation one has and so φ = φ d N dφ d N. N d dz + + ( ) = Le dz dz Le dz d φ d + N = dz dz φ = N + ( N ) Z + N d dz e = e ( N) N d + = Le dz ( N) N( Z)/ Le ( N) N / Le = z (39)

5 erican Journal o lied Matheatics 7; 5(): Perturation Solution (4) We now sueriose erturations on the asic solution. We write v = v ', = + ', = + ', = + ' (4) ϕ ϕ ϕ Sustitute in Eq (8) (33), and linearize y neglecting roducts o ried quantities. he ollowing equations are otained when Eq (39) and (4) are used.. v ' = (4) = ' v' + Ra ' eˆ Rnφ ' eˆ (43) z z (44) (45) w ' =, ' =, ϕ ' = at z = and at z = (46) It will e noted that the araeter R is not involved in these and susequent equations. It is just a easure o the asic static ressure gradient. For the case o a regular luid (not a nanoluid) the araeters Rn, N and N are zero, the second ter in Eq (45) is asent ecause φ dz and then Eq (45) is d ϕ = z / = satisied trivially. he reaining equations are reduced to the ailiar equations or the Horton Roger Lawood role. he six unknowns u ', v ', w ', ', ', ϕ ' can e reduced to three y oerating on Eq (43) with êz curl curl and using the identity curl curl grad div - together with Eq (4). he result is (47) here H is the two-diensional Lalacian oerator on the horizontal lane. he dierential Eq (47), (44), (45) and the oundary conditions (46) constitute a linear oundary-value role that can e solved using the ethod o noral odes. We write ( w', ', φ ') [ W( z), θ( z), ( z) ] ex ( st ilx iy) = Φ + + (48) and sustitute into the dierential equations to otain ' N ' ϕ' w = + Le z z NN ' Le z ' ' ( ) N ' ϕ' ' ϕ' + w = + σ ε Le Le w' = Ra ' Rn φ' H H D W Ra Rn ( α ) + α Θ α Φ = (49) Where (5) (5) (5) and (53) thus α is a diensionless horizontal wave nuer. For neutral staility the real art o s is zero. Hence we now writes = iω, where ω is real and is a diensionless requency. We now eloy a Galerkin-tye weighted residuals ethod to otain an aroxiate solution to the syste o Eq(49) (5). We choose as trial unctions (satisying the oundary conditions) write W = Θ = Φ = sin π z; =,,3... (54) N = N N. N W D D D s Le Le N D Φ = Le + ( + α ) Θ W= W N N, Θ = = Θ P, Φ = = C Θ (55) sustitute into Eq (49) (5), and ake the exressions on the lethand sides o those equations (the residuals) orthogonal to the trial unctions, there y otaining a syste o 3N linear algeraic equations in the 3N unknowns,, C, =,,... N. he vanishing o the deterinant o coeicients roduces the eigenvalue equation or the syste. One can regard Ra as the eigenvalue. hus Ra is ound in ters o the other araeters 5. Linear Staility nalysis 5.. Non-oscillatory Convection N ( D α ) W Θ ε Le Le σ ( s D α ) Φ = W =, Θ =, at Z = and at Z = d D dz α / ( l + ) First, we consider the case o non-oscillatory instaility, when ω = or a irst aroxiation we take N=.his roduces the result ( π + α ) Le Ra = N. (56) + Rn α ε Finding the iniu as a varies results in Le Ra = 4π N + Rn. (57) ε with the iniu eing attained at α = π. We recognize that in the asence o nanoarticles we recover the well-known =

6 6 Jada Pratha Kuar et al.: Convective ransort o Nanoluid Saturated with Porous Layer result that the critical Rayleigh nuer is equal to 4 π. Usually when one eloys a single-ter Galerkin aroxiation in this context one gets an overestiate y aout 3% (e.g. 75 instead o 78 in the case o the standard énard role) ut in this case the aroxiation haens to give the exact result. s we have noted, or a tyical nanoluid 3 Le is o order and N is not uch greater than. Hence the coeicient o Rn in Eq (57) is large and negative. hus under the aroxiations we have ade so ar we have the result that the resence o nanoarticles lowers the value o the critical Rayleigh nuer, usually y a sustantial aount, in the case when Rn is ositive, that is when the asic nanoarticle distriution is a to-heavy one. It will e noted that in Eq (57) the araeter N does not aear. he instaility is alost urely a henoenon due to uoyancy couled with the conservation o nanoarticles. It is indeendent o the contriutions o rownian otion and therohoresis to the theral energy equation. Rather, the rownian otion and therohoresis enter to roduce their eects directly into the equation exressing the conservation o nanoarticles so that the teerature and the article density are couled in a articular way and that results in the theral and concentration uoyancy eects eing couled in the sae way. It is useul to ehasize this y rewriting Eq (57) in the or Le Ra N Rn π ε + ( + ) = 4 (58) and noting that the let-hand side is the linear coination o the theral Rayleigh nuer Ra and the concentration Rayleigh nuer Rn. he role is analogous to the doule- diusive role discussed in Section 9.. o Nield and ejan [6]. It is also analogous to the ioconvection role discussed y Kuznetsov and vraenko [7]. We have deined Rn in a way so that it is ositive when the alied article density increases uwards (the destailizing situation). We note that Ra takes a negative value when Rn is suiciently large. In this case the destailizing eect o concentration is so great that the otto o the luid layer ust e cooled relative to the to in order to roduce a state o neutral staility. We ehasize that the sile exression in Eq (57) arises ecause the Lewis nuer has een assued to e large. In order to estiate the contriution o the ters involving N we have investigated the two-ter Galerkin results. he exression in the eigenvalue equation is colicated and it is diicult to ake a stateent that is siultaneously recise, sile and general. However, it is clear that the unctions o N are o second degree. We conclude that or ractical uroses Eq (58) is a good aroxiation. 5.. Oscillatory Convection We now consider the case ω.we conine ourselves to the one-ter Galerkin aroxiation. he eigen value equation now takes the or J iω N ( J + iω) Ra α ( + ) + Rnα ( + ) Le σ Le ε J iω = J ( J + iω)( + ) Le σ Where or shorthand we have written J = π +α 6. Non linear Staility nalysis (59) (6) For silicity, we consider the case o two diensional rolls, assuing all hysical quantities to e indeendent o y. Eliinating the ressure and introducing the strea unction we otain Ra x S Rn x Ψ + = (, ) + Ψ = + Ψ x ( x, z) (6) (6) (63) We solve Eq 3 sujecting the to stressree,isotheral,iso-nano concentration oundary conditions (64) o er ora local non-linear staility analysis, we take the ollowing Fourier exressions (65) (66) (67) Further, we take the odes (,) or strea unction, and (,) and (,) or teerature, and nanoarticle concentration, to get (68) (69) (7) Where the alitudes ( t), ( t), ( t), ( t), ( t) are unction o tie and are to e deterined. aking the N S Ψ + = S + σ ε x Le Le ( Ψ, S) + ε ( x, z) ψ ψ = = = φ = at z =, z ψ = n= = = n= = ( t) Sin(ax)sin( nπ z) n ( t)cos(ax)sin( nπ z) φ = n= = n C ( t)cos(ax)sin( nπ z) n ψ = ( t) Sin(ax)sin( πz) = ( t)cos(ax)sin( πz) + ( t)sin( πz) 3 S = ( t)cos(ax)sin( πz) + ( t)sin( πz) 4 5

7 erican Journal o lied Matheatics 7; 5(): -3 7 orthogonality condition with the eigen unctions associated with the considered inial odel, we get a ( t) = [ Rn 4( t) Ra ( t)] δ d( t) = [ a( t) + δ ( t) + aπ ( t) 3( t)] dt d3( t) aπ = + dt 4 π 3( t) ( t) ( t) a ( t) + ε ( ) ( ) dt σ Le Le + a ( t) 5 ( t) ε d4 t 4 t N = δ + t ( ) N 4 5 ( t ) 4 3 ( t ) d5 ( t) π + π Le Le = dt σ aπ ( t) 4( t) ε (7) (7) (73) (74) (75) π / a ( ) dx Z ( ) dx z = + π / a z= Sustituting exressions (45) and (73) in Eq (83) we get (83) (84) he nanoarticle concentration Nusselt nuer, Nu φ (t) is deined siilar to the theral Nusselt nuer. Following the rocedure adoted or arriving at Nu(t), one can otain the exression or Nu φ (t) in the or Nuφ ( t) = ( π ( t)) + N ( π ( t)) (85) 7. Results and Discussions Nu ( t) = π ( t) 5 3 he exressions o theral Rayleigh nuer or stationary and oscillatory convections are given y equations (56) and (59) resectively 3 thus we get a ( t) = [ Rn 4( t) Ra ( t)] δ D = [ a ( t) + δ ( t) + aπ ( t) ( t)] 3 aπ D3 = 4 π 3 ( t) + ( t) ( t) 4( t) N D4 = [ a ( t ) + δ [ + ( t )] σ ε Le Le + a ( t) 5 ( t)] ε N D5 = [ 4 π 5 ( t) + 4 π 3( t) σ Le Le aπ ( t ) 4( t )] ε (76) (77) (78) (79) (8) and = = = D = (8) D D3 D4 5 One ay also conclude that the trajectories o the aove equations will e conined to the initeness o the ellisoid. hus, the eect o the araeters Rn, Le, N on the trajectories is to attract the to a set o easure zero, or to a ixed oint to say 6.. Heat and Nanoarticle Concentration ransort he theral Nusselt nuer, Nu ( t) is deined as Nu ( t) = Heat transort y ( conduction + convection) Heat transort y conduction (8)

8 8 Jada Pratha Kuar et al.: Convective ransort o Nanoluid Saturated with Porous Layer Figure. Neutral curves on stationary convection or dierent values o, (a)nanoarticle concentration Rayleigh nuer Rn ()Lewis nuer Le (c)odiied diusivity ratio N (d)orsity ε. Figure a-d shows the eect o various araeters on the neutral staility curves or stationary convection or Rn = -., Le =, N = -5, ε =.9, with variation in one o the araeters. he eect o nanoarticle concentration Rayleigh nuer Rn is shown in Figure a. It is shown that the theral Rayleigh nuer decreases with increase in nanoarticle concentration Rayleigh nuer Rn, which eans that nanoarticle concentration Rayleigh nuer Rn destailizes the syste. It should e noted that the negative value o Rn indicates a otto-heavy case, while a ositive value indicates a to-heavy case. he eect o Lewis nuer Le on the theral Rayleigh nuer is shown in Figure.. One can see that the theral Rayleigh nuer increases with increase in Lewis nuer, indicating that the Lewis nuer stailizes the syste. he eect o odiied diusivity ratio N on the theral Rayleigh nuer is shown in Figure c, it is shown in Figure. c that as N increases Ra increase and hence N has a stailizing eect on the syste. Fro Figure d, one can oserve that as orosity ε increases, theral Rayleigh nuer decreases which eans that the orosity advances the onset o convection.

9 erican Journal o lied Matheatics 7; 5(): -3 9 Figure. Neutral curves on oscillatory convection or dierent values o (a) nanoarticle concentration Rayleigh nuer Rn () Lewis nuer Le (c )odiied diusivity ratio N (d )orosity ε (e) heral Caacity σ. Figure a-e dislays the variation o theral Rayleigh nuer or oscillatory convection with resect to various araeters. In Figure a it is seen that or negative values o Rn (otto heavy case). he theral Rayleigh nuer decreases as Rn increases which will delay the onset o convection. s the Lewis nuer Le increases the theral Rayleigh nuer Ra decreases as seen in Figure which ily that Lewis nuer Le destailizes the syste. he odiied diusivity ratio N do not show any eect on the oscillatory convection (Figure c). Fro the icture d, one can reveal that the orosity ε destailizes the syste or oscillatory convection, that is an increase in ε decreases the theral Rayleigh nuer. s the theral caacity ratio σ increases, the theral Rayleigh nuer also increases as can e oserved in Figure e, which ilies that σ has a stailizing eect on the syste or oscillatory convection. Figure 3. Variation o Nusselt nuer Nu with critical Rayleigh Nuer or dierent values o (a) Nanoarticle concentration Rayleigh nuer Rn,() Lewis nuer Le,(c) Modiied diusivity ratio. he nonlinear analysis rovides not only the onset threshold o inite alitude otions ut also the inoration o heat and ass transorts in ters o Nusselt Nu and Sherwood Sh nuers. he Nu and Sh are couted as the unctions o Ra (theral Rayleigh nuer), and the variations o these non-diensional nuers with Ra or dierent araeter values are deicted in Figures 3a-c and 4a-c resectively. In Figures 3a-c and 4a-c it is oserved that in each case, Sherwood nuer is always greater than Nusselt nuer and oth Nusselt nuer and Sherwood nuer start with the conduction state value at the oint o onset o steady inite alitude convection. When Ra is increased eyond Ra c, there is a shar increase in the values o oth Nu and Sh. However urther increase in Ra will not change Nu and Sh signiicantly. It is to e noted that the uer ound o Nu is 3 (siilar results were otained y Malashetty et al.). It should also e noted that the uer ound o Sh is not 3 (siilar results were otained y hadauria et al.). he uer ound o Nu reains 3 only or oth clear and nanoluid. Whereas, the uer ound or Sh or clear luid is 3 ut or nanoluid it is not ixed. N

10 Jada Pratha Kuar et al.: Convective ransort o Nanoluid Saturated with Porous Layer Figure 4. Variation o Sherwood nuer Sh with critical Rayleigh Nuer or dierent values o (a) Nanoarticle concentration Rayleigh nuer Rn, () Lewis nuer Le,(c) Modiied diusivity ratio. N In Figures. 3a and 4a we oserve that as the concentration Rayleigh nuer Rn increases, the value o Nu and Sh decreases, thus showing a decrease in the rate o heat and ass transort. Figures. 3 and 4 shows that as Lewis nuer increases oth Nu and Sh decreases, which ily that increasing the Lewis nuer suresses the heat and ass transort. In Figures. 3c and 4c we oserve that on increasing odiied diusivity ratio N, there is no eect on the Nusselt nuer whereas it increases the Sherwood nuer (which is siilar result oserved y hadauria et al [9]). Figure 5. Variation o Nusselt nuer Nu with Rayleigh Nuer or dierent values o, (a) Nanoarticle concentration Rayleigh nuer Rn, () Lewis nuer Le, (c) Modiied diusivity ratio N.

11 erican Journal o lied Matheatics 7; 5(): -3 ade using noral ode technique. However or weakly nonlinear analysis truncated Fourier series reresentation having only two ters is considered. We draw the ollowing conclusions. For stationary convection Lewis nuer Le, odiied diusivity ratio N, has a stailizing eect while nanoarticle concentration Rayleigh nuer Rn and orosity destailize the syste. For oscillatory convection theral caacity ratio σ, stailizes the syste where as nanoarticle concentration Rayleigh nuer Rn, Lewis nuer Le and orosity destailizes the syste. For steady inite alitude otions, the heat and ass transorted creases within crease in the values o nanoarticle concentration Rayleigh nuer Rn, Lewis nuer Le. he ass transort increases with increase in odiied diusivity ratio. he transient Nusselt nuer and Sherwood nuer increases with increase in Lewis nuer Le and odiied diusivity ratio N and decreases with nanoarticle concentration Rayleigh nuer Rn.he eect o tie on transient Nusselt nuer and Sherwood nuer is ound to e oscillatory when t is sall. However, when t ecoes very large oth the transient Nusselt and Sherwood value aroaches to the steady value. Figure 6. Variation o Sherwood nuer Sh with Rayleigh Nuer or dierent values o (a) Nanoarticle concentration Rayleigh nuer Rn, ()Lewis nuer Le,(c) Modiied diusivity ratio N. In Figure 5a it is oserved that as Rayleigh nuer Rn increases Nusselt nuer Nu decrease, thus showing a decrease in the heat transort. In Figure 5 as Lewis nuer Le increases the Nusselt nuer Nu decreases indicating there is retardation on heat transort. he odiied diusivity ratio N enhances the heat transort as see in Figure 5c. In Figure 6a as nanoarticle concentration Rayleigh nuer Rn increases the Sherwood nuer decreases, which ilies the suress o ass transort. In Figures 6 and 6c the ass transort is enhanced or Lewis nuer Le and odiied diusivity ratio N. 8. Conclusions We considered linear staility analysis in a horizontal orous ediu saturated y a nanoluid, heated ro elow and cooled ro aove, using Darcy odel which incororate the eect o rownian otion along with therohoresis. Further the viscosity and conductivity deendence on nanoarticle raction was also adoted ollowing iwari and Das [3]. Linear analys is has een Noenclature c Nanoluid s eciic heat at constant ressure c Seciic heat o the nanoarticle aterial ( ρ c) Eective heat caacity o the orous aterial d Nanoartical diaeter D rownian diusion coeicient ( s ), given y Eq.(4) D herohoretic diusion coeicient ( s ), given y Eq.(8) h Seciic enthaly o the nanoartical aterial H Diensional layer deth () j Diusion ass lux or the nanoarticales, given y Eq.(7) j herohoretic diusion, given y Eq.(7), k heral conductivity o the nanoluid (W/ K) k oltzann s constant k Overall theral conductivity o the orous ediu saturated y the nanoluid k heral conductivity o the article aterial K Pereaility ( )LeLewis nuer N Modiied diusivity ratio N Modiied article density increent Pressure (Pa) Diensionless ressure, K µα q Energy lux relative to a rae oving with the nanoluid velocity v

12 Jada Pratha Kuar et al.: Convective ransort o Nanoluid Saturated with Porous Layer Ra heral Rayleigh- Darcy nuer R asic-density Rayleigh nuer RnConcentration Rayleigh nuer t ie (s) t Diensionless tie, t α H Nanoluid teerature (K) Diensionless teerature, c h c c eerature at the uer wall (K) h eerature at the lower wall (K) v Nanoluid velocity (/s) v Darcy velocity εv D v DDientional Darcy velocity V heohoreatic velocity ( u, v, w) Diensionless Darcy velocity coonents ( ) u, v, w H α (/s) ( x, y, z) Diensionless Cartesian coordinate ( x, y, z ) ( x, y, z ) Greek Syols α H ; z is the vertically uward coordinate Cartesian coordinates heral diusivity o the orous ediu ɶ β Proortionality actor, given y Eq.(6) µ Viscosity o the luid µɶ Eective viscosity o the orous ediu ρ Fluid density ρ Nanoarticle ass density σ heral caacity ratio φ Nanoarticle volue raction φ φ φ Relative nanoarticle volue raction, φ φ Suerscrits Diensional variale ' Pertured variale Suscrits asic Solution Fluid Particle Reerences [] Choi, S.: Enhancing theral conductivity o luids with nanoarticles. In: Signier, D.., Wang, H. P. (eds.) Develoent and alication o Non-Newtonian lows, SME FED, Vol.3/MD Vol.66,.99-5 (995). [] Masuda, H., Eata,., eraae, K., Hishinua, N.: lteration o theral conductivity and viscosity o liquid y disersing ultra ine articles. Netsu ussei 7, 7-33 (993). [3] Eastan, J.., Choi, S. U. S., u, W., hosson, L. J.: noalously increased eective theral conductivities o ethylene glycol-ased nanoluids containing coer nanoarticles. l. Phys. Lett. 78, 78-7(). [4] Das, S. K., Putra, N., hiesen, P., Roetzel, W.: eerature deendence o theral conductivity enhanceent or nanoulids. SME J. Heat rans. 5, (3). [5] uongiorno, J.: Convective transort in nanoluids. SME J. Heat ranser. 8, 4-5 (6) [6] Pearlstein,. J: Eect o rotation on the staility o a douly diusive luid layer. J. Fluid Mech. 3, 389-4(98). [7] Chakraarti,, Guta,. S.: Nonlinear thero- haline convection in a rotating orous ediu. Mech. Res. Coun. 8, 9-(98). [8] Patil, P. R., Vidyanathan, G.: On setting u o convective currents in a ratoting orous ediu under the inluence o variale viscosity. Int. J. Eng. Sci.,3-3(983). [9] Vadasz, P: Free Convection in Rotating Porous Media, ransort Phenoena in Porous Media Elsevier, sterda (998). [] Horton, W., Rogers, F..: Convection currents in a orous ediu. J. l. Phys. 6, (945). [] Lawood, E. R.: Convection o a luid in a orous ediu. Proc. Ca. Phil. Soc. 44,58-5(948). [] Neild, D..: Onset o therohaline convection in a orous ediu. Water Resour. Res. 4, (968). [3] Rudraiah, N., Malashetty, M. S.: he inluence o coule olecular diusion on the doule diusive convection in a orous ediu. SME J. Heat rans. 8, (986). [4] Murray,.., Chen, C. F.: Doule diusive convection in a orous ediu. J. Fluid Mech., (989). [5] hadauria,. S.: Doule diusive convection in a orous ediu with odulated teerature on the oundaries. rans. Porous Med. 7, 9- (7a). [6] Vaai, K.: Handook o Porous Media. aylor and Francis, London (5). [7] Neild, D.., ejan,.: Convection in Porous Media. 3rd edn. Sringer, New York (6). [8] Neild, D.., Kuznetsov,. V.: heral instaility in aorous ediu layer saturated y nonoluid. Int. J. Heat Mass rans. 5, (9a). [9] Neild, D.., Kuznetsov,. V.: he cheng-minkowycz role or natural convective oundary-layer low in a orous ediu saturated y a nanoluid. Int J. Heat Mass rans. 5, (9). [] Neild, D.., Kuznetsov,. V.: he onset o convection in a horizontal nanoluid layer o inite det. Eur. J. Mech. 9, 7-3 ().

13 erican Journal o lied Matheatics 7; 5(): -3 3 [] garwal, S., hadauria,. S., Siddheshwar, P. G.: heral instaility o a nanoluid saturating a rotating anisotroic orous ediu. Sec. o. Rev. Porous Media egell House Pul. (), (). [] hadauria,. S., garwal, S., Kuar,.: Non-linear twodiensional convection in a nanoluid saturated orous ediu. rans. Porous Media 9(), 65-65(). [3] Uavathi, J. C.: Rayleigh enard convection suject to tie deendent wall teerature in a orous ediu layer saturated y a nanoluid. Meccanica. 5,98-984(5) [4] Uavathi, J. C. and Monica Mohite.: Convective transort in a orous ediu layer satuated with a Maxwell nanoluid. Jour o King Saud University-Engineering sciences. 8,56-68(6). [5] Uavathi, J. C. and Pratha Kuar, J.: Onset o Convection in a orous ediu layer saturated with an Oldoyd nanoluid, SME J. Heat rans. 39, 4-44(6). [6] Nield, D.., ejan,.: Convection in orous Medial, third ed., Sringer, New York, 6. [7] Kuznetsov,. V., vraenko,..: Eect o sall articles on the staility o ioconvection in a susension o gyrotactic icroorganiss in a layero inite length, Int. Coun. Heat Mass ranser 3, - (4). [8] uongiorno, J. and Hu, L. W. Nanoluid Coolants or dvanced Nuclear Power Plants, Paer 575, Proceedings o ICPP 5, Seoul, May 5-9. (5). [9] hadauria. S, garwal S: Natural convection in a nanoluid saturated rotating orous layer, a nonlinear study. rans Porous Media 87(), 585-6(). [3] iwari R. K, Das M. K: Heat transer augentation in a twosided lid-driven dierentially heated square cavity utilizing nanoluid. Int. J Heat Mass rans, 5, -8(7).