Effect of Modulation on the Onset of Thermal Convection in a Viscoelastic Fluid-Saturated Nanofluid Porous Layer

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1 J.C. Uavathi Int. Journal o Engineering Research and Alications ISSN : 48-96, Vol. 3, Issue 5, Se-Oct 3, RESEARCH ARTICLE OPEN ACCESS Eect o Modulation on the Onset o Theral Convection in a Viscoelastic Fluid-Saturated Nanoluid Porous Layer J.C. Uavathi Deartent o Matheatics, Gulbarga University, Gulbarga, Karnataka-585 6, India Abstract The stability o a viscoelastic luid-saturated by a nanoluid in a horizontal orous layer, when the boundaries o the layer are subjected to eriodic teerature odulation, is analyzed. The Darcy-Brinkan-Oldroyd-B luid odel is eloyed and only ininitesial disturbances are considered. The odel used or the nanoluids incororates the eect o Brownian otion. The theral conductivity and viscosity are considered to be deendent on the nanoarticle volue raction. Three cases o the oscillatory teerature ield were exained (a) syetric, so that the wall teeratures are odulated in hase, (b) asyetric corresonding to out-o hase odulation and (c) only the botto wall is odulated. Perturbation solution in owers o the alitude o the alied ield is obtained. The eect o the requency o odulation on the stability is clearly shown. The stability o the syste characterized by a correction Rayleigh nuber is calculated as a unction o the viscoelastic araeters, the concentration Rayleigh nuber, orosity, Lewis nuber, heat caacity ratio, Vadász nuber, viscosity and conductivity variation araeters and requency o odulation. It is ound that the onset o convection can be delayed or advanced by the actors reresented by these araeters. The nanoluid is ound to have ore stabilizing eect when coared to regular luid. The eect o all three tyes o odulation is ound to be destabilizing as coared to the unodulated syste. I. Noenclature c nanoluid seciic heat at constant ressure c seciic heat o the nanoarticle aterial c eective heat caacity o the orous ediu d nanoarticle diaeter g gravitational acceleration D B Brownian diusion coeicient ( s ) h seciic enthaly o the nanoarticle aterial H diensional layer deth ( ) j diusion ass lux or the nanoarticles j T, therohoretic diusion k theral conductivity o the nanoluid k Boltzan s constant k B k Le N N A B eective theral conductivity o the orous ediu theral conductivity o the article aterial Lewis nuber odiied diusivity ratio odiied article-density increent ressure diensionless ressure K q R energy lux relative to a rae oving with the nanoluid velocity v theral Rayleigh- Darcy nuber Rn concentration Rayleigh nuber t tie t diensionless tie t H T nanoluid teerature T diensionless teerature T T / Th T T c teerature at the uer wall T h teerature at the lower wall T reerence teerature R u, v, w diensionless Darcy velocity coonents u v w H,, v nanoluid velocity v D Darcy velocity v v diensionless Darcy velocity u, v, w D Va Vadász nuber x, y, z diensionless Cartesian coordinate x, y, z H x, y, z Cartesian coordinates Greek letters theral diusivity o the orous ediu, k / c roortionality actor conductivity variation araeter 93 P a g e

2 J.C. Uavathi Int. Journal o Engineering Research and Alications ISSN : 48-96, Vol. 3, Issue 5, Se-Oct 3, a t Non diensional acceleration coeicient stress relaxation coeicient strain retardation coeicient orosity o the ediu alitude o the odulation viscosity o the luid viscosity variation araeter luid density nanoraticle ass density araeter nanoarticle volue raction relative nanoarticle volue raction / h c diensional requency diensionless requency H k hase angle (, syetric odulation;,antisyetric odulation; i, only lower wall teerature odulation) II. INTRODUCTION Interest in sustainable energy has created signiicant deand or new theral storage and theral anageent technologies, including technologies that eloy nanoluids (which are susensions o nanoarticles in liquids). There is also interest in increasing the eiciency o existing heat transer rocesses via iroveents in the transort roerties o heat transer edia such as nanoluids. The ability to tune the roerties o nanoluids oers any advantages in this resect. For exale, a 39% increase in the heat transer coeicient has been reorted by Xuan et al. [] when an aqueous nanoluid containing % (v/v) coer nanoarticles was eloyed in lace o water in orced convective heat transer exerients in a horizontal tube. Siilarly, ool boiling exerients with an aqueous nanoluid containing.5% (v/v) aluina nanoarticles have yielded Wen et al. [] a 4% enhanceent in the heat transer coeicient when coared with the exerients conducted with ure water. Maxwell [3] was the irst resenter o a theoretical basis to redict a susension s eective conductivity about 4 years ago and his theory was alied ro illieter to icroeter sized articles susensions but Choi and Eastan [4] introduced the novel concet o nanoluids by alying the unique roerties o nanoluids at the annual Mechanical Engineering eeting o Aerican Society in 995. Goldstein et al. [5] added the condition that the articles ust be in colloidal susension. Choi and his colleagues carried out exerients on heat transort in systes with CuO nanoarticles in water and AlO3articles in ethylene glycol and water. They ound that the articles irove the heat transort by as uch as %, and they interreted their result in ters o an iroved theral conductivity k/ k which they naed the eective conductivity [4]. A nanoluid is a luid roduced by disersion o etallic or non-etallic nanoarticles or nanoibres with a tyical size o less than n in a liquid. These nanoluids can be eloyed to cool the ies exosed to such high teerature o the order -35 C, while extracting the geotheral energy. Further when drilling, they can also be used as coolants or the achinery and equient working in high riction and high teerature environent. In the etroleu industry also, nanoluids can be used as coolants or as drilling luids. Also in the above ields, we coe across orous edia in the or o rocks inside the earth s crust, which is being aected by the rotational coonent o the earth s sin on its axis. Buongiorno and Hu [6] suggested the ossibility o using nanoluids in advanced nuclear systes. Another recent alication o the nanoluid is in the delivery o nano-drug as suggested by Kleinstreuer et al. [7] and Eastan et al. [8] and conducted a corehensive review on theral transort in nanoluids to conclude that a satisactory exlanation or the abnoral enhanceent in theral conductivity and viscosity o nanoluids needs urther studies. Buongiorno [9] conducted a corehensive study to account or the unusual behavior o nanoluids based on Inertia, Brownian diusion, therohoresis, diusohoresis, Magnus eects, luid drainage, and gravity settling, and roosed a odel incororating the eects o Brownian diusion and the therohoresis. With the hel o these equations, studies were conducted by Tzou [] and ore recently by Nield and Kuznetsov []. Quite recently non-newtonian luids housed in luid-based systes, with and without orous atrix, have been extensively used in alication situations and hence warrant the attention they have been duly getting. In the asthenoshere and the deeer antle it is well known now that viscoelastic behavior is an iortant rheological rocess []. The other alication areas o viscoelastic luid saturated orous edia are low through coosites, tiber wood, snow systes and rheology o ood transort. The roble housed in a orous ediu suggests an elastohydrodynaical odel or geohysical alications and the likes o it [3-5]. Herbertt [6] was the irst to study natural convection in a viscoelastic luid using the Oldroyd [7] odel and showed that the elasticity o the luid inluenced the onset o arginal convection only in the resence o initial inite elastic stress. Green [8], and Vest and Araci [9] studied the analogous roble o Rayleigh-Benard convection in Jerey and Maxwell luids, resectively, and concluded that the onset o arginal convection is indeendent o viscoelastic araeters, but the condition or the onset o oscillatory convection is inluenced by these araeters. Eltayeb [] studied linear and nonlinear Rayleigh-Benard convection in a visco-elastic luid 94 P a g e

3 J.C. Uavathi Int. Journal o Engineering Research and Alications ISSN : 48-96, Vol. 3, Issue 5, Se-Oct 3, using the Oldroyd odel, and showed that or Prandtl nuber tending to ininity the results obtained by Green [8] using Jerey s odel, violate the criterion required or the onset o oscillatory convection. Further, he showed that the criterion or onset o oscillatory convection in the Maxwell luid used by Vest and Araci [9] is always satisied or larger values o the Prandtl nuber. One o the eective ethods to control convection is by aintaining a non unior teerature gradient. Such a teerature gradient ay be generated by (i) an aroriate heating or cooling at the boundaries [], (ii) injection o luid at one boundary and reoval o the sae at the other boundary [, (iii) an aroriate distribution o heat sources [3], and (iv) radiative heat transer [4]. These ethods are ainly concerned with only sacedeendent teerature gradients. However, in any o the ractical situations cited earlier, the non unior teerature gradient inds its origin in transient heating or cooling at the boundaries, so the basic teerature roile deends exlicitly on osition and tie. This has to be deterined by solving the energy equation under suitable tie-deendent teerature boundary conditions, called theral odulation. Rudraiah et al. [5] studied the eect o odulation on the onset o theral convection in a viscoelastic luid saturated sarsely acked orous layer. Malashetty et al. [6] analyzed the cobined eect o anisotroy o the orous ediu and tie deendent wall teerature on the onset o convection in a horizontal orous layer saturated with Oldroyd luid. Although the roble o Rayleigh-Bénard roble has been extensively investigated or non- Newtonian luids, relatively little attention has been devoted to the theral convection o nanoluids. The corresonding roble in the case o the eects o conductivity and viscosity ratio has also not received uch attention until recently. Kuznetsov and Nield [7] investigated the onset o Double-Diusive nanoluid convection in a layer o saturated orous ediu. Agarwal et al. [8] studied the non-linear convective transort in a binary nanoluid saturated orous layer. Nield and Kuznetsov [9] studied the linear stability theory or the orous ediu saturated by nanoluid with theral conductivity and viscosity deendent on the nanoarticle volue raction. In the resent study, the eect o theral odulation on the onset o convection in a Oldryod-B luid saturated with nanoluid orous ediu is investigated. The boundary teerature odulation alters the basic teerature distribution ro linear to nonlinear which hels in eective control o convective instability. The diiculty in dealing with such instability robles is that one has to solve tiedeendent stability equations with variable coeicients, and to our knowledge no work has been initiated or such luids in this direction. The resulting eigenvalue roble is solved by regular erturbation technique with alitude o the teerature odulation as a erturbation araeter. In articular, it is shown that the onset o convection can be advanced by a roer tuning o the requency o the boundary teerature odulation. III. MATHEMATICAL FORMULATION We consider an ininite horizontal orous layer saturated with a viscoelastic nanoluid, conined between the lanes z and z H, with the vertically downward gravity orce acting on it. A Cartesian rae o reerence is chosen with the origin in the lower boundary and the z -axis vertically uwards. The Boussinesq aroxiation, which states that the variation in density is negligible everywhere in the conservation excet in the buoyancy ter, is assued to hold. For an Oldroyd-B luid, the extra-stress tensor T is given by the constitutive equation [3] DS DA T I S, S A () Dt Dt where is the hydrostatic ressure, I the identity tensor, the viscosity o the luid, and S the extra stress tensor, and are constant relaxation and T retardation ties resectively. A q q is the strain-rate tensor, q is the velocity vector, is the gradient oerator, and DS T q. S S q q S () Dt t DA T q. A A q q A (3) Dt t It should be noted that this odel includes the classical viscous Newtonian luid as a secial case or, and to be the Maxwell luid when. It is well known that in low o viscous Newtonian luid at a low seed through a orous ediu the ressure dro caused by the rictional drag is directly roortional to velocity, which is the Darcy s law. By analogy with Oldroyd-B constitutive relationshis, the ollowing henoenological odel, which relates ressure dro and velocity or a viscoelastic luid in a orous ediu has been given by [3] D t K t q (4) where K is ereability, q D is Darcian velocity, which is related to the usual (i.e. volue averaged over a volue eleent consisting o luid only in the ores) velocity vector q by q q, ε is orosity o the orous ediu. We note that when, Equation (4) siliied to Darcy s law or low o viscous Newtonian luid through a orous ediu. Thus Equation (4) can be regarded as an aroxiate D 95 P a g e

4 J.C. Uavathi Int. Journal o Engineering Research and Alications ISSN : 48-96, Vol. 3, Issue 5, Se-Oct 3, or o an eirical oentu equation or low o Oldroyd-B luid through a orous ediu. Under consideration o the balance o orces acting on a volue eleent o luid, the local volue average balance o linear oentu is given by dq g. S r (5) dt where d is the aterial tie derivative, r is Darcy dt resistance or an Oldroyd-B luid in the orous ediu. Since the ressure gradient in Equation (4) can also be interreted as a easure o the resistance to low in the bulk o the orous ediu, and r is a easure o the low resistance oered by the solid atrix, thus r can be inerred ro Equation (4) to satisy the ollowing equation: t r K t q (6) Substituting Equation (6) into Equation (5), we obtain dq. t g S dt q K t (7) For Darcy odel, ignoring the advection ter q.q and the viscous ter. S, Equation (7) can be siliied to (ater droing the suix D on q or silicity) q g q t t K t (8) The conservation equations take the or. v. (9) Here D v is the nanoluid Darcy velocity. We write D v D u, v, w. The conservation equation or the nanoarticles, in the absence o therohoresis and cheical reactions, takes the or v D.. DB t () where is the nanoarticle volue raction, is the orosity, and D B is the Brownian diusion coeicient. We use the Darcy odel or a orous ediu, then the oentu equation or Oldryod-B nanoluid ollowing Equation (5) can be written as D e v g t t K t () Here is the overall density o the nanoluid, which we now assue to be given by D T T T () where is the article density, is a reerence density or the luid, and T is the theral voluetric v exansion. The theral energy equation or a nanoluid can be written as T c c vd. T k T c D B. T t (3) The conservation o nanoarticle ass requires that v D. DB. (4) t Here c is the luid seciic heat (at constant ressure), k is the overall theral conductivity o the orous ediu saturated by the nanoluid, and c is the nanoarticle seciic heat o the aterial constituting the nanoarticles (ollowing Nield and Kuznetsov [9]). Thus, k k ( ) k, (5) e s where is the orosity, k e is the eective conductivity o the nanoluid (luid lus nanoarticles), and k s is the conductivity o the solid aterial oring the atrix o the orous ediu. We now introduce the viscosity and the conductivity deendence on nanoarticle raction. Following Tiwari and Das[3], we adot the orulas, based on a theory o ixtures, e (6).5 ( ) k k k k k k k k k k e ( ) ( ) Here ( ) ( ) k and 96 P a g e (7) k are the theral conductivities o the luid and the nanoarticles, resectively. Equation (6) was obtained by Brinkan [33], and Equation (7) is the Maxwell-Garnett orula or a susension o sherical articles that dates back to Maxwell [34]. In the case where is sall coared with unity, we can aroxiate these orulas by e.5 (8) k k k k k k k e ( ) ( ) ( ) 3 ( k k ) ( k k ) k ( k k ) (9) We assue that the voluetric ractions o the nanoarticles are constant on the boundaries. Thus, the boundary conditions are w, at z () w, at z H () For theral odulation, the external driving orce is odulated haronically in tie by varying the teerature o lower and uer horizontal boundary. Accordingly, we take

5 J.C. Uavathi Int. Journal o Engineering Research and Alications ISSN : 48-96, Vol. 3, Issue 5, Se-Oct 3, T T z, t T t cos t at z () T T z, t T t cos t at z H (3) where t reresents a sall alitude o odulation (which is used as a erturbation araeter to solve the roble), the requency o odulation and the hase angle. We consider three tyes o odulation, viz., Case (a): Syetric (in hase, ) Case (b): asyetric (out o hase, ) and Case (c): only lower wall teerature is odulated while the uer one is held at constant teerature i. A. Basic State The basic state o the luid is quiescent and is given by g (4) b b T c k T t (5) d b dz (6) The solution o Eq. (5) satisying the theral conditions as given in Eqs. () and (3) is T T z T z t b t, where T z T z TR H (, ) Re z H z H it T z t b e b e e with i c H k, (7) (8) i T e e b (9) e e and Re stands or real art. We do not record the exressions o b and b as these are not exlicitly required in the reaining art o the aer. B. Linear Stability Analysis Let the basic state be distributed by an ininitesial erturbation. We now have v v ', b ', T Tb T ', b ' (3) where rie indicates that the quantities are ininitesial erturbations. Substituting Eq. (6) in Eqs. (9)-(5) and linerising by neglecting roducts o ried quantities, we have, s RTeˆ Rn eˆ s v s v z z a (3) T ' Tb T NB Tb T Tb w' k t z z Le z z z z (3) ' w' = ' t Le (33) w', T', ' at z, (34) Ater using the transorations,,,, x y z x y z H, t t H, u, v, w u, v, w H,, T, T T T c h Tc H,, K k ( c ), ( c ) ( c ), e, k s k s, k k k. k k k, k The diensionless grou that aear are Pr is K the Prandtl nuber, Da is the Darcy nuber, H Pr Va is the Vadász nuber, is the Da H relaxation araeter and is the H retardation araeter. a is the acceleration Va coeicient, Le is the nanoluid Lewis nuber, D R g K T T H R is the nanoarticle Rayleigh nuber, N B c is a c odiied article-density increent. In deriving Eq. (3) the ter roortional to the roduct o and T (Oberbeck-Boussinesq aroxiation) is neglected. This assution is likely to be valid in the case o sall teerature gradients in a dilute susension o nanoarticles. For the case o regular luid (not a nanoluid), the araeters Rn and N B are zero. We eliinate ressure by oerating on Eq. (3) with eˆz curl curl and using the identity curl curl grad div results in a ' srh Rn H ' s s s w (35) Here H is the two-diensional Lalacian oerator on the horizontal lane. Cobining Eqs. (3)-(34), we obtain equations or the vertical coonent o velocity w in the or (droing rie) (36) 97 P a g e

6 J.C. Uavathi Int. Journal o Engineering Research and Alications ISSN : 48-96, Vol. 3, Issue 5, Se-Oct 3, t t Le a s Rn s s s w where Re A A e A e e i e e e e z z it and i w IV. Method o Solution t We seek the eigen unctions w and eigen values Ra o Eq. (36) or the basic teerature Tb s R w gradient given by Eq. (38) that dearts ro the linear z t Le T roile b by quantities o order t. We where.5, and z thereore assue the solution o Eq. (36) in the or 3 k ( ) k s. w, R w, R t w, R t w, R... (4) k Substituting Eq. (4) into Eq. (36) and equating the It is worth noting that the actor coes ro the coeicients o various owers o t on either side o ean value o z over the range [, ] and the the resulting equation, we obtain the ollowing syste o equations u to the order o actor is the ean value o k t : z over the sae Lw (4) range. That eans that when evaluating the critical Rayleigh nuber it is a good aroxiation to base R G R R Lw s that nuber on the ean values o the viscosity and w Le Le conductivity based in turn on the basic solution or the (4) nanoluid raction. The boundary condition (34) can also be exressed in G R Lw s ters o w in the or R w Le Le d w w at z, (37) G R dz s R w Le Le Using Eq. (7), the diensionless teerature (43) gradient aearing in Eq. (3) ay be written as where T b (38) z Rn L a t t Le t t t t t R t Le. Rn Le and w, w, w are required to satisy the boundary R c 4 (46) conditions o Equation (37). We now assue the solutions or Eq. (4) in the or n w w z.ex i lx y, where w z w z sin n z n =,, 3, and l, are the wave nubers in the x y lane such that l. The corresonding eigen values are given by n Rn Le R (44) For a ixed value o the wave nuber, the least eigen value occurs at n and is given by Rn Le R (45) R assues the iniu value when c or (39) These are the values reorted by Horton and Rogers [35] in the absence o concentration Rayleigh nuber Rn. The equation or w then takes the or D Lw R i G sin z Le (47) d where D. dz Thus D sin z sin z ' cos z (48) with ' R. P. Ae A e e Using Eq. (48), Eq. (47) becoes z z it (49) 98 P a g e

7 J.C. Uavathi Int. Journal o Engineering Research and Alications ISSN : 48-96, Vol. 3, Issue 5, Se-Oct 3, Gsin z Lw R i ' L sin z cos z Le i where L. Le We solve Eq. (49) or w by exanding the right hand side o it in Fourier series exansion and inverting the oerator L. For this we need the ollowing Fourier series exansions n z 4 n e z gn e sin zsin n z dz n n (5) z cos cos n z n e n e z n z dz n n (5) so that z e sin z g sin n z (5) n n z e cos z n z (53) Now, where cos n n L n A ib (54) a 3 a Rn a n n n Le Le A n 3 n n Le Le and a 3 a Rn n n a n Le Le B n 3 n n Le Le It is easily seen that sin i t i t L n z e L, n sin n z e it L cos n z e L, n cos n z e and Eq. (49) now becoe it Lw i R so that w i R it I. P. An sin n z e L R. P. An sin n z e n n it R. P. Bn cos nz e Le n n Bn n, n A A it I. P. sin nz e L R. P. sin nz e n L, n n L, n it R. P. cos n z e Le L n it it (55) (56) 99 P a g e

8 J.C. Uavathi Int. Journal o Engineering Research and Alications ISSN : 48-96, Vol. 3, Issue 5, Se-Oct 3, where An A gn A gn B A A., and n n n To siliy Eq. (43) or w we need G R w Le Lw s R. w Le The equation or w then can be written as G D Dw Lw i R Ln w Le R Le where i n Ln. Le (57) (58) n n Le L, n where L, n is the colex conjugate o L, n We shall not require the solution o this equation but erely use it to deterine R. The solvability condition requires that the tieindeendent art o the right hand side o (58) ust be sin z. Multilying Eq. (58) by orthogonal to sin z and integrating between and we obtain LeR i G R wsin z dz Le (59) where an uer bar denotes the tie average. We have the Fourier series exansions i t sin z R. P. A sin n z e, n D sin z R. P. i t Cn sin n z e where C A g A g (6) n n n Using Eq. (6) in Eq. (59) we obtain A n L n L R. P. L, n i i LeR L, n R (6) 4 n B R. P. C L, n i i, sign o R c characterizes the stabilizing or 4 6n destabilizing eect o odulation. A ositive R c An. The indicates that the odulation eect is stabilizing while n n a negative R c indicates that the odulation eect is critical value o R, denoted by R c, is obtained at the destabilizing coared to the syste in which the wave nuber given by equation or the odulation is absent. c ollowing three dierent cases. We evaluate the ollowing cases. R c or (a) when the oscillating teerature ield is syetric so that the wall teeratures are odulated in hase (with ), (b) when the wall teerature ield is antisyetric corresonding to out-o-hase odulation (with ) (c) when only the teerature o the botto wall is odulated, the uer wall being held at a constant teerature ( with i ). V. Results and Discussion The roble o linear convection in a sarsely acked Oldryod-B luid saturated with nanoluid layer subject to dierent eriodic teerature boundary conditions is investigated. The solution is obtained on the assution that the alitude o the alied teerature odulation is sall. The exression or the critical correction Rayleigh nuber R c is couted as a unction o requency o the odulation or dierent araeter values and the results are deicted in Figs. -8. The The variation o critical correction theral Rayleigh nuber R c with requency or syetric odulation or dierent governing araeters (Figs -8) or both regular Rn and nanoluids Rn show that or sall requencies the critical correction Rayleigh nuber is negative indicating that the eect o syetric odulation is destabilizing while or oderate and large values o its eect is stabilizing. Figure reveal that an increase in the value o is to increase the agnitude o R c. On the other hand the eect o odulation diinishes as the stress relaxation araeter becoes saller and saller. The eak negative or ositive value o R c is ound to increase with or both regular and nanoluids. The eect o strain retardation araeter is ound to be oosite o the stress relaxation araeter as seen in Fig.. Thereore one can conclude that the stress relaxation araeter is ore ronounced in aiding the onset o convection coared to the eect o strain retardation araeter or both regular and nanoluids. Siilar 93 P a g e

9 J.C. Uavathi Int. Journal o Engineering Research and Alications ISSN : 48-96, Vol. 3, Issue 5, Se-Oct 3, results were observed by Malashetty et al. [6] or regular luid. Figure 3 shows the variation o R c with or dierent values o concentration Rayleigh nuber Rn. As Rn increases the agnitude o correction theral Rayleigh nuber Rc decreases indicating that the eect o Rn is to delay the onset o convection. However, R c is negative or sall requencies indicating that the syetric odulation has destabilizing eect while or oderate and large values o requency its eect is stabilizing. This is a siilar result obtained by Uavathi [36] or Newtonian luid. The eect o orosity or syetric odulation is shown in Fig. 4. This igures reveal that as increases the value o R becoes sall indicating that larger values o decreases the eect o odulation. Here also it is observed that as increases Rc increases to its axiu value initially and then starts decreasing with urther increase in. When is very large all the curves or dierent orosity coincide and R c aroaches to zero or both regular and nanoluid. Figure 5 deicts the variation o R c with requency or dierent values o Lewis nuber Le or the case o syetric odulation. Lewis nuber shows the siilar eect as that o orosity. That is to say that, an increase in the value o Lewis nuber decreases the value o R indicating that the eect o increasing Le is to c reduce the eect o theral odulation or both regular and nanoluid which is a siilar result observed Uavathi 36. The eect o theral caacity ratio is to increase R or both regular and nanoluids as seen in Fig. 6. Here also as increases R c increases to its axiu initially and then starts decreasing with urther increase in. When is very large all the curves or dierent theral caacity ratio coincide and R c aroaches to zero or both regular and nanoluids. The eect o Vadasz nuber Va shows the siilar nature as that o theral caacity ratio as seen in Fig. 7. That is to say that, increasing the value o Vadasz nuber is to decrease the theral odulation or both regular and nanoluid. The eect o viscosity variation araeter and conductivity variation araeter is shown in Figs. 8 and 9 resectively or syetric odulation. Their eect is ound to be siilar to the eect o stress relaxation araeter, theral caacity ratio and Vadasz nuber Va. That is to say that, and delay the onset o convection or both regular and nanoluids. The results obtained or the case o asyetric odulation are resented in Figs. -8. c c All these igures show that or all araeters sall requencies has destabilizing eect while or oderate and large values o requency their eects are stabilizing or both regular and nanoluid. It is seen ro Fig. that an increase in the value o is to increase the agnitude o R c. The eect o strain retardation araeter is to decrease the agnitude o R c as seen in Fig.. The eect o concentration Rayleigh nuber Rn, orosity, Lewis nuber Le, theral caacity ratio, Vadasz nuber Va, viscosity and conductivity variation araeters and show the sae eect as in the case o syetric odulation and hence a detailed exlanation is not required. The nature o all the grahs or lower wall teerature odulation is ound to be qualitatively siilar to the asyetric odulation and thereore we oit a grahical reresentation o the sae. VI. Conclusions The eect o theral odulation on the onset o convection or Oldroyd luid saturated with nanoluids orous layer is studied using a linear stability analysis and the ollowing conclusion are drawn.. The increase in stress relaxation araeter enhances the eect o odulation while increase in strain retardation araeter suresses the eect o odulation or all three tyes o odulation.. The concentration Rayleigh and Vadasz nuber suresses the eect o odulation. The orosity suresses the eect o odulation or regular luid whereas it enhances the eect o odulation or nanoluids. The theral caacity ratio, Vadasz nuber, viscosity and conductivity ratio enhances the eect o odulation or both regular and nanolid. 3. The eak values o theral correction Rayleigh nuber is obtained or regular luid when coared to nanoluids or all theral odulations. 4. The eect o all three tyes o odulation naely, syetric, asyetric, and only lower wall teerature odulations is ound to be destabilizing as coared to the unodulated syster. 5. The eect o theral odulation disaears or large requency in all the cases. 6. The onset o convection is delayed or nanoluids when coared to regular luid. 7. The eect o stress and strain relaxation araeter or syetric odulation or regular luid were also obtained by Malashetty et al. [6]. 93 P a g e

10 J.C. Uavathi Int. Journal o Engineering Research and Alications ISSN : 48-96, Vol. 3, Issue 5, Se-Oct 3, Rn = Rn = Syetric teerature odulation =.4 Le = = 5 = Va = =. = = R c x -3 Fig. Variation o R c with or dierent values o Rn and 5 9 Rn = Rn = Syetric teerature odulation Le = = 5 = Va = =. = = 6 3 =.,.,.4 =.,., R c x -3 Fig. Variation o R c with or dierent values o Rn and 93 P a g e

11 J.C. Uavathi Int. Journal o Engineering Research and Alications ISSN : 48-96, Vol. 3, Issue 5, Se-Oct 3, Syetric teerature odulation Le = = 5 = Va = =. =. = = Rn = R c x -3 Fig. 3 Variation o R c with or dierent values o Rn 5 Rn = Rn = Syetric teerature odulation Le = = Va = =. =. = = 5 = R c x -3 Fig. 4 Variation o R c with or dierent values o Rn and 933 P a g e

12 J.C. Uavathi Int. Journal o Engineering Research and Alications ISSN : 48-96, Vol. 3, Issue 5, Se-Oct 3, Rn = Rn = Syetric teerature odulation = 5 = Va = =. =. = = Le = R c x -3 Fig. 5 Variation o R c with or dierent values o Rn and Le 5 Rn = Rn = Syetric teerature odulation Le = = Va = =. =. = = = R c x -3 Fig. 6 Variation o R c with or dierent values o Rn and 934 P a g e

13 J.C. Uavathi Int. Journal o Engineering Research and Alications ISSN : 48-96, Vol. 3, Issue 5, Se-Oct 3, Rn = Rn = Syetric teerature odulation Le = = = 5 =. =. = = Va = R c x -3 Fig. 7 Variation o R c with or dierent values o Rn and Va 5 Rn = Rn = Syetric teerature odulation Le = = = 5 =. =. Va = = 5 = R c x -3 Fig. 8 Variation o R c with or dierent values o Rn and 935 P a g e

14 J.C. Uavathi Int. Journal o Engineering Research and Alications ISSN : 48-96, Vol. 3, Issue 5, Se-Oct 3, Rn = Rn = Syetric teerature odulation Le = = = 5 =. =. Va = = 5 =. - 3 R c x -3 Fig. 9 Variation o R c with or dierent values o Rn and 5 Asyetric teerature odulation Rn = Rn = Le = = 5 = Va = =. = = 5. = R c x -4 Fig. Variation o R c with or dierent values o Rn and 936 P a g e

15 J.C. Uavathi Int. Journal o Engineering Research and Alications ISSN : 48-96, Vol. 3, Issue 5, Se-Oct 3, Asyetric teerature odulation Rn = Rn = Le = = 5 = Va = =. = = = R c x -4 Fig. Variation o R c with or dierent values o Rn and 5 Asyetric teerature odulation Le = = 5 = Va = =. =. = = Rn = R c x -3 Fig. Variation o R c with or dierent values o Rn 937 P a g e

16 J.C. Uavathi Int. Journal o Engineering Research and Alications ISSN : 48-96, Vol. 3, Issue 5, Se-Oct 3, Asyetric teerature odulation Rn = Rn = Le = = Va = =. =. = = = 5 5 = R c x 6 Fig. 3 Variation o R c with or dierent values o Rn and 5 Asyetric teerature odulation Rn = Rn = = 5 = Va = =. =. = = 5 Le = R c x -3 Fig. 4 Variation o R c with or dierent values o Rn and Le 938 P a g e

17 J.C. Uavathi Int. Journal o Engineering Research and Alications ISSN : 48-96, Vol. 3, Issue 5, Se-Oct 3, Asyetric teerature odulation Rn = Rn = = 5 Le = Va = =. =. = = = R c x -3 Fig. 5 Variation o R c with or dierent values o Rn and 5 Asyetric teerature odulation Rn = Rn = = 5 Le = = =. =. = = 5 Va = 5 5 5,, R c x -3 Fig. 6 Variation o R c with or dierent values o Rn and Va 939 P a g e

18 J.C. Uavathi Int. Journal o Engineering Research and Alications ISSN : 48-96, Vol. 3, Issue 5, Se-Oct 3, Asyetric teerature odulation Rn = Rn = = 5 Le = = Va = =. =. = 5 = R c x -3 Fig. 7 Variation o R c with or dierent values o Rn and Asyetric teerature odulation Rn = Rn = = 5 Le = = Va = =. =. = 5 =. = R c x -3 Fig. 8 Variation o R c with or dierent values o Rn and Reerences [] Xuan YM, Li Q: Investigation on convective heat transer and low eatures o nanoluids. J Heat Tran-Trans ASME 3, 5:5-55. [] Wen DS, Ding YL: Exeriental investigation into the ool boiling heat transer o aqueous based gaa-aluina nanoluids. J Nanoart Res 5, 7: [3] Maxwell JC: Electricity and agnetisi. Clarendon Press Oxord P a g e

19 J.C. Uavathi Int. Journal o Engineering Research and Alications ISSN : 48-96, Vol. 3, Issue 5, Se-Oct 3, [4] Choi SUS, Eastan JA: Enhancing theral conductivity o luids with nanoarticles. in: Conerence: 995 International Mechanical Engineering Congress and Exhibition, San Francisco, CA (United States),. -7 Nov 995, ASME, San Francisco, [5] Goldstein RJ, Joseh DD, Pui DH: Convective Heat Transort in Nanoluids roosal. Faculty o Aerosace Engineering and Mechanics University o Minnesota, Minnesota, Seteber (). [6] Buongiorno J, Hu W. Nanoluid coolant or advanced nuclear ower lants. Paer No. 575, In: Proceedings o ICAPP 5, Seoul, 5 9 May, 5. [7] Kleinstreuer C, Li J, Koo J. Microluidics o nano-drug delivery. Int J Heat Mass Trans 8, 5: [8] Eastan JA, Choi SUS, Yu W, Thoson LJ. Theral transort in nanoluids. Annu Rev Mater Res 4, 34:9 46. [9] Buongiorno J. Convective transort in nanoluids. ASME J Heat Trans. 6, 8:4 5. [] Tzou DY. Instability o nanoluids in natural convection. ASME J Heat Trans 8, 3:74. [] Nield DA, Kuznetsov AV. The onset o convection in a horizontal nanoluid layer o inite deth. Euro J Mech B, 9:7 3. [] Lowrie W. Fundaentals o Geohysics. Cabridge University Press, Cabridge, 997. [3] Rao BK. Internal heat transer to viscoelastic lows through orous edia. Ex Heat Transer, 3(4): [4] Siddheshwar PG, Srikrishna CV. Rayleigh- Benard convection in a viscoelastic luid illed high orosity ediu with nonunior basic teerature gradient. Int J Math and Math Sci, 5(9): [5] Yoon DY, Ki MC, Choi CK. Onset o oscillatory convection in a horizontal orous layer saturated with viscoelastic liquid. Transort Porous Media 4, 55: [6] Herbert DM. On the stability o viscoelastic liquids in heated lane Couette low. J Fluid Mech 963, 7: [7] Oldroyd JG. Non-Newtonian Eects in Steady Motion o Soe Idealized Elastico- Viscous Liquids. Proc R Soc London A 958, 45: [8] Green T. Oscillating convection in an elastico-viscous liquid. Phys Fluids 968, :4 43. [9] Vest CM, Araci VS. Overstability o a viscoelastic luid layer heated ro below. J Fluid Mech 969, 36: [] Eltayeb IA. Nonlinear theral convection in an elasticoviscous layer heated ro below. Proc R Soc London A 977, 356:6-76. [] Rudraiah N, Sriani PK, Friedrich R. Finite alitude convection in a two coonent luid saturated orous layer. Int J Heat Mass Transer 98,5:75-7. [] Nield DA: Convective instability in orous edia with throughlow. AIChE J 987, 33:-4. [3] Soerton CW, Catton I: On the theralinstability o suerosed orous and luid layers. J Heat Transer 98, 4:6-65. [4] Vorteyer D, Rudraiah N, Sasikuar TP. Eect o radiative transer on the onset o convection in a orous ediu. Int J Heat Mass Transer 989, 3: [5] Rudraiah N, Radhadevi PV, Kaloni PN. Eect o odulation on the onset o theral convection in a viscoelastic luid-saturated sarsely acked orous layer. Can J Phy 99, 68:4-. [6] Malashetty MS, Siddeshwar PG, Mahantesh Sway: Eect o theral odulation on the onset o convection in a visco-elastic luid saturated orous layer. Trans Porous Mediu 6, 6: [7] Kuznetsov AV, Nield DA. The onset o double-diusive nanoluid convection in a layer o a saturated orous ediu. Trans Porous Media, 85: [8] Agarwal S, Bhadauria BS, Sacheti NC, Chandran P, Singh AK. Non-linear convective transort in a binary nanoluid saturated orous layer. Trans Porous Media, 93:9 49. [9] Nield DA, Kuznetsov AV. The onset o convection in a layer o a orous ediu saturated by a nanoluid: eects o conductivity and viscosity variation and cross-diusion. Trans Porous Media, 9: [3] Tan W, Masuouka T. Stokes irst roble or an Oldryod-B luid in a orous hal sace. Phy Fluids 5, 7:3-37. [3] Khuzhayorov B, Auriault J, Royer P. Derivation o acroscoic iltration law or transient linear viscoelastic luid low in orous edia. Int J Engng Sci, 38: [3] Tiwari RK, Das MK. Heat transer augentation in a two-sided lid-driven dierentially heated square cavity utilizing nanoluids. Int J Heat Mass Trans 7, 5: P a g e

20 J.C. Uavathi Int. Journal o Engineering Research and Alications ISSN : 48-96, Vol. 3, Issue 5, Se-Oct 3, [33] Brinkan HC. The viscosity o concentrated susensions and solutions. J Che Phy 95, : [34] Maxwell JC. A Treatise on Electricity and Magentis, nd edn. Oxord University Press, Cabridge 94. [35] Horton CW, Rogers FT. Convective currents in a orous ediu. J Al Phys 945, 6: [36] Uavathi JC: Eect o theral odulation on the onset o convection in a orous ediu layer saturated by a nanoluid. Trans Porous Mediu 3, 98: P a g e