The effect of Soret parameter on the onset of double diffusive convection in a Darcy porous medium saturated with couple stress fluid

Transcription

1 Availale olie at Advaes i Applied iee Researh, 1, 3 (3): IN: CODEN (UA): AARFC he effet of oret parameter o the oset of doule diffusive ovetio i a Dary porous medium saturated with ouple ress fluid rava N. Gaikwad *1 ad hrava. Kamle 1 Departmet of Mathematis, Gularga Uiversity, Jaa Gaga Campus, Gularga, Karataka, Idia Departmet of Mathematis, Govermet Fir Grade College, Chittapur, Karataka, Idia _ ABRAC he oset of doule diffusive ovetio i a ouple ress fluids saturated horizotal porous layer i presee of oret effet is udied aalytially usig liear aility aalyses. he modified Dary equatio is used to model the mometum equatio. he liear theory is ased o the usual ormal mode tehique. he effet of the ouple ress parameter, the solute Rayleigh umer, the Vadasz umer, the diffusivity ratio, the oret parameter o atioary ad osillatory ovetio is show graphially. Keywords: Doule diffusive ovetio, oret parameter, Couple ress fluid, Vadasz umer. _ INRODUCION he prolem of doule diffusive ovetio i porous media has attrated osiderale itere durig the la few deades eause of its wide rage of appliatios, from the solidifiatio of iary mixtures to the migratio of solutes i water-saturated soils. Other examples ilude geophysial syems, eletrohemiry ad the migratio of moiure through air otaied i firous isulatio. A omprehesive review of the literature oerig doulediffusive atural ovetio i a fluid saturated porous medium may e foud i the ook y Nield ad Beja [1]. Useful review artiles o doule-diffusive ovetio i porous media ilude those y Mojtai ad Charrier- Mojtai [7, 8]. Early udies o the pheomea of doule-diffusive ovetio i porous media are maily oered with the prolem of ovetive iaility i a horizotal layer heated ad salted from elow. he prolem of doulediffusive ovetio i a fluid saturated porous layer was iveigated y may authors (see e. g. aslim ad Narusawa [17]), revisia ad Beja [18], Murray ad Che [9], hivakumara ad Vekatahalappa [11], traugha ad Hutter [14] iveigated doule-diffusive ovetio with the oret effet i a porous layer usig the Dary- Brikma model ad derived a priori ouds. Bahloul et al. [1] arried out a aalytial ad umerial udy o doule-diffusive ovetio i a shallow horizotal porous layer uder the ifluee of the oret effet. Hill [] performed liear ad oliear aility aalysis of doule-diffusive ovetio i a fluid saturated porous layer with a oetratio-ased iteral heat soure usig Dary s law. Although the prolem of Rayleigh Beard ovetio has ee extesively iveigated for Newtoia fluids, relatively little attetio has ee devoted to the thermal ovetio of o-newtoia fluids. he orrespodig prolem i the ase of a porous medium has also ot reeived muh attetio util reetly. With the growig importae of o-newtoia fluids with suspeded partiles i moder tehology ad iduries, the iveigatios of suh fluids are desirale. he udy of suh fluids has appliatios i a umer of proesses that our i idury, suh as the extrusio of polymer fluids, solidifiatio of liquid ryals, oolig of a metalli plate i a ath, exoti luriatio, ad olloidal ad suspesio solutios. I the ategory of o-newtoia fluids oupleress fluids have diit features, suh as polar effets. he theory of polar fluids ad related theories are models 146

2 rava N. Gaikwad et al Adv. Appl. i. Res., 1, 3(3): for fluids whose miroruture is mehaially sigifiat. he oitute equatios for ouple ress fluids were give y tokes [13]. he theory proposed y tokes is the simple oe for mirofluids, whih allows polar effets suh as the presee of ouple ress, ody ouple ad o-symmetri tesors. uil et al. [16] iveigated the effet of suspeded partiles o doule diffusive ovetio i a ouple ress fluid saturated porous medium. hey reported that for the ase of atioary ovetio, the ale solute gradiet ad ouple ress have ailizig effets, whereas the suspeded partiles ad medium permeaility have deailizig effets. idheshwar ad Praesh [1] udied aalytially liear ad oliear ovetio i a ouple-ress fluid layer. Malashetty et al. [5] udied the oret effet o doule diffusive ovetio i a ouple ress liquid usig oth liear ad oliear aalyses. Reetly, Malashetty et al. [6] iveigated the loal thermal o-equilirium effet o the oset of ovetio i a ouple-ress fluid-saturated porous layer. he prolem of doule-diffusive ovetio i a porous medium saturated with Newtoia fluids has ee extesively udied. However, attetio has ot ee give to the udy of doule-diffusive ovetio i a porous layer saturated with No- Newtoia fluids suh as ouple-ress fluids with oret effet. he ojetive of this paper is to udy the effet of oret parameter i the presee of ouple ress fluid.. Mathematial Formulatio Fig. a: Physial ofiguratio We osider a horizotal porous layer saturated with a ouple-ress fluid ofied etwee two parallel ifiite ress-free oudaries, z, d, heated ad salted from elow. he temperature ad oetratio differee etwee the oudig plaes are ad respetively. A Cartesia oordiate syem is used, with the z-axis vertially upward i the gravitatioal field as show i aove Fig.a. We assume that the Oerak-Boussiesq approximatio is valid ad that the flow i the porous medium is govered y the modified Dary s law. he goverig equatios for the udy of doule- diffusive ovetio i a ouple ress-fluid saturated horizotal porous layer are (Hill [1], Malashetty et al [6] ). q (1) ρ q 1 1 ( + q q p + ρ g µ µ ) q, ε t ε k () γ + ( q. ) κ, t (3) ε + ( q. ) κ + D1, t (4) ρ ρ [1 β ( - ) + β ( - )], (5) where q ( u, v, w) is the veloity; is the temperature; is the solute oetratio; p is the pressure; ρ is the desity;, ad ρ are the referee temperature, oetratio ad desity respetively; g is the aeleratio due to gravity; µ is the fluid visosity; µ is the ouple- ress visosity; k is the permeaility of the porous medium; β are the thermal ad solute expasio oeffiiets respetively; ε is the porosity; β ad κ 147

3 rava N. Gaikwad et al Adv. Appl. i. Res., 1, 3(3): ad κ are the effetive thermal diffusivity ad solute diffusivity respetively. Here, D 1 quatifies the otriutio to the mass flux due to temperature gradiet. ( ρ) (1 ε ) K + ε K m f γ, κ, ( ρ p ) f ( ρ p ) f ( ρ) m (1 ε )( ρ) s + ε ( ρ p ) f. Here, K is the thermal odutivity; p is the speifi heat of the fluid, at oat pressure; is the speifi heat of the solid; ad the susripts f, s ad m deote fluid, solid ad porous medium values respetively. Basi ate he asi ate of the fluid is quieset ad is give y q (,,), p p ( z), ρ ρ ( z ), ( z ), ( z). (6) he temperature ( z ), solute oetratio ( z ), pressure P ( z ) ad desity ( z ) equatios: dp d d ρ g,,, ρ ρ[1 β ( ) + β ( )]. dz dz dz (7) ρ satisfy the followig Pertured ate Let the asi ate e pertured y a ifiitesimal perturatio so that q q + q, p p ( z) + p ', ρ ρ ( z ) + ρ ', ( ) ', z + ( z) + ', (8) where primes idiate that the quatities are ifiitesimal perturatios. uitutig eq. (8) ito eqs. (1) - (5) ad usig asi ate eqs. (7) ad (8) ad elow trasformatios x z,, d d, * * * * t t /( γd / κ ), ψ ψ / κ, /, /, (9) * * ( x z ) to reder the resultig equatios dimesioless, ad usig the ream futio ψ defied y ψ ψ, z x ' ' ( u, w ) (, ) we otai (after droppig the aerisks for simpliity) 1 1 (, ) 1 C ψ + ψ Ra Ra γva t +, Va ( x, z) x x (1) ψ ( ψ, ) + t x ( x, z ), (11) ε ψ ( ψ, ) Ra + τ r γ t x ( x, z ) Ra, (1) where εν d β g dk Vadsaz umer Va, thermal Rayleigh umer Ra, solute Rayleigh umer κ k νκ Ra β g dk, ouple-ress parameter νκ C µ µ d, ad diffusivity ratio τ κ. κ 148

4 rava N. Gaikwad et al Adv. Appl. i. Res., 1, 3(3): ( f, g) f g f g he symol is the Jaoia. he aerisks have ee dropped for simpliity. Further, to ( x, z) x z z x rerit the umer of parameters, we set ε ad γ equal to uity. Eqs. (1) - (1) are solved for ress-free, isothermal, vaishig ouple-ress oudary oditios, amely ψ ψ z at z,1. (13) he ress-free oudary oditios are hose for mathematial simpliity, without qualitatively importat physial effets eig lo. he use of ress-free oudary oditios is useful mathematial simplifiatio ut is ot physially soud. he orret oudary oditios are rigid-rigid oudary oditios, ut the the prolem is ot tratale aalytially. 3. Liear aility aalysis I this setio, we disuss the liear aility aalysis. o make this udy, we eglet the Jaoias i eqs. (1) - (1) ad assume the solutios to e periodi waves of the form ψ ψ si( π x) σt e θ os( π x) si( π z) ( 1,,3,...) φ os( π x), (14) where σ is the growth rate, whih is i geeral a omplex quatity ( σ σ + iσ ), ad is horizotal waveumer. uitutig eq. (14) i the liearized versio of eqs. (1) - (1), we otai σ + η δ ψ π ( θ φ ), V a σ + δ θ π ψ (16) ( ), Ra ( σ + τδ ) φ π ψ r δ θ, (17) Ra where η 1 + Cδ, δ π ( + 1). he parameter η is represetative of the ouple ress visosity of the fluid. I the ase of Newtoia fluid, we haveη 1. For o trivial solutio of ψ, θ ad φ we require r i (15) Ra σ σ + δ σ + τδ + η δ + Ra π σ + δ Va π ( σ + τδ + rδ ) ( )( )( ) ( ). (18) As usual, we assume that the mo uale mode orrespods to 1 (fudametal mode) ad rerit our aalysis to this ase (see e.g. Chadrasekhar [15] ). Aordigly, we set aalysis. tatioary ate If σ is real, the margial aility ours wheσ Ra at the margi of aility, i the form 4 η τδ + π τ + r τ + r ( ) ( ). δ π ( + 1) ad η 1+ Cδ i our further. he eq. (18) gives the atioary Rayleigh umer (19) 149

5 rava N. Gaikwad et al Adv. Appl. i. Res., 1, 3(3): he miimum value of the Rayleigh umer Ra ours at the ritial waveumer where satisfies the equatio C ( ) + (1 + C ) (1 + C ). () π π π It is importat to ote that the ritial waveumer depeds o the ouple-ress parameterc. I the ase of sigle ompoet syem, Ra, eq. (19) gives η τδ (1 + C δ ) τδ 4 4 ( + r ) ( + r ) π τ π τ I the presee of ouple resses, eq. (1) gives the value of the Rayleigh umer. (1) { π (1 + ) τ [1 + C π (1 + )]} ( τ r ) +. () he ritial waveumer is to e otaied from eq. (). For a sigle-ompoet ouple ress fluid syem whe oret parameter is aset i.e., r, the eq. () gives { π (1 + ) [1 + C π (1 + )] }.. (3) hese are exatly the values give y idheshwar ad Paresh []. Further, i the asee of ouple resses, i.e., whe C, the eq. (3) gives s t π ( + 1)., (4) whih is the lassial Horto ad Rogers [3] ad Lapwood [4] result with ritial values give y 1 ad 4π for Newtoia fluid through a Dary porous layer heated from elow. Osillatory ate It is well kow that the osillatory motios are possile oly if some additioal oraits like rotatio, saliity gradiet ad mageti field are preset. For the osillatory mode, suitutig σ r ad σ i iω (ω is real) i os eq. (18) ad rearrage the terms, we otai a expressio for osillatory Rayleigh umer Ra at the margi of aility i the form 4 τδ 6 (1 + τ ) δ η Va + η (1 + τ ) + δ Ra π ( τδ ηva) Va + + os Ra, (5) π [ δ 1 r + ηva] { ( ) } ad the o-dimesioal frequey ω i the form ω + [ Ra Ra ( + r )] 6 τδ η π δ τ 4 δ [ η δ + (1 + τ )] Va (6) he ritial Rayleigh umer for osillatory ate is omputed from eq. (5) for differet values of the parameter ad the result disussed i setio

6 rava N. Gaikwad et al Adv. Appl. i. Res., 1, 3(3): C Ra Ra atioary osillatory τ atioary osillatory Va1.,τ.5, Ra Fig 1 Neutral aility urves for differet values of ouple ress parameter C. Va 1.,C 1., Ra Fig Neutral aility urves for differet values of diffisuvities ratio τ 18 5,1,15, Va.5 Ra Ra Ra,15,1, atioary osillatory atioary osillatory Va1.,C 1.,τ Fig. 3: Neutral aility urves for differet values of solute Rayleigh umer Ra. Ra 15., C1.,τ Fig. 4: Neutral aility urves for differet values of Vadasz umer Va. 1431

7 rava N. Gaikwad et al Adv. Appl. i. Res., 1, 3(3): τ.5, Va 1., Ra 15 r Ra Log Ra,C.8 r -.1, -.1,.5,., atioary Va 1.,C 1., Ra 15., τ Fig. 5: Neutral aility urves for differet values of oret parameter r.. osillatory Log C Fig. 6: Variatio of ritial Rayleigh umer Ra,C with ouple ress parameter C for differet values of oret parameter r 5.5 Va 1., τ.5, Ra Va 1., C 1., Ra Log Ra,C Log Ra,C 4. τ.7,.5, τ.3,.5,.7 3. C atioary osillatory atioary osillatory Log Ra Fig. 7: Variatio of ritial Rayleigh umer Ra,C with solute Rayleigh umer Ra for differet values of C Log Ra Fig. 8: Variatio of ritial Rayleigh umer Ra,C with solute Rayleigh umer Ra for differet values of τ. 143

8 rava N. Gaikwad et al Adv. Appl. i. Res., 1, 3(3): REUL AND DICUION he oset of doule-diffusive ovetio i a porous layer saturated with a ouple-ress fluid i the presee of oret effet is aalyzed usig a liear theory. he liear theory is ased o the usual ormal mode tehique. Expressios for the atioary ad osillatory modes for differet values of parameters suh as ouple ress parameter, diffusivity ratioτ, solute Rayleigh umer Ra, Vadasz umer Va ad oret parameter r are omputed ad the results are depited i the figures 1-8. Fig.1 shows the eutral aility urves for differet values of the ouple-ress parameter C for fixed values of Va 1., τ.5, Ra s 15.. We oserve from this figure that the miimum value of the Rayleigh umer for oth atioary ad osillatory modes ireases with a irease i the value of the ouple-ress parameter C, idiatig that the effet of the ouple-ress parameter is to ailize the syem. he effet of diffusivity ratio τ o the eutral aility urves for fixed values of Va 1., C 1., Ra s 15. is show i fig.. We fid that the miimum value of Rayleigh umer for the atioary mode dereases with a irease i the value of diffusivity ratio τ. O the other had, the osillatory Rayleigh umer ireases with a irease i the value of diffusivity ratio τ. hus, the diffusivity ratio has a otraig effet o the aility of the syem i oth atioary ad osillatory modes. Fig. 3 displays the effet of the solute Rayleigh umer Ra o the eutral aility urves for oth atioary ad osillatory modes for fixed values of Va 1., C 1., τ.5. his figure idiates that the miimum Rayleigh umer for oth atioary ad osillatory modes ireases with a irease i the value of the solute Rayleigh umer, implyig that the effet of solute Rayleigh umer is to ailize the syem. he effet of Vadasz umer Va o eutral aility urves for oth atioary ad osillatory modes for fixed values of Ra 15., C 1., τ.5 is show i fig. 4. We oserve from this figure that the miimum value of the Rayleigh umer for oth atioary ad osillatory modes ireases with a irease i the value of the Vadasz umerva, idiatig that the effet of the Vadasz umer is to ailize the syem. Fig. 5 shows the eutral aility urves for differet values of oret parameter r (oth positive ad egative) for fixed values of Va 1., C 1., Ra 15., τ.5. We fid that as the oret parameter r ireases positively, the Rayleigh umer dereases. However, the effet of ireasig egative oret parameter is to irease the Rayleigh umer for oth atioary ad osillatory modes. his is due to the fat that for egative oret parameter, the heavier ompoet migrates towards the hotter regio. hus, outeratig the desity gradiet aused y temperature. Fig.7 depits the variatio of the ritial Rayleigh umer Ra. with ouple ress parameter C for differet values of oret parameter r ad for fixed values of τ.5, Va 1., Ra 15.. We fid that a irease of oret parameter r, dereases the ritial Rayleigh umer for the atioary mode ad ireases for osillatory mode. O the other had, the ritial Rayleigh umer for oth atioary ad osillatory modes ireases with a irease of ouple ress parameter idiatig that effet of ouple ress parameter is to ailize the syem. he variatio of the Rayleigh umer for oth atioary ad osillatory modes with the solute Rayleigh umer for differet values of the ouple-ress parameter C ad fixed values of Va 1., τ.5 is show i fig. 7. We oserve that the ritial Rayleigh umer for oth atioary ad osillatory modes ireases with irease of ouple ress parameter C. he ritial Rayleigh umer for the atioary ad osillatory modes ireases with a irease i the value of the solute Rayleigh umer, idiatig that the solute Rayleigh umer ailizes the syem. Further, we fid that the oset of ovetio is through the atioary mode for small ad medium values of the solute Rayleigh umer. However, whe the solute Rayleigh umer is ireased eyod a ertai ritial value that depeds o the other parameters, ovetio fir sets i through the osillatory mode. We also fid that for large solute Rayleigh umer, the ifluee of the ouple-ress parameter is isigifiat. Fig.8 depits the variatio of the ritial Rayleigh umer for atioary ad osillatory modes with solute Rayleigh umer for differet values of diffusivity ratio τ. We fid that the ritial Rayleigh umer dereases with a 1433

9 rava N. Gaikwad et al Adv. Appl. i. Res., 1, 3(3): irease i diffusivity ratio τ for the atioary mode. O the other had, the ritial Rayleigh umer ireases with a irease i the value of diffusivity ratio τ up to a ertai value of Ra ad the the tred reverses, idiatig that the diffusivity ratio has a dual effet o the osillatory mode whe the Vadasz umer ad oupleress parameter are fixed. CONCLUION he oset of doule diffusive ovetio i a porous medium saturated with ouple ress fluid i the presee of oret effet is iveigated usig the liear theory. 1. he diffusivity ratio τ deailizes the syem for atioary mode while ailizes the syem for osillatory mode.. he solute Rayleigh umer Ra ailizes the syem for oth atioary ad osillatory modes. 3. he effet of Vadasz umer Va is to deailize the syem i osillatory mode oly ad its effet is isigifiat i atioary mode. 4. he positive oret parameter r deailizes the syem ad egative oret parameter r ailizes the syem i oth atioary ad osillatory ovetio. Akowledgemet his work is supported y the Uiversity Grats Commissio, New Delhi uder the Major Researh Projet with No. F (R) dated REFERENCE [1] A. Bahloul, N. Boutaa, P. Vasseur, J. Fluid Meh., 3, 491, [] A. A. Hill, Pro. R. o., 5, A 461, [3] C. W. Horto, F.. Rogers, J. Appl. Phys., 1945, 16, [4] E. R. Lapwood, Pro. Com. Phil. o., 1948, 44, [5] M.. Malashetty,. N. Gaikwad, M. wamy, It. J. herm. i., 6, 45, [6] M.. Malashetty, I.. hivakumara,. Kulkari, Phys. Lett., 9, A 373, [7] A. Mojtai, M. C. Mojtai, Hadook of Porous media, Dekker, New York,. pp [8] A. Mojtai, M. C. Mojtai, Hadook of Porous media, aylor ad Frais, New York, 5, pp [9] B.. Murrary, C. F. Che, J. Fluid Meh., 1989, 1, [1] D. A. Nield, A. Beja, Covetio i porous media, priger, New York, 6. [11] I.. hivakumara, M. Vekathalappa, Advaes i fluid mehais (a series of olleted works of Prof. N. Rudraiah), ata MGraw-Hill, New Delhi, 4. [1] P. G. iddheshwar,. Praesh, It. J. No-Liear Meh., 4, 39, [13] V. K. tokes, Phys. Fluids, 1966, 9, [14] B. traugha, K. Hutter, Pro. R. o., 1999, A 455, [15]. Chadrashekhar; Hydrodyami ad Hydromageti aility, Dover, New York, [16] uil, R. C. harma, R.. Chadel, J. Porous Media, 4, 7, [17] M. F. aslim, U. Narusawa, J. Heat mass ras., 1986, 18, 1-4. [18] O. V. revisa, A. Beja, It. J. Heat Mass ras., 1986, 9,