Stability Analysis of Double-diffusive Convection in a Couple Stress Nanofluid

Transcription

1 DOI:.435/U.OVGU-8-3 ECHISCHE MECHIK, 38, 3, (8), Sumitted: anuary 8, 8 Staility nalysis o Doule-diusive Convection in a Coule Stress anoluid G. C. Rana,*, R. Chand, V. Sharma 3 Staility o doule-diusive convection in a horiontal layer o nanoluid is studied. he coule-stress luid model is emloyed to descrie the rheological ehavior o the nanoluid. Staility o nanoluid has een inluenced y the eatures o coule-stress luid, susended nanoarticles and examined under the consideration o momentum and thermal sli oundary conditions. y alying normal mode analysis method and linear staility theory, the disersion relation descriing the eect o various arameters is derived. We have assumed that the nanoarticle concentration lux is ero on the oundaries which neutralies the ossiility o oscillatory convection and only stationary convection occurs. he imact o the hysical arameters, like the coule stress arameter, solutal-rayleigh umer, thermo-nanoluid wis numer, thermo-solutal wis numer, Soret arameter and Duour arameter have also een oserved and comared with the ulished work. very good agreement is ound etween the resent aer and earlier ulished results. Introduction he technology o nanoluid ecomes a new challenge or the heat transer luid due to their higher thermal conductivity. he word nanoluid was irst roosed y Choi (995). he main imortant eature o nanoluid is the enhancement o thermal conductivity. uongiorno (6) roosed a mathematical model or nanoluid ased on the eects o rownian motion and thermohoresis o susended nanoarticles ater analying the eect o seven slis mechanism, he concluded that in the asence o turulent eddies, rownian diusion and thermohoresis are the dominant sli mechanisms. Stokes (966) roosed and ostulated the theory o coulestress luid. One o the alications o coule-stress luid is its use to the study o the mechanism o lurication o synovial joints, which has ecome the oject o scientiic research. human joint is a dynamically loaded earing which has articular cartilage as the earing and synovial luid as luricant. When luid ilm is generated, squeee ilm action is caale o roviding considerale rotection to the cartilage surace. he shoulder, knee, hi and ankle joints are the loaded-earing synovial joints o human ody and these joints have low-riction coeicient and negligile wear. ormal synovial luid is clear or yellowish and is a viscous non- ewtonian luid. ccording to the theory o Stokes, coule-stresses are ound to aear in noticeale magnitude in luids. Since the long chain hylauronic acid molecules are ound as additives in synovial luid. Walicki and Walicka (999) modeled synovial luid as coule-stress luid in human joints. Sharma and hakur () studied the coule-stress luid heated rom elow in hydromagnetics and ound that coule stress arameter has stailiing eect on the stationary convection. Doule-diusive convection in nanoluid is an imortant henomenon that has various alications in the ields o chemical science, ood rocessing, engineering and nuclear industries, geohysics, ioengineering and cancer theray, movement o iological luid, oceanograhy etc. he ase luid o the nanoluid is a coule-stress elastico-viscous luid. anoarticles do not aect the solute concentration. In nanoluids, the ase luid does not satisy the roerties o ewtonian luids in the real situations. he onset o convection in a horiontal layer heated rom elow (énard rolem) or a nanoluid was studied y Choi (995), ou (8), lloui et al. (), ield and Kunetsov (9, ), Wang and an (), Chand and Rana () and Rana et al. (4a, ). More realistic oundary conditions are used in this aer as discussed y ield and Kunetsov (4). We assume that there is no lux at the late and the nanoarticle lux value adjust accordingly. here is a need o changing the scale o dimensionless arameters. he asic solution o nanoarticle volume raction is changed. he oscillatory convection does not exist and only stationary convection occurs. Shivakumara et al. (3) studied the electrohydrodynamic instaility o a rotating coule stress luid and ound that the rotating luid layer ecomes destailiing in the resence o coule stress or all the oundary 46

2 condions considered. Rana (4) studied the thermal convection in coule-stress luid in hydromagnetics saturating a orous medium and ound that coule-stress arameter has stailiing eect on the system. Keeing in view o various alications o coule stress and nanoluid as mentioned aove, our main aim in the resent aer is to study the doule-diusive convection in a horiontal layer o coule-stress nanoluid. Mathematical Model and Governing Equations We consider an ininite horiontal layer o a coule stress elastico-viscous nanoluid o thickness d, ounded y the lanes and d as shown in Fig.. he layer is heated and soluted rom elow, which is acted uon y a gravity orce g (,, -g) aligned in the direction. he temerature,, concentration, C and the volumetric raction o nanoarticles, φ, at the lower (uer) oundary is assumed to take constant values, C and φ (, C and φ ), resectively. We know that keeing a constant volume raction o nanoarticles at the horiontal oundaries will e almost imossile in a realistic situation., C, φ Z d y Coule-stress nanoluid layer g g (,, -g) x, C, φ Z Fig.. Physical coniguration y alying oussinesq aroximation, the equations o conservation o mass and momentum or coule-stress [Kunetsov and ield (), Shivakumara (3), Chand and Rana (), Rana et al. (4a, )] nanoluid are q ()., dq ρ µ µ dt where ρ,, µ c, ( ) q g φρ ρ( φ) α ( ) α ( C C ) ( {( )}), c C µ and q(u, v, w), denote resectively, the density, viscosity, the material constant resonsile or coule stress roerty known as the coule stress viscosity, ressure, and Darcy velocity vector, where φ is the volume raction o nano articles, ρ is the density o nanoarticles, α is the coeicient o thermal exansion and α is analogous to solute concentration. C he equation o conservation o mass or the nanoarticles [uongiorno (6)] is dϕ D dt D φ. Heated rom elow t c, c, k and D e the luid seciic (at constant ressure), the heat caacity o the material constituting nanoarticles, the thermal conductivity and the diusivity o Duour tye, resectively. hen the thermal energy equation or a nanoluid is d D ρc k ρ c D φ.. ρcd C dt (4) he conservation equation or solute concentration [ield and Kunetsov ()] is () (3) 47

3 dc dt D C D. S (5) where D and D are resectively, the solute diusivity and diusivity o Soret tye. S ssuming the temerature to e constant and the thermohoretic nanoarticles lux to e ero at the oundaries [ield and Kunetsov (4)]. ow the oundary conditions are w w,, w, ϕ D, C C, D w, C C,, D ϕ D We introduce non-dimensional variales as x, y, u, v, w tκ d ( x, y,,), (u, v, w,) d, d κ t,, φ d μκ ( ) ' ( C C ) C C C ( ), ( ), at, (6) at d. (7) ( φ φ ), where k is the thermal diusivity o the luid. hereater droing the dashes ( ' ) or convenience. k ( ρc) Equations () - (8) in non-dimensional orm can e written as q (9)., Rs ( ) q Rm eˆ Ra eˆ Ceˆ Rnφ eˆ, q q. q Pr φ. φ φ, Ln Ln q. φ.. C, Ln Ln C. C φ, w ϕ w,,, C, w ϕ w,,, C, η () q () q (3) where we have dimensionless arameters as: κ D S κ μcκ Ln η D μd ( ) ρ φ ρ - φ Rm μκ D κ ( C C ) ( ) gd 3 Rn D κ ϕ (8) () at, (4) at. (5) 3 gρa d Ra μκ ( ρ) 3 ( ) gρα d ( C C ) C Rs ρ 3 ϕ gd D ( ) µκ D ϕ ( ) ( C C ) μd ( ρc) ( ρc) Here the arameter is a thermo-solutal wis numer, Ln is a thermo-nanoluid wis numer, η is the coule-stress arameter, Ra is the thermal Rayleigh umer, Rs is the solutal-rayleigh umer, Rm Density Rayleigh numer, Rn is the nanoarticle Rayleigh numer, is the modiied diusivity ratio, is the modiied article-density ratio, D is the diusion constant o nanoarticles, is the Duour arameter and is the Soret arameter. S ϕ 48

4 . asic Solutions he time-indeendent quiescent solution o equations () (5) with temerature, concentration and nanoarticle volume raction varying in the -direction only [ield and Kunetsov (, 4)], and Sheu () is (6) ( ), C C ( ), ( ), φ φ ( ). u v w, hereore, equations () (3) reduce to ( ) d Rs Rm RD d d φ ( ) C ( ) Rnφ ( ), (7) ( ) d ( ), ( ) dφ ( ) d ( ) d ( ) d C ( ) d C Ln ( ) d ( ),, Using oundary conditions (4) and (5), the solution o equation (8) is given y dφ ( ) d ( ), On sustituting this value in equation (9), we get d ( ) d C ( ), () On solving equations (8)-() y alying oundary conditions (4) and (5), we otain -, C - and φ ϕ. (3) It is oserved that the coule stress arameter has no eect on the asic solution. hese results are identical with the results otained y ield and Kunetsov (4).. Perturation Solutions o study the staility o the system, we suerimosed ininitesimal erturations on the asic state. We write '' '' '' '' '' ( u, v,w) q ( u,v,w),, C C C, φ ϕ φ,. q (4) Using equations given in (4) into equations (9) (5), lineariing the resulting equations y neglecting nonlinear terms that are roduct o rime quantities and droing the rimes ('') or convenience, we otained. q, (5) q Pr Rs ( ) q Ra eˆ Ceˆ Rnφ eˆ, η (6) (8) (9) () () w Ln φ Ln φ, (7) w Ln φ C. Ln (8) C w C. (9) 49

5 ϕ w,, C, at and at. (3) ote that as the arameter Rm is not involved in equations (5)-(3) it is just a measure o the asic static ressure gradient. he seven unknowns u, v, w,,, C and φ can e reduced to our y oerating equation (6) with eˆ curl curl and using the identity curl curl grad div together with equation (5) which yields Pr where w w Rs C 4 η w Ra H H Rn H φ, (3) H is the two-dimensional Lalace oerator on the horiontal lane, that is H. x y 3 ormal Modes nalysis Method Exress the disturances into normal modes o the orm [,,C, ] ( ), ( ), Γ( ), ( ) ex( ) w φ W Θ Φ irxisy t, (3) where r, s are the wave numers in the x and y direction, resectively, and is the growth rate o the disturances. Sustituting equation (3) into equations (3) and (7)-(3), we otain the ollowing oundary-value rolem Rs ( D ω ) η( D a ) ( D a ) W ω RaΘ ω Γ ω RnΦ, Pr W ( D ω ) Θ ( D ω ) Γ, (34) W D D D ω Θ DΦ ( D ω ) Γ, Ln Ln Ln (35) W ( D ω ) Θ ( D ω ) Φ, Ln Ln (36) w,, C, D DΘ where d ϕ at and at. (37) D and ω r s is the dimensionless horiontal wave numer. Considering solutions W, Θ, Γ and Ф o the orm W Wsin( π), Θ Θsin( π), Γ Γ sin( π), Φ Φsin( π). (38) Sustituting (38) into equations (33) (36) and integrating each equation rom to, we otain the ollowing matrix equations η Pr π 4 a ω Ra Ln Rs ω ω Rn W Θ Γ Ln Φ where is the total wave numer. he linear system (37) has a non-trivial solution i and only i, (33) (39) 5

6 Ra ω Rs η Pr [ - ( ω) ] 4 4 [( ) ] - Rn Ln ( ) ( ( ) Ln ) ( Ln )) Equation (38) is the disersion relation reresenting the eect o medium orosity, thermo-solutal wis numer, thermo-nanoluid wis numer, solutal Rayleigh umer, nanoarticle Rayleigh numer, kinematic viscoelasticity arameter, modiied diusivity ratio, Soret and Duour arameter on doule-diusive convection in a layer o coule stress nanoluid saturating a orous medium. 4 he Stationary Convection Due to the asence o oosing uoyancy orces, the oscillatory convection does not exist. So we consider only the case o stationary convection. For stationary convection, utting in equation (4) reduces it to Ra Rn 3 ( π ω ) η( π ω ) ( )( ) Rs( ) ω (4) [ Ln ( Ln )] Equation (4) reresents the thermal Rayleigh numer as a unction o the non-dimensional wave numer ω corresonding to the arameters F,,, Rs, Ln Rn,,. Since Pr vanishes with, thus, the stationary convection does not deend on the value o Prandtl numer Pr. Equation (4) is identical to that otained y ield and Kunetsov (), Chand and Rana () and Rana et al. (4a, ). lso in equation (37) the article increment arameter does not aear and the diusivity ratio arameter aears only in association with the nanoarticle Rayleigh numer Rn. his imlies that the nanoluid cross-diusion terms aroach to e dominated y the regular cross-diusion term. In the asence o the Duour and Soret arameters and equation (39) reduces to 3 ( π ω ) ( η( π ) Rs ( Ln ), Ra ω Rn ω which is identical with the result derived y Kunetsov and ield and Rana et al. In the asence o the stale solute gradient arameter Rs, equation (4) reduces to 3 ( π ω ) ( η( π ) ( Ln ), Ra ω Rn ω Equation (4) is identical with the results derived y Sheu (), Chand and Rana () and Rana et al. (4a, ). he critical cell sie at the onset o instaility is otained y minimiing Ra with resect to a. hus, the critical cell sie must satisy Ra ω ωω c, Equation (39) which gives ω c. (43) π.3.. (4) (4a) (4) 5 Results and Discussions he disersion relation (4) is analyed numerically and grahs have een lotted to deict the staility characteristics. ccording to the deinition o nanoarticle Rayleigh numer Rn, this corresonds to negative. In the ollowing discussion, negative values o Rn (indicates a value o Rn or heavy nanoarticles ( ρ > ρ) ottom heavy case) are resented. he variations o thermal Rayleigh numer Ra with the wave numer ω or dierent values o the coule stress arameter η., η.4 and η.6 is lotted in Fig. and it is noticed that the thermal Rayleigh numer Ra 5

7 increases with the increase o coule stress arameter. hus coule stress arameter stailies the stationary convection. η.6 η.4 η. Fig.. he variations o Rayleigh numer Ra with the wave numer ω or dierent values o the coule stress arameter η., η.4 and η.6 Rs 9 Rs 5 Rs ` Fig. 3. he variations o Rayleigh numer Ra with the wave numer a or dierent values o the solute concentration Rs, Rs 5 and Rs 9 In Fig. 3, the variations o thermal Rayleigh numer Ra with the wave numer ω or three dierent values o the solutal-rayleigh numer, namely, Rs, 5 and 9 is lotted and it is oserved that the thermal Rayleigh numer slightly increases with the increase in solutal Rayleigh numer so the solutal Rayleigh numer stailies the system slightly. In Fig. 4, the variations o thermal Rayleigh numer Ra with the wave numer a or three dierent values o the thermo-nanoluid wis numer, namely, Ln 3, 6 and 9 which shows that thermal Rayleigh numer increases with the increase in thermo-nanoluid wis numer. hus thermonanoluid wis numer has stailiing eect on the system. 5

8 Ln 9 Ln 6 Ln 3 Fig. 4. he variations o Rayleigh numer Ra with the wave numer a or dierent values o the thermonanoluid wis numer Ln 3, Ln 6 and Ln Fig. 5. he variations o Rayleigh numer Ra with the wave numer a or dierent values o the thermosolutal wis numer 5, 55 and 95 he variations o thermal Rayleigh numer Ra with the wave numer ω or three dierent values o the thermosolutal wis numer, namely, 5, 55 and 9 is lotted in Fig. 5 and it is noticed that thermal Rayleigh- Darcy numer increases slightly with the increase in thermosolutal wis numer so the thermosolutal wis numer has slight stailiing eect on the system. In Fig. 6, the variations o thermal Rayleigh numer Ra with the wave numer ω or three dierent values o the Soret arameter, namely,, 3 which shows that thermal Rayleigh numer increases with the increase in Soret arameter. hus Soret arameter has stailiing eect on the system. 53

9 3 Fig. 6. he variations o Rayleigh numer Ra with the wave numer a or dierent values o the Soret arameter, and Fig. 7. he variations o Rayleigh numer Ra with the wave numer a or dierent values o the Duour arameter 5, 45 and 85 he variations o thermal Rayleigh numer Ra with the wave numer ω or three dierent values o Duour arameter, namely 5, 45 and 85 is lotted in Fig. 7 and it is oserved that thermal Rayleigh numer increases with the increase in Duour arameter so the Duour arameter has stailiing eect on the onset o stationary convection in a layer o coule stress nanoluid. he system ecomes more stale when the values o Soret and Duour arameters are equal. he results otained in igures to 7 are in good agreement with the result otained y ield and kunetsov (, 4) Chand and Rana (), Rana et al. (4a, ), and Sheu (). 6 Conclusions he onset o doule-diusive convection in a layer o coule stress anoluid in a more realistic oundary conditions has een investigated which comrises the eects o thermohoresis and rownian motion. We have assumed that there is no lux at the oundary and the nanoarticle lux value adjust accordingly. It is ound that the coule stress arameter has no eect on the asic solution. he coule stress arameter, solutal Rayleigh umer, thermo-nanoluid wis numer, thermosolutal wis numer, Soret arameter and Duour arameter have stailiing eects on the stationary convection as shown in igures, 3, 4, 5, 6 and 7 resectively. Oscillatory convection does not exist under the more realistic oundary conditions. 54

10 cknowledgement uthors would like to thank the learned reeree or his valuale comments and suggestions or the imrovement o quality o the aer. Reerences lloui, Z. Vasseur, P. Reggio, M.: atural convection o nanoluids in a shallow cavity heated rom elow. Int.. o hermal Science, 5, (), uongiorno: Convective transort in nanoluids. SME. o Heat ranser, 8, (6), 4-5. R. Chand, G. C. Rana: Duour and Soret eects on the thermosolutal instaility o Rivlin-Ericksen elasticoviscous luid in orous medium. Zeitschrit ür aturorschung, 67a, (), S. Choi: Enhancing thermal conductivity o luids with nanoarticles. In: D.. Siginer and H. P. Wang (Eds), Develoments and lications o on-ewtonian Flows, SME FED, 3/MD, 66, (995), D.. ield,. V. Kunetsov: hermal instaility in a orous medium layer saturated y a nanoluid. Int.. Heat Mass rans., 5, (9), D.. ield,. V. Kunetsov: he onset o doule-diusive convection in a nanoluid layer. Int.. o Heat and Fluid Flow, 3, (), D.. ield,. V. Kunetsov: hermal instaility in a orous medium layer saturated y a nanoluid: revised mode. Int.. heat Mass ranser, 68, (4), -4. G. C. Rana: he onset o thermal convection in coule-stress luid in hydromagnetics saturating a orous medium. ulletin o the Polish cademy o sciences-echnical sciences, 6, (4), G. C. Rana, R.C. hakur and S. K. Kango: On the onset o thermosolutal instaility in a layer o an elasticoviscous nanoluid in orous medium. FME ransactions, 4, (4), -9. G. C. Rana, R. C. hakur and S. K. Kango: On the onset o doule-diusive convection in a layer o nanoluid under rotation saturating a orous medium. ournal o Porous Media, 7, (4), R. C. Sharma, K. D. hakur: Coule-stress luid heated rom elow in hydromagnetics. Cech.. Phys., 5, (), L.. Sheu: hermal instaility in a orous medium layer saturated with a viscoelastic nanoluid. rans Porous Med. 88, (), I. S. Shivakumara, M. kkanagamma, Chiu-On g: Electrohydrodynamic instaility o a rotating coule stress dielectric luid layer. Int.. Heat and Mass ranser, 6, (3), V. K. Stokes: Coule-stress in luids. Phys. Fluids, 9, (966), D. Y. ou: hermal instaility o nanoluids in natural convection. Int.. o Heat and Mass ranser, 5, (8), E. Walicki,. Walicka: Inertial eect in the squeee ilm o coule-stress luids in iological earings. Int.. l. Mech. Engg., 4, (999), S. Wang, W. an: Staility analysis o doule-diusive convection o Maxwell luid in a orous medium studied. Int.. o Heat and Fluid Flow, 33, (), ddress: Deartment o Mathematics, SCM Govt. College, Hamirur-775, Himachal Pradesh, IDI drgcrana5@gmail.com Deartment o Mathematics, Govt. College, Sugh-hatoli, Himachal Pradesh, IDI rameshnahan@yahoo.com 3 Deartment o Mathematics and Statistics, Himachal Pradesh University, Shimla-75, Himachal Pradesh, IDI veena_math_hu@yahoo.com 55