JMEH Vol. 47 No. June 6 pp 67-77 OI:.59/jcamech.6.5956 On the onset o triple-diusive convection in a layer o nanoluid Gian. Rana Ramesh hand Veena harma 3 hilasha harda 4 stract epartment o Mathematics Govt. ollege Nadaun-77 33 istrict Hamirpur Himachal Pradesh INI epartment o Mathematics Govt. ollege Nurpur-77 3 istrict Kangra Himachal Pradesh INI 3epartment o Mathematics and tatistics Himachal Pradesh University himla Himachal Pradesh INI 4epartment o Mathematics Govt. ollege m istrict Una Himachal Pradesh INI Received: 8 Mar. 6 ccepted: 4 May 6 On the onset o triple-diusive convection in a horiontal layer o nanoluid heated rom elow and salted rom aove and elow is studied oth analytically and numerically. The eects o thermophoresis and Brownian diusion parameters are also introduced through Buongiorno model in the governing equations. By using linear staility analysis ased on perturation theory and applying normal modes analysis method the dispersion relation accounting or the eect o various parameters is derived. The inluences o solute-rayleigh numer analogous solute-rayleigh numer thermo-nanoluid wis numer modiied diusivity ratio and nanoparticle Rayleigh numer on the staility o stationary convection are presented analytically and graphically. The solute Rayleigh numer and analogous solute Rayleigh numer have stailiing eects on the onset o stationary convection or oth top-heavy and ottom-heavy arrangements. The thermo-nanoluid wis numer and diusivity ratio have stailiing eects on the onset o stationary convection while nanoparticle Rayleigh numer has destailiing eect on the onset o stationary convection. The necessary conditions or the existence o oscillatory modes are also otained. very good agreement is ound etween the results o present paper and earlier pulished results. Keywords: onvection Triple-iusive Nanoluid Nanoparticles Rayleigh. Introduction oule-diusive convection is a natural phenomenon that has various applications in dierent areas such as geophysics soil sciences ood processing oil reservoir modeling oceanography limnology and engineering among others. oule-diusive orresponding uthor. Email ddress: drgcrana5@gmail.com convection is a mixing process o two luid components which diuse at dierent rates. Brakke [] explained a doule-diusive instaility that occurs when a solution o a slowly diusing protein is layered over a denser solution o more rapidly diusing sucrose. oule-diusive convection prolems related to dierent types o luids and geometric 67
Gian. Rana et al. conigurations have een extensively studied [-]. ll the aove researchers have considered only two luid component systems. However there are many situations in which more than two luid components involved. Examples o such multiple diusive convection luid systems include the solidiication o molten alloys geothermally heated lakes and sea water etc. Griiths [] Turner [3] Pearlstein et al. [4] and Lope [5] studied the triply diusive convection luid (where the density depends on three independently diusing agencies with dierent diusivities). These researchers ound that small concentrations o a third component with a smaller diusivity can have a signiicant eect upon the nature o diusive instailities and oscillatory and direct salt inger modes are simultaneously unstale under a wide range o conditions when the density gradients due to components with the greatest and smallest diusivity are o the same signs. ome undamental dierences etween the doule and triple diusive convection are noticed y these researchers. The presence o more than one chemical dissolved in luid mixtures is very oten requested or descriing natural phenomena (contaminant transport underground water low acid rain eects worming o stratosphere) (see Rionero [6]). Recently hand [7] studied Linear staility o triple-diusive convection in micropolar erromagnetic luid saturating porous medium while triple-diusive convection in Walters (model B ) luid with varying gravity ield saturating a porous medium studied y Kango et al. [8]. In recent years much research has ocused on the study o nanoluids with a view to applications in several industries such as the automotive pharmaceutical and energy supply industries. nanoluid is a colloidal suspension o nano sied particles. ommon luids such as water ethanol or engine oils are typically used as ase luids in nanoluids. mong the variety o nanoparticles that have een used in nanoluids it can e ound oxide ceramics such as l O 3 or uo nitride ceramics like ln or in and several metals such as l or u. hoi [9] was irst who coined the term nanoluid. Nanoluids are eing looked upon as great coolants due to their enhanced thermal conductivities and suspensions o nanoparticles are eing developed medical applications including cancer therapy. Buongiorno [] proposed that the asolute nanoparticle velocity can e viewed as the sum o the ase luid velocity and a relative slip velocity. Thus convection o nanoluids ased on Buongiorno s model has attracted great interest. considerale numer o doule-diusive convection prolems in a horiontal layer saturated y a nanoluid have also een numerically and analytically investigated [-7]. In this paper the study is extended to triple-diusive convection in a layer o nanoluid heated rom elow and salted rom aove and elow y salt and respectively. To the est o researchers knowledge this original study has not een pulished yet.. Mathematical Model and ormulation We consider an ininite horiontal layer o nanoluid o thickness d ounded y the planes and d heated rom elow and salted rom aove and elow y salt and respectively as shown in Figure. Each oundary wall is assumed to e impermeale and perectly thermal conducting. The layer is acted upon y a gravity orce g ( -g) aligned in the direction. The temperature T concentrations and the volumetric raction o nanoparticles at the lower (upper) oundary is assumed to take constant values T and (T and ) respectively. T Y T Z O g g(-g) Heated rom elow and soluted rom elow y salt and aove y salt Figure. Physical coniguration Z d Z Nanoluid Layer.. Governing Equations t p and q (u v w) denote respectively the density viscosity pressure and arcy velocity vector. Then the governing equations o conservation o mass and momentum or nanoluid (Boungiorno [] Nield and Kunetsov [] hand [7] Kango et al. [8] and Rana and hand [7]) in a triple-diusive convection are q () d q p q + dt () T T T p g X 68
Vol. 47 No. June 6 d where q stands or convection dt t derivative is the volume raction o nano particles p is the density o nano particles and is the density o ase luid is the uniorm temperature gradient T and uniorm solute gradients and we approximate the density o nanoluid y that o ase luid (i.e. ) (Boungiorno [] Nield and Kunetsov [] heu [] and Rana and hand [7]). The researchers approximate the density o the nanoluid y that o the ase luid that is to e considered (Boungiorno [] Nield and Kunetsov [] heu [] and Rana and hand [7]). The continuity equation or the nanoparticles (Buongiorno [] ) is T + q. B + T. (3) t T The thermal energy equation or a nanoluid is c. T T T q t T T c p B. T T. T (4) where (c) is heat capacity o luid (c ) p is heat capacity o nano particles. The conservation equation or solute concentrations (Kunetsov and Nield[]) are t t + q.. (5) + q.. (6) where and are the solute diusivities. The oundary conditions w T T w w at (7) w T T w w at. We introduce non-dimensional variales as (x* y* * ) x y d u v w t (u* v* w* ) d t* m d pd p* * T* T T T T * * (8) c There ater dropping the dashes ( * ) or convenience. Eqs. ()-(6) in non-dimensional orm can e written as. q (9) q q q p q Pr t R Rm e + RaT e Rn e e ˆ ˆ ˆ ˆ R ˆ e () N + q. + T () t T N + B q. T T+. T+ t NN B T. T t t + +.. () q (3).. q (4) where we have dimensionless parameters as: 69
Gian. Rana et al. Prandtl numer Pr ; Thermosolutal wis numer (5) ; 6) nalogous thermosolutal wis numer ; (7) Thermo-nanoluid wis numer Thermal Rayleigh Numer 3 g Td T T Ra ; olute Rayleigh Numer B g Rs () nalogous solute Rayleigh Numer g Rs R m Rn p ; ensity Rayleigh numer - gd Nanoparticle Rayleigh numer 3 p - gd Modiied diusivity ratio (4) N B N 3 ; ; T T Modiied particle- density ratio c p - ; c B ; (8) (9) ; () () (3) T T ; The dimensionless oundary conditions are (5) w T w w at w T w w at. (6) (7).... Basic olutions Following Nield and Kunetsov [] heu [] Rana et al. [5 6] and Rana and hand [7]. We assume a quiescent asic state that veriies uvw p p TT. (8) Thereore when the asic state deined in (8) is sustituted into Eqs. (9) (4) we get dp Rm+RT d Rs Rs Rn d d +N d T d N dt d B dt + + d d d NN dt B d d d d d (9) (3) (3) (3) (33) Using oundary conditions given in Eqs. (6) and (7) in the Eqs. (9) (33) the solution is given y T() - (34) () - () -. ccording to Buongiorno ( 6 ) or most nanoluid investigated so ar L n / is large o order 5-6 and since the nanoparticle raction 7
Vol. 47 No. June 6 decrement in not smaller than -3 which means L n is large. Typical value o N is not greater than aout. Then the exponents in equation (35) are small. Using oundary conditions given in Eqs. (6) and (7) in the Eqs. (9) (33) y expanding and retaining up to the irst order is negligile and so to a good approximation or the solution T() - () - () -. (35) These results are identical with the results otained y Nield and Kunetsov [] heu [] Rana et al. [5 6] and Rana and hand [7]..3. Perturation olutions To study the staility o the system the researchers superimposed ininitesimal perturations on the asic state so that q uvw + q uvw T +T + p p +p. (36) Using Eq. (36) into Eqs. (9) (4) lineariing the resulting equations y neglecting nonlinear terms that are product o prime quantities and dropping the primes ( ) or convenience the ollowing equations are otained as ollows: q (37) q p q + R ˆ T e Pr t Rs R e ˆ Rn ˆ ˆ e e R ˆ e N + w + t T (38) (39) T N B T w T+ t N N B T. t t w. w. Boundary conditions or Eqs. (37) - (4) are (4) (4) (4) w T w w (43) at w T w w (44) at. The parameter Rm is not involved in Eqs. (37)-(4) it is just a measure o the asic static pressure gradient. The eight unknown s u v w p T and can e reduced to ive y operating Eq. (38) with which yields e.curl curl 4 w w R HT Pr t (45) Rs Rs RnH H H where H +. is the two-dimensional x y Laplace operator on the horiontal plane and eˆ.curl q is the -component o vorticity. 3. Normal Modes nalysis Method The disturances into normal modes o the orm are expressed as wt (46) W ( ) exp ilx+imy+ t where l m are the wave numers in the x and y direction respectively and is the growth rate o the disturances. ustituting Eq. (46) into Eqs. (45) and (39)-(4) we otain the ollowing eigen value prolem 7
Gian. Rana et al. a W a W Pr Rs ar arn a Rs a W + W + a (47) (48) a (49) NB NNB W a (5) NB N W a a W W at and (5) (5) W W (53) at. d where and a l + m is the dimensionless d horiontal wave numer. 4. Linear taility nalysis and ispersion Relation onsidering solutions W and o the orm WWsin sin sin (54) sin sin. ustituting (54) into Eqs. (47) (5) and integrating each equation rom to the researchers otain the dispersion relation J Ra PrJ J. a Pr Rs Rs J (55) J J J NaJ Rn J Eq. (56) is the required dispersion relation accounting or the eect o Prandtl numer thermosolutal wis numer analogous thermo-solutal wis numer thermo-nanoluid wis numer solute Rayleigh Numer analogous solute Rayleigh Numer nanoparticle Rayleigh numer and modiied diusivity ratio on the onset o triple diusive convection in a layer o nanoluid. To examine the staility o the system the real part o is set to ero and we take ii in Eq. (56) then we otain 4 J J Pr 4 i J i Ra Rs 4 a Pr J i 4 4 J i J Ni 4 Rs 4 (56) J J i i i i where Pr J 4 J Rs 4 a Pr J i J J N Rn. J J Rs 4 i 4 i (57) Ra must e real as it is a physical quantity. Thus it ollows rom Eq. (57) that either i (exchange o stailities steady state) or i overstaility or oscillatory onset). 5. The tationary onvection For stationary convection putting i in equation (56) we otain Ra s a a Rs - N Rn. 3 Rs (58) 7
Vol. 47 No. June 6 Eq. (58) expresses the thermal Rayleigh numer as a unction o the dimensionless resultant wave numer a and the parameters Rs Rs Rn N. Eq. (58) is identical to that otained y Nield and Kunetsov [] heu [] Rana et al. [5 6] and Rana and hand [7]. lso in Eq. (58) the particle increment parameter N B does not appear and the diusivity ratio parameter N appears only in association with the nanoparticle Rayleigh numer Rn. This implies that the nanoluid cross-diusion terms approach to e dominated y the regular cross-diusion term. In the asence o the solute gradient parameter Rs Eq. (58) reduces to a Ra s a N Rn 3 Rs (59) Equation (59) is the same as the results derived y Nield and Kunetsov [] heu [] Rana et al. [5 6] and Rana and hand [7]. The critical cell sie at the onset o instaility is otained y minimiing Ra with respect to a. Thus the critical cell sie must satisy Ra a aa c Equation (58) which gives a c. (6) nd the corresponding critical thermal Rayleigh numer Ra c on the onset o stationary convection is given y 4 Ra Rs c 7 4 Rs - N Rn. (6) It is noted that i Rn is positive then Ra is minimied y a stationary convection. The result given in equation (6) is a good agreement with the result derived y heu [] and Rana and hand [7]. In order to study the eect o solute Rayleigh numer ( Rs ) analogous solute Rayleigh numer ( Rs ) thermo-nanoluid wis numer () diusivity ratio ( N ) and nanoparticle Rayleigh numer ( Rn ) on the stationary convection the ehaviour o and Ra s Rs Ra s Rs Ra s N Ra Ra s analytically are examined y the Rn researchers. From Eq. (58) we otain Ra s Rs (6) which is positive; thereore solute Rayleigh numer ( Rs ) inhiits the onset o triple-diusive stationary convection implying therey solute Rayleigh numer ( Rs ) has stailiing eect on the system which is an agreement with the results derived y Nield and Kunetsov [] heu [] Rana et al. [5 6] and Rana and hand [7]. Figure. Variation o stationary thermal Rayleigh numer Ra s with the wave numer a ordierent values o solute Rayleigh numer ( Rs ) s shown in Figure the stationary thermal Rayleigh numer Ra s is plotted against dimensionless wave numer a dierent values o solute Rayleigh numer ( Rs ). This shows that as Rs increases the stationary thermal Rayleigh numer ( Ra) s also increases. Thus solute Rayleigh numer ( Rs ) has stailiing eect on stationary convection which is in good agreement with the result otained analytically rom Eq. (6). It is evident rom Eq. (58) that Ra s Rs (63) which is positive; thereore analogous solute Rayleigh numer ( Rs ) inhiits the onset o triple-diusive stationary convection implying therey solute Rayleigh numer ( Rs ) has stailiing eect on the s 73
Gian. Rana et al. system which is an agreement with the results derived y hand [7] and Kango et al. [8]. Figure 3. Variation o stationary thermal Rayleigh numer Ra s with the wave numer a ordierent values o analogous solute Rayleigh numer ( Rs ) Figure 3 depicts that the stationary thermal Rayleigh numer Ra s is plotted against dimensionless wave numer a dierent values o solute Rayleigh numer ( Rs ). This shows that as (Ra) Rs increases the thermal Rayleigh numer s also increases. Thus solute Rayleigh numer ( Rs ) has stailiing eect on stationary convection which is in good agreement with the result otained analytically rom Eq. (63). From Eq. (58) we otain Ra s Rn (64) implying therey thermo-nanoluid wis numer ( ) inhiits the onset o triple-diusive stationary convection. Thus thermo-nanoluid wis numer ( ) has stailiing eect on the system i Rn < (i.e. ottom heavy arrangement) which is a good agreement with the results derived y Nield and Kunetsov [] heu [] Rana et al. [5 6] and Rana and hand [7]. s shown in Figure 4 the stationary thermal Rayleigh numer Ra s is plotted against dimensionless wave numer a dierent values o thermo-nanoluid wis numer (). This shows that as increases the thermal Rayleigh numer Ra s also increases or ottom-heavy arrangements. Thus o thermo-nanoluid wis numer () has stailiing eect on stationary convection which is in good agreement with the result otained analytically rom Eq. (64). Figure 4. Variation o stationary thermal Rayleigh numer Ra s with the wave numer a ordierent values o thermo-nanoluid wis numer () From Eq. (58) we otain N Ra s Rn (65) implying therey diusivity ratio ( N ) inhiits the onset o triple-diusive stationary convection. Thus diusivity ratio ( N ) has stailiing eect on the system i Rn < (i.e. ottom heavy arrangement) which is a good agreement with the results derived y Nield and Kunetsov [] heu [] Rana et al. [56] and Rana and hand[7] Figure 5. Variation o stationary thermal Rayleigh numer Ra s with the wave numer a ordierent values o diusivity ratio ( N ) In Figure 5 the stationary thermal Rayleigh numer Ra s is plotted against dimensionless wave numer a dierent values o diusivity ratio ( N ) as shown. This shows that as increases slightly the thermal Rayleigh numer Ra s also increases or ottom-heavy arrangements. Thus diusivity ratio ( N ) has low stailiing eect on stationary convection which is in good agreement with the result otained analytically rom Eq. (65). N 9 N 74
It is evident rom Eq. (58) that Ra Rn s N Vol. 47 No. June 6 (66) which is negative implying therey nanoparticle Rayleigh numer ( Rn ) hastens the triple-diusive convection implying therey nanoparticle Rayleigh numer ( Rn ) has destailiing eect on the system which is a good agreement with the results derived y Nield and Kunetsov [] heu [] Rana et al. [5 6] and Rana and hand [7]. Figure 6. Variation o stationary thermal Rayleigh numer Ra s with the wave numer a or dierent values o nanoparticle Rayleigh numer ( Rn ) Figure 6 shows the stationary thermal Rayleigh numer Ra s is plotted against dimensionless wave numer a or dierent values o nanoparticle Rayleigh numer ( Rn ). This shows that as increases the thermal Rayleigh numer Ra s decreases or ottom-heavy arrangements. Thus nanoparticle Rayleigh numer ( Rn ) has destailiing eect on stationary convection which is in good agreement with the result otained analytically rom Eq. (66). 6. OILLTORY ONVETION The oscillatory Rayleigh numer is given y 4 J J Pr R osc a Pr J J J J 4 J N. 4 J 4 4 Rs 4 4 Rs 3 a a a (67) a3 i i i (68) where 4 3 a a Pr a Pr Rs a Pr Rs N a a Pr Rn 3 a Pr a Pr Rs N a Pr Pr a Rs a a Rn a 3 a Pr Pr a Pr Rs N a Pr Rn a3 a. a Pr Rs 3 ince is o order N and so N. Thus Eq. (68) does not admit positive value o i i. Hence the necessary conditions or the occurrence o oscillatory convection are. 7. onclusions Triple-diusive convection in a layer o Nanoluid heated rom elow and soluted rom elow and aove is investigated y using a linear staility analysis method. The main conclusions are as ollows: The solute Rayleigh numer ( Rs ) and 75
Gian. Rana et al. analogous solute Rayleigh numer ( Rs ) have stailiing eects on the onset o stationary convection or oth top-heavy and ottom-heavy arrangements as shown in igures and 3 respectively. The thermo-nanoluid wis numer () and diusivity ratio ( N ) have stailiing eects on the onset o stationary convection or ottomheavy arrangements as shown in igures 4 and 5 respectively. Nanoparticle Rayleigh numer ( Rn ) has destailiing eect on the onset o stationary convection as shown in igure 6. Necessary conditions or the occurrence o oscillatory convection are otained and are given y. KNOWLEGEMENT: The authors are grateul to the learned reerees or their technical comments and valuale suggestions or the improvement o the paper. Reerences: Brakke MK (955) Zone electrophoresis o dyes proteins and viruses in density gradient columns o sucrose solutions. rch. Biochem. Biophys. 55:75-9 Buongiorno J (6) onvective transport in nanoluids. ME J. o Heat Transer 8:4-5 arpenter JR ommer T Wüest () taility o a doule-diusive interace in the diusive convection regime. J. Phys. Oceanogr. 4:84 854 arpenter JR ommer T Wüest () imulations o a doule-diusive interace in the diusive convection regime. Journal o Fluid Mechanics. 7:4-436 hand (3) Linear staility o triple-diusive convection in micropolar erromagnetic luid saturating porous medium. ppl. Math. Mech. 34:39-36 hand R Rana G () uour and oret eects on the thermosolutal instaility o Rivlin-Ericksen elastic-viscous luid in porous medium. Z. Naturorsch. 67a:685-69 hoi (995) Enhancing thermal conductivity o luids with nanoparticles In:.. iginer and H.P. Wang (Eds) evelopments and pplications o Non-Newtonian Flows ME FE 3/M. 66:99-5 Gaikwad N Biradar B (3) The onset o doulediusive convection in a Maxwell luid saturated porous layer. pecial Topics and Review in Porous Media. 4: 8-95 Griiths RW (979) The inluence o a third diusing component upon the onset o convection. J Fluid Mech. 9:659-67 Kango K Rana G hand R (3) Triple-diusive convection in Walters (model B) luid with varying gravity ield saturating a porous medium. tudia Geotechnica et Mechanica XXXV:45-56 Lope R Romero L Pearlstein J (99) Eect o rigid oundaries on the onset o convective instaility in a triply diusive luid layer. Physics o Fluids :897-9 Nield (967) The thermohaline Rayleigh-Jereys prolem. J Fluid Mech 9:545-558 Nield Kunetsov V () The onset o doulediusive convection in a nanoluid layer. Int. J. o Heat and Fluid Flow. 3:77-776 Pearlstein J Harris RM Terrones G (989) The onset o convective instaility in a triply diusive luid layer. J Fluid Mech. :443-465 Rana G () Thermosolutal instaility o compressile Rivlin-Ericksen elastico-viscous rotating luid permeated with suspended dust particles in porous medium. J. dvance Res. ppl. Math. 4:6-39 Rana G harma V () Hydromagnetic Thermosolutal instaility o Walters (Model B) rotating luid permeated with suspended particles in porous medium. Int. J. o Multiphysics. 5:35-338 Rana G Thaku R (3) oule-iusive convection in compressile Rivlin- Ericksen luid permeated with suspended particles in a Brinkman porous medium. Int. J. ppl. Math. Mech. 9:58-73 Rana G Thakur R Kango K (4) On the onset o thermosolutal instaility in a layer o an elastico-viscous nanoluid in porous medium. FME Transactions. 4:-9 Rana G Thakur R Kango K (4) On the onset o oule-iusive convection in a layer o nanoluid under rotation saturating a porous medium. J. o Porous Media. 7:657-667 Rana G hand R (5) taility analysis o doulediusive convection o Rivlin-Ericksen elastico-viscous nanoluid saturating a porous medium: a revised model. Forsch Ingenieurwes. 79:87 95 Rionero (3) Triple diusive convection in porous media. cta Mech. 4:447-458 heu LJ () Linear staility o convection in a viscoelastic nanoluid layer. World cademy o cience Engineering and Technology. 58:89-95 tern ME (96) The salt ountain and thermohaline convection. Tellus. :7-75 Tham L Naar R Pop I (3) Mixed convection oundary layer low past a horiontal circular cylinder emedded in a porous medium saturated y a nanoluid: Brinkman model. J. o Porous Media. 6:445-457 Turner J (968) The ehaviour o a stale salinity gradient heated rom elow. J Fluid Mech. 33:83- Turner J (985) Multicomponent convection. nn Rev Fluid Mech. 7:-44 76
Vol. 47 No. June 6 Yadav grawal G Bhargava R (3) On the onset o doule-diusive convection in a layer o saturated porous medium with thermal conductivity and viscosity variation. J. o Porous Media. 6:5-77