EFFECTS OF PARABOLIC AND INVERTED PARABOLIC SALINITY GRADIENTS ON DOUBLE DIFFUSIVE MARANGONI CONVECTION IN A COMPOSITE LAYER AN EXACT STUDY

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1 EFFECTS OF PRBOLIC ND INVERTED PRBOLIC SLINITY GRDIENTS ON DOUBLE DIFFUSIVE MRNGONI CONVECTION IN COMPOSITE LYER N EXCT STUDY 1 R. Suihr n B. Kol 1 Depren of Mheic, Governen Science College, Bnglore 6 1, Krnk, INDI. Depren of Mheic, BTLITM, Bnglore-6 99, Krnk, INDI. BSTRCT The Effec of Prbolic n Invere prbolic Sliniy grien on he one of Double Diffuive Mrngoni Convecion in wo-lyer ye, copriing n incopreible wo coponen flui ure porou lyer over which lie lyer of he e flui in he icrogrviy coniion, re inveige. The upper bounry of he flui lyer i free n he lower bounry of he porou lyer i rigi n boh he bounrie re inuling o he n. he inerfce, he velociy, her re, norl re, he, he flux, n flux re ue o be coninuou conucive for Drcy-Brinkn oel. The reuling Eigen vlue proble i olve excly. The Therl Mrngoni nuber for liner, prbolic n invere prbolic liniy profile re obine. The effec of ifferen phyicl preer on he one of ouble iffuive Mrngoni convecion re inveige for bove profile in eil. Keywor: Double iffuive convecion, Sliniy grien, Therl Mrngoni nuber, Drcy-Brinkn oel 1. INTRODUCTION In he generion of echno-vvy worl, he chip e up of pure cryl re of gre en leing o he enorou cope for he evoluion n explorion in he inury of cryl growh. There re ny eho of growing cryl n hee cn be clifie on he bi of eho of proucing uper urion Ioherl eho (conn eperure eho) ex., Hyroherl growh n Non ioherl eho (eperure vriion eho) In he ce of Ioherl eho, ny propery of he cryl h i eperure epenen will be uner beer conrol. Hyroherl growh i cryl growh fro queou oluion high eperure n preure. Even uner hyroherl coniion o of he eril grown hve very low olubiliie in olven. Thu o chieve reonble olubiliie, lrge quniie of oher eril re e which o no rec wih he eril being grown. Thee eril re clle inerlizer. The ppru coni of n uoclve coniing wo lyer. Nurien in he lower pr (nurien region) of he uoclve iolve in he flui (olven + Minerlizer + cryl eril), which i kep o hoer hn he upper pr (growh region) of he uoclve. The eperure ifference cue convecion fro he nurien region o he growh region n he upper flui i upercoole which rive he cryllizion. Since he flui h ore hn one iffuive coponen of ifferen oleculr iffuiviie (he, concenrion of inerlizer) he convecion i uli iffuive n he eril in he nurien chip cn be regre porou eiu. Thi eho of growing cryl excly iule he ouble (if one inerlizer i e), riple (if wo inerlizer re e) n uli coponen (if ore hn wo inerlizer re e) iffuive convecion in horizonl copoie lyer( flui lyer overlying flui ure porou lyer ). In he ce of non ioherl eho of growing cryl h i, when he eperure grien i ipoe on he ye, he in vnge i h he iffuion ph i uully horer, o reonble re re chieve wihou elbore conrol or ppru inveen. In hee iuion, inining unifor eperure n liniy grien i liiion n occurrence of non-unifor eperure n liniy grien i reliy. The uy of non unifor grien i no given uch enion n he non unifor liniy grien re rrely ouche. Though oe lierure i vilble on he uy of non unifor eperure grien, bu he non unifor liniy grien i crce. Recenly Subbr Prneh e l (1) hve inveige he effec of non unifor bic concenrion grien on he one of ouble iffuive convecion in icropolr flui lyer hee n lue fro below n coole fro bove. The Eigen vlue re obine uing Glerkin eho for free-free, rigi-free, rigi-rigi velociy bounry cobinion wih ioherl on pin-vnihing pereble Volue 3, Iue 1, Ocober 1 Pge 8

2 bounrie. One liner n five non liner concenrion profile re coniere n heir coprive influence on one i icue n reul re epice grphiclly. I i oberve h he flui lyer wih upene pricle hee n lue fro below i ore ble copre o he clicl flui lyer wihou upene pricle. Here we ke n ep o uy he effec of wo non unifor liniy grien (prbolic n invere prbolic profile) on he one of urfce enion riven ouble iffuive convecion in horizonl copoie lyer by Exc eho [6]. Here we give oe lierure on he effec of non unifor eperure grien on Mrngoni convecion in ingle horizonl flui n porou lyer eprely. Nnjunpp Rurih n Preep G Sihehwr () hve inveige he effec of non-unifor bic eperure grien on he one of Mrngoni convecion in horizonl lyer of Bouineq flui wih upene pricle. I i oberve h he flui lyer wih upene pricle hee fro below i ore ble copre o he clicl flui lyer wihou upene pricle. The proble h poible pplicion in icrogrviy iuion []. Shivkur e l () hve inveige he effec of ifferen bic eperure grien on he one of ferro convecion riven by cobine urfce enion n buoyncy force re uie. The reul inice h he biliy of Ryleigh-Bernr-Mrngoni Ferro convecion i ignificnly ffece by bic eperure grien n he echni for uppreing or ugening he e i icue in eil. I i hown h he reul obine uner he liiing coniion copre well wih he exiing one [1]. Melvin Johnon Fu e l (9) hve uie he effec of ix ifferen non-unifor bic e eperure grien on he one of Mrngoni convecion in horizonl icropolr flui lyer boune below by rigi ple n bove by non-eforble free urfce ubjece o conn he flux. They ue Ryleigh Riz echnique o olve he reuling Eigen vlue proble n icue he influence of he vriou preer on he one of Mrngoni convecion [3]. Sii Suzillin Puri Mohe I e l (9) hve inveige he effec of ix ifferen non-unifor bic eperure grien on he one of Mrngoni convecion in horizonl lyer wih free-lip boo hee fro below n coole fro bove. They olve he reuling he Eigen vlue proble uing ingle-er Glerkin expnion proceure n hve icue he effec of he vriou preer on he one of Mrngoni convecion []. Coing o he ingle porou lyer, Shivkur e l (1) hve inveige he effec of ifferen for of bic eperure grien on he crierion for he one of convecion in lyer of n incopreible couple re flui ure porou eiu i inveige. I i hown h he principle of exchnge of biliy i vli, n he Eigen vlue proble i olve nuericlly uing he Glerkin echnique. The prbolic n invere prbolic bic eperure profile hve he e effec on he one of convecion []. Suihr n Mnjunh (1) hve inveige he n exc uy of Mgneo-Mrngoni-convecion in wo lyer ye copriing n incopreible elecriclly conucing flui ure porou lyer over which lie lyer of he e flui in he preence of vericl gneic fiel in he icrogrviy coniion. The lower rigi urfce of he porou lyer n he upper free urfce re coniere o be inuling o eperure perurbion. he upper free urfce, he urfce enion effec epening on eperure re coniere. he inerfce, he norl n ngenil coponen of velociy, he n he flux re ue o be coninuou. The reuling Eigen vlue proble i olve excly for boh prbolic n invere prbolic eperure profile n nlyicl expreion of he Therl Mrngoni Nuber re obine. Effec of vriion of ifferen phyicl preer on he Therl Mrngoni Nuber for boh profile re copre [7].. FORMULTION OF THE PROBLEM We conier horizonl wo coponen flui ure, ioropic, prely pcke porou lyer of hickne unerlying wo coponen flui lyer of hickne, in he icrogrviy coniion. The lower urfce of he porou lyer i rigi n he upper urfce of he flui lyer i free wih he urfce enion effec epening on boh eperure n concenrion. Boh he bounrie re kep ifferen conn eperure n concenrion. Crein coorine ye i choen wih he origin he inerfce beween porou n flui lyer n he z xi, vericlly upwr. The coninuiy, oenu, energy n concenrion equion re, q (1) q q q P q () T q T T (3) C q C C () For he porou lyer, q () Volue 3, Iue 1, Ocober 1 Pge 9

3 q P q q K T q T T C q. C C q Where he ybol in he bove equion hve he following ening u, v, w ie, i he flui vicoiy, i he flui eniy, p f (6) (7) (8) i he velociy vecor, i he Cp i he rio of he cpciie, C p i he C pecific he, K i he perebiliy of he porou eiu, T i he eperure, i he herl iffuiviy of he flui, C i he concenrion, i he olue iffuiviy of he flui, i he poroiy, n he ubcrip n f refer o he porou eiu n he flui repecively. The bic ey e i ue o be he quiecen n we conier he oluion of he for, u, v, w, P, T, C,,, Pb z, Tb z, Cb z (9) in he flui lyer n in he porou lyer u, v, w, P, T, C,,, Pb z, Tb z, Cb ( z) (1) where he ubcrip b enoe he bic e. The eperure iribuion Tb z, Tb z, re foun o be Tu T z Tb z T in z (11) Tl T z Tb z T in z (1) Tu Tl T i he inerfce eperure. The concenrion iribuion Cb z, Cb z, re foun o be Cb C Cu h z in z z (13) Cb CL C h z in z z (1) Here h( z), h ( z ) re liniy grien in flui n porou lyer repecively uch h h z z n inerfce h z h z h z z.the ubcrip b enoe he bic e. he n noe h C C u l C i concenrion he inerfce. In orer o inveige he biliy of he bic oluion, infinieil iurbnce re inrouce in he for, q, P, T, C, Pb z, Tb z, Cb z q, P,, S (1) n q, P, T, C, Pb z, Tb z, Cb z q, P,, S (16) where he prie quniie re he perurbe one over heir equilibriu counerpr. Now Equion (1) n (16) re ubiue ino he Equion (1) o (8) n re linerize in he uul nner. Nex, he preure er i eliine fro () n (6) by king curl wice on hee wo equion n only he vericl coponen i reine. Volue 3, Iue 1, Ocober 1 Pge 6

4 The vrible re hen non-ienionlize uing,, velociy, eperure, n he concenrion in he flui lyer n, T T u n C Cu he uni of lengh, ie,,, T, l T Cl C he correponing chrceriic quniie in he porou lyer. Noe h he epre lengh cle re choen for he wo lyer o h ech lyer i of uni eph. In hi nner he eile flow fiel in boh he flui n porou lyer cn be clerly obine for ll he eph rio ˆ. The ienionle equion for he perurbe vrible re given by, For he flui lyer Pr lyer, Pr rio, 1 w w Pr (17) w (18) S w h( z) S (19) w ˆ w w Pr () w (1) S w h ( z ) S () i he Prnl nuber, K D i he Drcy nuber, ˆ i he Prnl nuber, i he iffuiviy rio in he flui lyer. For he porou i he vicoiy i he iffuiviy rio in he porou lyer. We ke he norl oe expnion n eek oluion for he epenen vrible in he flui n porou lyer ccoring o w W z n z f x, y e (3) S S z n w W z n z f x, y e () S S z Wih f f n f f, where n re he non-ienionl horizonl wve nuber, n n n re he frequencie. Since he ienionl horizonl wve nuber u be he e for he flui n porou lyer, we u hve n hence ˆ (17) o () n enoing he ifferenil operor n z z proble coniing of he following orinry ifferenil equion, i obine,. Subiuing Equion (3) n () ino he Equion by D n D repecively, n Eigen vlue Volue 3, Iue 1, Ocober 1 Pge 61

5 In z 1, n () Pr D n W (6) D D W ( ) (7) In 1 D n S W h z z n D ˆ 1 D W Pr (8) D n W (9) D n S W h ( z ) I i known h he principle of exchnge of inbiliie hol for Double Diffuive Mrngoni convecion in boh flui n porou lyer eprely for cerin choice of preer. Therefore, we ue h he principle of exchnge of inbiliie hol even for he copoie lyer. In oher wor, i i ue h he one of convecion i in he for of ey convecion n ccoringly, we ke. We ge, In z 1 D W ( z) D ( z) W ( z) D S( z) W ( z) h z ( ) ( ) D S ( z ) W ( z ) h z In z 1 n n (3) (31) (3) D ˆ 1 D W ( z ) (33) (3) D z W z (3) (36) Thu o olve he bove orinry ifferenil equion we nee 16 bounry coniion. 3. BOUNDRY CONDITIONS The boo bounry i ue o be rigi n inuling o eperure n concenrion o he bounry coniion z re w T S (37) w,,, z z z The upper bounry i ue o be free, inuling o eperure n concenrion o he pproprie bounry w T S coniion z re w, ˆ T S,, where z T C z z T T S S i he urfce enion, here T, S T S T T C C he inerfce z z ), he norl coponen of velociy, ngenil velociy, eperure, he flux, n (i.e.,, flux re coninuou n repecively yiel (following Niel (1977)), w w T T S S w w,, T T,, S S, (38) z z z z z z We noe h wo ore velociy coniion re require z. Since we hve ue he Drcy-Brinkn equion of oion for he flow hrough he porou eiu, he phyiclly feible bounry coniion on velociy re he following, he inerfce z z n Volue 3, Iue 1, Ocober 1 Pge 6

6 P w w P z z which will reuce o w w w 3 3 z z K z z z The oher pproprie velociy bounry coniion he inerfce z, z cn be, w w w w z () z ll he Sixeen bounry coniion (3) o () re non-ienionlie n re ubjece o Norl oe expnion n re given by W (1), D W (1) M (1) M S (1), D(1), DS (1) Where (39) ˆ ˆ ˆ ˆ ˆ TW () W (1), TDW () D W (1), T D W () ˆ D W () M ˆ () 3 () ˆ ˆ3 3 3 T D W DW D W D W D W () Tˆ (1), D() D (1), S() SS ˆ (1), DS() D S (1), w, D w (), D (), D S () (1) T Tu T i he herl Mrngoni nuber, v Tˆ T T / T T C Cu M S v Sˆ C C / C C, n ˆ / Mrngoni nuber, while, L U L U i he olue i he eph rio. We ee h ˆ / ˆ / ˆ T nˆ / ˆ / ˆ S becue he ey e he n fluxe re coninuou cro he inerfce. The Equion (31) o (36) re o be olve wih repec o he bove bounry coniion (1).. EXCT SOLUTION z, S z z, S z n hu The oluion of he Equion (31) n (3) re inepenen of, cn be olve n expreion for W n W cn be obine, W z C Coh z C zcoh z C Sinh( z) C zsinh( z) () 1 3 W z C Coh z C Sinh z C Coh z C Sinh z (3) where 1, n he expreion for W ( z ) n W ( z) re ˆ 1 1 ( ) 3 ( ) W z C Coh z zcoh z Sinh z zsinh z () W z C1 Coh z Sinh z 6 Coh z 7 Sinh z (),,,,,, re conn which re eerine uing correponing velociy he bounry where coniion (1),(1),(1),(1),(1),(1),(1) Volue 3, Iue 1, Ocober 1 Pge 63

7 ,, ˆ 1 ˆ T Sinh( ) Coh( ) 9 T ˆ,, Sinh( ) Coh( ) 9 T n ' i re ˆ ˆ3 T ˆ ˆ Sinh( ) Sinh( ) Sinh( ) Sinh( ) ˆ Coh( ) Coh( ) ˆ 3 3 Coh( ) Coh( ) ˆ ˆ T ( ) ( ) ˆ ˆ Sinh Sinh T1, ( Coh( ) ( )) ˆ ˆ Coh T T ˆ ˆ 1 ˆ Coh( ) Coh( ) T ˆ ˆ 1 ˆ 6 ( ) ( ) ˆ ˆ Sinh Sinh T Coh( ) Sinh( ) Sinh( ), Coh( ) Sinh( ) Sinh( ) T ˆ ˆ Coh Coh 1 Sinh Sinh, The Teperure iribuion re obine fro he Equion (3) n (3) by ubiuing expreion for W n W, re below 1 z z z C 3 1 1{ 8Coh z 9 Sinh z [ Sinh( z)( z 1 z ) Coh( z)( z 3 z )]} z z C1{ 1Coh z 11Sinh z Sinh( z ) z Coh( z ) Coh z Sinh z } 6 7 The conn 8, 9, 1, 11 re eerine uing eperure bounry coniion 1 (1) n re obine below. Where TCoh ˆ, Sinh ' i re Sinh Coh 1, ˆ TCoh Sinh Coh Sinh (1),(1), (1), Volue 3, Iue 1, Ocober 1 Pge 6

8 Coh 1 Sinh Sinh 3 Coh 3 Sinh Coh Coh Sinh Sinh Coh Coh 1 7 ˆ TSinh ˆ T Coh 3.1 Liner Sliniy Profile We conier liner liniy profile of he for h z h z 1 S z n S z re obine expreion for 6 7 Sinh Sinh Coh, ubiuing hi in eq.(33) n (36) he 1 3 z z 1 S z C1 1 Coh z 13 Sinh z Sinh z z z 1 Coh z z 3 z S z C Coh z Sinh z z z 6 7 Sinhz Cohz Coh z Sinh z p,,, re eerine uing liniy bounry coniion The conn re obine below. ˆ 6 Sinh 1 7 S Coh 9 p (1),(1),(1),(1) n 6 Coh Sinh 3, 1, 1 9 p 9 p Where, for i = 6 o 1 re ' i Sinh 1 Coh Coh 3 Sinh 3 Volue 3, Iue 1, Ocober 1 Pge 6

9 Sinh Coh Coh Sinh p Sinh Coh Coh Sinh 9 1 p 6 7 Sinh Coh SSinh ˆ Coh CohSinh SSinh ˆ Sinh SSinh ˆ Coh Coh 7 8Coh p p The Therl Mrngoni nuber for Liner Sliniy Profile i obine by he bounry coniion M M (1) Where 11 C 1 Coh 1 Sinh Coh Sinh 3 Coh Sinh C1 1Coh 13Sinh Sinh 1 Coh C { Coh Sinh [ Sinh( )( 1 ) Coh( )( 3 )]}. Prbolic Sliniy Profile We conier prbolic liniy profile of he for expreion for S z n S z re obine h z z, h z z, ubiuing hi in eq.(33) n (36), he 3 z 1 z 1 z z 3 z S z C Coh z Sinh z Sinh z 1 3 z 1 z z 3 z 3 z Cohz 3 1 z z z z S z C1 18 Coh z 19 Sinh z Sinh z Coh z p 6 z 7 7 z 6 Coh z Sinh z The conn 16, 17, 18, 19 re eerine uing he liniy bounry coniion 16 (1) n re obine below (1),(1),(1), Volue 3, Iue 1, Ocober 1 Pge 66

10 Where ˆ p7 p 7 p7 16 S Coh Sinh p3, 17 Sinh pcoh p6, p8 p8 p, p p8 p, for i = 1 o 8 re ' i 1 3 p p Sinh Coh Coh Sinh 3 1 p3 Sinh Coh p Coh Sinh p Coh Sinh Sinh p Coh Coh Sinh Sinh Coh p, p p p p p p7 p p6coh pcoh Coh p3ssinh SSinh Sinh ˆ ˆ p SSinh ˆ Coh Coh Sinh The Therl Mrngoni nuber for Prbolic Sliniy Profile i obine by he bounry coniion M M (1) Where 11 C 1 Coh 1 Sinh Coh Sinh 3 Coh Sinh 1 C1 16Coh 17 Sinh Volue 3, Iue 1, Ocober 1 Pge 67

11 Sinh Coh C { Coh Sinh [ Sinh( )( 1 ) Coh( )( 3 )]}.3 Invere Prbolic Sliniy Profile h z 1 z, h z 1 z, ubiuing hi in eq.(33) We conier prbolic liniy profile of he for n (36) he expreion for S z n S z re obine 3 z 1 z z 1 z 1 z 3 z z 3 z S z C Coh z Sinh z Sinh z 1 z 3 z z 3 z 3 z 1 z z 1 z Coh z 3 1 S z C Coh z Sinh z 1 3 z z z z z z Sinh z Coh z p 6 6 z z 6 Coh z Sinh z The conn, 1,, 3 re eerine uing liniy bounry coniion (1),(1),(1),(1) n re obine below. Where ˆ I S Coh Sinh I, Sinh I Coh I I I, I I7 I, for i 1o 7 re ' i I I Sinh Coh Coh Sinh I3 Sinh Coh p 7 6 Coh Sinh Volue 3, Iue 1, Ocober 1 Pge 68

12 I Coh Sinh p Coh 6 Sinh 7 Sinh 7 Coh I p I I I I Coh I Coh Coh I SSinh SSinh Sinh I Coh ˆ ˆ I SSinh ˆ Coh Coh Sinh The Therl Mrngoni nuber for invere Prbolic Sliniy Profile i obine by he bounry coniion M M (1) Where 11 C 1 Coh 1 Sinh Coh Sinh 3 Coh Sinh C1 Coh 1Sinh Sinh Coh C { Coh Sinh [ Sinh( )( 1 ) Coh( )( 3 )]}. INTERPRETTIONS The Therl Mrngoni nuber for he profile, nely, liner, prbolic n invere prbolic profile for ifferen preer re preene grphiclly funcion of eph rio ˆ by fixing he oher preer. The effec of he vriion of he preer like Horizonl Wve nuber, Vicoiy rio ˆ, Solue Mrngoni nuber M, Diffuiviy rio, n he Drcy nuber D on he herl Mrngoni nuber i iplye in figure,3,, n PRBOLIC PROFILE LINER PROFILE INVERTED PRBOLIC PROFILE Fig.1: The vriion of herl Mrngoni nuber for Liner, Prbolic n Invere prbolic profile wih repec o he eph rio. Figure1 how he vriion of herl Mrngoni nuber for ifferen profile wih repec o he Volue 3, Iue 1, Ocober 1 Pge 69

13 eph rio for fixe vlue of D 1, Sˆ 1, Tˆ 1, 1, 1,, M 1,. Here he herl Mrngoni nuber for he profile iffer only for ller vlue of eph rio. Grphiclly i i evien h he prbolic liniy profile i he o ble one n he invere liniy profile i he unble one, o by chooing he pproprie liniy profile one cn conrol he one of ouble iffuive Mrngoni convecion in copoie lyer in icrogrviy coniion. p 1 = 1 1 = 1 = = = = =3 8 6 = = Fig. () Fig (b) Fig. (c) Fig. The effec of horizonl wve nuber 3,, on he Therl Mrngoni nuber M The effec of ' ' horizonl wve nuber on he Therl Mrngoni nuber in liner, prbolic n invere prbolic profile re hown in Figure (), (b) n (c) repecively for fixe vlue of D 1, Sˆ 1, Tˆ 1, 1, p 1, M 1,. The line curve i for 3 n he ll oe line curve i for, he big oe curve i for. The curve for ll he profile re converging, inicing h for lrger vlue of eph rio, he correponing herl Mrngoni nuber coincie. The effec of horizonl wve nuber i e for ll he profile, h i he incree in he vlue of he horizonl wvenuber, he vlue of he herl Mrngoni nuber incree, o he one of ouble iffuive Mrngoni convecion i elye n hence he ye i bilize = 3 = = 3 = = = = 3 = = Fig. () Fig. (b) Fig. (c) Fig.3. The effec of ˆ 3,, on he Therl Mrngoni nuber M The effec of he vicoiy rio ˆ which i he rio of he effecive vicoiy of he porou eiu o he flui vicoiy re iplye in Figure (), (b) n (c) repecively for fixe vlue of D 1, Sˆ 1, Tˆ 1, 1, 1,, M 1. The line curve i for 3, he big oe curve i for n he ll oe line p curve i for. The curve for he prbolic profile re converging boh he en, (fig. 3(b)) h i, he effec. ˆ 1, o he effec of he vicoiy rio i liie o of he vicoiy rio i only for he vlue of eph rio hi rnge of eph rio. In hi rnge, for fixe vlue of eph rio, he incree in he vlue of vicoiy Volue 3, Iue 1, Ocober 1 Pge 7

14 rio ˆ incree he herl Mrngoni nuber. Where he effec of he vicoiy rio i oppoie o h for liner n invere prbolic liniy profile. The curve for he liner n invere prbolic profile re iverging n he effec of he vicoiy rio i lrger for lrger vlue of he eph rio. For fixe vlue of eph rio, he incree in he vlue of vicoiy rio ecree he herl Mrngoni nuber n o ebilize he ye n hence he one of he ouble iffuive Mrngoni convecion i erlier. 1 1 M = M = 1 M = M = 1 M = 1 M = 1 1 M = M = 1 M = Fig. () Fig. (b) Fig. (c) Fig.. The effec of M,1,1 on he Therl Mrngoni nuber M The effec of he Solue Mrngoni nuber M re iplye in Fig.,b n c for he liner, prbolic n invere prbolic liniy profile repecively for fixe vlue of D 1, Sˆ 1, Tˆ 1, 1, 1,,. The line curve i for M, he big oe curve i for M 1 n he ll oe line curve i for M 1 curve for he prbolic profile re converging boh he en, h i, he effec of he vicoiy rio i only for he rnge of vlue of eph rio nuber 1 ˆ 9. The increing vlue of p.the M incree he vlue of he Therl Mrngoni M i.e., o bilize he ye, o h, one of urfce riven ouble iffuive convecion i elye. Where he curve for he liner n invere prbolic profile, re converging for lrger vlue of eph rio n he incree in he olue Mrngoni nuber h no uch effec of he herl Mrngoni nuber for lrger vlue of he eph rio. For fixe vlue of eph rio ˆ, he incree in he vlue of of he olue Mrngoni nuber ecree he herl Mrngoni nuber, o he ouble iffuive Mrngoni convecion e in erlier n hence ebilize he ye. 1 1 =1 =. = =. =. = =1 =.7 = Fig. () Fig. (b) Fig. (c) Fig.. The effec of.,.7,1 on he Therl Mrngoni nuber M Volue 3, Iue 1, Ocober 1 Pge 71

15 The effec of he iffuiviy rio in flui lyer re iplye in Figure (), (b) n (c) for liner, prbolic n invere prbolic liniy profile repecively for fixe vlue of D 1, Sˆ 1, Tˆ 1, 1,, M 1,. p The line curve i for., he big oe curve i for.7 n he ll oe line curve i for 1. The curve for he prbolic profile re converging boh he en, h i, he effec of 1 ˆ 1, where he curve for he liner n invere he vicoiy rio i only for he vlue of eph rio prbolic profile re converging. The incree in he vlue of incree he vlue of he Therl Mrngoni nuber M for he liner n invere prbolic liniy profile, o bilize he ye, o he one of urfce riven ouble iffuive convecion i elye, where he e ecree he correponing herl Mrngoni nuber for he prbolic liniy profile. 3 D = 1 D = 9 D = D = 1 D = 9 D = 8 1 D =1 D = 9 D = The effec of he Drcy nuber Fig. () Fig.(b) Fig.(c) Fig.6. The effec of D 8,9,1 on he Therl Mrngoni nuber M D K, re iplye in Figure (), (b) n (c) for liner, prbolic n invere prbolic liniy profile repecively for fixe vlue of Sˆ 1, Tˆ 1, 1, 1,, M 1, The line curve i for D 8, he big oe curve i for D 9 n he ll p oe line curve i for D 1. The effec of The Drcy nuber i viible only for ll vlue of eph rio. The effec of Drcy nuber i e for ll he profile. For fixe vlue of eph rio, incree in he vlue of Drcy nuber incree he herl Mrngoni nuber for ll he profile, h i hi bilize he ye, o he one of urfce riven ouble iffuive convecion i elye, hi y be ue o he preence of econ iffuing coponen. 6. CONCLUSIONS 1. The prbolic liniy profile i he o ble one n he invere liniy profile i he unble one.. For vriou vriion of he preer, he effec of he prbolic liniy profile i oppoie o hoe of liner n invere prbolic liniy profile excep for h of Drcy nuber. 3. The vriion of Drcy nuber h e effec on he one on he ouble iffuive Mrngoni convecion for ll he profile.. The incree he vlue of horizonl wve nuber, he vicoiy rio n olue Mrngoni nuber n he ecree in he vlue of iffuiviy rio in he flui lyer bilize he ye for prbolic liniy profile, where he e ebilize he ye for he liner n invere prbolic liniy profile. cknowlegeen I expre y griue o Prof. N. Rurih n Prof. I.S. Shivkur, UGC-CS in Flui echnic, Bnglore Univeriy, Bnglore, for heir help uring he forulion of he proble. REFERENCES [1] I.S. Shivkur, N. Rurih n C.E. Nnjunpp Effec of non-unifor bic eperure grien on Ryleigh-Bern- Mrngoni convecion in ferroflui Journl of Mgnei n Mgneic Meril, 8,379-39,. Volue 3, Iue 1, Ocober 1 Pge 7

16 [] I.S.Shivkur, S.Surehkur n Devrju N Effec of non-unifor eperure grien on he one of convecion in couple re Flui ure porou eiu Journl f pplie Flui Mechnic, Vol., No.1, 9-, 1. [3] Melvin Johnon Fu, Norihn M. rifin, Moh Noor S n Rolin Moh Nzr Effec of Non-Unifor Teperure grien on Mrngoni Convecion in Micropolr Flui Europen Journl Scienific Reerch, ISSN 1-16X Vol. 8, No., 61-6,9. [] Nnjunpp Rurih n Preep G. Sihehwr Effec of non-unifor bic eperure grien on he one of Mrngoni convecion in flui wih upene pricle eropce cience echnology,,17-3,. [] Sii Suzillin Puri Mohe I, Norihn M. rifin, Moh Noor S n Rolin Moh Nzr Effec of Non-Unifor Teperure grien on Mrngoni Convecion wih Free Slip Coniion ericl Journl of Scienific Reerch ISSN 1-3X iue 1, 37-,9. [6] Subbr Prneh, run Kur Nrynpp Effec of Non-Unifor Bic Concenrion Grien on he One of Double-Diffuve Convecion in Micropolr Flui, Scienific Reerch n ceic publiher, Vol.3,No.,My 1. [7] Suihr R n Mnjunh N Effec of prbolic n invere prbolic eperure profile on gneo Mrngoni convecion in copoie lyer, Inernionl Journl curren reerch(ijcr), vol.6, Iue 3, pp 3-, Mrch 1. Volue 3, Iue 1, Ocober 1 Pge 73