On the analytic solution for the steady drainage of magnetohydrodynamic (MHD) Sisko fluid film down a vertical belt

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1 Available a hp://pvamu.edu/aam Appl. Appl. Mah. IN: Vol. 10, Issue 1 (Jue 015), pp Applicaios ad Applied Mahemaics: A Ieraioal Joural (AAM) O he aalyic soluio for he seady draiage of mageohydrodyamic (MHD) isko fluid film dow a verical bel A. M. iddiqui 1, Hameed Ashraf, T. Haroo, ad A. Walai 1 Deparme of Mahemaics Pesylvaia ae Uiversiy York Campus, 1031 Edgecomb Aveue, York, PA 17403, UA ams5@psu.edu; Deparme of Mahemaics COMAT Isiue of Iformaio Techology Park Road Chak hehzad Islamabad, Pakisa hameedashraf09@yahoo.com; ahiraharoo@comsas.edu.pk; ahsawalai@gmail.com Received: February18, 014; Acceped: March 4, 015 Absrac This paper preses a aalyic sudy for he seady draiage of mageohydrodyamic (MHD) isko fluid film dow a verical bel. The fluid film is assumed o be elecrically coducig i he presece of a uiform rasverse mageic field. A aalyic soluio for he resulig o liear ordiary differeial equaio is obaied usig he Adomia decomposiio mehod. The effecs of various available parameers especially he Harma umber are observed o he velociy profile, shear sress ad voriciy vecor o ge a physical isigh of he problem. Furhermore, he shear hiig ad shear hickeig characerisics of he isko fluid are discussed. The physical quaiies discussed for he isko fluid film have also bee discussed for he Newoia fluid film ad compariso bewee hem made. Keywords: MHD; Thi film flow; isko fluid model; draiage; Adomia decomposiio mehod MC 010 No.: 76A0, 76D08 67

2 68 A. M. iddiqui e al. 1. Iroducio The pheomeo of hi film flow is ivolved i may aural ad idusrial problems, some of which o a rigid surface are drive by graviy. Alog wih he udersadig ad developmes i he fudameal issues relaig o such flows, a wide variey of idusrial, biological ad medical applicaios have beefied from scieific research. The recurrig feaure i his pheomeo is ha whe a fluid is disposed o a verical rigid objec, i adheres o i ad drais dow he objec uder codiios such ha he graviaioal ad viscous forces domiae he ierial forces due o which he fluid film formed i coac wih he free surface drai dow uder he acio of graviy oly. This ype of draiage is ermed as free draiage ad i cosiss of a exeio of fluid bouded by a objec ad wih a free surface (usually air). The fluid film hickess, is much shorer ha he legh of he objec so ha he flow akes place predomialy i he loger dimesios uder he acio of graviy oly. The flow velociy perpedicular o he plae is much smaller ha he mai flow velociy. Hece, we ca cosider i as a oe dimesioal flow, Jeffreys (1930), Gree (1936), Deso (1970), Raghurama (1971), Muso ad Youg (1994), O Brie ad chwarz (00), Mayers (005). The sudy of hi film flow of a elecrically coducig fluid i he presece of a rasverse applied mageic field has become he basis of umerous scieific ad egieerig applicaios. I rece imes, grea ieres has bee show by may researchers owards he sudy of mageohydrodyamic (MHD) hi film flows due o he effec of mageic fields o he fluid film hickess ad o he performace of may sysems ivolvig he pheomeo of hi liquid films of elecrically coducig fluids, Dua (1973), Hameed ad Ellahi (011), Alam e al. (01). We will cosider he paricular example of he moio of MHD isko fluid film dow a ifiiy, log verical bel which ca be exeded o ay frame of coaigs ad lubricaio process. I rece years, he flow of o-newoia fluids have gaied cosiderable imporace because of is promisig applicaios i various fields of egieerig, echology, bioscieces, paricularly i maerial processig, geophysics, chemical ad uclear idusry, food idusry ad polymer processig. Drillig mud, ooh pase, greases, polymer mels, ceme slurries, pais, blood, clay coaigs ec., are some examples of o-newoia fluid. I is difficul o sugges a model for such broad ad complex class of fluids ha ca sigle hadily describe all he properies of o-newoia fluids. Therefore, several cosiuive equaios have bee proposed o characerize ad predic he physical srucure ad behaviour of such fluids for differe maerials, Mayers (005). The flow characerisics of he above meioed fluids iclude shear-hiig, shear-hickeig, viscoplasiciy, viscoelasiciy ec., which are usually aalyzed wih he help of he power law fluid model. ome of hese flow characerisics were o fully described by he power law model. I view of his siuaio, isko proposed a cosiuive equaio (amed afer him isko fluid model ) which icludes he Newoia ad he power law fluids as special case o characerize hese ypes of flows. isko fluids are capable of describig shear hiig ad shear hickeig pheomea, ad have may impora ad well kow idusrial applicaios. I is he mos appropriae model for describig he flow of greases havig high viscosiies a low shear raes ad low viscosiies a high shear raes. Waerbore coaigs ad meallic auomoive basecoaig where polymeric suspesios are used, ceme slurries, lubricaig greases, mos psueodoplasic

3 AAM: Ier. J., Vol. 10, Issue 1 (Jue 015) 69 fluids ad drillig fluids are some of is applicaios i idusry, isko (1958), iddiqiui e al. (007), (009), (013), Mekheimer ad El Ko (01). The Adomia Decomposiio mehod has prove o be a valuable aleraive aalyic ool for solvig liear, oliear ordiary as well as parial differeial equaios, which occur i egieerig ad applied scieces, Adomia (1987). This mehod does o require ay small parameer, liearizaio, perurbaio ad oher similar resricios raher i provides a direc scheme. A advaage of his mehod is ha i provides a soluio i he form of a ifiie coverge series i which each compoe ca easily be deermied by recursio. Hosseii ad Nasabzadeh (006) have discussed he rapid covergece of he series soluio obaied by ADM. The firs several compued erms of he series soluio obaied, usually provides a good approximaio wih a high degree of accuracy whe compared wih oher echiques, Alam e al. (01). Wazwaz (1999), (001), (009) ad iddiqiui e al. (010), (01) have applied his mehod for differe ypes of liear ad oliear differeial equaios. As i our case, for he draiage of mageohydrodyamic (MHD) isko fluid film dow a verical bel, a exac soluio seems o be difficul, so a rucaed umber of erms will be used for he soluio purpose. The purpose of he prese paper is o aalyically sudy he seady draiage of mageohydrodyamic (MHD) isko fluid film dow a verical bel. We exed he work of iddiqiui e al. (013) o observe he effecs of he uiform rasverse mageic field. We shall solve his problem for he firs ime by ADM. We shall fid he physical expressios like he velociy profile, he volume flow rae, he average film velociy, he shear sress, he force exered by he fluid film o he bel surface ad he voriciy vecor. We shall also discuss he iflueces of he fluid behaviour idex, The sisko fluid parameer, okes umber ad he Harma umber o he velociy profile, he shear sress, he flow rae ad he voriciy vecor via ables ad o velociy profile via graphs. The res of he paper is orgaized as follows: he basic goverig equaios ad he cosiuive equaio for he icompressible isko fluid model are give i secio, i secio 3 he problem is formulaed, secio 4 coais he soluio of he problem usig he ADM ad icludes he volume flow rae, he average film velociy, he shear sress, he force exered by he fluid film o he bel surface ad he voriciy vecor, i secio 5 he iflueces of he available parameers ad dimesioless umbers are discussed hrough ables ad graphs ad fially cocludig remarks are give i secio 6.. Basic Equaios The coiuous flow behaviour of a elecrically coducig isko fluid film is govered by he Maxwells' equaios ad Ohm's law ogeher wih he equaio of coiuiy, momeum balace ad he cosiuive equaio for he icompressible isko fluid model: The Maxwells' equaios i simplified form are 1 E B B = 0J,. B = 0, E =,. E =, (1) c

4 70 A. M. iddiqui e al. where B is he oal mageic field, 0 is he mageic permeabiliy, J is he elecric curre desiy, E is he elecric field, is he permiiviy of free space (a elecric cosa) ad is he charge desiy. A elecrically coducig isko fluid film movig wih velociy V i he presece of a exeral mageic field B, i addiio o a elecric field E, is subjeced o a exra erm VB, ha accous for he curre iduced by he Lorez force o he charge carriers. Ohm's law professes ha he curre desiy is proporioal o he oal elecric field, mahemaically i geeralized form, i ca be wrie as J = E VB, () where is he elecric coduciviy. The mageic iducio equaio for he mageic flux B ca easily be derived from Maxwells' equaios (1) ad Ohm's law (). I suggess ha a applied mageic field be iduces a mageic field b i he medium ad oal field B is he sum of he applied ad iduced mageic fields, i.e., B= B0 b. If he mageic Reyolds umber ad he effec of he polarizaio of he ioized fluid, are egligibly small. The he so ha iduced mageic field b ad elecric field E respecively are assumed o be zero. Fially, i is assumed acily ha a mageic field wih a cosa mageic flux desiy B 0 is applied perpedicular o he velociy field. Hece, he fial form of he MHD body force caused by he exeral mageic field is JB = B V. (3) 0 The equaio of coiuiy ad momeum balace respecively are divv = 0, (4) DV = f p div JB, (5) D where V is he velociy vecor, is he cosa desiy, f is he body force per ui mass, p is he dyamic pressure, is he exra sress esor ad D is he maerial ime derivaive. The cosiuive equaio for icompressible isko fluid model, isko (1958), iddiqiui e al. (009), (013), Mekheimer ad El Ko (01) is give by D 1 1 = a b r A1 1, A (6) where a, b are maerial cosas ad is he fluid behaviour idex. If a = 0 he equaios for

5 AAM: Ier. J., Vol. 10, Issue 1 (Jue 015) 71 he power law fluid model ad if b = 0 for Newoia fluid are obaied ad A 1 is he Rivli- Erickse esor: = T A, =, 1 L L L V (7) superscrip T deoes he raspose ad V is he velociy gradie. 3. Problem Formulaio We cosider a seady, lamiar ad parallel flow of a elecrically coducig icompressible isko fluid flowig dow a ifiie verical bel. As a resul, a hi uiform fluid film of hickess is formed i coac wih he saioary air. We choose a xz -coordiae sysem such ha x -axis is ormal o he bel ad z -axis alog he bel i dowward direcio as show i he Fig. 1. We eglec he hermal effecs ad assume ha he fluid compleely wes he bel, he bel exeds o ifiiy i he y -direcio so ha d = 0, here is o applied pressure drivig he flow ad fluid film fall uder he acio of graviy oly. Therefore, he oly velociy compoe is i z -direcio. Accordigly we assume ha dy V = [0,0, w( x)], = ( x). (8) A uiform rasverse mageic field wih a cosa mageic flux desiy B 0 is applied perpedicular o he bel i he direcio of posiive x -axis. Therefore, he elecromageic force per ui volume becomes JB = [0,0, B w( x)]. (9) 0 Figure 1. The geomery of he problem

6 7 A. M. iddiqui e al. Profile (8) ideically saisfies he equaio of coiuiy (4). Equaio (6) upo usig Equaio (7) ad profile (8) yields he followig o zero compoes of exra sress esor: xz dw = a b dx 1 dw = dx zx. (10) The momeum balace (5) wih he help of Equaios (8)-(10) ad assumpios we made, will lead o d w b d dw B0 g w =. dx a dx dx a a (11) The boudary codiios associaed wih Equaio (11) are w = 0 a x = 0, ( o slip ), (1) = 0 a x =, ( free surface). (13) xy The free surface codiio (13) afer makig use of Equaio (10) akes he form dw = 0 a x =. dx (14) Cosiderig o-dimesioal parameers w = w, g x x =, io Equaio (11) ad boudary codiios (1) ad (14), afer omiig ` ', we obai d w d dw M w =, (15) dx dx dx w = 0 a x = 0, (16) where dw = 0 a x dx g = a g =1, (17)

7 AAM: Ier. J., Vol. 10, Issue 1 (Jue 015) 73 is he okes umber, = a( b ) g 1 is he isko fluid parameer ad M = B0 a is he Harma umber. Equaio (15) is a secod order o-liear ad ihomogeeous ordiary differeial equaio. I seems o be difficul o have exac soluio for (15). I he ex secio, we will apply ADM o solve (15) subjec o boudary codiios (16) ad (17). 4. oluio of he Problem Keepig i mid he mai seps of ADM (for referece see Adomia (1987), Wazwaz (009)), we rewrie (15) i operaor form as where L xx ( w) = L N( w) M w, (18) x d L xx = ad dx respecively, are wo ad oe fold liear operaors ad d L x =, dx dw N( w) = dx is a oliear erm. ice Operaig L xx is iverible, he he wo folds iverse operaor L 1 (*) = (*) dxdx. xx 1 L xx o boh sides of Equaio (18), we ge 1 Lxx is defied as 1 1 w( x) = A Bx x L xx LxN( w) M L xx w( x), (19) where A ad B are cosas o be deermied. We prese he decomposiio of ukow

8 74 A. M. iddiqui e al. w (x) by he decomposiio series w( x) = wk ( x), (0) ad he expasio of o liear erm N (w) by a ifiie series of Adomia polyomials k=0 N( w) = A, (1) where he compoes wk ( x), k 0 ad he Adomia polyomials Ak, k 0 ca easily be compued. ubsiuig he decomposiio series (0) ad ifiie series of Adomia polyomials (1) io (19), we have k=0 k 1 1 wk ( x) = A Bx x Lxx Lx Ak M Lxx wk ( x). k=0 k=0 k=0 () Boudary codiios (16) ad (17), afer makig use of decomposiio series (0), yield w0 (0) = w1 (0) = w (0) = = 0, ' ' ' w0 (1) = w1 (1) = w () = = 0. (3) Equaio () follows wih he followig recursive relaio ( ) =, 0 w x ABx x (4) 1 L A M L w ( x), k 0. 1 wk 1( x) = Lxx x k xx k (5) Equaio (4) wih he help of boudary codiios (3) leads o x w ( ) = x (6) ubsiuig he decomposiio series (0) io he ifiie series of Adomia polyomials (1), he usig biomial expasio, we acquire ' w x), A 0 = 0 ( (7) A ' 1 ' w ( x) w ( ), 1 = 0 1 x (8)

9 AAM: Ier. J., Vol. 10, Issue 1 (Jue 015) 75 where `dash over w repres he derivaive wih respec o x. By makig use of boudary codiios (3), Equaio (6) ad Adomia polyomials A k from Equaios (7) ad (8); he recursive relaio (5) gives he followig compoes of w (x) : w ( x) = 1 1 M 4 1 (1 x) 6 x 1x 1 (1 ), 1 x w ( x) = M 31 (1 x) 1 (1 x) 6 1 (9) (1 x) M x 1 (1 x) x 3 1 ( )( 3) M x 60x 15x 90x 1 (1 x), (30) 70 Iserig compoes w 0( x), w1 ( x) ad w ( x) from Equaios (6), (9) ad (30) io he decomposiio series (0), we ge he soluio of Equaio (15) of he followig form: 1 1 w( x) = 1 (1 x) 1 (1 x) 1 (1 x) 1 31 (1 x) (1 x) M 6x 1x 1 (1 x) (1 ) 1 1 (1 ) 3 x 1 (1 ) 3 x x x x ( )( 3) M x 60x 15x 90x 1 (1 x), 70 (31) which is he velociy profile for he seady draiage of mageohydrodyamic (MHD) isko fluid film dows a verical bel. Remark: We recover he soluios for (i) he isko fluid film [iddiqui e al. (013)] whe M 0, (ii) he MHD Newoia fluid film whe 0 ad (iii) he Newoia fluid film [Va Rossum (1958)] whe boh M ad 0 i (31). The o-dimesioal volume flow rae Q ad average film velociy w are defied by

10 76 A. M. iddiqui e al. which upo usig (31) leads o 1 Q = w = w( x) dx, (3) M 65 17M Q = w = M ( 6 8) 315 (33) The shear sress (10) exered by he bel o fluid film i dimesioless form is give by where XZ = dw dw, (34) dx dx = xz g. XZ a M XZ = (1 x) (1 x) 3(1 x) (1 x) (1 x) 6 M M (1 x) 3(1 x) (1 x) (1 x) M x 30x 5x 15 (1 x) (1 x) (1 x) 10 M M 3(1 x) (1 x) (1 x) 3(1 x) (1 x) M (1 x) M 3 5 (1 x) 10 x 30x 5x 15 (1 x) ]. (35) Equaio (35) a x = 0 gives he shear sress exered by he bel o fluid film a he bel surface: XZ x=0 = M 3 1 M 3 M 4 M 10 4 M 1 M M M (36) The force exered by he fluid film o he bel surface, i dimesioless form durig draiage is defied by

11 AAM: Ier. J., Vol. 10, Issue 1 (Jue 015) 77 F z 1 = XZ x= 0 0 dx, (37) which afer makig use of (36) yields F z = M 3 1 M 3 M 4 M 10 4 M 1 M M M , (38) which provides he force exered by he fluid film a he bel surface. I dimesioless form, he voriciy vecor is calculaed as M = (1 x) (1 x) 3(1 x) (1 x) (1 x) M M (1 x) 3(1 x) (1 x) (1 x) M x 30x 5x 15 (1 x), 10 j (39) where j is he ui vecor i he y -direcio. 5. Resuls ad Discussio I his secio, we shall observe he quaiaive effecs of fluid behaviour idex, isko fluid parameer, okes umber ad Harma umber M ivolved i he prese aalysis. For he umerical evaluaios of he aalyical resuls obaied i he previous secio, we developed umerical codes i Mahemaica ad displayed some of he impora resuls via ables (see Tables 1-1) ad graphs (see Figures -6). Tables 1-4 show he disribuio of velociy profile of he MHD isko fluid film, draiig from he verical bel. From hese ables, i is oed ha as we move wihi he domai i.e., x [0,1] he velociy icreases. We delieaed he decrease i velociy wih icrease i he Harma umber M. This is because of he fac ha he iroducio of rasverse mageic field damps he hi film of isko fluid, flowig dow he bel. Moreover, hese ables disclosed he slower draiage of isko fluid film whe compared wih he Newoia fluid film. The slower draiage of shear hiig fluid film as compared o he shear hickeig fluid film has also bee evide. This idicaes ha he rheology of he fluid has sigifica effecs o he hi film flow. Tables 5-8 are abulaed o observe he effec of Harma umber M o he shear sress. From hese ables we observed ha shear sress decreases as we move from bel surface o he free

12 78 A. M. iddiqui e al. surface. Wih icrease i he Harma umber M shear sress decreases. I is also observed ha shear sress experieced by he isko fluid film is greaer i amou ha ha of he Newoia fluid film. hear hickeig fluid film bears more shear sress as compared o he shear hiig fluid film which decreases wih icreasig hearma umber M. Disribuio of voriciy vecor is show i ables 9-1. From hese ables, we ifer ha as we proceed wihi he domai x [0,1], he voriciy effec decreases, i.e., maximum ear he bel ad miimum a he free surface. Negaive sig idicaes ha film has clockwise roaioal effecs. Decrease i voriciy effec wih icrease i he Harma umber M is see. Furhermore, we also oed ha he isko fluid film has lesser clockwise roaioal effecs ha ha of he Newoia fluid film. hear hickeig fluid film have more roaioal effecs as comparig o he shear hiig fluid film which is agai he evidece of he rehological effecs of isko fluid. Table 1. Velociy disribuio for he hi film flow of MHD isko fluid whe = 1. ad = 0.0 Newoia fluid film MHD Newoia fluid film M = 0.0 M = 0.4 x w (x) w (x) Table. Velociy disribuio for he hi film flow of MHD isko fluid whe = 0., = 1. ad M = 0 hear hiig fluid film hear hickeig fluid film = 0.5 = 1.5 x w (x) w (x)

13 AAM: Ier. J., Vol. 10, Issue 1 (Jue 015) 79 Table 3. Velociy disribuio for he hi film flow of MHD isko fluid whe = 0., = 1. ad M = 0. hear hiig fluid film hear hickeig fluid film = 0.5 = 1.5 x w (x) w (x) Table 4. Velociy disribuio for he hi film flow of MHD isko fluid whe = 0., = 1. ad M = 0.4 hear hiig fluid film hear hickeig fluid film = 0.5 = 1.5 x w (x) w (x) Table 5. hear sress disribuio for he hi film flow of MHD isko fluid whe = 0 ad = 1. Newoia fluid film MHD Newoia fluid film M = 0.0 M = 0.4 x XZ (x) XZ (x)

14 80 A. M. iddiqui e al. Table 6. hear sress disribuio for he hi film flow of MHD isko fluid whe = 0., = 1. ad M = 0 hear hiig fluid film hear hickeig fluid film = 0.5 = 1.5 x XZ (x) XZ (x) Table 7. hear sress disribuio for he hi film flow of MHD isko fluid whe = 0., = 1. ad M = 0. hear hiig fluid film hear hickeig fluid film = 0.5 = 1.5 x XZ (x) XZ (x) Table 8. hear sress disribuio for he hi film flow of MHD isko fluid whe = 0., = 1. ad M = 0.4 hear hiig fluid film hear hickeig fluid film = 0.5 = 1.5 x XZ (x) XZ (x)

15 AAM: Ier. J., Vol. 10, Issue 1 (Jue 015) 81 Table 9. Voriciy vecor disribuio for he hi film flow of MHD isko fluid whe = 0 ad = 1. Newoia fluid film MHD Newoia fluid film M = 0.0 M = 0.4 x (x) (x) Table 10. Voriciy vecor disribuio for he hi film flow of MHD isko fluid whe = 0., =1. ad M = 0 hear hiig fluid film hear hickeig fluid film = 0.5 = 1.5 x (x) (x) Table 11. Voriciy vecor disribuio for he hi film flow of MHD isko fluid whe = 0., =1. ad M = 0. hear hiig fluid film hear hickeig fluid film = 0.5 = 1.5 x (x) (x)

16 8 A. M. iddiqui e al. Table 1. Voriciy vecor disribuio for he hi film flow of MHD isko fluid whe = 0., = 1. ad M = 0.4 hear hiig fluid film hear hickeig fluid film = 0.5 = 1.5 x (x) (x) Figures -6 are showig he effecs of fluid behaviour idex, isko fluid parameer, okes umber ad Harma umber M o he velociy profile of he MHD isko fluid film. I is observed i Figure ha he velociy icreases wih he icrease i ad shear hickeig fluid film drais dow faser ha he shear hiig fluid film. From Figure 3 (draw o observe he effec of o velociy profile), we depiced ha he velociy decreases wih he icrease i. The compariso of Newoia fluid film ( = 0) ad isko fluid film ( > 0) shows oable icrease i magiude of he velociy from isko fluid film o Newoia fluid film. Figure 4 shows he effec of o velociy profile, icrease i velociy wih he icrease i ca be see i i. I order o observe he effec of M o velociy, Figures 5 ad 6 are ploed. I is evide ha he velociy decreases wih he icrease i M, i.e., fluid drais dow slowly i he presece of mageic field. Figure. The effec of variaio i fluid idex o velociy profile, for =1., = 0. ad M = 0.4

17 AAM: Ier. J., Vol. 10, Issue 1 (Jue 015) 83 Figure 3. The effec of variaio i isko fluid parameer o velociy profile, for = 1., = 0.4 (a) = 0.5 ad (b) = 1.5 M, Figure 4. The effec of variaio i okes umber = 0.5 ad (b) = 1.5 o velociy profile for = 0., M = 0.4, (a) Figure 5. The effec of variaio i Harma umber M o velociy profile for = 1. ad = 0., (a) = 0.5 ad (b) = 1.5

18 84 A. M. iddiqui e al. Figure 6. The effec of variaio i Harma umber M o velociy profile for =1. ad = 0 6. Cocludig Remarks I he prese heoreical work, we aalyically ivesigaed he flow behaviour durig he seady draiage of mageohydrodyamic (MHD) isko fluid film dow a verical bel. Modelig of he problem yielded o-liear ordiary differeial equaio which has bee aalyzed usig ADM. We foud ADM much easier o proceed ad cocluded ha i ca provide ay desired higher order soluio recursively wih efficiecy ad high accuracy. ADM does o require liearizaio, perurbaio or ay oher similar resricive assumpios which is he mai advaage of his mehod for obaiig he approximae soluios as is show i our prese work. We observed he effec of Harma umber M o velociy profile, shear sress ad voriciy vecor via ables ad he effecs of fluid behaviour idex, isko fluid parameer, okes umber ad Harma umber M o velociy profile via graphs. The resuls obaied idicae he followig fidigs: The velociy decreases moooically wih he icrease i Harma umber M. The decrease i velociy wih icreasig M discloses he fac ha rasverse mageic field damps he draiage of hi film flow of isko fluid dow he bel. The velociy icreases wih he icrease i ad ad decreases wih icreasig. hear sress exered by he bel o isko fluid film decreases wih icrease i he Harma umber M. Wih he icrease i he Harma umber M, he voriciy effec decreases. isko fluid film has clockwise roaioal effecs ad hese effecs are maximum ear he bel ad miimum a he free surface. These fidigs show ha presece of a uiform rasverse mageic field sressed he sysem.

19 AAM: Ier. J., Vol. 10, Issue 1 (Jue 015) 85 Ackowledgemes The auhors wish o express heir very sicere haks o he reviewers for heir valuable suggesios ad commes. REFERENCE Adomia, G. (1987). Aalyical soluios for ordiary ad parial differeial equaios, Ceer for Applied Mahemaics, Uiversiy of Georgia, Ahes, Georgia 3060, U.. A. Alam, M. K., iddiqui, A. M. ad Rahim, M.T. (01). Thi film flow of mageohydrodyamic (MHD) Johso-egelma fluid o verical surfaces usig he Adomia decomposiio mehod, Applied Mahemaics ad Compuaio, 19, pp Deso, C. D. (1970). The draiage of Newoia liquids eeraied o a verical surface, Id. Eg. Chem. Fudam., 9(3), pp Dua, D. K. (1973). Draiig of a powerlaw liquid dow a verical plae i he presece of a raseverse mageic field, Idia Joural of Theoreical Physics, Vol. 1(1), pp Gree, G. (1936). Viscous moio uder graviy i a liquid film adherig o a verical plae, Philos. Mag. er. 6 (148), pp Hameed, M. ad Ellahi, R. (011). Thi film flow of o-newoia MHD fluid o a verically movig bel, I. J. Numer. Meh. Fluids, 66, pp Hosseii, M. M. ad Nasabzadeh, H. (006). O he covergece of Adomia decomposiio mehod, Applied Mahemaics ad Compuaio, 18, pp Jeffreys, H. (1930). The draiig of a verical plae, Joh s College. Mekheimer, Kh.. ad El Ko, M. A. (01). Mahemaical modellig of useady flow of a isko fluid hrough a aisoropically apered elasic areries wih ime-varia overlappig seosis, Appl. Mah. Modell, doi: /j.apm Muso, B. R. ad Youg, D. F. (1994). Fudameals of Fluid Mechaics, Joh Wiley ad os, New York. Myers, T. G. (005). Applicaio of o-newoia models o hi film flow, Physical Review E 7, pp O Brie,. B. G. ad chwarz, L. W. (00). Theory ad modelig of hi film flows, Ecyclopedia of urface ad Colloid ciece, Copyrigh D by Marcel Dekker, Ic. Raghurama, J. (1971). Aalyical sudy of draiage of cerai o-newoia fluids from verical fla plaes, I. E (I) Joural CH, U. D. C, pp iddiqui, A. M., Ahmed, M. ad Ghori, Q. K. (007). Thi film flow of o-newoia fluids o a movig bel, Chaos, olios ad Fracals 33, pp iddiqui, A. M., Asari, A. R., Ahmad, A. ad Ahmad, N. (lae), (009). O Taylor's scrapig problem ad flow of isko fluid, Mahemaical Modelig ad Aalysis Vol. 14(4), pp iddiqui, A. M., Hameed, M., iddiqui, B. M. ad Ghori, Q. K. (010). Use of Adomia decomposiio mehod i he sudy of parallel plae flow of a hird grade fluid, Commu Noliear ci Numer imula 15, pp

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