Finite Element Solution of MHD Transient Flow past an Impulsively Started Infinite Horizontal Porous Plate in a Rotating Fluid with Hall Current

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1 Jurnal f Appled Flud Mechancs, Vl. 5,. 3, pp , 01. Avalable nlne at ISS , EISS Fnte Element Slutn f MHD Transent Flw past an Impulsvel Started Infnte Hrzntal Prus Plate n a Rtatng Flud wth Hall Current J. Anand Ra 1, R. Srnvasa Rau and S. Svaah 3 1 Department f Mathematcs, Unverst Cllege f Scence, Osmana Unverst, Hderabad, , Andhra Pradesh, Inda. Department f Basc Scence and Humantes, Padmasr Dr. B. V. Rau Insttute f Technlg, arsapur, Meda (Dt), 50313, Andhra Pradesh, Inda. 3 Department f Mathematcs, Gtam Unverst, Hderabad Campus, Hderabad, 5039, Andhra Pradesh, Inda. Crrespndng Authr emal: srvass999@gmal.cm (Receved Octber 1, 010; accepted Februar 0, 011) ABSTRACT The prblem f a transent three dmensnal MHD flw f an electrcall cnductng vscus ncmpressble rtatng flud past an mpulsvel started nfnte hrzntal prus plate tang nt accunt the Hall current s presented. It s assumed that the flud rtates wth a cnstant angular velct abut the nrmal t the plate and a unfrm magnetc feld appled alng the nrmal t the plate and drected nt the flud regn. The magnetc Renlds number s assumed t be s small that the nduced magnetc feld can be neglected. The nn-dmensnal equatns gvernng the flw are slved b Galern fnte element methd. The epressns fr the prmar and secndar velct felds are btaned n nn-dmensnal frm. The effects f the phscal parameters le M (Hartmann number), Ω (Rtatn parameter) and m (Hall parameter) n these felds are dscussed thrugh graphs and results are phscall nterpreted. Kewrds: MHD, Hall current, Rtatn, Transent, Hrzntal prus plate 1. ITRODUCTIO MHD s the scence f mtn f electrcall cnductng flud n presence f magnetc feld. There are numerus eamples f applcatn f MHD prncple. Engneers appl MHD prncple n fusn reactrs, dspersn f metals, metallurg, desgn f MHD pumps, MHD generatrs and MHD flw meters etc. The dnam and mtr s a classcal eample f MHD prncple. Gephscs encunters MHD characterstcs n the nteractn f cnductng flud and magnetc feld. MHD cnvectn prblems are als ver sgnfcant n felds f Stellar and Planetar magnetspheres, aernautcs and chemcal and electrcal Engneerng. The MHD prncple als fnds ts applcatn n Medcne and Blg. Applcatn n bmedcal engneerng ncludes cardac MRI, ECG etc. The prncple f MHD s als used n stablzng a flw aganst the transtn frm lamnar t turbulent flw. MHD, n ts present frm s due t the pneer cntrbutns f several ntable authrs. It was emphaszed b Cwlng (1957) that when the strength f the magnetc feld s suffcentl large, Ohm s law needs t be mdfed t nclude Hall current. The Hall effect s due merel t the sdewas magnetc frce n the drftng free charges. The electrc feld has t have a cmpnent transverse t the drectn f the current denst t balance ths frce. In man wrs n plasma phscs, the Hall effect s gnred. But f the strength f magnetc feld s hgh and the number denst f electrns s small, the Hall effect cannt be dsregarded as t has a sgnfcant effect n the flw pattern f an nzed gas. Hall effect results n a develpment f an addtnal ptental dfference between ppste surfaces f a cnductr fr whch a current s nduced perpendcular t bth the electrc and magnetc feld. Ths current s termed as Hall current. Mdel studes n the effect f Hall current n MHD cnvectn flws have been carred ut b man authrs due t applcatn f such studes n the prblems f MHD generatrs and Hall acceleratrs. Sme f them are Datta and Jana (1976) studed scllatr magnet hdrdnamc flw past a flat plate wth Hall Effect. Das et al. (1996) studed the radatn effects n flw past an mpulsvel started vertcal plate an eact slutns. Muthucumaraswam and Ganesan (1998) nvestgated the unstead flw past an mpulsvel started vertcal plate wth heat and mass transfer. Sngh (000) studed an scllatr hdrmagnetc cuette flw n a rtatng sstem.

2 J. Anand Ra et al. / JAFM, Vl. 5,. 3, pp , 01. Abeldahab and Elbarbar (001) nvestgated hall current effect magnethdrdnamcs free cnvectn flw past a sem nfnte vertcal plate wth mass transfer. Achara et al. (001) dscussed Hall Effect wth smultaneus thermal and mass dffusn n unstead hdrmagnetc flw near an accelerated vertcal plate. Knanu et al. (001) presented magnethdrdnamc free cnvectn heat and mass transfer f a heat generatng flud past an mpulsvel started nfnte vertcal prus plate wth hall current and radatn absrptn. Ganesan and Palan (00) studed the effects f mass transfer n the MHD flw past an mpulsvel started sthermal nclned plate. Tahar et al. (00) studed MHD flw ver a mvng plate n a rtatng flud wth magnetc feld, hall currents and free stream velct. Abdul Maleque and Abdul Sattar (005) studed the effects f varable prpertes and Hall current n stead MHD lamnar cnvectve flud flw due t a prus rtatng ds. Prasad et al. (006) reprted transent radatve hdrmagnetc free cnvectn flw past an mpulsvel started vertcal plate wth unfrm heat and mass flu. Prasad et al. (007) studed the radatn and mass transfer effects n tw-dmensnal flw past an mpulsvel started nfnte vertcal plate. Sharma et al. (007) studed the Hall Effect n MHD med cnvectve flw f a vscus ncmpressble flud past a vertcal prus plate mmersed n prus medum wth heat surce/sn. Chaudhar and Kumar Jha (008) dscussed the heat and mass transfer n elastc vscus flud past an mpulsvel started nfnte vertcal plate wth hall effect. Heat and Mass transfer. Palan and Abbas (009) studed the free cnvectn MHD flw wth thermal radatn frm an mpulsvel started vertcal plate. The rtatng flw f an electrcall cnductng flud n presence f a magnetc feld s encuntered n Gephscal flud dnamcs. It s als mprtant n the slar phscs dealng wth the sunspt develpment, the slar ccle and the structure f rtatng magnetc stars. It s well nwn that a number f astrnmcal bdes pssess flud nterrs and magnetc felds. Changes that taes place n the rate f rtatn, suggest the pssble mprtance f hdrmagnetc spn-up. Man authrs have studed ths prblem f spn-up n MHD under dfferent cnductns f whm the names f Sngh (000) and Tahar et al. (00) are wrth mentnng. Prasad et al. (011) dscussed fnte dfference analss f radatve free cnvectn flw past an mpulsvel started vertcal plate wth varable heat and mass flu. Vasu et al. (011) studed the radatn and mass transfer effects n transent free cnvectn flw f a dsspatve flud past sem-nfnte vertcal plate wth unfrm heat and mass flu. Due t mprtance f studng MHD flw prblems n rtatng flud, we have prpsed n the present paper t nvestgate the prblem f the unstead MHD flw f an electrcall cnductng vscus flud past a suddenl started nfnte hrzntal prus plate tang nt accunt the effect f Hall current. Here ur man bectve s t stud the effects f the magnetc feld, rtatn f the flud and Hall current n the velct felds and sn frctns at the plate.. MATHEMATICAL FORMULATIO The equatns gvernng the mtn f an ncmpressble vscus electrcall cnductng rtatng flud n presence f a magnetc feld are Equatn f cntnut:. q 0 (1) Mmentum equatn: q q q. q pj B t q Krchhff s law:. J 0 General Ohm s law: e e 1 J J B EqB pe B e e Gauss s law f magnetsm:. B 0 () (3) (4) Where s the velct vectr, the angular velct f the flud, the flud denst, p the pressure, the current denst, the magnetc nductn vectr, the ceffcent f vscst, the electrcal cnductvt, the tme, the strength f the appled magnetc feld, the electrn frequenc, the electrn cllsn tme, e the electrn charge, the number denst f electrn, the electrn pressure, the electrc feld and the ther smbls have ther usual meanngs. We nw cnsder an unstead flw f an ncmpressble vscus electrcall cnductng rtatng flud past a suddenl started nfnte hrzntal prus plate wth cnstant suctn n presence f a unfrm transverse magnetc feld tang nt accunt the effect f Hall current. Our nvestgatn s restrcted t the fllwng assumptns. ) All the flud prpertes are cnstants and the buanc frce has n effect n the flw. ) The plate s electrcall nn-cnductng. (5) ) The flud s rtatng wth angular velct abut the nrmal t the plate. v) The magnetc Renlds number s s small that the nduced magnetc feld can be neglected. v) s cnstant v) 0 v) The flud awa frm the plate s undsturbed due t rtatn Intall the plate was at rest and the flud was rtatng abut the nrmal t the plate. At tme 0, the plate s suddenl mved n ts wn plane wth velct and whch s thereafter mantaned cnstant. 106

3 J. Anand Ra et al. / JAFM, Vl. 5,. 3, pp , 01. We ntrduce a crdnate sstem (,,, ) wth X- as hrzntall n the drectn f the plate velct, Y-as hrzntall n the drectn f the secndar velct and Z-as alng the nrmal t the plate whch s the as f rtatn. Let be the flud velct, be the current denst at the pnt,,, and be the appled magnetc feld,,, beng the unt vectrs alng X-as, Y-as and Z-as respectvel. As the plate s nfnte n X-drectn and Y-drectn, therefre all the quanttes ecept pssbl the pressure are ndependent f and. w The Eq. (1) gves 0 (6) z whch s trvall satsfed b ww (7) w beng the suctn velct. Therefre the velct vectr s gven b ˆ ˆ ˆ q u v w The Eq. (5) s satsfed b B B ˆ J The Eq. (3) reduces t z 0 whch shws that z J z 0 (10) (as the plate s electrcall nn-cnductng) Hence the current denst s gven b ˆ J J ˆ J (11) Under the assumptns (v) and (v), the Eq. (4) taes the frm: m J J B qb (1) B (8) (9) Where s the Hall parameter. The Eqs. (8), (9), (11) and (1) elds, B J mu v 1 m B J mv u 1 m (13) (14) Wth fregng assumptns and under the usual bundar laer and Bussnesq s apprmatn the Eq. () reduces t u u u B mv u w v (15) t z z 1 m v v v B mu v t z z 1 m w v (16) Where s the cnstant suctn velct and s the nematc vscst. The relevant ntal and bundar cndtns are u 0, v 0 fr u U v at z, 0 0 t u 0, v 0 at z t 0 z (17) 0 (18) We ntrduce the fllwng nn-dmensnal varables and parameters. wz wt u v z, t, u, v, M v U U B w ( Hartmann number), The nn-dmensnal frm f the Eqs. (15) and (16) are u u u M v mvu (19) t z z 1m v v v M u muv (0) t z z 1m Subect t the ntal and bundar cndtns u 0, v 0 fr t 0 z (1) u 1, v 0 at z 0 t 0 u 0, v 0 at z () 3. METHOD OF SOLUTIO B applng Galern fnte element methd fr Eq. (19) ver the element (e), z z z s: z u u u ( et ) z t z dz 0 z Au P (3) M Where A=, P = Amv v ; 1 m Integratng the frst term n Eq. (3) b parts ne btans z ( et ) u z z u z z ( et ) z u u dz z ( et ) t z Au P eglectng the frst term n Eq. (4), ne gets: z z ( et ) u z z dz 0 ( et ) u u Au P t z 0 (4) Let u be the lnear pecewse apprmatn slutn ver the element (e) 107

4 J. Anand Ra et al. / JAFM, Vl. 5,. 3, pp , 01. where, u u and z z z z, are the bass functns. One z z z z btans: z z u dz u dz u z u z A z z z u dz u u dz u z z P dz z Smplfng we get u 1 1 u l 1 1u 6 1 u 1 1 1u A 1u l 1 1u 6 1 u P 1 1 where prme and dt dentes dfferentatn w.r.t z and tme t respectvel. Assemblng the element equatns fr tw cnsecutve elements z 1 z z] and z z z 1 fllwng s btaned: u u u () e u u l u u u () e l u 1 1 0u 1 1 A P u u 1 1 T (5) w put rw crrespndng t the nde t zer, frm Eq. (5) the dfference schemes wth l hs: 1 1 u 1 u u 1 u 1 4u u 1 h 6 1 u 1 u 1 h (6) A u 1 4 u u 1 P 6 Applng the trapezdal rule, fllwng sstem f equatns n Cran-chlsn methd are btaned: n1 n1 n1 Au 1 1 Au Au 3 1 (7) n n n * A u Au Au P Applng the same prcedure t the Eq. (0) then the fllwng equatn s btaned: n1 n1 n1 Bv 1 1 Bv Bv 3 1 n n n * Bv Bv Bv Q (8) where A 1 = + A + 3rh 6r; A = 4A + 1r + 8; A 3 = + A 3rh - 6r; A 4 = - A 3rh + 6r; A 5 = 8 4A 1r; A 6 = - A + 3rh + 6r; B 1 = - A + 3rh 6r; B = 8 + 1r - 4A; B 3 = - Ah 3rh - 6r; B 4 = + A 3rh + 6r; B 5 = 8 + 4A 1r; B 6 = + A + 3rh + 6r; * P 1Amv 1 v ; * Q 1Amu 1 u ; Here r= and h, are mesh szes alng -drectn h and tme-drectn respectvel. Inde refers t space and refers t the tme. In the Eqs. (7) and (8) tang = 1(1) n and usng bundar cndtns (1) and (), then the fllwng sstem f equatns are btaned: AX B 1(1) (5) where A s are matrces f rder n and X, B s are clumn matrces havng n-cmpnents. The slutns f abve sstem f equatns are btaned b usng Thmas algrthm fr prmar velct and secndar velct. Als, numercal slutns fr these equatns are btaned b C prgramme. In rder t prve the cnvergence and stablt f Galern fnte element methd, the same C prgramme was run wth smaller values f h and and n sgnfcant change was bserved n the values f u and v. Hence the Galern fnte element methd s stable and cnvergent. 4. SHEAR STRESS The sn frctn at the plate n the drectn f the prmar velct s gven b u z z 0 (6) 108

5 J. Anand Ra et al. / JAFM, Vl. 5,. 3, pp , 01. The sn frctn at the plate n the drectn f the secndar velct s gven b v z z 0 5. RESULTS AD DISCUSSIOS (7) In rder t stud the effect f magnetc feld fr flw cnfguratn shwn n Fg. 1, Hall current and rtatn f the flud, we have carred ut numercal calculatns fr the dmensnless prmar velct u, secndar velct v and sn frctns and at the plate due t the prmar and secndar velct felds respectvel fr dfferent values f the rtatn parameter Ω, Hartmann number M and Hall parameter m eepng the value f tme t fed at t = 1. Fgures, 3 and 4 ehbt the varatn f the prmar velct u versus z under the nfluence f the rtatn parameter Ω, Hartmann number M and the Hall parameter m. These three fgures shw that the prmar velct u falls when Ω and M are ncreased, t rses due t ncreasng value f the Hall parameter m. That s the prmar mtn s retarded under the effects f the transverse magnetc feld and the rtatn f the flud where as ths mtn s accelerated under the Hall Effect. Ths phenmenn s clearl supprted b the phscal realt. The same fgures further establsh the fact that the prmar velct asmpttcall decreases frm mamum value u = 1 t ts mnmum value u = 0 as z ncreases. Fg. 1. Flw cnfguratn Fg. 4. The prmar velct u versus z fr M = 1.0, Ω = 0.4, t = 1 Fg.. The prmar velct u versus z fr M = 1.0, m = 0.5, t = 1 Fg. 5. The secndar velct v versus z fr M = 1.0, m = 0.5, t = 1 Fg. 3. The prmar velct u versus z fr Ω = 0.4, m = 0.5, t = 1 Fg. 6. The secndar velct v versus z fr Ω = 0.4, m = 0.5, t = 1 109

6 J. Anand Ra et al. / JAFM, Vl. 5,. 3, pp , 01. The change f behavur f the secndar velct v aganst z due t varatns f Ω, M and m s dsplaed n Fgs. 5, 6 and 7. It s nferred frm these fgures that are velct feld frst ncreases n a ver thn laer adacent t the plate and after ths laer t asmpttcall decreases t ts zer value as z ncreases. It s als mared frm these fgures that an ncrease n the value f the parameters Ω, M and m results n a stead grwth n v. n ther wrds t ma be stated that the secndar mtn s accelerated under the effects f Hall current and rtatn and due t the applcatn f the transverse magnetc feld. The prfles fr sn-frctn due t prmar velct under the effects f Ω, M and m are presented n the Fgs. 8 and Fg. 9. We see frm these fgures that when the strength f the appled magnetc feld and the angular velct f rtatn f the flud are ncreased the sn-frctn ncreases and falls due t Hall effect. Frm ths bservatn we ma nterpret that the vscus drag n the plate due t prmar mtn s reduced under Hall Effect, but ths frctnal frce ncreases under the effects f the magnetc feld and rtatn. Fg. 7. The secndar velct v versus z fr M = 1.0, Ω = 0.4, t = 1 Fg. 10. The sn frctn τ versus m fr M = 1.0, t = 1.0. Fg. 8. The sn frctn τ versus m fr M = 1.0, t = Fg. 11. The sn frctn τ versus m fr Ω = 0.4, M = 1.0, t = 1.0 u Chaudhar (008) Present Result Fg. 9. The sn frctn τ versus m fr Ω = 0.4, t = z Fg. 1. Cmparsn f prmar velct u versus z fr m = 0.5, M = 1.0, Ω = 0.4 and t =

7 J. Anand Ra et al. / JAFM, Vl. 5,. 3, pp , 01. Fgures 10 and 11 demnstrate the varatn f snfrctn due t the secndar velct under the effects f Ω, M and m. It s seen that rses under the effects f M and Ω. These fgures further predct that fr small and mderate values f m, rses as m ncreases but fr large values f m ths behavur taes reverse trend. In rder t ascertan the accurac f the Galern fnte element methd, the present results are cmpared wth the prevus results f Chaudhar (008) fr M = 1.0, Ω = 0.4, t = 1 n Fg. 1. The are fund t be n an ecellent agreement. 6. COCLUSIO Our results f nvestgatn ma be summarzed t the fllwng mprtant cnclusns. 1. The prmar mtn s retarded under the effects f transverse magnetc feld and the rtatn f the flud where as ths mtn s accelerated under the Hall Effect.. The secndar mtn s accelerated under the effects f the Hall current and rtatn and due t the applcatn f the transverse magnetc feld. 3. The vscus drag due t prmar mtn s reduced under Hall Effect, but ths frctnal frce ncreases under the effects f the magnetc feld and rtatn. 4. The sn-frctn due t secndar velct rses under the effects f the Hartmann number M and the angular velct f rtatn Ω. REFERECES Abdul Maleque, Kh. and Md. Abdur Sattar (005). The Effects f Varable prpertes and Hall current n stead MHD lamnar cnvectve flud flw due t a prus rtatng ds. Int. Jurnal f Heat and Mass Transfer 48, Abeldahab, E.M. and E.M.E. Elbarbar (001). Hall current effect magnet hdrdnamcs free cnvectn flw past a sem nfnte vertcal plate wth mass transfer, Int. J. Engg. Sc. 39, Achara, M., G.C. Dash and L.P Sngh (001). Hall effect wth smultaneus thermal and mass dffusn n unstead hdrmagnetc flw near an accelerated vertcal plate. Ind. J. Phscs. B. 75B, Chaudhar R. C. and A. Kumar Jha (008). Heat and mass transfer n elastc vscus flud past an mpulsvel started nfnte vertcal plate wth Hall Effect. Latn Amercan Appled Research 38, Cwlng, T.G. (1957). Magnet Hdrdnamcs. Wle Inter Scence, ew Yr. Crammer, K. P. and S.L. Pa (1978). Magnet-flud Dnamcs fr Engneers and Appled Phscs. Mc- Graw Hll b C ew Yr. Das, U., R. Dea and V.M. Sundalgear (1996). Radatn effects n flw past an mpulsvel started vertcal plate-an eact slutns. J. The. Appl. Flud Mech. 1(), Datta,. and R.. Jana (1976). Oscllatr magnethdrdnamc flw past a flat plate wlth hall effects. J. Phs. Sc. Japan 40, Ferrar, V.C.A. and C. Plumptn (1966). An Intrductn t Magnet Flud Mechancs. Clarandn Press, Ofrd. Ganesan, P. and G. Palan (00). The effects f mass transfer n the MHD flw past an mpulsvel started sthermal nclned plate. Jurnal f Energ, Heat and Mass Transfer 4, 1 1. Knanu, M., J.K. Kwanza and S.M. Uppal (001). Magnethdrdnamc free cnvectn heat and mass transfer f a heat generatng flud past an mpulsvel started nfnte vertcal prus plate wth Hall current and radatn absrptn. Energ Cnservatn and Management. 4, Muthucumaraswam, R., and P. Ganesan (1998). Unstead flw past an mpulsvel started vertcal plate wth heat and mass transfer. Jurnal f Energ, Heat and Mass Transfer 34, Palan, G. and I.A. Abbas (009). Free cnvectn MHD flw wth thermal radatn frm an mpulsvel started vertcal plate. nlnear Analss: Mdellng and Cntrl 14(1), Prasad, V.R.,. Bhasar Redd and R. Muthucumaraswam (011). Fnte Dfference analss f radatve free cnvectn flw past an mpulsvel started vertcal plate wth varable heat and mass flu. Jurnal f Appled Flud Mechancs 4(1), Prasad, V.R.,. Bhasar Redd and R. Muthucumaraswam (007). Radatn and mass transfer effects n tw- dmensnal flw past an mpulsvel started nfnte vertcal plate. Int. J. Thermal Scences 46(1), Prasad, V.R.,. Bhasar Redd and R. Muthucumaraswam (006). Transent radatve hdrmagnetc free cnvectn flw past an mpulsvel started vertcal plate wth unfrm heat and mass flu, Theretcal Appled Mechancs 33(1), Sharma, B.K., A.K. Jha and R.C. Chaudhar (007). Hall Effect n MHD Med Cnvectve Flw f a Vscus Incmpressble Flud Past a Vertcal Prus Plate Immersed n Prus Medum wth Heat Surce/Sn. Rm. Jurnal Phs 5(5),

8 J. Anand Ra et al. / JAFM, Vl. 5,. 3, pp , 01. Sherclff, J.A. (1965). A Tet B f Magnet Hdrdnamcs. Pergamn Press, Lndn. Sngh, K.D. (000). An scllatr hdrmagnetc Cuette flw n a rtatng sstem. ZAMM 80, Suneetha, S.,. Bhasar Redd and V. Ramachandra Prasad (011). Radatn and Mass transfer effects n MHD free cnvectve Dsspatve flud n the presence f heat surce/sn. Jurnal f Appled Flud Mechancs 4(1), Tahar, H.S., A. J. Chamha and G. ath (00). MHD flw ver a mvng plate n a rtatng flud wth magnetc feld, Hall currents and free stream velct. Int. J. Engng Sc. 40(13), Vasu B, V. Ramachandra Prasad and. Bhasar Redd (011). Radatn and mass transfer effects n transent free cnvectn flw f a dsspatve flud past sem-nfnte vertcal plate wth unfrm heat and mass flu. Jurnal f Appled Flud Mechancs 4(1),