JJMIE Jordan Journal of Mechanical and Industrial Engineering

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1 JJMIE Jodn Jonl of Mhnil nd Indsil Engining Volm Nmb Jny. 7 IN gs 4-49 Chmil Rion Eff on Unsdy MHD Flow s n Implsily d Inlind l wih Vibl Tmp nd Mss Diffsion in h sn of Hll Cn U.. Rjp * G m Dpmn of Mhmis nd sonomy Unisiy of Lknow Lknow U. Indi. Rid JN 6 pd MY 7 7 bs Th psn sdy is id o o xmin h ombind ffs of Hll n nd hmil ion on flow modl. Th modl onsiss of nsdy flow of isos inompssibl nd lilly onding flid. Th flow is long n implsily sd inlind pl wih ibl wll mp nd mss diffsion. Th mgni fild of nifom sngh is pplid ppndil o h flow. Th modl onins qions of moion diffsion qion nd qion of ngy. Th Goning qions inold in h psn nlysis sold by h Lpl-nsfom hniq. Th loiy pofil is disssd wih h hlp of gphs dwn fo diffn pms lik hml Gshof nmb mss Gshof nmb ndl nmb hmil ion pm Hll pm h mgni fild pm nd hmid nmb. Th nmil ls obind fo skin-fiion w bld. W fond h h ls obind fo loiy onnion nd mp in onn wih h l flow of h flid. ywods: 7 Jodn Jonl of Mhnil nd Indsil Engining. ll ighs sd. Inodion MHD flow poblms ssoid wih h nd mss nsf plys impon ols in diffn s of sin nd hnology lik hmil ngining mhnil ngining biologil sin polm ngining nd biomhnis. h poblms fqnly o in pohmil indsy hmil po dposiion on sfs ooling of nl os h xhng dsign fos fi dynmis nd gophysis. Th infln of mgni fild on isos inompssibl nd lilly onding flid is of g impon in mny ppliions sh s mgni mil possing glss mnfing onol posss nd pifiion of d oil. i long wih hmd ] sdid h Hll ff on nsdy MHD Co flow nd h nsf of Binghm flid wih sion nd injion. Fh i ] onsidd h ff of ibl popis on h nsdy Hmnn flow wih h nsf onsiding h Hll ff. Eff of mss nsf on flow ps n implsily sd infini il pl wih onsn h flx nd hmil ion ws insigd by Dk l. 3]. Chmilly ing MHD bondy ly flow of h nd mss nsf o moing il pl wih sion ws insigd by Ibhim nd Mkind 4]. Th ff of Hll n on h mgno hydodynmi bondy ly flow ps smi-infini fs pl ws sdid by gii 5]. Mhmswmy 6] onsidd ff of hmil ion on moing isohml il sf wih sion. Fh Mhmswmy long wih Gnsn 7] insigd fis od hmil ion on flow ps n implsily sd il pl wih nifom h nd mss flx. Mipl nd Nikoi 8] nlyd Hll ff on nsdy MHD f onion flow o shing sh wih ibl isosiy nd isos dissipion. MHD osilloy hnnl flow h nd mss nsf in physiologil flid in h psn of hmil ion ws dlopd by Mis nd dhiky 9]. op ] insigd h ff of Hll n on hydomgni flow n n ld pl. op nd Wnb ] fh sdid Hll ff on MHD bondy ly flow o oninos moing fl pl. Visos flow o non-linly shing sh in h psn of hmil ion nd mgni fild ws sdid by Rpis nd dikis ]. Rpis nd fosis 3] sdid flow of isos flid hogh poos mdim bondd by il sf. Th sdy of MHD flow ps n implsily sd il pl wih ibl mp nd mss diffsion ws don by Rjp nd m 4]. Fh Rjp nd km 5] wokd on ff of hmil ion on f onion MHD flow hogh poos mdim bondd by il sf. Ziyddin nd m 6] insigd MHD h nd mss nsf f onion flow n h low sgnion poin of n isohml ylind imbddd in * Cosponding ho -mil: jpglko@gmil.om.

2 4 7Jodn Jonl of Mhnil nd Indsil Engining. ll ighs sd - Volm Nmb IN poos domin wih h psn of diion. Combind ffs of diion nd Hll n on MHD flow ps n xponnilly ld il pl in h psn of oion ws sdid by Thmihsd nd ndngn 7]. Tiphy l. 8] nlyd hmil ion ff on MHD f oni sf o moing il pl hogh poos mdim. Eli w 9] sdid nsdy MHD flow ps n implsily sd inlind pl wih ibl mp nd mss diffsion in h psn of Hll n. Th min ppos of h psn insigion is o sdy h ffs of hmil ion on nsdy MHD flow ps n implsily sd inlind pl wih ibl wll mp nd mss diffsion in h psn of Hll n. Th sls shown wih h hlp of gphs nd bl.. Mhmil nlysis Th gomil modl of h poblm is shown in fig- Fig. hysil modl Th x xis is kn long h il pln nd noml o i. Ths h xis lis in h hoionl pln. Th pl is inlind n ngl α fom il. nss mgni fild B of nifom sngh is pplid on h flow. Th isos dissipion nd indd mgni fild ws ngld d o is smll ff. Iniilly i hs bn ssmd h h pl nd h flid h sm mp T. Fh h spis onnion is C ywh in h flid in siony ondiion. im > h pl ss moing wih loiy in is own pln nd mp of h pl is isd o ; lso h onnion ll n h pl is isd linly wih sp o im. D o h Hll ff h will b wo omponns of h momnm qion. Th flow modl is s nd: 3. Eqions of moion gcos T T * B m g Cos C C m T w B m m Diffsion qion C C D Eqion of ngy T T C C C C p k 4 Th iniil nd bondy ondiions : T T C C fo y : T T Tw T υ C C Cw C υ 3 = 5 T T C C s wh is h pimy loiy - h sondy loiy g- h lion d o giy β - olmi offiin of hml xpnsion -im m - Hll pm T-mp of h flid β * - olmi offiin of onnion xpnsion C- spis onnion in h flid - h kinmi isosiy - h dnsiy C p - h spifi h onsn pss k- hml ondiiy of h flid D- h mss diffsion offiin Tw - mp of h pl = C w - spis onnion h pl = B - h nifom mgni fild - hmil ion nd is lil ondiiy. H m wih - yloon fqny of lons nd - lon ollision im. Th following non-dimnsionl qniis inodd o nsfom qions 3 nd 4 ino dimnsionlss fom: υ μ ρυ υ D σb μ υ C C p M C ρ Cw C k g Tw T C G 3 υ * gβ Cw C T T Gm θ 3 T T Th symbols in dimnsionlss fom s nd: - h pimy loiy - h sondy loiy - im θ - h mp C - h onnion G - hml Gshof nmb Gm - mss Gshof nmb μ- h offiin of isosiy h hmil ion w 6

3 7Jodn Jonl of Mhnil nd Indsil Engining. ll ighs sd - Volm Nmb IN pm - h ndl nmb - h hmid nmb M- h mgni pm. Th flow modl in dimnsionlss fom is G M m m M m m C C Cos G Cos C C Th osponding bondy ondiions 5 bom: m : θ C fo y : θ C = θ C s. Dopping bs in h bo qions w g: G M m m M m m C Cos G Cos C C C Th bondy ondiions bom m : θ C fo ll : θ C = 6 θ C s. Wiing h qions nd 3 in ombind fom q q GCosα θ GmCosα C q C C C Finlly h bondy ondiions bom : q θ C fo y : q θ C = q θ C s. M im H q= + i m Th dimnsionlss goning qions 7 o 9 sbj o h bondy ondiions sold by h sl Lpl nsfom hniq. Th solion obind is s nd: C 4 f ] { f f 4 ] ] } 5 G Cos q { 4 G } m Cos ] 4 G Cos G ] m Cos ] Th xpssions fo h symbols inold in h bo qions gin in h ppndix. 4. kin Fiion Th dimnsionlss skin fiion h pl is dq x i. y d Th nmil ls of τ x nd τ y fo diffn pms gin in bl-.

4 44 7Jodn Jonl of Mhnil nd Indsil Engining. ll ighs sd - Volm Nmb IN Rsls nd Disssion In h psn pp w sdid h ffs of Hll n nd hmil ion on nsdy MHD flow. Th loiy pofil fo diffn pms lik hml Gshof nmb G mgni fild M Hll pm m hmil ion ndl nmb hmid nmb nd im shown in figs fom. o.9. Th onnion pofil fo diffn pms lik hmil ion hmid nmb nd im shown in figs fom 3. o 3.3. Th nmil ls of skinfiion psnd in Tbl. D o giy omponn gcosα h flid flows wih high loiy whn pl is il s ompd o flow whn pl is hoionl. I is obsd in figs. nd. h h pimy nd sondy loiis of flid ds whn h ngl of inlinion is insd. Fom figs. nd. w obs h if mss Gshof nmb is insd hn h loiis go insd. Fom figs.3 nd.3 i is ddd h whn hml Gshof nmb G is insd hn h loiis insd. If Hll n pm m is insd hn is insd nd is dsd figs.4 nd.4. Th infln of mgni fild on flow is obsd fom figs.5 nd.5. I is sn h h ff of insing ls of h pm M sls in dsing nd insing. I is in gmn sin h mgni fild sblishs fo whih s gins h min flow sling in slowing down h loiy of flid. If h hmil ion pm is insd hn h loiis insd hogho h bondy ly gion figs.6 nd.6. Fh i is obsd h loiis ds whn ndl nmb is insd figs.7 nd.7. Whn h hmid nmb is insd hn h loiis g dsd figs.8 nd.8. Fh fom figs.9 nd.9 i is obsd h loiis ins wih im. If ion pm nd hmid nmb insd hn onnion is dsd figs 3. nd 3.. hysilly h ins of mns ds of moll diffsiiy D. Th is h poss of diffsion will ds. I is obsd h loiis ins wih im figs 3.3. kin fiion is gin in bl. Th l of τ x inss wih h ins in ngl of inlinion of pl hml Gshof nmb nd Hll ns pm; nd i dss wih ngl of inlinion of pl mss Gshof Nmb mgni fild hmil ion pm ndl nmb hmid nmb nd im. Th l of τ y inss wih h ins in ngl of inlinion of pl hml Gshof nmb nd h mgni fild; nd i dss wih mss Gshof nmb Hll n pm h hmil ion pm ndl nmb hmid nmb nd im M m G m G Fig.: loiy fo diffn ls of α M G m 5 m.7. 3 G Fig.: loiy fo diffn ls of Gm G 5 M m.7. G m Fig.3: loiy fo diffn ls of G M m G m G Fig.4: loiy fo diffn ls of m

5 7Jodn Jonl of Mhnil nd Indsil Engining. ll ighs sd - Volm Nmb IN M m.7. G m G M m.7 3 G m G Fig.5: loiy fo diffn ls of M Fig.9: loiy fo diffn ls of M m.7. G m G G.7. G m m. M Fig.6: loiy fo diffn ls of Fig.: loiy fo diffn ls of α M m 3..4 G m 5.6 G m G M G.7. 3 m Fig.7: loiy fo diffn ls of Fig.: loiy fo diffn ls of Gm M m G m G Fig.8: loiy fo diffn ls of G 5 3 M G m m Fig.3: loiy fo diffn ls of G

6 46 7Jodn Jonl of Mhnil nd Indsil Engining. ll ighs sd - Volm Nmb IN G. M 3.7. G m m G m G. 3 M m Fig.4: loiy fo diffn ls of m Fig.8: loiy fo diffn ls of.8.6 G m 3 G m M m M G G m Fig.5: loiy fo diffn ls of M Fig.9: loiy fo diffn ls of G m G m M Fig.6: loiy fo diffn ls of Fig 3.: fo diffn ls of o G m G. M m Fig.7: loiy diffn ls of Fig 3.: fo diffn ls of

7 7Jodn Jonl of Mhnil nd Indsil Engining. ll ighs sd - Volm Nmb IN Fig 3.3: fo diffn ls of Tbl: kin fiion fo diffn ms. α in dg M m Gm G x y

8 48 7Jodn Jonl of Mhnil nd Indsil Engining. ll ighs sd - Volm Nmb IN Conlsion Th onlsions of h psn sdy s follows: imy loiy inss wih h ins in hml Gshof nmb mss Gshof Nmb Hll n pm nd im. imy loiy dss wih h ngl of inlinion of pl h mgni fild hmil ion pm ndl nmb nd hmid nmb. ondy loiy inss wih h ins in hml Gshof nmb mss Gshof Nmb h mgni fild nd im. ondy loiy dss wih h ngl of inlinion of pl Hll ns hmil ion pm ndl nmb nd hmid nmb. x inss wih h ins in ngl of inlinion of pl G nd m; nd i dss wih ngl of inlinion of pl Gm M nd. y inss wih h ins in ngl of inlinion of pl G nd M nd i dss wih Gm m nd. Connion of h flid n h pl inss wih im nd i dss wih nd. Rfns ] i Hm li nd hmd Mohmd Eiss yd Th Hll ff on nsdy MHD Co flow wih h nsf of Binghm flid wih sion nd injion pplid Mhmil Modlling 8 pp ] i H Th ff of ibl popis on h nsdy Hmnn flow wih h nsf onsiding h Hll ff ppl. Mh. Modl. 7 7 pp ] Ds U N Dk R nd ondlgk V M Effs of mss nsf on flow ps n implsily sd infini il pl wih onsn h flx nd hmil ion. Fosh. Ingniws 6 pp ] Ibhim Y nd Mkind O D Chmilly ing MHD bondy ly flow of h nd mss nsf o moing il pl wih sion inifi Rsh nd Essys 59 pp ] gii M Th ff of Hll n on h mgno hydodynmi bondy ly flow ps smi-infini fs pl. Jonl of h hysil oiy of Jpn Vol. 74 pp ] Mhmswmy R Effs of hmil ion on moing isohml il sf wih sion Mh.55 pp ] Mhmswmy R nd Gnsn Fis od hmil ion on flow ps n Implsily sd il pl wih nifom H nd Mss Flx. Mhni Vol. 47 No. -4 pp ] Mipl inis nd Nikoi ishn Hll ffs on nsdy MHD f onion flow o shing sh wih ibl isosiy nd isos dissipion Wold pplid ins Jonl 33 6 pp ] Mis J C dhiky D MHD osilloy hnnl flow h nd mss nsf in physiologil flid in psn of hmil ion lxndi Engining Jonl55 pp ] op I. Th ff of Hll ns on hydomgni flow n n ld pl J. Mh. hys. i. 5 pp ] op I nd Wnb T Hll ffs on mgnohydodynmi bondy ly flow o oninos moing fl pl Mh.Vol. 8 pp ] Rpis nd dikis C Visos flow o non-linly shing sh in h psn of hmil ion nd mgni fild. Innionl Jonl of Non- Lin Mhnis 4 pp ] Rpis nd dikis C Flow of isos flid hogh poos mdim bondd by il sf In. J. Eng. i. pp ] Rjp U nd m nd MHD flow ps n implsily sd il pl wih ibl mp nd mss diffsion pplid Mhmil ins Vol. 5 no. 3 pp ] Rjp U nd m nd Eff of hmil ion on f onion MHD flow hogh poos mdim bondd by il sf Elixi ppl. Mh 4 pp ] Ziyddin nd m Mnoj MHD h nd mss nsf f onion flow n h low sgnion poin of n isohml ylind imbddd in poos domin wih h psn of diion Jodn Jonl of Mhnil nd Indsil Engining JJMIE Volm 5Iss pp ] Thmihsd M nd ndngn J Combind ffs of diion nd Hll n on MHD flow ps n xponnilly ld il pl in h psn of oion Innionl Jonl of Innoi Rsh in Comp nd Commniion Engining Vol. Iss Dmb 4. 8] Tiphy R Dsh G C Mish R nd Bg Chmil ion ff on MHD f oni sf o moing il pl hogh poos mdim lxndi Engining Jonl 5 54 pp ] Rjp U nd m G Unsdy MHD Flow s n Implsily d Inlind l wih Vibl Tmp nd Mss Diffsion in h psn of Hll n ppliions nd pplid Mhmis Vol. Iss pp ppndix f f ] 7 f ]

9 7Jodn Jonl of Mhnil nd Indsil Engining. ll ighs sd - Volm Nmb IN f f ] f ] 3 f ] 4 f ] 5 f ] 6 f ] 7 f ] 8 f ] 9 f ] 3 f