Natural Convection Magnetohydrodynamic Flow of a Micro Polar Fluid past a Semi infinite Vertical Porous Flat Moving Plate

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1 Intnational Jonal of pplid Envionntal Scincs ISSN 9- Vol, Nb 9 (), pp. 9- Rsach India blications Natal Convction Magntohdodnaic Flow of a Mico ola Flid past a Si infinit Vtical oos Flat Moving lat S. anda, S. S. Das and N. C. Ba Dpatnt of hsics, KISS, KIIT Caps-, atia, Bhbanswa- (Odisha), India. Dpatnt of hsics, KBDV Collg, Niakap, Khodha- 9 (Odisha), India. Dpatnt of hsics, KIIT Univsit, atia, Bhbanswa- (Odisha), India. bstact Th objctiv of this pap is to analz an nstad natal convction flow of a viscos incopssibl lcticall condcting ico pola flid past a si-infinit vtical poos flat oving plat in psnc of tansvs agntic fild. Th vtical poos plat is sbjctd to ov with a nifo vlocit in th pwad diction in its own plan and th f sta vlocit follows an xponntiall incasing o dcasing sall ptbation law. Th poos plat absobs th pola flid with a sction vlocit vaing with ti. Th govning qations fo vlocit, angla vlocit and tpat of flow fild hav bn solvd sing ptbation tchniq. Th ffcts of th ipotant flow paats sch as Gashof nb fo hat tansf G, agntic paat M, viscosit atio, plat vlocit Up and andtl nb tc. on th vlocit, angla vlocit and tpat pofils of th flow fild a discssd with th hlp of figs. This pap has so lvanc in gothal and ocanic ciclation. Kwods: Unstad, agntohdodnaic, f convction, ico pola, oving plat

2 9 S. anda, S.S. Das and N.C. Ba NOMENCLTURE sction vlocit paat B agntic flx dnsit Cp spcific hat at constant pss G Gashof nb fo hat tansf g acclation d to gavit k thal condctivit M agntic fild paat N Nsslt nb n dinsionlss xponntial indx andtl nb T tpat t dinsionlss ti U scal of f sta vlocit,v coponnts of vlocitis along and ppndicla to th plat, spctivl V scal of sction vlocit x, distancs along and ppndicla to th plat, spctivl Gk sbols Flid thal diffsivit Dinsionlss viscosit atio f Cofficint of voltic thal xpansion of th woking flid Spin-gadint viscosit Scala constant () Elctical condctivit Flid dnsit Λ Cofficint of go-viscosit Flid dnaic viscosit Flid kinatic viscosit Flid kinatic otational viscosit Dinsionlss tpat ngla vlocit vcto Spscipts diffntiation with spct to dinsional poptis Sbscipts p plat w wall condition f sta condition

3 Natal Convction Magntohdodnaic Flow of a Mico ola Flid past 9. INTRODUCTION Th phnonon of flow and hat tansf in an lcticall condcting pola flid past a vtical poos plat nd th inflnc of a agntic fild has bn a sbjct of intst of a good nb of sachs bcas of its applications in sval filds of scinc and tchnolog fo xapl in th bonda la contol in th fild of aodnaics (o t al. []), gothal ng xtactions, oil xploation and in plasa stdis. hsicall, ico pola flids psnt flids consisting of andol ointd paticls sspndd in a viscos di (Dp [] and Lkaszwicz []). Sval Dacian poos MHD stdis hav bn caid ot considing th ffcts of agntic fild on lcticall condcting flid with o withot hat tansf in vaios configations. Chakha [] invstigatd th nstad convctiv hat and ass tansf flow of an lcticall condcting flid in psnc of tansvs agntic fild past a si-infinit vtical pabl oving plat with hat absoption. Mbldog and Ogl [] analzd th ffct of hat and ass tansf on nstad natal convction flow of a otating lcticall condcting flid past a vtical poos flat plat in th psnc of adiativ hat tansf and tansvs agntic fild. Das and his co-woks [] stiatd th ffct of ass tansf on agntohdodnaic flow and hat tansf past a vtical poos plat thogh a poos di nd oscillato sction and hat soc and obsvd that a gowing agntic paat dcs th vlocit of th flow fild at all points of th flow fild. Th tansint f convction MHD flow past an infinit vtical poos flat plat in psnc of ass tansf has bn stdid b anda and his ta []. Das t al. [] potd th f convction ffcts on nstad viscos flow of an lcticall condcting flid past an infinit vtical poos plat with hat soc/sink in psnc of agntic fild. atil and Klkani [9] obsvd th ffct of chical action on f convctiv flow of a pola flid thogh a poos di in th psnc of intnal hat gnation. Das and his gop [] analzd nstad ixd convctiv flow of a pola flid past a si-infinit vtical poos oving plat in psnc of tansvs agntic fild and noticd that an incas in agntic paat nhancs th vlocit as wll as angla vlocit of th flow fild at all points of th flow fild. Rcntl, Das and his associats [] discssd th agntohdodnaic convctiv ass tansf flow of a pola flid past a si infinit vtical poos flat oving plat bddd in a poos di and potd that thi slts a in good agnt with thos of th pvios gop []. Th psnt std invstigats an nstad natal convction flow of a viscos incopssibl lcticall condcting ico pola flid past a si-infinit vtical poos flat oving plat which is sbjctd to ov with a nifo vlocit in pwad diction in its own plan in psnc of a tansvs agntic fild. Th plat absobs th pola flid with a sction vlocit vaing with ti and th f

4 9 S. anda, S.S. Das and N.C. Ba sta vlocit follows an xponntiall incasing o dcasing sall ptbation law. Th ffcts of th flow paats on th flow fild acoss th bonda la hav bn discssd and analzd with th aid of figs. This pap has so lvanc in gothal and ocanic ciclation.. MTHEMTICL FORMULTION OF THE ROBLEM Consid an nstad viscos incopssibl flow of an lcticall condcting ico pola flid past a si-infinit vtical poos oving plat bddd in a poos di and sbjctd to a tansvs agntic fild in th psnc of a pss gadint. Th tansvsl applid agntic fild and agntic Rnolds nb a v sall and hnc th indcd agntic fild is ngligibl (Cowling []). Viscos and Dac s sistanc ts a takn into accont with constant pabilit of th poos di. Th hol siz of th poos plat is assd to b significantl lag than a chaactistic icoscopic lngth scal of th poos di. Th got of th pobl is shown in Fig. Following Yaaoto and Iwaa [], w ass th poos di as an assbl of sall idntical sphical paticls fixd in spac. Th flow vaiabls a fnctions of and t onl. Fig. Got of th pobl

5 Natal Convction Magntohdodnaic Flow of a Mico ola Flid past 9 Und th abov conditions, th govning qations fo ass, ont and ng consvation in Catsian coodinats a givn b: Continit: v, () Lina ont: t v p x g f T T B () ngla ont: t j v, () Eng: T t v T T, () wh,,, g, f,, B, j,,, T and a spctivl th dnsit, kinatic viscosit, kinatic otational viscosit, acclation d to gavit, cofficint of voltic thal xpansion of th flid, lctical condctivit of th flid, agntic indction, ico-intia dnsit, coponnt of th angla vlocit vcto noal to x-plan, spin-gadint viscosit, tpat and ffctiv flid thal diffsivit; x and ppndicla to th plat spctivl and vlocitis along x and a th dinsional distancs longitdinal and dictions spctivl., v a th coponnts of dinsional It is assd that th poos plat ovs with a constant vlocit ( p ) in th longitdinal diction and th f sta vlocit ( U ) follows an xponntiall incasing o dcasing sall ptbation law. W, fth ass that th sction vlocit ( v ) and th plat tpat (T) va xponntiall with ti. Fo sall vlocitis, th hat d to viscos dissipation is nglctd in th ng qation (). With th abov assptions, th bonda conditions fo vlocit and tpat filds a n t p,t Tw T w T, at

6 9 S. anda, S.S. Das and N.C. Ba n U U t, T T, as, () wh U is a scal of f sta vlocit and n is a scala constant. It is clal sn fo th continit qation () that th sction vlocit noal to th plat is a fnction of ti onl and thfo, w shall tak it in th fo: n t v V, () wh is a al positiv constant, and a sall and lss than nit and V is a scal of sction vlocit which is a non-zo positiv constant. Otsid th bonda la, qation () givs dp dx du dt B U W now intodc th following non-dinsional vaiabls: U v V U p t V,v,,U,U p,,t V U U U V T T, T T w () n, n, V j V j, C p B is th andtl nb, M is th agntic k V f g(tw T ) paat, G is th Gashof nb fo hat tansf. () U V gain, th spin-gadint viscosit which givs th lationship btwn th cofficints of viscosit and ico-intia is dfind as ( ) j j ( ), (9) wh dnots th dinsionlss viscosit atio which is dfind as () H, Λ is th cofficint of go-viscosit o votx viscosit. Using qations ()-(), th govning qations ()-() a dcd to th following non-dinsional fo: t nt du G M U dt, ()

7 Natal Convction Magntohdodnaic Flow of a Mico ola Flid past 99 t nt, () t nt, () wh j () Th cosponding bonda conditions now tak th fo nt U p,, at =, U,, as (). METHOD OF SOLUTION In od to solv th qations ()-() b dcing ths patial diffntial qations to a sst of odina diffntial qations in non-dinsional fo, w ass th lina vlocit, angla vlocit and tpat as nt O nt O () () nt O Sbstitting qations ()-() in qations ()-() and qating th haonic and non-haonic t, nglcting th cofficint of O( ), w gt th following pais of qations fo,,. and,, " M M G () (9) " M n M n G () () n () () n ()

8 S. anda, S.S. Das and N.C. Ba H th pis dnot diffntiation with spct to. Th cosponding bonda conditions now dc to:,,,,, U p at =,,,,, as () Th soltions of qations (9)-() satisfing bonda conditions () a givn b () 9 ) ( () () (9) () () wh,,, n, n, M, M, n M, n M,, M, G,, p U, p U,, n M,, 9, G,,, 9 9,

9 Natal Convction Magntohdodnaic Flow of a Mico ola Flid past 9. ().. Skin fiction Th skin fiction at th wall of th plat is givn b w w U V () Using Eqations (), () and () in Eqation (), th skin fiction bcos w nt () 9.. Rat of hat tansf Th at of hat tansf o th hat flx at th wall in ts of Nsslt nb is givn b N () Using Eqations (), () and () in Eqation (), th hat flx bcos N nt (). DISCUSSIONS ND RESULTS Th std potd hin invstigats an nstad natal convction flow of a viscos incopssibl lcticall condcting ico pola flid past a si-infinit vtical poos flat oving plat which is sbjctd to ov with a nifo vlocit in pwad diction in its own plan in psnc of a tansvs agntic fild. Th govning qations fo lina vlocit, angla vlocit and tpat of th flow fild hav bn solvd ploing ptbation tchniq and th ffcts of th flow paats on th lina vlocit, angla vlocit and tpat of flow fild acoss th bonda la hav bn discssd and analzd with th aid of vlocit pofils (-), angla vlocit pofils (-) and tpat pofil () as dtaild blow... Vlocit fild Th flow paats affcting th vlocit of th flow fild a Gashof nb fo hat tansf G, agntic paat M, viscosit atio and th plat vlocit Up. Th vlocit of th flow fild is fond to chang o o lss with th vaiation of th

10 S. anda, S.S. Das and N.C. Ba abov paats. Th ffcts of ths paats on th vlocit fild a discssd with th hlp of Figs -. G= - G= G= G= Fig. Effct of Gashof nb fo hat tansf G on Vlocit pofils against Th ffct of Gashof nb fo hat tansf G on th vlocit fild is psntd in Fig. Th Gashof nb fo hat tansf is fond to acclat th vlocit of th flow fild na th plat. Fig dpicts th ffct of agntic paat M on th vlocit fild. gowing agntic paat is noticd to nhanc th vlocit of th flow fild at all points. M= M=. M= M= Fig. Effct of agntic paat M on Vlocit pofils against

11 Natal Convction Magntohdodnaic Flow of a Mico ola Flid past Fig. Effct of viscosit atio on Vlocit pofils against Th ffct of viscosit atio on th vlocit of th flow is shown in Fig. Th viscosit atio is obsvd to dclat th vlocit of th flow fild at all points. Fig, lcidats th ffct of th plat vlocit Up on th vlocit fild. Fo th cvs of th said fig, it is clal obsvd that an incas in plat vlocit has an acclating ffct on th vlocit of th flow fild na th plat. Th slts shown in vlocit pofils, and a in good agnt with thos of Das t al. [] and Fig closl atchs with thos of Das and his gop []. Up= Up= Up=. Up= Fig. Effct of plat vlocit Up on Vlocit pofils against

12 S. anda, S.S. Das and N.C. Ba.. ngla vlocit fild Th ffct of agntic paat M and th viscosit atio on th angla vlocit of th flow fild has bn shown in Figs -. Th pofils of Fig clal show that a gowing agntic paat M lads to nhanc th angla vlocit of th flow fild at all points. Fig lcidats th ffct of plat vlocit Up on th angla vlocit of th flow fild. It is obsvd that an incas in of plat vlocit lads to dcas th angla vlocit of th flow fild. Th angla vlocit pofils a in good agnt with thos of Das t al. [, ]. M= M=. M= M= Fig. Effct of agntic paat M on ngla vlocit pofils against Up= Up= Up=. Up= Fig. Effct of plat vlocit Up on ngla vlocit pofils against

13 Natal Convction Magntohdodnaic Flow of a Mico ola Flid past.. Tpat fild Th plot of tpat of th flow fild against fo diffnt vals of th andtl nb is shown in Fig. Copaing th cvs of th said fig, it is obsvd that an incas in andtl nb dcs th tpat of th flow fild at all points. Th tpat pofils shown in Fig closl atchs with thos of Das t al. []..... =. = =.. Fig. Effct of andtl nb on Tpat pofils against. CONCLUSIONS W saiz th conclsions dawn fo th abov analsis on th lina vlocit, angla vlocit and tpat of th flow fild.. Th Gashof nb fo hat tansf G acclats th vlocit of th flow fild na th plat.. gowing agntic paat M nhancs th lina vlocit as wll as angla vlocit of th flow fild at all points.. Th ffct of incasing viscosit atio is to tad th lina vlocit of th flow fild at all points.. Th plat vlocit Up is obsvd to nhanc th lina vlocit of th flow fild na th plat whil it shows vs ffct in cas of angla vlocit.. n incas in andtl nb dcass th tpat of th flow fild at all points.

14 S. anda, S.S. Das and N.C. Ba REFERENCES [] o, E.L., Blgin,.N., and Kvshinskii, E.V., 9, stic hdochanics, J. ppl. Math., 9(), pp.-. [] Dp, N. V., 9, Eqations of a flid bonda la with copl stsss, J. ppl. Math. Mch., (), pp. -. [] Lkaszwicz, G., 999, Micopola Flids- Tho and pplications, Bikhas, Boston. [] Chakha,. J.,, Unstad MHD convctiv hat and ass tansf past a si-infinit vtical pabl oving plat with hat absoption, Int. J. Engng. Sci., (), pp. -. [] Mbldog I.U., and Ogl,.,, Hat and ass tansf of an nstad MHD natal convction flow of a otating flid past a vtical poos flat plat in th psnc of adiativ hat tansf, Int. J. Hat Mass Tansf, (9-), pp.9-9. [] Das, S. S., Satapath,., Das, J. K., and anda, J.., 9, Mass tansf ffcts on MHD flow and hat tansf past a vtical poos plat thogh a poos di nd oscillato sction and hat soc, Int. J. Hat Mass Tansf,. (-), pp [] anda, S., Das, S. S., and Ba, N. C.,, MHD f convction tansint flow past an infinit vtical poos flat plat in psnc of ass tansf, Int. J. ppl. Engng. Rsach, (), pp [] Das, S. S., Misha, S., and Tipath,.,, Effct of f convction on nstad viscos flow of an lcticall condcting flid past an infinit vtical poos plat with hat soc/sink in psnc of agntic fild, Int. J. ppl. Engng. Rsach, (), pp.-. [9] atil.m., and Klkani,.S.,, Effcts of chical action on f convctiv flow of a pola flid thogh a poos di in th psnc of intnal hat gnation, Int. J. Thal Scincs, (), pp.-. [] Das, S. S., Biswal, S. R., and anda, J.., 9, Hdoagntic nstad ixd convctiv flow of a pola flid past a si-infinit vtical poos oving plat, Fa East J. Ocan Rsach, (), pp. -. [] Das, S. S., Saan M. R., and adhan, B.,, MHD convctiv ass tansf flow of a pola flid past a si infinit vtical poos flat oving plat bddd in a poos di, Int. J. ppl. hs., (), pp.-. [] Cowling, T. G., 9, Magntohdodnaics, Intscinc blishs, Nw Yok. [] Yaaoto, K., and Iwaa, N., 9, Flow with convctiv acclation thogh a poos di, J. Eng. Math.,, pp. -.