MHD Oscillatory Flow of a Visco Elastic Fluid in a Porous Channel with Chemical Reaction
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1 Innonl Jonl of ngnng n Invnon IN (Onln): IN (n): Vol Iss ǁ Fb. ǁ.6-5 MHD Osllo Flow of Vso ls Fld n oos hnnl wh hl Ron B.Dvk.V. Nn nd.vnkn Dn of Mhs Vnksw Unvs T 57 5 A. Ind Fld Dns Dvson A VIT Unvs Vllo 6 4 T.N Ind ABTRAT: An nlss of fs od hoognos hl on nd h so on MHD osllo flow of vso-ls fld hogh hnnl flld wh sd oos d od. Th sn vso-ls fld odl s wokng o sggs hologl lqds nond n bohnolog (dl s) nd hl ngnng. Ths hologl odl nods ddonl s no h on qon. I s ssd h h fld hs sll l ondv nd h logn fo odd s v sll. Th dnsonlss govnng qons solvd nlll sng gl bon hod. Th ffs of vos s on vlo nd onnon flds snd ghll nd dsssd. Kwods: MHD osllo flow vso-ls fld hl on hl don h so. I. INTRODUTION Th sd of non Nwonn fld flows hs gnd h non of ngns nd sns n n s d o s on lon n vos bnhs of sn ngnng nd hnolog: ll n hl nd nl ndss l ossng gohss nd bo-ngnng. In vw of hs lons n xnsv ng of hl odls hs bn dvlod o sl h dvs hdodn bhvo of hs non-nwonn flds. In l dffn vso-ls fld odls (lk h Rvln-ksn sond od odl. Oldod odl Johnson-gln odl) hv bn snd b n nvsgos n dvs ng of gos sng vos s of nll nd oonl shs (s fo nsn [-4]). Th Flds whh xhb h ls o of solds nd vsos o of lqds lld vso-ls flds. Ths fld flows nond n nos s of ohl bodl nd nvonnl ngnng nldng ololn olsn snng [5] dnll-lodd jonl bngs [6] blood flow [7] nd gologl flows [8]. Th sd of vso-ls flds hogh oos d hs bo h bss of n snf nd ngnng lons. Ths of flow s of g on n h ol ngnng onnd wh h ovn of ol gs nd w hogh svo of ol o gs fld o h hdologs n h sd of h gon of ndgond w nd o h hl ngnng fo h fon nd flon osss nd n h ss lk dg on hogh hn skn. Th nls of hs sbj v sfl n ovng h w fo dnkng nd gon oss. Mn sh woks [9-4] hv d h non owds h lon of vso-ls fld flow of dffn go hogh oos d n hnnls of vos oss-sons. Rjgol [5] nlzd h ok s obl fo non-nonn fld. Hossn nd Tkh [6] sdd h don ffs on xd onvon long vl l wh nfo sf. H nsf nhnn fo h ow-lw flds hogh lll-l dobl-ss h xhngs wh xnl l ws xnd b wo.l [7]. sl [8] sdd h hologl ffs of non-nwonn bhvo of dslng flds on sbl of ovng nf n dl ol dsln hns n oos d. Nk nd Ko [9] xnd h bon-ndd flow of non- Nwonn flds ov non sohl bod of b sh n fld-sd oos d. Mh nd Ro [-] sdd h bon-ndd flow of non-nwonn flds ov non sohl hozonl/vl l bddd n oos d. Th flow of h ondng fld s ffvl hngd b h sn of h gn fld nd h gn fld s lso bd d o h oon of h ondng fld. Ths hnonon s hfo nlokng n h nd h dsln of hs bnh of sn s lld Mgno hdodns (MHD). Infln of h gn fld on h non-nwonn fld flow hs wd lons n hl ngnng llgl ngnng nd vos ndss. Rshs hv onsdbl ns n h sd of flow hnonon bwn wo lll ls. Bs of s on n ho xns o dn h onsv os of h fld n lbon ngnng nd n nsoon nd ossng nond 6 g
2 Mhd osllo flow of vso ls fld n oos hnnl wh hl on n hl ngnng h flow on non-nwonn vso-ls fld s wohwhl o nvsg. Mn sh woks [-5] hv d h non owds h lon of vso-ls fld flow hogh vos s of hnnl nd h nfln of gn fld. Mknd nd Osls [6] hv bn dsssd MHD sd flow n hnnl wh sl bl bonds. ngh [7] obnd n x solon of n osllo MHD flow n hnnl flld wh oos d. Rhn nd k [8] nlzd h nsd MHD flow of ds vso-ls Oldod fld nd vng bod fo hogh ngl hnnl. ngh nd ngh [9] sdd MHD flow of ds vso-ls (Oldod B- lqd) hogh oos d bwn wo lll ls nlnd o h hozon. Msh l. [] nvsgd flow nd h nsf of MHD vso-ls fld n hnnl wh shng wlls. Msood Khn [] sdd h Rlgh oks obl fo n dg n vsols fld wh fonl dvv odl. Th on odd n oos d ws xodnl n oon sh s h o of M fl lls odls nd h olld ndgond w bs of dshgng h ox sbsn. Rnl n shs [-5] sdd h ffs of hl on on vos flow flds b sng nll hods. Mhj nd nvs [6] xnd h wok of fn [7] b onsdng h nsf ffs on MHD osllo flow n n s wv hnnl. Th n objv of hs s o xnd h wok of fn [7] n h dons: () o onsd h MHD osllo flow of vso-ls fld n hnnl () o onsd h h so ff () o nld h hl on. Th govnng qons of h flow solvd nlll nd h ffs of vos flow s on h flow fld hv bn dsssd. Th ognzon of h nns of h s s follows. In. w dsb h odl wh s govnng qons nd bond ondons. H w lso dsb h solon hod bfl. In. w sn sls nd dssson. Fnll n.4 w sz o sls nd sn o onlsons. II. MATHMATIA ANAYI W onsd n nsd osllo flow of vso-ls fld hogh ln hnnl wh hl on hl don nd h so n h sn of nsvs gn fld. I s ssd h h fld hs sll ll ondv nd h logn fo odd s v sll. Also s ssd h h s no ld volg so h h l fld s bsn h onnon of h dffsv ss n h bn x s ssd o b v sll n oson wh h oh hl ss whh sn nd hn o nd Dfo ffs nglgbl. Th x-xs s kn long h low l nd sgh ln ndl o h s h -xs. Assng Bonssnsq nossbl fld odl h qons govnng oon gvn s B g T T T g () x k T k T q T T Q D K ( ) Th bond ondons fo vlo nd ss onnon flds gvn s follows T Tw w on (5) T T on H w ss h h fld s oll hn wh lv low dns nd dv h flx s odng o h Rf [8] s gvn b q 4 ( T T w ) W now nod h non dnsonlss vbls s follows. U x T T B U R x H U Tw T (6) K gt ( Tw T) g ( w ) D U U U () (4) 7 g
3 Mhd osllo flow of vso ls fld n oos hnnl wh hl on 8 g. 4 U D U K J k Q D k N k U w Th dnsonlss govnng qons ogh wh h o bond ondons (nglng b sbols) n b wn s H x R ) ( (7) (8) J T (9) Th dnsonlss bond ondons on () on III. MTHOD OF OUTION To solv qons (7) (8) (9) fo l osllo flow. x () bsng qon () n qons (7) (8) nd (9) w g ) ( () d d () d d (4) bj o h bond ondons on (5) on Th nlzd solons of qons ()-(4) wh ssf bond ondons (5) gvn b osh osh sn sn sn sn
4 Mhd osllo flow of vso ls fld n oos hnnl wh hl on 9 g Th non dnsonl sss nso wll s gvn b U sn os osh sn Th of h nsf oss h hnnls wll s gvn s N N os sn Th of ss nsf oss h hnnls wll s gvn s h h os Wh N H J I shold b nond h n h bsn of h hl on h so nd vso ls h lvn sls obnd ddd s h sls obnd b Mknd nd Mhon [7]. IV. NOMNATUR H B logn ndon f h onsn ss d Mn hlf wdh of h hnnl D D nb H so g vonl fo shof nb Modfd shof nb H Inn of gn fld H Hnn nb J hl on
5 Mhd osllo flow of vso ls fld n oos hnnl wh hl on k Thl ondv K N Rdon N Nssl nb h wll l nb ss q Rdv h flx R oos d bl offn Rnolds nb hd nb oos d sh fo T T Fld T T w Wlls of Axl vlo U Flow n vlo k bols Fld dns T Mgn bl Fld offn of hl xnson offn of ss xnson Mgn bl ondv of h fld Fld dns Wv lngh Fqn of h osllon kn fon h wll Kns vsos offn Mn don bsoon offn V. RUT AND DIUION Th folon of h ffs of hl on hl don nd h so on MHD osllo flow of vso-ls fld n oos hnnl hs bn fod n h dng sons. Ths nbls s o o h nl llons n h dsbon of h qons of on ng nd dffson whh govn h flow fld solvd b sng gl bon hod. Th ffs of vos vbls lk hl on J h hd nb Hnn nb H vso-ls oos d sh fo on vlo nd onnon ofls hs bn dsssd wh h hl of ghs lod gns. Th vlo ofls fo dffn vls of shof nb nd Modfd shof nb dsbd n fgs. nd. I s obsvd h n nsng n lds o s n h vls of vlo. In ddon h vs show h k vl of vlo ns dl n h wll of h oos l s Modfd shof nb ns nd hn ds o h lvn f s vlo. Fgs. nd 4.svl sn h dnsonlss vlo nd ofls fo dffn vls of don N. I s sn fo hs fgs h boh h vlo nd ofls ns wh ns of don N. Th nfln of hl on J on vlo nd ss onnon ofls oss h bond l snd n fgs. 5 nd 6. W no fo fg. 5. h h s g
6 Mhd osllo flow of vso ls fld n oos hnnl wh hl on ds n hozonl vlo ofls wh ns n hl on J. Th ns of h hl on J lds o ds h bond l hknss nd o nhn h h nsf n h sn of hl nd soll bon fo. I s lso obsvd h h k vl ns n h oos bond sf. W nod fo fg. 6. h h s kd ff of nsng h vl of hl on on onnon dsbon n h bond l. Fh s obsvd h nsng vl of h hl on ds h onnon of ss n h bond l hs s d o h f h dsv hl ds h soll bond l hknss nd nss h ss nsf. Fo dffn vls of hd nb of nsng vls of h vlo ofls lod n fg. 7. I s obvos h h ff sls n dsng vlo dsbon oss h bond l. Fg. 8 shows h onnon ofls oss h bond l fo vos vls of hd nb. Th fg shows h n nsng n sls n n nsng h onnon dsbon. Th nfln of l nb on vlo nd ofls hv bn llsd n fgs. 9 nd. I s obsvd h n ns n sls ds n vlo. B n ofl s ns wh nsng vls of. Fgs. nd shows h von of vlo nd ofls fo dffn vls of. I s sn fo hs fgs. Th vlo nd ofls ns wh n nsng of h so. Th ffs of vos s on lol skn fon offn of h nsf N nd shwood nb h. Fg. shows h lol skn fon offn fo dffn vls of nd kng ll h s fxd. Fo hs fg w s h fo fxd dss s nss. On h oh hnd s nss skn fon offn nss. Th ff of h so on h of h nsf fo h l fo dffn vls of N s llsd n fg 4. Fo h w s h fo fxd h of h nsf fo h hd sf dss wh h nss of N. On h oh hnd s vl nss h of h nsf dss. Fg. 5 dls wh h von of hwood nb fo dffn vls of nd J. Ths fg vls h fo fxd h nss wh h ns of J. On h oh hnd s vl nss h hwood nb offn nss. Fgs. 6 nd 7 sn h ff of Hnn nb H on vlo nd skn fon offn ofls fo boh Nwonn nd Non-Nwonn fld ondons svl. Insng h Hnn nb H hs h ndn o ds h vlo nd skn fon. In ddon s sn h h fld vlo nd skn fon low fo Non Nwonn flds hn fo Nwonn flds. VI. ONUION W nlz h ffs of hl on hl don nd h so on MHD osllo vso-ls flow n hnnl flld wh oos d. Th qons of on ng nd dffson whh govn h flow fld solvd b sng gl bon hod. Th bhvos of vlo onnon skn fon Nssl nb nd hwood nb hs bn dsssd fo vons n h govnng s. Fo h sn llons w onld h followng onlsons.. Th gn H ds h vlo of h flow fld ll ons d o h gn ll of h onz fo ng on h flow fld.. Th fld oon s dd d o hl on. Hn h onson of hl ss ss fll n h onnon fld whh n n dnshs h bon ffs d o onnon gdns. Hn h flow fld s dd.. D o hl on h onnon of h fld dss. Ths s bs h onson of hl ss lds o fll n h ss onnon fld. 4. Th gnd of skn fon h l s fond o ds d o nsng H. 5. Th of ss nsf h ns d o ns n J nd of h nsf N ds d o ns n N. 6. Th nll sls obnd n hs wok o gnlzd fo of Mknd nd Mhon [7] nd n b kn s lng s b kng J γ. g
7 Mhd osllo flow of vso ls fld n oos hnnl wh hl on I s hod h h sls obnd wll no onl ovd sfl nfoon fo lons b lso sv s oln o h vos sds. RFRN []. J.. Oldod: On h folon of hologl qons of s. o Ro o (ond) A. Vol.No (95). []. J. Ro: Flow of Johnson-gln fld bwn ong oxl lnds wh nd who son. In. J. Nonln Mh Vol (999) []. T. H M. jd I. o: Th-dnsonl flow ov shng sf n vsols fld Nonln Anlss: Rl Wold Alons Vol (8). [4]. hng-mng hng Ko-Hsng hn Hn-Mng hn h-hn Wng Thol sd of osllo hnon n hozonl losd-loo lsng h wh sl d n hnnl Innonl onons n H nd Mss Tnsf Vol (). [5].. bbn D. Bd nd. Wo: Th ol of nsn holog n ol snng. Rholog A Vol (6). [6]. J.A. Th: Non-Nwonn lbon wh h onvd Mxwll odl.amj. Tbol. Vol (996). [7].. A. Vlsos: Th vsols bhvo of blood nd blood-lk odl flds wh hss on h soson of sd nd osllo sh. lnl.hoholog nd Molon. Vol (998). [8]. Z. Wo M. Hndks nd M.T. H: A vlo-bsd oh o vso-ls flow of ok. Mh. ol. Vol (5). [9]. B. dhk nd. Vnkn: MHD flow of vso ls fld s bl bd. Rg. ngg. H Mss Tnsf. Vol. No (988). []. R.. hdh And. Jn: Hll ff on MHD xd onvon flow of vsols fld s n nfn vl oos l wh ss nsf nd don. Thol nd Ald Mhns Vol (6). [].. os Mohddn V. R. sd. V. K. V nd O. Anw Bég Nl sd of nsd f onvv h nd ss nsf n wls-b vsols flow long vl on In. J. of Al. Mh nd Mh. 6 (5): 88-4 () []. A. A. Rs And H.. Tkh: H Tnsf Fo Flow Of An lso-vsos Fld In. o. H nd ss nsf Vol (989). []. Aj K ngh: H so nd don ffs on gno-onvon flow of vsols fld s shng sh: Anlss wh K's fnons Innonl onons n H nd Mss Tnsf Vol (8). [4]. M. Khn. Hd Al Ho Q: On ld flows of vsols fld wh h fonl Bgs odl Nonln Anlss: Rl Wold Alons Vol (9). [5]. K. R. Rjgol: On ok s obl fo non-nonn fld A Mh. Vol (98) [6]. M. A. Hossn nd H.. Tkh: Rdon ff on xd onvon long vl l wh nfo sf H nd Mss Tnsf Vol (996). [7]. wo-ng n h-dong Ho Jng-Jng Hng Y-R hn: H nsf nhnn fo h ow-lw flds hogh lll-l dobl-ss h xhngs wh xnl l Innonl onons n H nd Mss Tnsf Vol (). [8]. H. sl Rhologl ffs of non-nwonn bhvo of dslng flds on sbl of ovng nf n dl ol dsln hns n oos d. In. J. ng.. Vol (986). [9]. A. Nk nd H. Ko Bon ndd flow of non-nwonn flds ov non-sohl bod of b sh n fld sd oos d. Al.. Rs. Vol (99). []. K. N. Mh nd K. N. Ro Bon ndd flow of non-nwonn flds n oos d s vl l wh nonnfo sf h flx In. J. ng.. Vol. 97 (994). []. K. N. Mh nd K. N. Ro Bon ndd flow of non-nwonn flds ov non-sohl hozonl l bddd n oos d In. J. ng.. Vol (994). []. N. Nnoss: Unsd MHD flows n ong ls-vsos fld. Asohss nd s sn Vol (99). []. T. R Mh. Dhol nd A.. : Mon nd h nsf n h MHD sgnon on flow of vso-ls fld owd shng sf Mhn Vol (7). [4]. J.. Ro And N. K. hdh: H nsf b ln flow of n lso-vsos lqd long ln wll wh od son zh. J. hss Vol (98). [5]. H. Vn V. K. vnk. And.. dsh: Non-sl solons n Vsols hd flow nd h nsf ov oos shng sh wh nnl h gnon nd sss wok In. J. Modn hs Vol (7). [6]. OD. Mknd. Osls: MHD sd flow n hnnl wh sl bl bonds Ron. J. hs. Vol (6). [7]. K. D. ngh: x solon of n osllo MHD flow n hnnl flld wh oos d In. J. of Ald Mhns nd ngnng Vol (). [8]. M. M. Rhn nd M.. A. k: Unsd MHD flow of vso-ls oldod fld nd vng bod fo hogh ngl hnnl Blln of l Mhl o 96: 46 (4). [9]. A. K. ngh nd N.. ngh: MHD flow of ds vso-ls lqd hogh oos d bwn wo nlnd lll ls odng of Nonl Ad of ns Ind 66A: 4 (996). []. J.. Ms.. h H. J Rh : Flow nd h nsf of MHD vso-ls fld n hnnl wh shng wll: so lon o hodns. o Flds Vol. 7. (8). []. Msood Khn: Th Rlgh oks obl fo n dg n vsols fld wh fonl dvv odl Nonln Anlss: Rl Wold Alons Vol (9). [].. V. nn: hl on nd hl don ffs on n sd hd f onvon flow s n nfn vl l wh vbl son nd h so o snk IJMMA. Vol (). [].. V. nn D.h. Ksvh nd. Vnkn: Vsos sson nd hl don ffs on n nsd MHD onvon flow s s- nfn vl bl ovng oos l. Innonl Jonl of Mhl Ahv. Vol (). g
8 Mhd osllo flow of vso ls fld n oos hnnl wh hl on [4]..V. Nn nd.vnh Infln of vbl bl on nsd MHD onvon flow s s-nfn nlnd l wh hl don nd hl on Jonl of ngh nd Mss Tnsf Vol () [5]. Dll l Bbll Tlkd: Infln of flng hl nd ss dffson on nsd MHD bon-dvn onvon s vl sf wh hl on nd o ffs NN Vol () [6]. R. Mhj. nvs: A no on h nsf o MHD osllo flow n n s wv hnnl Innonl onons n H nd Mss Tnsf. Vol ( ). [7]. O. D. Mknd nd. Y. Mhon: H nsf o MHD osllo flow n hnnl flld wh oos d. Ro. J. hs. Vol (5). [8]. A... ogl W.. Vnvn nd.. ls: Dffnl oxon fo dv h nsf n non-ln qonsg gs n qlb. An Ins of Aons nd Asons Vol (968). RAH.5.4. = = N=.=5.=.=.J=4.=.g=. =.7=.H=. =. =. = Fg: ff of.4. N=.=.=.=.J=4.=.=. =.7=.H=.5 =. =. = on vlo Fg: ff of on vlo.4. N=5.=.=.=.J=. =.6=. =.7=.H=. =. =.=...5 N= N= N=.=.=.=. = Fg: ff of N on vlo Fg: 4 ff of N on T N=5.=.=.=.J=5. =.6=. =.7=.H=. =. =.=...8 J=. =.6=. =.7= J= J= Fg: 5 ff of J on vlo Fg: 6 ff of J on onnon = = J=. =.6=.=. = N=5.=.=.=.J=4.=.=. =.7=.H=.5 =. =. = g
9 Mhd osllo flow of vso ls fld n oos hnnl wh hl on Fg: 7 ff of on vlo Fg: 8 ff of on onnon.4.5. = = N=.=5.=.=.J=4.=.=. =.7=.H=. =. =. = Fg: 9 ff of.5 N=.=6.5=.5=. = on vlo Fg: ff of on T.6.4 H= = N=.=5.=.=.J=. =.6=. =.7=.H=. =. =.= Fg: ff of H on vlo.4. N=.=.=.=.J=. =.6=. =.7=.H=. =. =.= Fg: ff of on vlo.5 = N=.5=.=.=.J=5 =.6=. =.7=.H=. =. =.= N=.=5.=.=...4 = Fg: ff of on T Fg: 4 kn fons fo vss N=.=.=.=.J=5. =.6=. =.7=.H=. =. =.=...44 N=.=.=.=.J=5. =.6=. =.7s=.H=. =. =.= H= Fg: 5 kn fons fo H vss.4 = Fg: 6 kn fons fo vss 4 g
10 N h Mhd osllo flow of vso ls fld n oos hnnl wh hl on J= N=.=.=.=. = N= J=. =.6=.= Fg: 7Nssl nb fo N vss Fg: 8 hwood nb fo J vss H= Nwonn fld Non-Nwonn fld Nwonn fld Non-Nwonn fld H= Fg: 9 ff of H on vlo Fg: kn fons fo H vss 5 g
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