The radiation effect on the unsteady MHD convection flow through a nonuniform horizontal channel

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1 Availabl onlin a Plagia Rsarch Librar Advancs in Applid Scinc Rsarch :-8 ISSN: CODEN USA: AASRFC h radiaion ffc on h nsad MHD convcion flow hrogh a nonniform horizonal channl * K. Jaarami Rdd K. Sniha and M. Jaabharah Rdd Dp. of Mahmaics Priadarsini Insi of chnolog irpai Chioor Dis. A.P.India Dp. of Mahmaics Malla Rdd Enginring Collg for Womn Maisammagda Hdrabad India Dp. of Mahmaics SKI Srikalahashi Chioor Dis. A.P. India _ ABSRAC In his chapr w discss h radiaion ffc on h nsad MHD convcion flow hrogh a non-niform horizonal channl. h nsadinss is d o h imposd oscillaor fl on h convcion flow hrogh h nonniform channl. h prrbaion analsis is carrid o wih h slop of h bondar as h prrbaion paramr. h vloci and mprar profils wr plod and hir bhavior is discssd in dail. h Srss and h avrag Nssl nmbr ar also calclad and ablad for hs ss of paramrs. K Words: MHD Poros Mdim Prandl Nmbr Grashof Nmbr and Radiaion Paramr. _ INRODUCION Unsad convcion flows pla an imporan rol in arospac chnolog rbo-machinr and chmical Enginring. Sch flows aris d o ihr nsad moion of bondar or bondar mprar. Unsadinss ma also b d o oscillaor fr sram vloci or mprar. hs oscillaor fr convciv flows ar imporan from chnological poin of viw. Nanda and Sharma [ ] hav discssd h nsad fr convciv flow pas a smi-infini pla wih oscillaor wall mprar and shown h isnc of similari solion. Lar Sondalgkar and Pop [] hav solvd his problm sing momnm-ingral mhod. Kllhr and Yang [] hav sdid diffrn aspcs of his problm. h obaind similar solions of h laminar fr convcion bondar lar qaions for h innr and h or sad flow along a vrical had pla whos mprar oscillas whn h man srfac mprar varis as powr n of disanc from h lading dg. h corrsponding smi-infini horizonal pla whos mprar oscillas abo a consan man has bn sdid b Mhri and Mai [5] and Mial [6]. Mrkin [7] and Zonian [8] hav also analzd fr convcion ffcs on an infini horizonal clindr whn is mprar oscillas harmonicall wih im. Rcnl wo problms on fr convcion hav bn solvd b Pop [9 ]. Mhri and Mai [5] hav considrd h fr convcion flow and ha ransfr along a smi-infini horizonal pla whn pla mprar oscillas abo a consan man. Vrma and Singh [] hav analsd h fr convcion flow along a horizonal pla oscillaor in is own plan. h ffcs of srfac mprar oscillaions on h skin fricion and h ha ransfr from a srfac o h srronding flow is of spcial inrs o h ha ransfr nginring. h ffc of pla mprar oscillaions on fr convcion flow along h smi-infini horizonal pla has bn considrd b Sharma and Mishra [] basd on Lighhill s chniq and h sad sa solions wr obaind sing Karman-Polhasn mhod. Vajravl and Nafh [] hav invsigad h inflnc of h wall wavnss on fricion and prssr drop of h gnralizd co flow. Vajravl and Sasr [] hav analsd h fr convciv ha ransfr in a viscos incomprssibl flid confind bwn long vrical wav wall and a paralll fla wall in h prsnc of a consan ha sorc. Lar Vajravl and Dbnah [5] hav ndd his sd o convciv flow in a vrical wav channl in for diffrn gomrical configraions. Plagia Rsarch Librar

2 K. Jaarami Rdd al Adv. Appl. Sci. Rs. :-8. Formlaion of h problm W considr h nsad moion of a viscos incomprssibl lcricall condcing flid hrogh a poros mdim in a horizonal channl bondd b wav walls in h prsnc of a consan ha sorc /sink. A niform magnic fild of srngh Ho is applid normal o h walls. h Bossinsq approimaion is sd so ha h dnsi variaion will b considrd onl in h boanc forc. h viscos Darc and Ohmic dissipaions ar nglcd in comparison o h flow b condcion and convcion. Also h kinmaic viscosi ν h hrmal condcing k ar rad as consans. W choos a rcanglar Carsian ssm O wih -ais in h dircion of moion and -ais in h vrical dircion and h walls ar akn a ± Lfδ/L whr L is h disanc bwn h walls f is a wic diffrniabl fncion and δ is a small paramr proporional o h bondar slop. A linar dnsi mprar variaion is assmd wih ρ and ar h dnsi and mprar in h qilibrim sa. h flow is mainaind b an oscillaor volm fl ra for which a characrisic vloci is dfind as iϖ q k L Lf Lf d h qaions govrning h nsad magno hdrodnamic flow and ha ransfr in Carsian coordina ssm Oz in h absnc of an inp lcric fild ar Eqaion of conini. v. Eqaion of linar momnm p σµ H ρ v µ ρ o v v v p v v ρ v µ ρg o.. Eqaion of nrg ρ C qr v λ Q p.5 Eqaion of sa ρ ρ ρ.6 whr ρ is h dnsi of h flid in h qilibrim sa is h mprar and in h qilibrim sa v ar h vloci componns along O dircions p is h prssr is h mprar in h flow rgion ρ is h dnsi of h flid µ is h consan cofficin of viscosi In h qilibrim sa Whr p ρ g p p p p.7 bing h hdrodnamic prssr and in his sa h mprar gradin balancs h ha fl gnrad b sorc Q. D D h bondar condiions for h vloci and mprar filds ar v on -L fδ/l Plagia Rsarch Librar

3 K. Jaarami Rdd al Adv. Appl. Sci. Rs. :-8 Plagia Rsarch Librar v on L fδ/l.8 Invoking Rossland approimaion Brwsra for h radiaiv fl w g q R r σ.9 panding in alor sris abo and nglcing highr ordr rms 9a. In viw of h conini qaion. w dfin h sram fncion as v. Eliminaing prssr p from qaions. &. and sing. h qaions govrning h flow in rms of ar ] [ H g o ρ σµ ν. 6 Q k C R p σ θ ρ. Inrodcing h non-dimnsional variabls in. &. as ql L L Ψ Ψ / / / θ ϖ. h corrsponding bondar condiions i k ϖ f on sa h δ θ f on δ θ a θ.5 Whr ν ql R h Rnolds nmbr ν L g G h Grashof nmbr k µ C p Ρ h Prandl nmbr k QL α h Ha sorc paramr k N R σ h radiaion paramr

4 K. Jaarami Rdd al Adv. Appl. Sci. Rs. :-8. Solion of h problm Inrodc h ransformaion sch ha δ δ hn ~ O δ ~ O For small vals of δ<< h flow dvlops slowl wih aial gradin of ordr δ and hnc w ak ~ O. W adop h prrbaion schm and wri i i k δ k. i i θ θ k θ δ θ k θ h corrsponding bondar condiions ar θ θ ± ± θ ± ± ± h local ra of ha ransfr cofficin Nssl nmbr N on h walls has bn calclad sing h formla N whr m θ ± f θ θ m w θ.5 θ d and h corrsponding prssions ar DISCUSSION OF HE NUMERICAL RESULS h primar aim of or analsis is o invsiga h radiaion ffc on h bhavior of h mprar indcd boanc forc aking in o accon h ffc of srfac gomr and wall mprar raio. h flow is analsd for diffrn ss of h paramrs GRMαγ and N govrning h flow. I shold b nod ha h flow is basicall asmmric d o disinc srfac mprars. For compaion prpos w assm h bondaris o b ± f ± and > corrsponds o dilad channl and < corrsponds o consricd rdc h bondaris o ±. W confin or anion o dilad f channl. h ransformaion channl. h non-niformi in h bondar givs ris o h scondar ransvrs flow and hnc h gnral parn of h flow can b jdgd b h bhavior of h Rsslan of primar and scondar vlociis. h compaion of h individal vloci componns wold nabl s o invsiga h ffc of ach bod forc acing on h flow and is rlad inflnc on h primar and scondar flows. Plagia Rsarch Librar

5 K. Jaarami Rdd al Adv. Appl. Sci. Rs. : Fig [] Variaion of wih G R5 M α.5 N π/ π/ I II III IV V VI G R5 R7 R Fig [] Variaion of wih R G5 M α.5 N π/ π/ I II III R 5 7 Plagia Rsarch Librar 5

6 K. Jaarami Rdd al Adv. Appl. Sci. Rs. : M M5 M Fig [] Variaion of wih M G5 R5 α.5 N π/ π/ I II III M Fig [] Variaion of wih α G R5M.5 N π/ π/ I II III IV V VI α Plagia Rsarch Librar 6

7 K. Jaarami Rdd al Adv. Appl. Sci. Rs. : Fig [5] Variaion of wih G5 R5 Mα N π/ π/ I II III IV As in h cas of rslan flow h primar vloci is posiiv for all G>. In h cas of cooling of h channl walls w find ha h vloci changs from posiiv o ngaiv nar h lowr bondar - hr b hibiing a rvrsal flow for G and for highr vals of G w noic h rvrsal flow in h nir flow rgion. his rgion nlargs wih incras in G < wih maimm occrring a -.. For G> h maimm of occrs a.6 and his poin of maimm vloci drifs owards h mid rgion for highr G fig.. Fig. [] shows h variaion of wih Rnolds nmbr R. I is fond ha for a smallr val of R5 hr is no rvrsal flow b for highr R7 h rvrsal occrs in h midrgion and for sill highr vals of R h rvrsal flow appars in h nir flow rgion. rdcs wih R 7 and nhancs for highr R. Fig. [] indicas ha h rvrsal flow occrs in h nir flow rgion for highr vals of M 5 and his nlargs wih incras in M. h variaion of wih h ha sorc/sink paramr is hibid in fig.[]. I is noicd ha for α> hr is no rvrsal flow an whr in h flid rgion whil for α< w noic rvrsal flow in h nir flow rgion and his nlargs wih incras in α <. h inflnc of srfac gomr on h flow phnomna is hibid in fig.[5]. h rvrsal flow which appars in nir flow rgion for. disappars for highr.6 and rappars in h flid rgion for.9. Highr h dilaion of h channl walls largr h magnid of. REFERENCES []. Nanda R.S.Sharma V.P. J. Flid Mch []. Nanda R.S.Sharma V.P. AIAA.J []. Sondalgkar V.M. Pop I In. J. Ha Mass ransfr []. Kllhar M.D. Yang K.. ZAMP [5]. Mhri P.K. Mai M.K. In. J. Ha Mass ransfr [6]. Mial V. In. Pr & Appl. Mahs [7]. Mrkin J.H. J. Flid Mch [8]. Zonian R. Kh. J. Mch [9]. POP I. C.R. Acad. Sci Plagia Rsarch Librar 7

8 K. Jaarami Rdd al Adv. Appl. Sci. Rs. :-8 []. POP I. Rozpr. Inz []. Vrma R.L. Singh P. J. Phs []. Sharma V.P. Mishra K.P. A.M []. Vajravl K. Alih Nafh. In J. Mahs. & Mah. Sci. 98 V. 85. []. Vajravl K. Sasr K.S. J. Flid. Mch 978 V [5]. Vajravl K. Dbnah l. Aca. Mch. 986 V.59. Plagia Rsarch Librar 8