[Lavana t al. (9): Stmbr ] ISSN: 9-9 Imact Factor (PIF):. MHD FLOW OVER N INFINIE VERIL OSILING POROS PLE WIH RDIION HEMIL REION ND DFOR EFFE B Lavana * P.Srivni P. Ramirdd * Dartmnt of Mathmatics Priadarshini ollg of Enginring. Nllor cadmic consltant Vikrama Simhari nivrsit kavali Rsarch scholar of Krishna nivrsit Machaliatnam P. orrsondnc thor: lavanab@gmail.com Kords: Dfor ffct Radiation chmical raction hat transfr MHD vrtical lat. bstract his ork analss th non linar MHD flo hat and mass transfr charactristics of an incomrssibl viscos lctricall condcting and Bossinsq flid ovr a vrtical oscillating oros rmabl lat in rsnc of homognos chmical raction of first ordr thrmal radiation and Dfor ffcts. h roblm is solvd analticall sing th rtrbation tchniq for th vlocit th tmratr and th concntration fild. h xrssion for th skin friction Nsslt nmbr and Shrood nmbr ar obtaind. h ffcts of varios thrmo-hsical aramtrs on th vlocit tmratr and concntration as ll as th skinfriction cofficint Nsslt nmbr and Shrood nmbr has bn comtd and discssd qalitativl. Introdction Magntohdrodnamics (MHD) is th branch of continm mchanics hich dals ith th flo of lctricall condcting flids in lctric and magntic filds. Man natral hnomna and nginring roblms ar orth bing sbjctd to an MHD analsis. Magntohdrodnamic flos hav alications in mtorolog solar hsics cosmic flid dnamics astrohsics gohsics and in th motion arth s cor. In addition from th tchnological oint vi MHD fr convction flos hav significant alications in th fild of stllar and lantar magntoshr aronatical lasma flos chmical nginring and lctronics. n xcllnt smmar of alications is givn b Hgs and Yong[]. Ratis[] stdid mathmaticall th cas of tim varing to dimnsional natral convctiv flo of an incomrssibl lctricall condcting flid along an infinit vrtical oros lat mbddd in a oros mdim. Hlm[] analzd MHD nstad fr convction flo ast a vrtical oros lat mbddd in oros mdim. Elabashbsh[] stdid hat and mass transfr along a vrtical lat in th rsnc of magntic fild. hamkha[] analzd an nstad MHD convctiv viscos incomrssibl hat and mass transfr along a smi-infinit vrtical oros lat in th rsnc of transvrs magntic fild thrmal and concntration boanc ffcts. h radiation ffcts hav imortant alications in hsics and nginring articlarl in sac tchnolog and high tmratr rocsss. Bt vr littl is knon abot th ffcts of radiation on th bondar lar. hrmal radiation ffcts on th bondar lar ma la imortant rol in controlling hat transfr in olmr rocssing indstr hr th qalit of th final rodct dnds on th hat controlling factors to som xtnt. High tmratr lasmas cooling of nclar ractors liqid mtal flids or gnration sstms ar som imortant alications of radiativ hat transfr. England and Emr [] hav stdid th radiation ffcts of an oticall thin gra gas bondd b a stationar lat. Ratis and Massalas [] invstigatd th ffcts of radiation on th oscillator flo of a gra gas absorbing-mitting in rsnc indcd magntic fild and analtical soltions r obtaind ith hl of rtrbation tchniq. h fond ot that th man vlocit dcrass ith th Hartmann nmbr hil th man tmratr dcrass as th radiation incrass. h hdrodnamic fr convctiv flo of an oticall thin gra gas in th rsnc of radiation hn th indcd magntic fild is takn into accont as stdid b Ratis t al. [] sing rtrbation tchniq. h concldd that th vlocit and indcd magntic fild incras as th radiation incrass. Hossain t al. [9] dtrmind th ffct of radiation on th natral convction flo of an oticall dns incomrssibl flid along a niforml hatd vrtical lat ith a niform sction. Magntohdro-dnamic mixd fr forcd hat and mass convctiv stad incomrssibl laminar bondar lar flo of a gra oticall thick lctricall condcting viscos flid ast a smi-infinit vrtical lat for high tmratr and concntration diffrncs hav stdid b Emad and Gamal []. Orhan and Kaa [] invstigatd th mixd convction hat transfr abot a rmabl vrtical lat in th rsnc of magnto and thrmal radiation ffcts sing th Kllr box schm an fficint and accrat finit-diffrnc schm. h concldd that an incras in th radiation aramtr dcrass th local skin friction aramtr and incrass th local hat transfr aramtr. Ghosh t al. [] considrd an xact soltion for th hdromagntic natral convction bondar lar flo ast an infinit vrtical flat lat ndr th inflnc of a transvrs magntic fild ith magntic indction ffcts and th transformd ordinar diffrntial qations ar solvd xactl. s th imortanc of radiation in th filds of arodnamics as ll as sac scinc tchnolog th rsnt std is motivatd toards this dirction. htt: //.ijrsm.com () Intrnational Jornal of Rsarch Scinc & Managmnt []
[Lavana t al. (9): Stmbr ] ISSN: 9-9 Imact Factor (PIF):. It has bn fond that nrg flx can b gnratd not onl b tmratr gradint bt also b concntration gradint as ll. h nrg flx casd b concntration gradint is calld Dfor ffct. hs ffcts ar vr significant hn th tmratr and concntration gradint ar vr high. nghl t al. [] stdid th Dfor and Sort ffcts on fr convction bondar lar ovr a vrtical srfac mbddd in a oros mdim. Postlnic [] analzd th inflnc of magntic fild on hat and mass transfr from vrtical srfacs in oros mdia considring Sort and Dfor ffcts. lam t al. [] invstigatd th Dfor ffcts on stad MHD mixd convctiv and mass transfr flo ast a smi-infinit vrtical lat. hamkha and Bn- Nakhi [] analzd MHD mixd convction-radiation intraction along a rmabl srfac immrsd in a oros mdim in th rsnc of Dfor s ffcts. In man chmical nginring rocsss a chmical raction btn a forign mass and th flid dos occrs. hs rocsss tak lac in nmros indstrial alications sch as th olmr rodction th manfactring of cramics or glassar th food rocssing and so on Singh t.al[] analzd th ffcts of chmical raction and radiation absortion on MHD fr convctiv hat and mass transfr flo ast a smi-infinit vrtical moving lat ithtim dndnt sction. Ibrahim t.al[] rsntd th ffct of chmical raction and radiation absortion on MHD flo ast a continosl moving rmabl srfac ith hat sorc and tim dndnt sction. h main objctiv of this ar as to xlor th ffcts of Dfor radiation chmical raction on MHD flo flid ovr an infinit vrtical oscillating oros lat. h magntic fild is imosd transvrsl to th lat. h tmratr and concntration of th lat is oscillating ith tim abot a constant nonzro man val. h dimnsionlss govrning qations involvd in th rsnt analsis ar solvd sing a closd analtical mthod and discssd qalitativl and grahicall. Mathmatical formlation. nstad MHD flo of a viscos incomrssibl flid ast along a vrtical oscillating lat ith hrmal radiation and mass transfr ffcts on variabl tmratr and also ith variabl mass diffsion in th rsnc of transvrs alid magntic fild has bn stdid. h axis is takn along th lat in th vrtical ard dirction and th -axis is takn normal to th x lat. Initiall it is assmd that th lat and flid ar at th sam tmratr lvl htt: //.ijrsm.com () Intrnational Jornal of Rsarch Scinc & Managmnt [] in th stationar condition ith concntration at all th oints. t tim t> th lat is givn an oscillator motion in its on lan ith vlocit cos( t th sam tim th lat tmratr is raisd linarl ith tim and also mass is diffsd from th lat linarl ith tim. transvrs magntic fild of niform strngth B is assmd to b alid normal to th lat. h indcd magntic fild and viscos dissiation is assmd to b ngligibl as th magntic Rnold s nmbr of th flo is takn to b vr small. h flid considrd hr is gra absorbing/mitting radiation bt a non-scattring mdim. hn b sal Bossinsq s aroximation th nstad flo is govrnd b th folloing qations. t t t K g D K * ( ) g ( ) B k ( c c q r ) DmK S P r () h bondar conditions for th vlocit tmratr and concntration filds ar: t : t Whr cos( t ) is th vlocit in th ( as x ( dirction ) n t ( Kis th rmabilit aramtr () ) () n t at ). t () is th volmtric cofficint of thrmal * xansion is th volmtric cofficint of xansion for concntration is th dnsit is th lctrical condctivit K- is th thrmal condctivit g is th acclration d to gravit is th tmratr is th flid tmratr at h lat is th flid tmratr in th fr stram is th scis concntration is th scific hat at constant rssr is th
[Lavana t al. (9): Stmbr ] ISSN: 9-9 Imact Factor (PIF):. scis concntration in th fr stram th radiativ hat flx. h radiant absortion for th cas of an oticall thin gra gas is xrssd as q r a ( a ) is th scis concntration at srfac D is th chmical molclar diffsivit Whr an ar th Stfan-Boltzmann constant and th Man absortion cofficint rsctivl. assm that th tmratr diffrncs ithin th flo ar sfficintl small so that aftr sing alor s sris to xand folloing aroximations. t K abot th fr stram tmratr a ( ) () can b xrssd as a linar fnction of and nglcting highr ordr trms. his rslts in th In ordr to rit th govrning qations and th bondar conditions in dimnsionlss formth folloing non dimnsional () q r is () qantitis ar introdcd. t t v v K K v * vg Gm v Pr k Sc v D ) K rv a v Kr R k v B v vg( M Gr DmK D V ) v S t P () sing th transformations () th non dimnsional forms of () () and () ar Gr Gm ( M ) t K t Pr R D Pr Kr t Sc (9) () () h corrsonding bondar conditions ar; cos( t) t t at = as () Whr Gr Gm M K R Pr Kr Sc D ar th magntic aramtr rmabilit Grashof nmbr for hat transfr Grashof nmbr for mass transfr Prandtl nmbr chmical raction aramtr Schmidt nmbr radiation aramtr and dfor rsctivl. htt: //.ijrsm.com () Intrnational Jornal of Rsarch Scinc & Managmnt []
[Lavana t al. (9): Stmbr ] ISSN: 9-9 Imact Factor (PIF):. Mthod of soltion In ordr to rdc th abov sstm of artial diffrntial qations to a sstm of ordinar diffrntial qations in dimnsionlss form assm th trail soltion for th vlocit tmratr and concntration as: t) ( ) ( t) ( ) ( t) ( ) ( it it it () () () Sbstitting Eqations () () and () in qations (9) () and () obtain: Gr Gm o DPr o o " Hr th rims dnot th diffrntiation ith rsct to. h corrsonding bondar conditions can b rittn as cos( ) it t i t t i t t at = as (9) h analtical soltions of qations () - () ith satisfing th bondar conditions (9) ar givn b ( ) ( 9 ) ( ) ( ) ( ) ( t ) it it it In vi of th abov soltions th vlocit tmratr and concntration distribtions in th bondar lar bcom ( ) 9 ( ) ( ) t It is no imortant to calclat th hsical qantitis of rimar intrst hich ar th local all shar strss th local srfac hat and mass flx. Givn th vlocit fild in th bondar lar can no calclat th local all shar strss (i.. Skinfriction) is givn b * f and in dimnsionlss form obtain 9 From tmratr fild no std th rat of mass transfr hich is givn in non-dimnsional form as: N From concntration fild no std th rat os mass transfr hich is givn in non-dimnsional form as: Sh t () () () () () () () () () htt: //.ijrsm.com () Intrnational Jornal of Rsarch Scinc & Managmnt []
[Lavana t al. (9): Stmbr ] ISSN: 9-9 Imact Factor (PIF):. Whr Sc( Kr i i d Pr t t ) i i t 9 cost Pr R M i t it K Gr Gr Gm Rslts and discssions: o find ot th rslts analtical comtation has bn carrid ot sing th mthod dscribd in th rvios aragrah of varios govrning aramtrs naml thrmal Grashof nmbr Gr. Modifid Grashof nmbr Gm th magntic fild aramtr M rmabilit aramtr K randtl nmbr Pr radiation aramtr R D Dfor aramtr Schmidt nmbr Sc and Kr chmical raction aramtr. in rsnt std th folloing dfalt aramtr ar adotd for comtations Gr= Gm= M= K=. Pr=. Sc=. D=. Kr=. indicatd on th aroriat grah. In ordr to gt a hsical insight in to th roblm th ffct of varios govrning aramtrs on th hsical qantitis ar comtd and rrsntd in figrs - and discssd in dtail. For th cas of diffrnt vals of thrmal Grashof nmbr th vlocit rofils on th bondar lar ar shon in Fig- shos th Vlocit rofils for diffrnt val of magntic aramtr. s th magntic aramtr incrass vlocit dcrass Fig.. s xctd it is obsrvd that an incras in Grashof nmbr lads to incras in th vals of vlocit d to nhancmnt in boanc forc. Hr th ositiv vals of Grashof nmbr corrsond to cooling of th srfac. Fig- shos th vlocit rofil for radiation aramtr as radiation aramtr incrass vlocit dcrass. of rsisting forc slos don th flid vlocit as shon in this figr. th vlocit rofils for diffrnt vals of th radiation aramtr clarl as radiation aramtr incrass th ak vals of th vlocit tnds to incrass. Fig. rrsnts tical vlocit rofils in th bondar lar for varios vals of th modifid Grashof nmbr hil all othr aramtrs ar kt at som fixd vals. h vlocit distribtion attains a distinctiv maximm val in th vicinit of th lat srfac and thn dcras rorl to aroach th fr stram val. s xctd th flid vlocit incrass and th ak val mor distinctiv d to incras in th concntration boanc ffcts rrsntd b modifid Grashof nmbr. his is vidnt in th incras in th val of vlocit as modifid Grashof nmbr incrass. For diffrnt vals of th Schmidt nmbr th vlocit rofils ar lottd in Fig.. It is obvios that an incras in th Schmidt nmbr rslts in dcras in th vlocit ithin th bondar lar. Fig. illstrats th bhavior vlocit for diffrnt vals of chmical raction aramtr. It is obsrvd that an incras in lads to a dcras in th vals of vlocit Kr. vals of th vlocit tnds to incras. Fig. shos th vlocit rofils for diffrnt vals of th rmabilit aramtr clarl as rmabilit aramtr incrass th ak vals of th vlocit tnds to incras. For diffrnt vals of tim on th vlocit rofils ar shon in Fig.. It is noticd that an incras in th vlocit ith an incrasing tim t. Fig.9 illstrats th tmratr rofils for diffrnt vals of Prandtl nmbr. It is obsrvd that th tmratr dcras as an incrasing th Prandtl nmbr. h rason is that smallr vals of Prandtl nmbr ar qivalnt to incras in th thrmal condctivit of th flid and thrfor hat is abl to diffs aa from th hatd srfac mor raidl for highr vals of Prandtl nmbr. Hnc in th cas of smallr Prandtl nmbr th thrmal bondar lar is thickr and th rat of hat transfr is rdcd. Fig. has bn lottd to dict th variation of tmratr rofils for diffrnt vals of radiation aramtr b fixing othr hsical aramtrs. From this Grah obsrv that tmratr dcras ith incras in th radiation aramtr R. for diffrnt vals of dfor aramtr on th tmratr rofils ar shon in Fig-. it is noticd that an incras in th tmratr ith th incras in dfor armtr. Fig. dislas th ffct of Schmidt nmbr Sc on th concntration rofils rsctivl. s th Schmidt nmbr incrass th concntration dcras. Fig. illstrats th concntration rofils for diffrnt vals of chmical raction aramtr. It is obsrvd that concntration dcrass ith th incras in chmical raction aramtr. From abl. shos th incras in magntic fild aramtr incras in th skin friction. abl. indicats th incras in radiation aramtr shos th incras in th skin friction and Nsslt nmbr.abl. Dislas th incras in Prandtl nmbr dislas th incras in skin friction and Nsslt nmbr. abl-. Effcts of dfor shos th dcras in Nsslt nmbr.. t=i/. ll grahs thrfor corrsond to ths vals nlss scicall htt: //.ijrsm.com () Intrnational Jornal of Rsarch Scinc & Managmnt []
[Lavana t al. (9): Stmbr ] ISSN: 9-9 Imact Factor (PIF):. M=... Fig -. Vlocit rofils for diffrnt vals of magntic aramtr. 9 Gr=... Fig-. Vlocit rofils for diffrnt vals of Grash of nmbr R =... Fig-. Vlocit rofils for diffrnt vals of radiation aramtr. Gm =... Fig-. Vlocit rofils for diffrnt vals of modifid Grashof nmbr htt: //.ijrsm.com () Intrnational Jornal of Rsarch Scinc & Managmnt []
[Lavana t al. (9): Stmbr ] ISSN: 9-9 Imact Factor (PIF):. Sc=....... Fig-. Vlocit rofils for diffrnt vals of Schmidt nmbr. Kr =....... Fig-. Vlocit rofils for diffrnt vals of hmical raction aramtr. K =....... Fig-. Vlocit rofils for diffrnt vals of rmabilit aramtr. 9 t =....... Fig-. Vlocit rofils for diffrnt vals of tim. htt: //.ijrsm.com () Intrnational Jornal of Rsarch Scinc & Managmnt []
[Lavana t al. (9): Stmbr ] ISSN: 9-9 Imact Factor (PIF):. r =....... Fig-9. mratr rofils for diffrnt vals of Prandtl nmbr. R=... Fig-. mratr rofils for diffrnt vals of Radiation aramtr. D=... Fig-. mratr rofils for diffrnt val of Dfor ffct... sc=............ Fig-. concntration rofils for diffrnt vals Schmidt nmbr. htt: //.ijrsm.com () Intrnational Jornal of Rsarch Scinc & Managmnt []
[Lavana t al. (9): Stmbr ] ISSN: 9-9 Imact Factor (PIF):..... Kr =........... Fig-. concntration rofils for diffrnt vals chmical raction aramtr. abl-. Effcts of magntic aramtr on skin friction. M.... f.9...9 abl-. Effcts of Radiation aramtr on skin friction and nsslt nmbr. R f N........9...9. abl-. Effcts of Prandtl nmbr on skin friction and Nsslt nmbr. Pr f N.... -.9 -.9....9.. abl-. Effcts of dfor nmbr on Nsslt nmbr. D.... N.9...9 htt: //.ijrsm.com () Intrnational Jornal of Rsarch Scinc & Managmnt [9]
[Lavana t al. (9): Stmbr ] ISSN: 9-9 Imact Factor (PIF):. Rfrncs. Hgs WF and ong FJ. h lctro-magnto dnamics of flids. Jhon il and sons N York.. Ratis. Flo throgh a oros mdim in th rsnc of magntic fild. Intrnational jornal of nrg Rsorcs vol..9-.. Hlm K.MHD nstad fr convction flo ast a vrtical oros lat.zmm vol..-.. Elabashbsh EM. Hat and mass transfr along a vrtical lat ith variabl tmratr and concntration in th rsnc of magntic fild. Intrnational jornal of nginring scincs vol..-.. hamka J. nstad MHD convctiv hat and mass transfr ast a smi infinit vrtical rmabl moving lat ith hat absortion. Intrnational jornal of Enginring scincs vol..-.. England W. G. and Emr. F. (99): hrmal radiation ffcts on th laminar fr convction bondar lar of an absorbing gas J. of Hat ransfr Vol. 9. -.. Ratis. and Massalas. V. (99): Magntohdrodnamic flo ast a lat b th rsnc of radiation Hat and Mass transfr Vol.. -9.. Ratis.. Prdikis. and Lontitsis. (): Effcts of radiation in an oticall thin gra gas floing ast a vrtical infinit lat in th rsnc of a magntic fild Hat and mass ransfr Vol. 9. -. 9. Hossain M.. lim M.. and Rs D.. S. (999): h ffct of radiation on fr convction from a oros vrtical lat Int. J. Hat and Mass ransfr Vol. No.. -9.. Emad M.. and Gamal El-Din.. (): hrmal radiation ffcts on magntohdrodnamic flo ast a smi-infinit vrtical lat in th rsnc of mass diffsion an. J. Phs. Vol. No.. P... Orhan. and hmt K. (): Radiation ffct on MHD mixd convction flo abot a rmabl vrtical lat Hat Mass ransfr Vol.. 9... S. K. Ghosh Bég O.. and Zco J. (9): Hdromagntic fr convction flo ith indcd magntic fild ffcts Phsics and stronom Volm Nmbr -... M. nghl H.S. akhar I. Po () Dfor and Sort ffcts on fr convction bondar lar ovr a vrtical srfac mbddd in a oros mdim Std. niv. Babs-Bolai Math...... Postlnic () Inflnc of a magntic fild on hat and mass transfr b natral convction from vrtical srfacs in oros mdia considring Sort and Dfor ffcts Int. J. Hat Mass ransfr... 9. M.S. lam M.M. Rahman M.. Malq M. Frdos () Dfor and Sort ffcts on stad MHD combind frforcd convctiv and mass transfr flo ast a smi-infinit vrtical lat hammasat Int. J. Sci. chnol. ().....J. hamkha. Bn-Nakhi () MHD Mixd convction-radiation intraction along a rmabl srfac immrsd in a oros mdim in th rsnc of Sort and Dfor s Effcts Hat Mass ransfr... Singh H Ram and Kmar R: std of th ffct of chmical raction and radiation absortion on MHD convctiv hat and mass transfr flo ast a smi-infinit vrtical moving lat ith tim dndnt sction. Intrnational jornal of aid Mathmatical and Mchanics Vol():-.. Ibrahim FS Elai M and Bakr : ffct of chmical raction and radiation absortion on nstad MHD fr convction flo ast a smi-infinit vrtical rmabl moving lat ith hat sorc/sction. ommnication in Non-Linar Scinc and Nmrical Simlation Vol.-. htt: //.ijrsm.com () Intrnational Jornal of Rsarch Scinc & Managmnt []