Casson Flow of MHD Fluid Moving. Steadily with Constant Velocity

Transcription

1 Alid atmatical Scincs Vol. 9 5 no HIKAI Ltd tt:d.doi.org.988ams Casson Flow of HD Flid oing Stadil wit Constant Vlocit Y. Immanl Baji Pll and C. K. Kirbasankar 3 3 Dartmnt of atmatics Satabama Unirsit Cnnai India Dartmnt of atmatics S Unirsit Cnnai India Corigt 5 Y. Immanl Baji Pll and C. K. Kirbasankar. Tis is an on accss articl distribtd ndr t Crati Commons Attribtion Licns wic rmits nrstrictd s distribtion and rrodction in an mdim roidd t original work is rorl citd. Abstract T aim of t rsnt ar is to std t Casson flow of iscos flid moing stadil wit constant locit ndr t inflnc of niform magntic fild alid rndiclarl to t flow of iscos liqid. Wit t l of similarit transformations t gorning artial diffrntial qations corrsonding to t momntm ar rdcd to linar ordinar diffrntial qations. Analtical soltion of t qation is obtaind. Kwords: HD Casson flid Similarit transformation Introdction In dro magntic flow w std of t flow of lctricall condcting flid in rsnc of awll lctromagntic fild. T flow of condcting flid is ffctil cangd b t rsnc of t magntic fild wr as t magntic fild is also rtrbd d to t motion of t condcting flid. Tis nomna is calld magnto drodnamics and in sort writtn as HD. Tr ar widr alications in nginring tcnolog cosmolog astrosics and otr alid scincs. On accont of arid ractical alications of HD flow roblms in cannls trog oros mdim man rsarcrs: Sing & am [] am & Sing [] Simomra [6] and otrs a aid tir attntions in tis dirction. Two dimnsional flow of flid btwn aralll lats tat ar moing smmtricall abot t lin of smmtr as its alications sciall in olmr

2 5 Y. Immanl t al. rocssing modling of snttics transortation insid liing bodis dromcanical macinr and comrssioninjction rocsss. an scintists a contribtd tir fforts towards t bttr ndrstanding of ts ts of flows. S Gans t al. [5] and A oamd Ismail t al.[] stdid t roblm for nstad flow of iscos flid btwn aralll oros lats. Unstad magnto drodnamic flow of Casson flid btwn aralll lat cannl wit at sorc as bn instigatd b Kirbasankar C K t al. [3]. Initiatd b t riosl stdid sr t Casson flow of an incomrssibl flid btwn two aralll lats and its alications in t rsnt ar flow of a Casson flid btwn aralll lats is instigatd. T gorning artial diffrntial qations ar conrtd into t linar ordinar diffrntial qations sing t sitabl similarit transformations. Formlation of t Problm Considr t two-dimnsional flow of an incomrssibl Casson flid or an onntiall srinkingstrtcing oros st at = wit t flow bing confind in >. T rological qation of stat for an isotroic and incomrssibl flow of a Casson flid as follows: c ij c B c ij B ij wr ij is t i j)-t comonnt of t strss tnsor = ij ij and ij ar t i j)-t comonnt of t dformation rat is t rodct of t comonnt of dformation rat wit itslf c is a critical al of tis rodct basd on t non- Nwtonian modl B is lastic dnamic iscosit of t non-nwtonian flid and is t ild strss of t flid. T qations gorning t flow ar: ) ) 3)

3 Casson flow of HD flid moing stadil wit constant locit 55 wr υ is t kinmatic flid iscosit ρ is t flid dnsit β = μb πc is t Casson aramtr. T bondar conditions for t flow roblm ar: ) ) at ) ) at ) If and w b t comonnt of t flid locit along t coordinat as ais ais and z ais rsctil tn assming t nidirctional flow i.. dirction w a w and T continit qation ilds w bt w z z i.. from wic w infr tat is indndnt of. Tis maks t non-linar trm q ) q in t Nair-Stoks qation zro. W nglct bod forcs fb wic ar mainl d to grait in t Nair-Stoks qations. Assming t flow ariabls ar indndnt of z dirction It mans tat t gorning qations for tis flow ar 5) 6) i.. is a fnction of onl. Assm t caractristic locit moing rndiclar to t flid at a constant rssr gradint. Now t gorning qations ar If t flow is inflncd b transrs magntic fild tn t gorning qation will b B 7) T bondar conditions for gnralizd flow ar

4 56 Y. Immanl t al. Introdcing t following non-dimnsional qantitis in qation 7) w gt Diffrntiat t qation wit rsct to d d i.. P d d wr P is a ositi constant. Now t gorning qation will b P 8) wit bondar conditions 3 Gnral Soltion of Formation of t Problm Eqation 8) rdcs to t form P D D ) ) 9) wr wit bondar conditions Soltion of qation 9) is sin sin ) sin sin ) ) P T olmtric flow r nit tim sin ) P d Q

5 Casson flow of HD flid moing stadil wit constant locit 57 slts and Discssions A similarit soltion of t Nair-Stoks qations for t Casson flid flow btwn two aralll lats as bn rsntd. Emloing rtrbation mtod w a obtaind aroimat analtical soltions for t locit of t flid flow of two dimnsional Casson flid btwn two aralll lats. Figr sows t ffcts of Casson aramtr β on locit rofils. Vlocit is fond to incras wit t incrasing als of nolds nmbr and Casson aramtr β. From Figr it is obsrd tat t aial locit dcrass wit incrasing t magntic fild. Figr. Vlocit Profils for diffrnt als of β and Figr. Vlocit Profils for diffrnt als of frncs [] A. oamd Ismail S. Gans and C. K. Kirbasankar Unstad agntodrodnamic flow btwn two Paralll Plats trog a oros mdim Intrnational Jornal on Dsign and anfactring Tcnologis 3; Vol.7No..-6. [] C. B. Sing P. C. am Unstad agnto drodnamic Flid Flow Trog a Cannl : Jornal of Scintific sarc. XXVIII No. 978). [3] C. K. Kirbasankar S. Gans Unstad HD Flow of a Casson Flid in a Paralll Plat Cannl wit Hat and ass Transfr of Cmical action. Pari Indian Jornal of sarc Amdabad Gjarat India. Volm: 3 Iss ISSN tt:d.doi.org fb3 [] P. C. am C. B. Sing U. Sing Hall ffcts on at and mass transfr flow trog oros mdim Astrosics Sac Scinc 98) 5-5. tt:d.doi.org.7bf6558

6 58 Y. Immanl t al. [5] S. Gans S. Krisnambal Unstad stoks flow of iscos flid btwn two aralll oros lats Int. J Information Scincs and Comting 7;Vol No [6] Y. Simomra Larg dd simlation of magnto drodnamic trblnt cannl flow ndr niform magntic fild Psics Flids A3 No. 99) 398. tt:d.doi.org cid: Janar 3 ; Pblisd: Fbrar 5 5