Heat Transfer in Hydromagnetic Fluid Flow: Study of Temperature Dependence of Fluid Viscosity

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1 Jounal of Appled Flud Mechancs, Vol. 7, No. 4, pp , 4. Avalable onlne at ISSN , EISSN Heat Tansfe n Hydomagnetc Flud Flow: Study of Tempeatue Dependence of Flud Vscosty S. K. Ghosh, G. C. Sht and J. C. Msa 3 Depatment of Mathematcs, Gahbeta College, Paschm Mednpoe-77, W.B., Inda Depatment of Mathematcs, Jadavpu Unvesty, Kolkata-73, Inda 3 Insttute of Techncal Educaton & Reseach, Sksha O Anusandhan Unvesty, Bhubaneswa 753, Inda Coespondng Autho Emal: gopal_tkgp@yahoo.co.n (Receved August 7, 3; accepted Decembe 8, 3) ABSTRACT Flow of a vscoelastc flud though a channel wth stetchng walls n the pesence of a magnetc feld has been nvestgated. The vscosty of the flud s assumed to vay wth tempeatue. Convectve heat tansfe s consdeed along wth vscous dsspaton and Ohmc dsspaton. The equatons that goven the moton of the flud and heat tansfe ae coupled and non-lnea. The govenng patal dffeental equatons ae educed to a set of odnay dffeental equatons by usng smlaty tansfomaton. The tansfomed equatons subject to the bounday condtons ae solved by developng a sutable fnte dffeence scheme. Numecal estmates of the flow and heat tansfe vaables ae obtaned by consdeng blood as the wokng flud. The computatonal values ae found to be n good ageement wth those of pevous studes. Keywods: MHD flow, Vaable vscosty, Vsco-elastc flud, Vscous dsspaton, Stetchng wall, Heat tansfe NOMENCLATURE A a b f K M T flud vscosty paamete channel half wdth constant of popotonalty (stetchng ate) non-dmensonal steam functon vscoelastc paamete themal conductvty magnetc paamete Pandlt numbe tempeatue of the flud wall tempeatue fee steam tempeatue u v ' velocty components along x- decton velocty components along y- decton smlaty vaable dynamc vscosty efeence vscosty knemetc vscosty non-dmensonal tempeatue densty of the flud mxed convecton paamete denotes dffeentaton w..to. INTRODUCTION Flow of a flud ove a movng contnuous sold suface has many applcatons n scence and engneeng. Heated flud movng between feedng olls and wnd up olls, coolng of polyme mateals by usng a contnuous movng sheet, glass and fbe poducton and manufactung of polymec sheets ae some examples that nvolves flow of a vscoelastc flud ove a stetchng sheet. Sddappa and Khapate (976) pefomed analytcal study of the flow of Rvln-Eckson flud ove a movng contnuous sold suface. Investgaton of the heat tansfe dung steady flow of an ncompessble vscous flud when tempeatue dffeence between the suface and the ambent flud s popotonal to some powe of the dstance was conducted by Caagphe et al.(98). Rajagopal et. al (984) pesented a numecal soluton fo small values of the vscoelastc paamete. Intoducng heat tansfe, Dandapat and Gupta (989) extended the poblem consdeed n

2 S. K. Ghosh et al. / JAFM, Vol. 7, No. 4, pp , 4. (Rajagopal et al. (984)) and found an exact soluton of the same poblem. Convectve flow of non-newtonan fluds was consdeed by Hakem and Abde (6). They examned the effect of vscosty vaaton on flow of a mcopola flud and found that t has substantal contbuton towads change of skn fcton and heat tansfe. Abel and Begum (8) caed out an analyss on MHD vscoelastc flud flow ove a stetchng sheet fo the case of lage Pandlt numbe. They obtaned a closed fom soluton by usng Rosseland appoxmaton and Kumme`s functon. Alhab et al. () pesented a study on vscoelastc flud flow wth heat tansfe ove a stetchng sheet. Takng nto consdeaton the effect of chemcal eacton and themal statfcaton, they made an obsevaton that wth se n tempeatue, thee s an ncease n the values of the non-dmensonal paametes lke, Gashof numbe, Pandlt numbe etc. They also found that due to chemcal eacton, the tempeatue and the velocty of the flud educe consdeable. Assumng the vscosty and dffusvty as functons of tempeatue, Mukhopadhyay (9) dscussed the unsteady flow and heat tansfe ove a stetchng sheet n the pesence of wall sucton. In ths study the classcal Runge-Kutta method was used fo solvng the nonlnea odnay dffeental equatons. A numecal study was pefomed by Shadan et al. (6) to nvestgate the unsteady bounday laye flow and heat tansfe ove a stetchng sheet, whle Cha (994) obtaned a soluton fo vscoelastc flud flow ove a stetchng sheet n the pesence of a magnetc feld wth the consdeaton of themal dffuson tem n the enegy equaton. Shama and Rao (998), Vajavelu and Rope (999), as well as Cotell (6a, 6b) nvestgated smla poblems to exploe the effects of elastc defomaton and the wok done theeby. In ode to evaluate the nfluence of elastc paamete and heat tansfe chaactestcs, Cotell (7) studed the flow and heat tansfe of a second ode vscoelastc flud by consdeng a non-unfom heat souce, vscous dsspaton and themal adaton. He epoted that the sad paametes affect qute consdeable the heat tansfe and flud flow. Elbashbeshy and Aldaqwody () analyzed numecally the unsteady mxed convectve flow and heat tansfe ove a poous stetchng suface. In ths study they nvestgated the effect of themal adaton and magnetc feld on flud flow. Snce haemoglobn contans on oxde, blood flow s lkely to be affected n the pesence of a magnetc feld. At the tme of vaous dagnostcs pocedues, such as MRI, magnetc theapy of the athts patents, magnetc theapy of cance patents, CT scan etc. human body s subject to magnetc felds of consdeable hgh stength. Apat fom ths, 634 nomal human bengs also sometmes have to wok unde the nfluence of electomagnetc felds when they ae equed to woks wth machnes havng electomagnetc component. Beng motvated towads physologcal flud flows, Msa et al. (8) and Msa and Sht (9a) epoted a study of the flow and heat tansfe n a vscoelastc flud unde the acton of a tansvese magnetc feld. They consdeed Waltes-B flud model to depct the flud vscoelastcty. It was not possble to teat the poblem analytcally and so they developed a sutable numecal method to solve the poblem. The flow of bomagnetc fluds n dffeent stuatons was studed extensvely by Msa and Sht (9a, 9b). Both these nvestgatons wee motvated towads studyng the haemodynamcal flow of blood n atees. A smla poblem was studed by Ray Mahapata and Gupta (4) that concen dffeent types of flud flow ove a vetcal plate/suface. Shama and Sngh (9) studed the effects of vaable themal conductvty, heat souce/snk on the flow of electcally conductng flud n the pesence of tansvese magnetc feld. They consde the heat tansfe n the flud whch s passng ove a stetchng sheet. The afoesad studes ae mpotant n the sense that no nvestgaton has been made wth the consdeaton of vscous as well as Ohmc dsspaton along wth the tempeatue dependent vscosty. In the pesent pape, consdeng Walte- B flud model of flud vscoelastcty, we have nvestgated the flud flow n a paallel plate channel unde the nfluence of a tansvese magnetc feld. In ode to study the poblem fom moe geneal platfom the flud vscosty has been consdeed as a functon of tempeatue. It has also been assumed that veloctes of plates of the channel ae vayng lnealy wth dstance fom the ogn and that the channel has a stetchng moton. To study the heat tansfe n the flow egon, the enegy equaton s consdeed such that t takes cae of vscous and Ohmc dsspaton as well as the stan enegy asng out of the elastc defomaton. Bounday laye appoxmaton and smlaty tansfomaton have been used to teat the coupled non-lnea patal dffeental equatons. Ths set of equatons s solved by the use of petubaton technque by takng the vscoelastc paamete as the petubaton paamete. The esultng non-lnea odnay dffeental equatons ae then solved by applyng Newton s method wth fnte dffeence technque. The numecal esults fo some physcal vaables whch ae mpotant fo havng an nsght to undestand the flow physcs ae pesented gaphcally. The study beas the potental of multfold applcatons to dffeent poblems of engneeng and ndustes as well as physology

3 S. K. Ghosh et al. / JAFM, Vol. 7, No. 4, pp , 4. and medcne. Hope, ths model wll help the medcal pacttones fo the dagnoss of the people suffeng fom feve (wth hgh tempeatue) and have to undego some scannng n the pesence of hgh stength magnetc feld such as n MRI. Moeove, any alment pesst n a physc whch s suounded by appled magnetc feld may also be nvestgated n tune of ths pape and unde the context of nvestgated esults.. FORMULATION AND ANALYSIS Let us consde the steady lamna flow and heat tansfe of an ncompessble vscoelastc flud though a channel havng stetchng moton, whch s bounded by two paallel plates. Takng the axs of the channel as efeence to x -axs, the uppe and lowe walls of the channel s desgnated as y a. X a Y B O Fg.. A sketch of the physcal poblem The vscosty s consdeed to be tempeatuedependent wth Ohmc dsspaton and the suface velocty of each wall to vay dectly wth dstance fom the channel axs. The moton of the flud s dven by the stetchng moton of the channel. Let us denote B the stength of the appled magnetc feld and the electcal conductvty of the flud. We shall consde hee the steady two-dmensonal moton of the flud. Takng nto account heat tansfe, the bounday laye equaton wth Boussnesq appoxmaton can be pesented as (cf. Msa et al (8)), u v () x y u u u u B u u v ( ) x y y y y k 3 u u v u ( u ) v 3 x y y x y and y () T T K T u C p u` v x y y y k u u u B u v u (3) y y x y whee the Eq.() epesents the mass consevaton of flud passng ove any coss-secton of the channel, the second tem on the ght hand sde of () appeas to epesent the effect of vaable vscosty whle the second and thd tems bea the effect of magnetc feld foce and vscoelastcty of the flud. The second, thd and fouth tem on the ght hand sde of (3) ae ntoduced to nvestgate the flow behavo due to the contbuton of the vscous, vscoelastc and Ohmc dsspaton espectvely. Also, uv, ae the components of velocty of flud along x - and y - dectons, k the vscoelastc paamete, the densty of the AT flud and e, beng the coeffcent of knematc vscosty. Consdeng the flow to be symmetc about the axs of the channel ou study can be estcted to the egon y a only. Hence the bounday condtons may be lsted as u bx, v, T T w at y a, b (4) and u, v, T at y (5) y y whee b epesents the stetchng ate of the channel walls. Let us ntoduce the smlaty vaables and nondmensonal vaables as follows u bxf () ; v baf ( ) ( ) ( ), n whch, x, y, a a T. T w u u ab, v v ab Usng these smlaty vaables n tems of the nondmensonal vaables n () - (3) we have f ff ( A ) f Mf A f ( A ) K ( f f ff v f ) ' ; (6) f f { A ( )} (7) (8) Pf P ( f f ) ( A ) Af k ( f f f f f ) M f (9) Equatons (6) to (9) consttute a system of coupled non-lnea odnay dffeental equatons. 635

4 S. K. Ghosh et al. / JAFM, Vol. 7, No. 4, pp , 4. Consdeng the vscoelastc paamete k to be small, we shall use the petubaton method, by takng k as the petubaton paamete. Thus we wte f f k f k f () ( ) ( ) ( ) ( )... Usng () n Eqs. (6) to (9), equatng the coeffcents of lke powes of k fom both sdes and neglectng squaes and hghe powes of k, we deve the followng set of equatons. f f f ( A ) f Mf A f () f f ( f f f f ) ( A ) f Mf () '' A ( A ) f ( f f f f v f ) ( f f f ) ( f f ( A ) Mf A ( A ) f ( f f f f f f f f f ) f f f f f f ) (3) whee dmensonless paametes appeang n the Eqs. () to (3) ae defned as B, b, C p M P, b kt w K k b K, A A T, w 3 A,,,. Moeove, the bounday condtons (4) and (5) educe to f, f, at (4) and f, f, at (5) In ode to detemne the flow feld and the tempeatue feld, we need to solve the Eqs.(8), (9) and () to (3) subject to the bounday condtons (4) and (5). It may be noted that the equatons n queston ae hghly non-lnea. So we have to develop a sutable numecal pocedue, descbe n the secton that follows. 3. COMPUTATIONAL RESULTS AND DISCUSSION We develop a fnte dffeence scheme whch s stable, moe accuate, effcent and elatvely smple. Ths method conssts of an teaton pocedue and use of Thomas Algothm. Thus t s vey much senstve to the ntal guess. Howeve, the poblem unde consdeaton has advantage fo makng an appopate ntal guess. Usng Newton s lneazaton technque, we now poceed to solve the Eqs.(), () and (3) subject to the bounday condtons (4) and (5). Usng cental dffeence scheme fo the devatves wth espect to, we can wte V V ( ) V (( ) ) and ( V ') V V V (( ) ( ) whee V stands fo dffeent vaables; s the gdndex n -decton wth * ; =,,,.,m and s the step sze along the -axs. Iteatons ae pefomed wth the followng elaton V ) n n n V ( V, whee n ( V ) epesents the eo at the n-th teaton, =,,, m. The stablty of the numecal model s checked by educng the step sze and conveges fo the accuacy level of O( - 5 ). Fo the numecal computaton we have used step length =.5, futhe decease n does not bng about any sgnfcant change. A detaled descpton of the method can also be found n Msa et. al. (8, 9a, 9b). Hee we consde an llustatve example of blood flow n atees, by takng the values of dffeent paametes as epoted n the pevous lteatues. In ode to get a physcal nsght of the poblem, we ntend to study the effects of vaous paametes lke A ϵ (.,.5), a paamete elated to the vaable vscosty, M ϵ (, 3) the magnetc paamete, the Pandlt numbe P ϵ (5., 5), K ϵ (.,.) the vscosty paamete and ϵ (., 5) the mxed convecton paamete on the vscosty, ts gadent and tempeatue. In the fst backet, ange of the paametes appea n the pesent nvestgaton ae atculated. The computatonal esults ae pesented gaphcally. f.4 A= A=.5 ) A=.8 A=. A=.8 A=.5 Fg.. The vaaton of steam functon fo dffeent A wth =., M=., P =. K =.5 Fgue. gves the dstbuton of the flud velocty n the channel fo dffeent values of the paamete A assocated wth the tempeatue dependent vscosty. Plots coespondng to Newtonan vscosty and vscoelastcty have been shown n ths fgue. Takng nto consdeaton that blood s a vscoelastc flud one may use the esults to obseve 636

5 S. K. Ghosh et al. / JAFM, Vol. 7, No. 4, pp , 4. the change n vscosty as the body tempeatue ased. It s seen that <A< poduce negatve values of the steam functon but n contast fo A>t s postve and slowly ncease as the values of A nceases. Thus the coeffcent appeas n the expesson of tempeatue dependent vaable vscosty can contol the flow phenomenon followng a cetan ule n a egula patten. So, the paamete A deceleates the moton by enhancng the fctonal esstance between the flud layes. By collectng dffeent blood samples fom dffeent people wokng n hot/modeate/cold envonment and meanng the vscosty of the blood samples one can have an dea of the tempeatue dependence of blood vscosty. The extenal foce (Loentz foce) that ceates etadaton n the moton of the flud because of ts chaactestcs of attactng electcally conductng lqud. Hemoglobn n blood havng on gets attacted by magnetc feld whch esults to the slow movement of flud along the stetchng sheet. Fgue. 3 shows the sgnfcant effect of magnetc paamete on the functon that evokes the velocty. It s futhe seen that blood encounte back flow fo M. The pesence of a magnetc feld can affect the flud consdeably. Fom ths fgue t s seen that the tendency of the back flow s enhanced as the stength of the magnetc feld s nceased. A compason between the esults of the pesent study ae n good ageement wth those of Msa et al. (8). f M= M=,,5,, Fg. 3. The vaaton of steam functon fo dffeent M when =., A =.5, P =. K =.5 The gaphcal pesentaton of the esults of the pesent study ndcates that owng to the nfluence of the magnetc feld etads the flow. Fom above two fgues we conceve the dea that the flow may be etaded due to the some ange of values of A and M. Howeve, the values fo A<, flud patcles dsplacement ate get enhanced. Hence, the paamete A has a cucal ole n the movement of flud ove a stetchng sheet and the moton of flud fo A< s faste than A> because fo the values of A<, vscosty decease.e. fctonal esstance decease. Fg. 4. Axal velocty dstbuton when A< ( =., M=., P =. K =.5) Fgue. 4 gves the dstbuton of velocty at dffeent locatons of the channel fo dffeent values of the vscosty paamete whose value changes wth the change n flud tempeatue. When the flud vscosty s low, the flud s faste. Ths may be attbuted to the fact that the less the vscosty of the flud, the less s the fctonal foce actng on the channel walls. Ths causes enhancement of the flow ate. Ths s a novel obsevaton n the case of vscoelastc fluds ove a stetchng suface. It s nteestng to note fom Fg. 4 that though the flud s vscoelastc the negatve values of the paamete A sgnfcantly affects the flow behavo and the paabolc pofle of Newtonan vscous flow s geatly dstubed. It may be obseved futhe that the vscous effect suppesses the stetchng effect on the flow. Ths suggests that the flow of ndustal fluds that possess vscoelastc behavo can be changed to Newtonan flow by applyng heat to the flud n the flow egme. Ths mechansm can also be used to contol the flow egon n stuatons when the flud has a tendency to coagulate leadng to fom a dense flud egon. f K =. K =. K = K =.5 K = Fg. 5. Axal velocty dstbuton fo dffeent values of K ( =., M=., P =. A =.5) Fgue. 5 depcts the vaaton of axal velocty along wth the heght of the channel fo dffeent 637

6 S. K. Ghosh et al. / JAFM, Vol. 7, No. 4, pp , 4. values of the vscoelastc paamete K. Vscoelastcty s a popety of the flud whch comes to the momentum equaton fom the consttutve equaton of the flud and to dentfy the effect of ths popety. It shows that the velocty of flud nceases as the values of the vsco-elastc paamete K nceases. The tendency of backwad flow may be attbuted to the stetchng moton of the suface on whch the flow takes place. The appecable change n velocty at dffeent axal statons fo each K s hghly senstve as long as the non-newtonan behavo of the flud s domnant. Upto ths dscusson we studed the appecable change of a functon whch s not the axal velocty. Now Fgs. 6 and 7 llustate the vaaton of axal velocty f along wth the heght of the channel fo dffeent values of the vscosty paamete A as well as the magnetc paamete M. Fg. 6 depcts the effects of vscosty paamete A on the gadent of steam functon. We obseve that ths fgue has a non-hythmc behavo n contast to the steam functon found n Fg.. It shows that the gadent of velocty n y-decton dmnshed wth the nceasng values of A. Moeove, fo the values of A.8 the velocty n y-decton s nea about zeo. f' A=. A=.5 A=.5 A=. A= Fg. 6. Gaphcal epesentaton of f fo dffeent values of A ( =., M=., P =..9 K =.5) M= It may be ponted out that fo hghe values of the vscosty paamete leads to the potental flow, whee the acceleaton tem contbuton n the computaton s neglgbly small. The effect of magnetc feld stength on the velocty gadent n y-decton s shown n Fg. 7. Fom ths fgue we obseve that as the magnetc paamete M nceases the axal velocty nea the wall deceases gadually. The esults found n ths fgue ae good ageement wth the pevous studes A=.7 A=.8 A=.5 A=.5 A= Fg. 8. Tempeatue dstbuton fo dffeent values of A when ( =., M=., P =. K =.5) As n the pesent nvestgaton the heat tansfe takes place n the flud, we nteestngly taken up the ventue to study the pofle due to some paametes assocated wth the enegy equaton as well as n the momentum equaton. Fgs. 8 to epesent the tempeatue dstbuton along the heght of the channel fo dffeent values of the paametes nvolved n the pesent study unde consdeaton. An abtay dstbuton of tempeatue s obseved fo vaous values of the vscosty paamete A as shown n Fg. 8. It s nteestng to note fom ths fgue that fo the hghe values of A tempeatue vaaton s nsgnfcant. The heat tansfe assocated wth the vaous values of s depcted n Fg. 9. It shows that the paamete has sgnfcant esponse on the tempeatue dstbuton and hence we obseved that the tempeatue deceases wth the deceasng values of. Wth a caeful obsevaton t s found that the values of beas same magntude fo all. f'.6.3 M=,,5,, =, 5,,.5, Fg.7. Gaphcal epesentaton of f vesus M ( =., K =.5, P =. A =.5) Fg. 9. Tempeatue dstbuton fo dffeent values ( A =.5, M=., P =. K =.5) 638

7 S. K. Ghosh et al. / JAFM, Vol. 7, No. 4, pp , M= M=5 M= M= Fg.. Dstbuton of tempeatue fo dffeent values of M ( =., K =.5, P =. A =.5) Fgue gves the tempeatue dstbuton along the heght of the channel fo dffeent magnetc feld stength n tems of magnetc paamete M. We obseved that the tempeatue gadually nceases wth the ncease of the magnetc paamete M. Fo hghe values of M, maxmum tempeatue occus at the cental lne of the channel. It may be ponted out that the tempeatue dffeence between M=5 and M= s lage, howeve between M= and M= the dffeence s neglgble n compae to the hghe values of M P=,, 8, Fg.. Pandtl numbes effect on tempeatue vaaton ( =., K =.5, M=. A =.5) In Fg. we pesent the tempeatue dstbuton fo dffeent values of the Pandatl numbe P, an mpotant paamete appeas n the poblem of flud flow and heat tansfe. We obseved that the tempeatue deceases sgnfcantly n the cental lne of the channel wth the ncease of the Pandatl numbe P. Howeve, the tempeatue vaaton wth Pandatl numbe P n the vcnty of the wall s nsgnfcant. K >. absobs heat and at dffeent channel postons the tempeatue emans same. It may pont out that the esults found n the above fgues ae nteestng due to ts ntegty n esponse of tempeatue to each values of K 4 3 K =. K =.5 K =. K =. K = Fg.. Tempeatue pofle fo dffeent values of the vscoelastc paamete K ( =., K =.5, P =. M =.5) 4. CONCLUDING REMARKS In ths pape, we have studed the effect of vaable vscosty coeffcent, Pandlt numbe, magnetc paamete and vsco-elastc paamete on the flud velocty as well as heat tansfe fo a vsco-elastc flud flow ove a stetchng sheet. The govenng non-lnea dffeental equatons ae solved by developng a numecal scheme, whch s stable and conveges fo a sutable choce of ntal values. Thee s a sgnfcant change n the velocty as well as velocty gadent fo paametes lke vaable vscosty coeffcent, magnetc paamete and vscoelastc paamete. Axal velocty of flud nceases wth the deceasng values of the paamete A and the magnetc paamete ecede the flow wth ts nceasng values. The smla chaactestc pepetuated though the velocty gadent also. The nteestng and sgnfcant esults found n heat tansfe ae fo paametes lke K,, P etc. The most mpotant concluson s that the hghe values of the paamete K made stagnant the heat flow n the flud and t kept the tempeatue same eveywhee n the channel space; howeve, ths popety s not synonymous fo. The tempeatue dstbuton n the channel wth the vscoelastc paamete K s shown n Fg.. It has been shown that the tempeatue dstbuton vaes n appecable manne and modeate magntude fo dffeent K. It s also found that ths change n s sgnfcant only fo small values of K. It may be mentoned hee that the values of 639 ACKNOWLEDGEMENTS Authos ae gateful to the esteemed evewes fo the valuable comments and suggestons fo mpovng the pape n ts cuent fom. The autho S. K. Ghosh s thankful to the UGC (ERO), Kolkata fo the fnancal suppot of ths nvestgaton unde Mno Reseach Poject (No. F.

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