Steady Stagnation Point Flow and Heat Transfer Over a Shrinking Sheet with Induced Magnetic Field

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1 Journal o Appled Flud Mehans, Vol. 7, No., pp. 73-7,. Avalable onlne at ISSN , EISSN Steady Stagnaton Pont Flow and Heat Transer Over a Shrnkng Sheet wth Indued Magnet Feld A. Snha Department o Mathemats Jadavpur Unversty, Jadavpur-73, Inda Emal: anruddha.snha7@gmal.om (Reeved November 8, 3; aepted Deember 8, 3) ABSTRACT The problem o the steady magnetohydrodynam (MHD) stagnaton-pont low o an nompressble, vsous and eletrally ondutng lud over a shrnkng sheet s studed. The eets o an ndued magnet eld and thermal radaton are taken nto aount. Veloty and thermal slp ondtons have also been norporated n the study. The nonlnear partal derental equatons are transormed nto ordnary derental equatons va the smlarty transormaton. The transormed boundary layer equatons are solved numerally usng Newton s lnearzaton method. Computatonal results or the varaton n veloty, temperature, skn-rton o-eent and Nusselt number are presented graphally and n tabular orm. Study reveals that the surae veloty gradent and heat transer are enhaned by dereasng magnet parameter. Keywords: Shrnkng sheet; Stagnaton low, Indued magnet eld; Slp eets; Boundary layer. INTRODUCTION A lass o low problems wth obvous relevane to numerous applatons n ndustral manuaturng proesses s the low ndued by the strethng moton o a lat elast sheet. Suh low stuatons are enountered, or example, n aerodynam extruson o plast and rubber sheets, melt-spnnng, hot rollng, wre drawng, glass-ber produton, polymer sheets, oolng o a large metall plate n a bath whh may be an eletrolyte, et. Durng ts manuaturng proess, a strethed sheet nterats wth the ambent lud both thermally and mehanally. The study o heat transer and low eld s neessary or determnng the qualty o the nal produts o suh proesses as explaned by Karwe and Jalura (988, 99). Crane (97) was the rst who studed the steady two-dmensonal nompressble boundary layer low o a Newtonan lud aused by the strethng o an elast lat sheet whh moves n ts own plane wth a veloty varyng lnearly wth the dstane rom a xed pont due to the applaton o a unorm stress. Ths problem s partularly nterestng sne an exat losed orm soluton o the two-dmensonal Naver-Stokes equatons has been obtaned. In reent years, some nterest has been gven to nvestgate the low over a shrnkng sheet, where the sheet s strethed toward a slot and t would ause a veloty away rom the sheet. However, n ertan stuatons, the shrnkng sheet solutons do not exst sne the vortty an not be onned n a boundary layer. A poneerng paper on ths problem has been publshed by Mklav and Wang (6). From the physal grounds vortty (rotaton or non-potental) low over the shrnkng sheet s not onned wthn a boundary layer, and the low s unlkely to exst unless adequate suton on the boundary s mposed (Mklav and Wang (6)). Fang and Zhang (9) obtaned an analytal soluton or the thermal boundary layers wth suton over shrnkng sheet. The shrnkng sheet problem has also been extended to mropolar lud (Ishak et al. (8)) as well as magnetohydrodynam lud (Sajd et al. (8)). Durng the past entury,many engneerng problems o lud mehans have been solved by usng the boundary-layer theory and the results ompare well wth the expermental observatons or Newtonan luds (Shlhtng and Gersten ()). An extenson o the boundary layer theory to non-newtonan luds s ound to be rather dult (Rajagopal et al. (98)). Ths dulty s aused by the dversty o non-newtonan luds n ther onsttutve behavour and smultaneous vsous and elast propertes. Consequently, most studes on non-newtonan boundary layers have used smple rheologal models suh that these two eets an be taken nto aount separately. Reently, several researhers (Sngh et al. (), Sharma and Sngh (9), Rashd and Eran (), Eran et al. ()) analytally as well as numerally studed the boundary layer low n the stagnaton pont regon o a two dmensonal body. Hemenz (9) was the rst to study the two-dmensonal (D)

2 A. Snha / JAFM, Vol. 7, No., pp. 73-7,. stagnaton-pont low aganst an nnte lat plate. He ound an exat soluton to the governng Naver-Stokes equatons. Mahapatra and Gupta () onsdered the boundary layer low near the stagnaton-pont on a strethng sheet, where tme dependene s also taken nto aount. Reently, Zhu et al. (9) presented the analytal solutons o stagnaton-pont low over a strethng sheet. These are ew examples or the problem where the ndued magnet eld s neglgble. To date, very lttle work has been done on the boundary layer low and heat transer wth the onsderaton o the ndued magnet eld. Kumar et al. (9) nvestgated the MHD low and heat transer over a strethng surae by onsderng the eet o the ndued magnet eld. Rashd et al. () nvestgated the smultaneous eets o partal slp and thermal-duson and duson-thermo on MHD low due to a rotatng dsk. Owng to the above mentoned studes, n ths paper, we have nvestgated a numeral soluton on a MHD stagnaton-pont low wth heat transer over a shrnkng sheet n the presene o magnet eld havng nterest to the engneerng ommunty and to the nvestgators dealng wth the problem n geophyss, astrophyss, eletrohemstry and polymer proessng. The eets o an ndued magnet eld and thermal radaton are taken nto aount. In ths paper, we have nvestgated the eets o slp veloty and thermal slp on an eletrally ondutng vsous nompressble lud. Boundary layer theory s appled to smply the equatons that govern the lud moton, ndued magnet eld and energy. Computatonal results presented through graphs put orward theoretal estmates o the nluene o varous parameters nvolved n the study. The results reported here are applable to a varety o boengneerng problems. MATHEMATICAL ANALYSIS Let us Consder a steady, two-dmensonal low and heat transer o an nompressble eletrally ondutng lud near the stagnaton pont on a heated shrnkng sheet n the presene o a ree stream ue ( x) and unorm ambent temperature T. The x-axs runs along the shrnkng surae n the dreton o moton and the y-axs s perpendular to t, as shown n Fg.. The wall shrnkng sheet veloty uw ( x) and the wall temperature Tw ( x) are are proportonal to the dstane (x) rom the stagnaton pont, where ue ( x )= ax and Tw ( x ) = T T ( x / l ), where a s a onstant, T s the reerene temperature. It s also assumed that an ndued magnet eld o strength H (say) ats n the dreton normal to the surae. It s supposed that at the wall, the normal omponent o the ndued magnet eld H has a vanshng value, whle the parallel omponent H assumes the value H. Fg.. Physal sketh o the problem Aordng to the boundary layer approxmatons, the bas equatons o the problem an be wrtten as ollows (Datt et al. (), Sharma and Sngh (9), Sngh et al. ()) u v = () x H H = x u u e H H ( du u v H H ) = ( u e e x x dx H dh e e e u ) dx H H u u H u v H H = x x T T k T q u v = r x y p p () (3) () (5) Where s the oeent o vsosty, s the lud densty, beng the knemat oeent o vsosty, represents magnet dusvty, k beng the thermal ondutvty, p s the spe heat at onstant pressure and permeablty. e s the Magnet The boundary ondtons or the present problem an be wrtten as H u u u v H = w s, =, = =, T = T T at y = w s u = u ( x ) = ax, H = H ( x ) = H ( x / l ), e e and (6) T = T at y (7) n whh uw = x, = u us N, Tw = T T( x / l ) and = T Ts K, a an beng postve onstants. N s the veloty slp ator and K s the thermal slp ator. The radatve heat lux term by usng the Rosseland 7

3 A. Snha / JAFM, Vol. 7, No., pp. 73-7,. approxmaton (Sngh et al. ()) s gven by * T q r = (8) 3k * where * s the Stean-Boltzmann onstant and k * beng the mean absorpton oeent. Assumng that the derenes n temperature wthn the low are suh that T an be expressed as a lnear ombnaton o the temperature, we expand T n Taylor s seres about T as ollows T = T T 3 ( T T ) 6 T ( T T ) and negletng hgher order terms beyond the rst degree n ( T T ), we get T = T 3 T 3T (9) Let us ntrodue the ollowng dmensonless varables: = ( ) / T T x ( ), =, = ( ) / y Tw T H = ( / ) ( ), = ( ) / ( ) H x l g H H g () l The veloty omponents u and v an be obtaned rom the stream unton as u = and v =. x It may be noted that the ontnuty Eq.() s automatally satsed. Then substtutng Eqs. (8) to () nto Eqs. (3) to (5) yelds the ollowng dmensonless equatons: a ( g gg ) = () = () g g g and ( Nr) = (3) Pr In the above equatons, prmes denote derentaton wth respet to. The non-dmensonal parameters appeared n Eqs () to (3) are dened as H = e ( ) s the magnet parameter, l = s the reproal magnet Prandtl number, å 3 6 T Nr = s radaton parameter and å 3k k = p Pr s Prandtl number. k The boundary ondtons (6) and (7) gve rse to () =, () = S (), g () = g() =, () = S t () a ( ) =, g( ) =, ( ) = The non-dmensonal veloty slp ator non-dmensonal thermal slp ator by S = N and S = K t. () (5( S and S t are gven The magnet parameter, whh gves the order o the rato o the magnet energy and the knet energy per unt volume, s related to the Hartman number Ha (Hartmann (937)) and the low Reynolds number Re and the magnet Reynolds number Re m as Ha =, Ha = H l ( ) /, Re = ReRe m (6) ( l ) l ( l ) l and Rem = U l = e where l s the haraterst length o the shrnkng surae omparable wth the dmensons o the eld. 3 NUMERICAL PROCEDURE Several authors suh as Andersson et al. (99) and Ay () used numeral tehnques or the soluton o two-pont boundary value problems n terms o the Runge-Kutta ntegraton sheme along wth the shootng method. Although ths method provdes satsatory results, t may al when appled to problems n whh the derental equatons are very senstve to the hoe o ts mssng ntal ondtons. Moreover, dulty arses n the ase n whh one end o the range o ntegraton s at nnty. The end pont o ntegraton s usually approxmated by replang a nte representaton to ths pont and t s obtaned by estmatng a value at whh the soluton wll reah ts asymptot state. On the ontrary to the above mentoned numeral method, we used n the present paper that has better stablty, smple, aurate and more eent. The essental eatures o ths tehnque s that t s based on a nte derene sheme wth entral derenng and based on the teratve proedure. We substtute = F and g = G n () and (), we get a F F F ( G gg ) = (7) and G G Fg = (8) whle the boundary ondtons () and (5) assume the orm 75

4 A. Snha / JAFM, Vol. 7, No., pp. 73-7,. () =, F () = S F(), g () = G () =, a F( ) =, G( ) = (9) Now usng the entral derene sheme or dervatves wth respet to, we wrte P P ( P) = O(( ) ) () and P P P ( P) = O(( ) ), () ( ) where P stands or F, G and ; s the grd ndex n -dreton wth = * ; =,,...m and s the nrement along the -axs. Newton s lnearzaton method an then be appled to lnearze the dsretzed equatons as ollows. When the values o the dependent varables at the n th teraton are known, the orrespondng values o these varables at the next teraton an be obtaned by usng the equaton P n = P n ( P ) n, () n whh ( P ) n represents the error at the n th teraton, =,,,..., n. It s worthwhle to menton here that the error ( P ) n at the boundary s zero, beause the values o P at the boundary are known. Usng () n (7) and droppng the quadrat terms n ( P ) n, we get a system o blok tr-dagonal equatons. To solve ths tr-dagonal system o equatons, we have used the "Tr-dagonal matrx algorthm", usually reerred as "Thomas algorthm". It may be mentoned here that nstead o ths, one ould use Gauss elmnaton method. But n that ase, the number o operatons would be m 3, whle n the method that we have employed here, the number o operatons s m, where m s the number o unknowns. Thus the error ommtted n our method s muh less than that n the method o Gauss elmnaton. In the proess o determnaton o the dstrbuton o the unton ( ), the auray an be dened as the derene between the alulated values o ( ) at two suessve operatons, say ( n ) th and nth. In the present ase, the error s equal to = n ( ) n ( ) and s estmated to be less than 6. ollowng values o the dmensonless parameters nvolved n the present problem under onsderaton. a =.5,.8,.5,.,.5; =.,.5,.,.5,.,8.; =.,5.,.,.; =.,.5,.5,.,.; S =.,.5,.,.5; S t Pr =.3,.5,.7,.,.5; Nr =.,.,.,3.,.. Numeral omputaton has been arred out by takng =.5 wth grd ponts. The results omputed have been presented graphally n Fgs. to. In order to perorm grd ndependent test or the hoe o 3 grd ponts, we repeated the omputatonal proedure by onsderng a number o mesh szes by alterng the values o. I we take =.8 (mess sze=5) or =. (mess sze=) then the gures wll be unaltered. It has also been asertaned that or any mesh sze less than 3, the results are naurate. Fgures to respetvely dsplay the eets o magnet parameter on axal veloty, ndued magnet eld and temperature proles when a =., Pr =.7, Nr =., =.5, S =.5 and S t =.. Fg. reveals that axal veloty ( ) dereases when the magnet parameter nreases. Ths s due to the at that an nrease n the value o sgnes the nrease o Lorentz ore and thereby magntude o veloty redues. It s also seen rom ths gure that or large value o magnet parameter, axal veloty vanshes to a ertan pont. Ths pont s alled pont o nlexon. From Fg. 3, t s seen that the ndued magnet eld strength dereases as nreases. It may be noted rom ths gure that or any value o, ndued magnet eld strength nreases wth the heght o the hannel. It s noted rom Fg. that the magnet parameter bears the potental to nrease the temperature n the boundary layer. Ths s beause o the at that the ndued magnet eld to an eletrally ondutng lud gves rse to a resstve type o ore known as Lorentz ore. Ths ore has the tendeny to nrease the lud temperature.. NUMERICAL ESTIMATES AND DISCUSSION The system o ordnary derental Eqs () to (3) subjet to the boundary ondtons () and (5) are solved numerally by employng a nte derene sheme wth Newton s lnearzaton method desrbed n the prevous seton. In order to aheve the numeral soluton, t s neessary to assgn 76

5 A. Snha / JAFM, Vol. 7, No., pp. 73-7,. Fg.. Axal veloty dstrbuton or derent values o temperature prole dereases n the boundary layer wth nreasng value o a /. Ths s beause o the at that the thermal boundary layer dereases wth nrease n stranng moton near the stagnaton regon. Fg. 3. Varaton o ndued magnet eld n x -dreton or derent values o The varaton o axal veloty, ndued magnet eld and temperature proles or derent values o the rato a / s shown n the Fgs. 5 to 7 respetvely. Fg. 6. Varaton o ndued magnet eld n x -dreton or derent values o a / Fg.. Temperature dstrbuton or derent values o Fg. 5. Axal veloty dstrbuton or derent values o a / It an be seen rom Fg. 5 that the boundary layer thkness dereases as a / nreases when we onsder a/ >. Physally ths phenomenon an be explaned as ollows: or xed value o orrespondng to the shrnkng o the surae, the nrease n a n relaton to mples the nrease n the stranng moton near the stagnaton regon that an nrease the aeleraton o the external stream. Thereore, an nrease n a / has the eet o thnnng the boundary layer. From Fg. 6, t an be seen that all the g proles nrease wth the nrease n a /. Fg. 7 shows that the the Fg. 7. Temperature dstrbuton or derent values o a / Fgures. 8 and 9 llustrate the eets o the veloty slp ator on the veloty and the ndued magnet eld proles when a/ =.5 and =.5. Fg. 8 reveals that veloty n the axal dreton monotonally nreases as slp veloty nreases. Ths gure urther ndates that or large values o slp veloty ator ( S ), the rtonal resstane between the vsous lud and the surae s elmnated and shrnkng o the sheet does no longer mpose any moton o the lud. It s also seen that or no slp ondton,e, or S =., low separaton s observed adjaent to the lower wall. We have a smlar observaton n respet o the varaton o the ndued magnet eld wth the nrease n slp ator (. Fg. 9). 77

6 A. Snha / JAFM, Vol. 7, No., pp. 73-7,. values o Pr Fg. 8. Nature o axal veloty dstrbuton or derent values o S Fg.. Temperature dstrbuton or derent values o Nr Fg. 9. Varaton o ndued magnet eld n x -dreton or derent values o S Fgures. to gve some haraterst temperature proles or derent values o Prandtl number ( Pr ), radaton parameter ( Nr ) and thermal slp ator S t respetvely. Fg. demonstrates the eet o Prandtl number ( Pr ) on temperature prole n the boundary layer. It s seen that the eet o Prandtl number s to derease the temperature prole n The boundary layer. Ths an be attrbuted to the at that the thermal boundary thkness dereases wth nrease n Prandtl number. Fg. shows the eet o thermal radaton on temperature. It s observed that the nrease n thermal radaton parameter ( Nr ) produes a sgnant nrease n the thkness o the thermal boundary layer o the lud and so the temperature nreases. Fg. gves the temperature dstrbuton or derent values o thermal slp ator ( S t ). Ths gure shows that temperature dereases as the thermal slp nreases. Fg.. Temperature dstrbuton or derent values o S t Fg. 3. Varaton o skn-rton wth or derent values o The skn-rton oeent, dened as: C w = = () / Re x U w u where w = ( ) y = Fg.. Temperature dstrbuton or derent s an mportant physal quantty that bears the potental to explore some vtal normaton regardng problems suh as the one under our present onsderaton. Fgure. 3 gves the varaton o skn-rton wth magnet parameter or derent values o the reproal magnet Prandtl number. It s seen that skn-rton nreases as nreases. It s also 78

7 A. Snha / JAFM, Vol. 7, No., pp. 73-7,. seen that skn-rton dereases as nreases. Another mportant haraterst o the present study s the loal Nusselt number Nu x, dened as Nu x xqw = = Re () k ( T T ) x w T where qw = k( ) y = The values o () on the strethng wall, omputed on the bass o the present study are presented n tabular orm. Table Values o -θ ()or derent values o, a /, Pr, Nr when λ =., S =.5 and S t =. a / Pr Nr -θ() From Table, one an have an dea o the varaton n loal Nusselt number or derent values o magnet parameter ( ), a /, radaton parameter ( Nr ) and Prandtl number ( Pr ). Ths table shows that nrease n a / or Pr, enhanes the loal Nusselt number, whle nrease n magnet parameter or radaton parameter leads to a reduton n loal Nusselt number. Wth an am to valdate our numeral model, we have ompared our results or the axal veloty dstrbuton wth those reported reently by Nadeem and Hussan (9) who arred out a smlar study under some smplyng assumptons and obtaned analytal soluton by usng the homotopy analyss method (HAM). We note that the results o our numeral model are n exellent agreement wth those reported by Nadeem and Hussan (9) (. Fg. ). 5. SUMMARY AND CONCLUSION We have obtaned an exat smlarty soluton o a steady two-dmensonal magnetohydrodynam stagnaton-pont low o an nompressble eletrally ondutng lud vsous lud over a shrnkng sheet n the presene o an ndued magnet eld. The numeral analyss o a problem that deals wth the nvestgaton o the nluene o veloty slp, thermal slp and thermal radaton on the low eld. The ollowng onlusons an be drawn as a result o the numeral omputatons: () In the presene o a magnet eld, the lud veloty dereases. Ths s aompaned by a reduton n the veloty gradent at the wall, and thus the loal skn-rton oeent dereases. () As Prandtl number nreases, the loal Nusselt number nreases and thus the temperature gradent at the surae redues. Ths mples that the ambent lud gans temperature rom the strethng sheet. (3) As thermal radaton nreases, thermal boundary layer thkness nrease. ACKNOWLEDGEMENTS The author s grateul to the NBHM, DAE, Mumba or the nanal support o ths nvestgaton. REFERENCES Ay, A. A. (). MHD ree-onvetve low and mass transer over a strethng sheet wth hemal reaton. Heat Mass Trans., Andersson, H. I., K. H. Beh and B. S. Dandapat (99). Magnetohydrodynam low o a Powerlaw lud over a strethng surae. Int. J. Non-Lnear Meh. 7, Cobble, M. H. (979). Free onveton wth mass transer under the nluene o a magnet eld. Nonlnear Analyss: Theory, Methods and Applatons 3, Crane, L. J. (97). Flow past a Strethng Plate. Zetshrt ur Angewandte Mathematk und Physk (ZAMP), Datt, P. S., K. V. Prasad, M. S. Abel and A. Josh (). MHD vso-elast lud low over a non-sothermal strethng sheet. Int. J. Eng. S., Fg.. Axal veloty dstrbuton n the absene o ndued magnet eld and veloty slp (when a / = ). (Comparson o the results o the present study wth the analytal soluton o Nadeem and Hussan (9)) Eran, E., M. M. Rashd and A. Basr parsa (). The Moded Derental Transorm Method or Solvng O-Centered Stagnaton Flow towards a Rotatng Ds. Internatonal Journal o Computatonal Methods Fang, T. and J. Zhang (9). Closed-orm exat solutons o MHD vsous low over a shrnkng sheet. Communatons n Nonlnear Sene and Numeral Smulaton,

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