Unsteady Formulations for Stagnation Point Flow Towards a Stretching and Shrinking Sheet with Prescribed Surface Heat Flux and Viscous Dissipation

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1 Internatonal Journal of Flud Mechancs & Thermal Scences 7; 3(): do:.648/j.jfmts.73. ISSN: (Prnt); ISSN: (Onlne) Unsteady Formulatons for Stagnaton Pont Flow Towards a Stretchng and Shrnng Sheet wth Prescrbed Surface Heat Flu and Vscous Dsspaton Oey Oseloa Onyejewe Computatonal Scence Program, Adds Ababa Unversty, Arat Klo Campus, Adds Ababa, Ethopa Emal address: ouzas@yahoo.com To cte ths artcle: Oey Oseloa Onyejewe. Unsteady Formulatons for Stagnaton Pont Flow Towards a Stretchng and Shrnng Sheet wth Prescrbed Surface Heat Flu and Vscous Dsspaton. Internatonal Journal of Flud Mechancs & Thermal Scences. Vol. 3, No., 7, pp do:.648/j.jfmts.73. Receved: December, 6; Accepted: December 7, 6; Publshed: Aprl 4, 7 Abstract: The unsteady stagnaton pont flow and heat transfer wth prescrbed flu towards a stretchng and shrnng sheet wth vscous dsspaton s studed. Smlarty transformaton s adopted to ntally convert the governng dfferental equatons nto nonlnear ordnary dfferental equatons. The two-pont boundary value ordnary dfferental equatons (ODE) are subsequently converted nto partal dfferental equatons by ntroducng a tme-marchng scheme. A Cran-Ncolson Newton- Rchtmeyer scheme s employed to dscretze the resultng equatons. Intal guesses are made for the dependent varables and the soluton advanced n tme untl temporal varatons of the scalar profle are dmnshed and the steady-state solutons satsfy the smlarty equatons. A varaton of the heat flu at one of the boundares produced notceable varatons n the temperature feld that can be related to the magntude of the Prandtl number and velocty rato parameter. Keywords: Stagnaton Pont Flow, Heat Transfer, Prescrbed Flu, Cran-Ncolson-Newton-Rchtmeyer, Tme Marchng Scheme, Steady State, Prandtl Number, Stretchng and Shrnng Sheet. Introducton The mpact of a vscous flud on a sold object (Fg. ) consttutes a prototypcal stagnaton pont flow. Stagnaton pont flow and heat transfer wth prescrbed flu have contnued to receve consderable attenton over the years because of ther wde applcaton. The heat transfer over a surface becomes qute nterestng when the heat flu boundary condton s taen nto account partcularly for those problems nvolvng the coolng of electrcal and nuclear components, space shuttle re-entry nto the earth s atmosphere, melt-spnnng processes etc. Ths class of problems often encounters rapd heat-flu changes, and possble meltdown hence overheatng and burnout consttute very mportant desgn consderatons. As a result, accurate predcton of the temperature profle becomes vtal (Bejan []). The study of vscous flow near a sold object can be traced bac to Hemenz [] smlarty analyss. That s why the stagnaton pont flow s sometmes referred to as Hemenz flow. He reduced the Naver-Stoes equatons governng the flow to thrd order ordnary dfferental equatons by usng smlarty transformatons. An etenson of hs wor was carred out by Homann [3]. Hemenz flow was further studed by Na [4] usng the method of fnte dfferences. Due to ts fundamental nature and n combnaton wth ts practcal mportance, the classcal problem has been generalzed n several ways to nclude dverse physcal effects. These comprse vscous or nvscd, forward or reverse flows over surfaces. For eample, the study of flow over a stretchng sheet has several ndustral applcatons especally n paper and glass fber producton, and n the etracton of polymer sheets drawn through a stagnant flud wth a controlled coolng system. One of the frst attempts to loo nto the flow of ncompressble flud over a lnearly stretchng sheet was conducted by Crane [5] who provded smlarty solutons for the case of a steady two-dmensonal ncompressble boundary layer flow produced by a stretchng sheet that moves n ts own plane. The heat transfer component of ths study was carred out by Carragher and

2 7 Oey Oseloa Onyejewe: Unsteady Formulatons for Stagnaton Pont Flow Towards a Stretchng and Shrnng Sheet wth Prescrbed Surface Heat Flu and Vscous Dsspaton Crane [6], whle the mass transfer effect ncludng sucton and blowng was nvestgated by Gupta and Gupta [7]. Cham [8] studed the stagnaton pont flow towards a stretchng sheet when the stretchng velocty s dentcal to the free stream velocty. In a smlar wor, Saads [9, ] consdered the case where the stretched surface was assumed to move wth unform velocty whle Mahapatra and Gupta [] found out that the profle of the boundary layer s a functon of the stretchng sheet parameters and the angle of ncdence. Several other authors can be credted for wor n ths area. Dulal and Hremath [] studed the heat transfer characterstcs n a lamnar boundary layer flow of an ncompressble vscous flud over an unsteady stretchng sheet placed n a porous medum n the presence of vscous dsspaton and nternal absorpton or generaton of heat. Relevant references n ths area can be found n the boos by Ingham and Pop [3], Neld and Bejan [4]. Fgure. Stagnaton pont flow. Relatng heat transfer to flud flow has provded a ln between the applcaton of flud numercal smulaton to the ndustry, Ths can be observed n such dverse areas as navgaton, avaton, rocet propulson, space eploraton, shp buldng, food processng, and auto manufacture to menton just a few. The heat transfer characterstcs of a stretchng sheet becomes sgnfcantly mportant when the heat flu boundary condton s taen nto consderaton. Dutta et al. [5] eamned the effects of unform heat flu on the temperature feld due to stretchng. A smlar study was carred out by All and Magyar [6] for power-law velocty and temperature dstrbutons. The effect of a magnetohydrodynamc (MHD) flow on a stretchng sheet was studed by Isha et al. [7]. Smlar wor was carred out by Anderson [8] hs wor was later etended wth the ncluson of mass transfer by Char [9]. Mahapatra and Nandy [] provded a momentum and heat transfer soluton nvolvng a shrnng sheet for the case of an MHD asymmetrc stagnaton pont flow. Further wor on hydromagnetc flows ncludes those of Mande and Charles [], Maonde []. Conjugate convectve flows apply n cases nvolvng nternal heat generaton. These often lead to obtanng useful desgn parameters [3, 4, and 5]. The am of ths study s to numercally nvestgate the problem of an unsteady stagnaton pont flow towards a stretchng and shrnng sheet wth prescrbed surface heat flu and vscous dsspaton. The governng dfferental equaton s obtaned from the vortcty transport equaton and converted nto ts thrd order stream functon analog. By usng a combnaton of analytc technques, nondmensonalzaton and approprate boundary condtons t was possble to arrve at a second order nonlnear transent partal dfferental equaton. The resultng system was solved to steady state by an mplct fnte dfference Cran- Ncolson-Newton-Rchtmeyer scheme. The effects of varous non-dmensonless parameters on the flow and heat transfer profles are presented graphcally and dscussed.. Mathematcal Formulatons We see the governng equaton for boundary layer flow towards a stagnaton pont flow (Fg. ). The and y aes are taen along the normal and along a shrnng stretchng surface. It s assumed that the free stream velocty s gven by U ς ( ) velocty uw ( ) = and that the plate s stretched or shrun wth a =l where ς andl constants are U, uw are the freestream and -component velocty. Under ths assumptons we can defne the velocty rato as ζ = υ. We l start by developng the stream functon equaton va the vortcty transport. The condton for ts estence s that the flow s ncompressble. Startng from the steady, two dmensonal form of the vortcty equaton u ω = ν ω and relatng the streamfuncton to the vortcty u =, v =, The vortcty vector s epressed as: ω = ψ and when substtuted n the vortcty transport equaton yelds a fourth order equaton partal dfferental equaton for the stream functon: ψ ψ ψ ν ψ ψ ψ ψ = + + Boundary Condtons: Two boundary condtons on ψ follow from the no-slp boundary constrant at the surface. The other two concur wth the approach of the vscous-flow soluton to nvscd flow far enough from the surface. at y = =, = (, y) as y A, Ay () (a) (b) where A = ψ and ψ s a constant n the nvscd flow soluton. Because the frst dervatve of the streamfuncton approaches a constant value A asymptotcally, we can deduce that ts second dervatve approaches zero as y ψ

3 Internatonal Journal of Flud Mechancs & Thermal Scences 7; 3(): approaches nfnty. Ths observaton follows the logc that once a dervatve of a functon approaches a constant value, all hgher order dervatves of that functon wll approach zero asymptotcally. We mae a further assumpton that the streamfuncton comprses both the vscous and nvscd components of flow and as a result; t can be epressed as ψ = ϕ F y. Substtutng ths assumed form of ψ nto ( ) ( ) equaton () yelds eplct equatons for the vscous and ϕ = A for the nvscd regon nvscd regons namely: ( ) and for the vscous component, we have the followng thrdorder ordnary dfferental equaton (ODE) 3 ν d F d F df + ϕ A 3 + = dy dy dy Wth accompanyng boundary condtons: ( ) ( ) (3) at y = F = F = (4a) ( ) as y F y (4b) We adopt Wlco [6] approach, and ntroduce the followng dmensonless ndependent and dependent varables η y A, f ( η ) A F ( y) = ν = ν nto equatons (3) and (4) to produce: 3 d f d f df + f 3 + = dy dy dy wth the followng boundary condtons ( ) ( ) (5) at y = f = f = (6a) ( ) ( ) as y f y, f y (6b) Equaton (5) s the so called Hemenz equaton. A slghtly dfferent approach has been adopted for ts dervaton. One of the reasons for ths s the relatve ease wth whch boundary condtons are handled as well as the avodance of an teratve procedure to arrve at an approprate ntal condton. Equaton (5) facltates the computaton of the velocty feld, but for many computatons, we also need the pressure feld. To ntate ths, we mae another assumpton that the pressure comprses the statc and dynamc components and can be descrbed as follows: p p = ρa χ ( ) + F ( y) (7) For the classcal case of stagnaton pressure, at = y =, ( ) y= F y =, and the dynamc component s completely obvated. But we rela the condtons a lttle bt by consderng the fact that as wth velocty we epect the pressure to approach the nvscd flow soluton n the lmt as y, gven ths crteron, the pressure should satsfy the condton: p p ρa + y as y Snce a clear concept of the pressure dynamcs has been determned, we need an eplct equaton for pressure to valdate our assumptons. Tang the dvergence of the momentum n a two-dmensonal form of the Naver-Stoes equaton and after rearrangng terms, we obtan the Posson equaton for pressure u v u v p = ρ + + Though t can be assumed that once the velocty feld has been calculated from the streamfuncton values, equaton(9) can be solved to yeld the pressure, nevertheless, t stll has to be put n a form that satsfes the physcs of the flow as dctated by equatons (7) and (8). Equaton (7) s further epanded as follows: and the functon Y ( y ) satsfes: (8) (9) χ ( ) = (a) d Y df = 4 dy dy wth the followng boundary condtons: (b) dy at y =, Y ( ) =, as y ( ) y () dy Equaton () guarantees that p be the true stagnaton pressure whle the pressure gradent matches that of the nvscd flow n a regon far above the surface. In order to complete our soluton of the pressure equaton, we need to recast equaton (b) nto ts dmensonless form. We defne: A P( η ) = Y ( y) and substtutng nto equaton (b) we η obtan: d P df = 4 dη dy accompaned by the followng boundary condtons () dp P( ) = η as η (3) dη Usng equaton (5), the squared gradent of the dmensonless streamfuncton n equaton () s elmnated. Integratng twce and usng equaton (3), we obtan a much smplfed form of the pressure profle:

4 9 Oey Oseloa Onyejewe: Unsteady Formulatons for Stagnaton Pont Flow Towards a Stretchng and Shrnng Sheet wth Prescrbed Surface Heat Flu and Vscous Dsspaton P ( η ) f ( η ) = df + (4) dη We note that the orgnal equaton (equaton ()) for the streamfuncton was fourth order and represents the vscous analog of the Laplace s equaton for the streamfuncton. In addton t satsfes the contnuty equaton automatcally. After ntegratng once, equaton () was reduced to thrd order ODE (equaton 5). The overrdng approach found n lterature s to reduce the governng thrd order ODE to systems of lower order Odes before applyng numercal technques le the Runge Kutta method. We adopt a dfferent approach here by reducng equaton (5) to ts second order analog before adoptng a tme marchng procedure. We recall that the dmensonless streamfuncton s related to the horzontal velocty u η = f Ths facltates the ( ). formulaton of a tme marchng scheme wth the boundary condtons: u f u = u + u t η η ( ) ( ) ( η ) (5) f =, u =, u = U as η (6a) For an ntal condton, a guess s made by assumng a quadratc profle for the velocty profle near the surface of the plate u ( η ) η = 3,, η 3 3 η (6b) Net we consder the boundary layer equaton for heat transfer T T T u ρ c p u v µ + = + (7a) where ρ s the densty, c p s the specfc heat, µ s the nematc vscosty, T s the temperature,, y are the ndependent varables. The boundary condtons for the velocty and temperature felds are at y = u = v = T = (7b) as y u U, T T (7c) To solve the energy equaton we obtan the velocty feld from the soluton of the momentum equaton. These are then substtuted nto equaton (6) n terms of the smlarty varables η and f. We ntate ths process by convertng equaton (6) and (7) nto dmensonless forms by ntroducng: ( ηf f ) T T u ν θ =, = f, =.5.5 U U νu c to fnally obtan a dmensonless energy equaton together wth the boundary condtons: ( f ) θ dθ Pr f Pr η + dη = (8) d d θ at η = =, as η θ η 3. Dscretzaton (9) For ths study, we have elected to convert equaton (8) to a tme marchng numercal scheme by ntroducng a temporal dervatve. We employ the Cran-Ncolson Newton-Rchtmeyer scheme for dscretzaton, whose general form for a onedmensonal partal dfferental equaton for nonlnear dffuson, convecton and source term s gven as: + φ φ φ ( ) ( ) 3 ( ) t = ϑ φ + ϑ φ + ϑ φ φ + () + φ φ ϑ ( φ ) + ϑ ( φ ) + ϑ3 ( φ ) φ where + ( ) φ ( ) φ ϑ φ ϑ φ t ϑ ( φ ) φ = + t (a) + φ φ ϑ ( φ ) = ϑ ( φ ) ϑ φ φ φ + t ϑ φ t + t Usng a forward dfference for the tme dervatve: Smlarly + (b) φ φ ϑ ( φ ) ( ) = ϑ φ (c) + ϑ φ + φ + φ ϑ φ + + ϑ + + = + + φ [ ] [ ] ϑφ ϑφ φ φ ϑ φ Wthout any loss n generalty; assume ( ) ϑ φ equaton () can be wrtten as: + ( φ ) φ φ φ φ φ φ φ = (d) = φ, then + (e)

5 Internatonal Journal of Flud Mechancs & Thermal Scences 7; 3(): 6-4 The dffuson term can now be obtaned as follows: ( ) ( ) + + φ φ φ φ φ φ φ φ = By the same toen ( ) ( ) φ+ φ+ φ φ φ φ φ φ = + φ φ = ( φ+ + φ )( φ+ φ ) ( φ + φ )( φ φ ) (f) (g) (h) Followng ths approach, a nonlnear reacton term s dscretzed as: ( φ ) χ χ ϒ + ϒ φ = ϒ φ + t t ( ϒφ ) = = ϒ ( ϒ ) φ t χ φ χ + t χ φ φ χ () where ϒ s the reacton coeffcent, and χ s the power to φ + whch the nonlnear term s rased. We note that s now the new dependent varable and s related to the desred + + dependent varable by φ = φ φ. A forward dfference dscretzaton of the transent term results n + φ φ = t t. Coeffcents of φ, φ, φ +, consttute the trdagonal coeffcent matr and together wth the nown rght hand sde nown vector are solved wth the Thomas algorthm for the dependent varables untl steady state s acheved. The dscretzaton of the momentum and energy equatons follow the stencl set out n equaton() and are much easer to apply for lnear terms. 4. Results and Dscussons We estmated the value of the far pont η by usng an ntal dstance to solve the momentum and heat transfer u and θ. The soluton process equatons to obtan ( ) ( ) s agan repeated wth another value of dstance η untl the two successve values of ( ) ( ) small predetermned value. u and θ dffer by a very Fg. shows the dmensonless streamfuncton, f ( η ), ts second dervatve f ( η ), the horzontal velocty u ( η ) as well as the pressure p( η ). Both the horzontal velocty as well as the streamfuncton are zero at the surface and the horzontal velocty approaches the nvscd value at the far- feld. On the other hand the pressure profle satsfes the boundary condton as specfed n equaton (). The flow results are n agreement wth those n page 57 Wlco [6]. Snce ths wor s heat flu based, we shall be focusng a lot of on temperature and temperature gradent profles for dfferent parameter values. Dependent Varables Stagnaton Pont Flow Fgure. Flow and pressure profles. f df/d d f/d pressure The effects of Prandtl number, negatve and postve velocty ratos for an sothermal left sde boundary condton on the temperature and temperature gradent profles are llustrated n Fgs (3-6). For Fg. 3, each temperature profle, ntally decreases and then ultmately goes to zero for large values of η. The temperature gradent profles for Fg. 4 ehbt an ntal decrease n gradent, then a rse followed by an asymptotc approach to zero for all the profles. Based on these observatons, t s worthwhle to note that consderng the values of Prandtl numbers, f the momentum dffusvty s hgh (.e. free movement of flud due to densty dfferences) and the thermal dffusvty s low (.e. for low thermal conductvty fluds), the overall heat transfer s facltated by natural convecton. Prandtl number s relatvely hgh for such cases. Ths s what obtans for natural convecton of certan fluds, for eample water velocty rato = -.8 Pr=. Pr=. Pr= Fgure 3. dstrbutons for zero flu, negatve velocty rato, dfferent values of Prandtl number.

6 Oey Oseloa Onyejewe: Unsteady Formulatons for Stagnaton Pont Flow Towards a Stretchng and Shrnng Sheet wth Prescrbed Surface Heat Flu and Vscous Dsspaton Gradent velocty rato = -.8 Pr =. Pr =. Pr= 6. The mpact of a postve velocty rato on the temperature and temperature gradent profles can be seen n Fg. 5 and Fg. 6. As epected, both the temperature and the temperature gradent decrease along the surface n accordance wth the specfed boundary condton. But the postve velocty rato effect on the zero flu boundary condton s profound. Ths s as a result of a sgnfcant drop n temperature and the alteraton of the shape of the boundary layer. Hence an ncrease n the velocty rato causes the temperature of the flud to decrease. For such a case, the thcness of the thermal boundary layer decreases due to the flud at the surface havng a temperature lower than that of the flat plate Fgure 4. gradent profles for zero flu, negatve velocty rato, dfferent values of Prandtl number velocty rato = -.8 Pr=. Pr=. Pr= velocty rato =.8 Pr=. Pr=. Pr= Fgure 7. dstrbutons for unt postve flu, negatve velocty rato, dfferent values of Prandtl number Fgure 5. profles for zero flu, postve velocty rato, dfferent values of Prandtl number. -5 velocty rato = -.8 Pr =. Pr =. Pr= 6..5 velocty rato =.8 Gradent - -5 Gradent Pr =. Pr =. Pr= Fgure 8. gradent dstrbutons for unt postve flu, negatve velocty rato, dfferent values of Prandtl number Fgure 6. gradent profles for zero flu, postve velocty rato, dfferent values of Prandtl number. Fgs. 7 and 8 show the profles resultng from specfyng a unt postve flu at the left boundary of the plate as well as a negatve velocty rato. The profles have the same shapes as those for the zero flu ecept that ther magntudes are dfferent. The temperature profles dsplay a steeper front and hgher magntude n ths case. The temperature gradents are

7 Internatonal Journal of Flud Mechancs & Thermal Scences 7; 3(): 6-4 also more promnent. Wth regards to the magntudes of the scalar profles and the assgned values of the Prandtl number, we note that f the momentum dffusvty s low (.e. there s not enough densty dfferences to enhance the movement of the flud) and the thermal dffusvty s low (.e. for fluds wth low values of thermal dffusvty), then the transfer of heat from the from the boundary wll not be facltated as the flud s not only ncapable of absorbng heat but also cannot move freely due to nsuffcent momentum from convecton currents. The profles we encounter here are better assessed f we compare Fgs. 7 and 3 to see how the ntroducton of a postve flu at the left boundary (wth all the other parameters remanng the same) results n an overall relatve ncrease n temperature wthn ths regon. It s obvous that we encounter a vector quantty here whose drecton nforms us that heat s drawn out of the plate. Fgs 9 and show the effect of a postve velocty rato on the temperature and flu profles. An ncrease n velocty rato results n a sgnfcant temperature loss mplyng that the thermal boundary layer decreases. Agan t should be remembered that because of ts dependence on the flud velocty, the temperature feld s affected by the velocty rato. As the velocty rato ncreases, the temperature of the flud decreases. Ths s because the postve value of the velocty rato promotes heat transfer to the flud. Fg. shows a monotonc rse n temperature gradent untl t gets to a pont where the temperatures of all the fluds become zero n agreement wth the specfed Drchlet boundary condton at the rght end of the plate velocty rato =.8 Pr=. Pr=. Pr= Fgure 9. dstrbutons for unt postve flu, postve velocty rato, Prandtl numbers. Fgs and 7 llustrate the effect of changng the sgn and hence the drecton of the flu at the left boundary for the same problem parameters. One should not loose sght of the Fourer s law and the mplcaton of ts vector nterprton concernng the drecton of the flu. Consequently Fg. 7 records a much hgher temperature than Fg.. However Fg. dsplays much steeper slopes than Fg. 7. Fg. gves values of the flu and reflects the heat transfer trend of Fg.. Fg. 3 dsplays the temperature profles for a unt negatve flu specfcaton at the left boundary, eepng the magntude of the velocty rato the same. Ths gves us a handle on the magntude of heat flow as well as the drecton. A comparson made between Fgs 3 and 9 yelds valuable nformaton concernng the ln between the temperature profles dsplayed n each case as well as magntude and sgns (drectons) of the flu specfcaton. Fgs and 4 gve further nformaton concernng the flu. Whle Fg. 9 shows a monotonc decrease n temperature for all values of Prandtl number, Fg. 3 dsplays an eactly opposte trend. Each of these cases s n consonance wth the vector nterprton of the Fourer s law. However, one thng they both have n common s that at the regon η > 3 all the profles move towards satsfyng the Drchlet boundary condton specfcaton at the rght boundary. Gradent velocty rato =.8 Pr =. Pr =. Pr= Fgure. gradent dstrbutons for unt postve flu, postve velocty rato, Prandtl numbers velocty rato = -.8 Pr=. Pr=. Pr= Fgure. dstrbutons for unt negatve flu, negatve velocty rato, Prandtl numbers.

8 3 Oey Oseloa Onyejewe: Unsteady Formulatons for Stagnaton Pont Flow Towards a Stretchng and Shrnng Sheet wth Prescrbed Surface Heat Flu and Vscous Dsspaton Gradent Fgure. gradent profles for unt negatve flu, negatve velocty rato, Prandtl numbers. velocty rato = -.8 velocty rato =.8 Pr =. Pr =. Pr= Conclusons In ths wor, we have presented unsteady numercal formulatons for stagnaton pont flow towards a sheet wth prescrbed surface heat flu and vscous dsspaton. Snce more emphass s devoted to the energy component of the formulaton, several graphcal plots are made to llustrate the temperature and the temperature gradent felds. The numercal results obtaned are graphcally dsplayed and dscussed for varous problem parameters consdered n the problem formulaton. Numercal eperments nvolvng flu varatons are made on the left of the problem doman. For each of these results, we can observe that as we move far from the sheet, the effects of vscous dsspaton vansh and the profles ntersect each other at some pont. Some of the basc features of ths model can be appled to study a varety of scenaros nvolvng more comple formulatons such as heat characterstcs for Newtonan and non-newtonan fluds, flows n porous meda and MHD slp flow to menton just a few Pr=. Pr=. Pr= 6. Acnowledgement We than the anonymous referee for hs/her useful comments. References [] A. Bejan, Convectve heat transfer nd ed. Wley NY Fgure 3. profles for unt negatve flu, postve velocty rato, Prandtl numbers. Gradent velocty rato =.8 Pr =. Pr =. Pr= Fgure 4. gradent profles for unt negatve flu, postve velocty rato, Prandtl numbers. [] K. Hemenz, De grenschcht an enem n den glechformngen flussgetsstrom enguchten ger-aden reszylnder, Dngl., Polytech. Journal vol. 3 pp. 3-4, 9. [3] F. Homann, Der enfluss grosset zahget be der stromung um den Zylnder and um de ugel, Zetschrft fur Angewandte Mathematc und mechanc, vol. 6 pp , 936. [4] T. Y. Na Computatonal methods n engneerng boundary value problem, Academc Press, NY 979. [5] L. J. Crane, Flow past a stretchng sheet, Zetschrft fur Angewandte Mathemat und Phys vol. pp , 97. [6] P. Caragher and L. J. Crane, Heat transfer n a contnuous stretchng sheet, ZAMM vol. 6 pp , 985. [7] P. S. Gupta and A. S. Gupta, Heat and mass transfer for a stretchng sheet wth unform sucton and blowng, The Canadan Journal of Chemcal Engneerng, vol. 55 pp , 977. [8] T. C. Cham, Stagnaton pont flow towards a stretchng sheet J. Phys. Soc. Jpn. Vol. 63 pp , 994. [9] B. C. Saads, Boundary-layer behavor on contnuous sold surfaces: I. Boundary-layer equatons for two-dmensonal and asymmetrc flow, Amercan nsttute of chemcal engneers(aiche) Journal vol. 7() pp. 6-8, 96.

9 Internatonal Journal of Flud Mechancs & Thermal Scences 7; 3(): [] B. C. Saads, Boundary layer behavor on contnuous sold surfaces: II The boundary layer on a contnuous flat surface, AICHE J. vol. 7 pp. -5, 96. Do:./ac.697 [] T. R. Mahapatra and A. S. Gupta, Heat transfer n stagnatonpont flow towards a stretchng sheet, Heat and Mass Transfer, vol. 38 pp. 57-5,. [] P. Dulal and P. S. Hremath, Computatonal modellng of heat transfer over an unsteady stretchng surface embedded n porous medum, Mecccanca, vol. 45 pp , 9. [3] D. B. Ingham and I. Pop, Transport n porous meda III Elsever, Oford 5. [4] D. A. Neld and A. Bejan, Convecton n porous meda, 3 rd ed. Sprnger scence and busness meda nc. N. Yor 5. [5] B. K. Dutta, P. Roy A. S. Gupta, feld n flow over a stretchng sheet wth unform heat flu, Int. Comm. Heat and Mass Trans vol.8 pp , 985. [6] M. E. Al and E. Magyar, Unstaedy flud and heat flow nduced by a submerged stretchng surface whle ts steady moton s slowng down gradually, Int. Jnl. Heat and Mass Trans. vol. 5 pp , 7. [7] A. Isha, R. Nazar, and I. Pop, Magnetohydrodynamcs stagnaton pont flow towards a stretchng vertcal sheet, Magneto-hydrodynamcs, vol. 4 pp.7-3, 6. [8] H. I. Anderson, MHD flow of vscoelastc flud past a stretchng surface, Acta Mechanca,, vol. 95 pp. 7-3, 99. [9] M. I. Char, Heat and mass transfer n a hydromagnetc flow of the vscoelstc flud past a stretchng surface, Physcs of Fluds vol. 7 pp , 994. [] T. R. Mahapatra and S. K. Nandy, Momentum and heat transfer n MHD asymmetrc stagnaton pont flow over a shrnng sheet, Jnl. Appled Flud Mechancs, vol. 6 pp. -9,. [] O. D. Mande and W. M. Charles, Computatonal dynamcs of hydromagnetc stagnaton flow towards a stretchng sheet, Appled, Comput. Math. vol. 9 () pp. 43-5, [] O. D. Mande, On MHD boundary layer flow and mass transfer past a vertcal plate n a porous medum wth constant heat flu, Int. Jnl. Num. Mthds. For Heat and Flud flow. Vol. 9 pp , 9. [3] K. Bhattacharyyya, Heat transfer n boundary layer stagnaton pont flow towards a shrnng sheet wth nonunform heat flu:, Chn. Physcs B vol. pp , 3. [4] I. Mealey and J. H. Mern, Free convecton boundary layers on a vercal surface n a heat generatng porous medum, IMA Journal of Appled Mathematcs vol. 73 pp. 3-53, 8. [5] A. Postelncu and I. Pop, Smlarty soluton of free convecton boundary layers over vertcal and horzontal surfaces n porous meda wth nternal heat generaton, Int. Comm. Heat and Mass Transfer, vol. 6 pp. 83-9, 999. [6] D. C. Wlco, Basc flud mechancs nd ed. DCW ndustres 3.