Mixed Convection of the Stagnation-point Flow Towards a Stretching Vertical Permeable Sheet

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1 Malaysan Mxed Journal Convecton of Mathematcal of Stagnaton-pont Scences 1(): Flow 17 - Towards 6 (007) a Stretchng Vertcal Permeable Sheet Mxed Convecton of the Stagnaton-pont Flow Towards a Stretchng Vertcal Permeable Sheet Anuar Ishak 1 *, Roslnda Nazar 1, Norhan M. Arfn & Ioan Pop 3 1 School of Mathematcal Scences, Unverst Kebangsaan Malaysa, UKM Bang, Selangor, Malaysa Department of Mathematcs, Unverst Putra Malaysa, UPM Serdang, Selangor, Malaysa 3 Faculty of Mathematcs, Unversty of Cluj, R-3400 Cluj, CP 53, Romana ABSTRACT An analyss was done for the steady two-dmensonal stagnaton-pont mxed convecton flow of an ncompressble vscous flud towards a stretchng vertcal permeable sheet n ts own plane. The stretchng velocty and the surface temperature are assumed to vary lnearly wth the dstance from the stagnaton-pont. Two equal and opposte forces are mpulsvely appled along the x-axs so that the wall s stretched, keepng the orgn fxed n a vscous flud of constant ambent temperature. The transformed boundary layer equatons were solved numercally for some values of the parameters consdered usng an mplct fnte dfference scheme known as the Keller-box method. Flow and heat transfer characterstcs were analyzed and dscussed. Both cases of the assstng and opposng flows were consdered and t was found that dual solutons exst for the opposng flow, whereas a unque soluton resulted for the assstng flow. Keywords: Boundary layer, heat transfer, mxed convecton, permeable sheet, stagnaton-pont flow, stretchng sheet INTRODUCTION The flow near a stagnaton pont has attracted many nvestgatons durng the past several decades because of ts wde applcatons n many practcal applcatons such as coolng of electronc devces by fans, coolng of nuclear reactors, and many hydrodynamc processes. Hemenz (1911) was the frst to study the two-dmensonal stagnaton-pont flow, and obtaned an exact smlarty soluton of the governng Naver-Stokes equatons. Snce then many nvestgators have consdered varous aspects of such flow, and obtaned closedform analytcal as well as numercal solutons. Ramachandran et al. (1988) studed lamnar mxed convecton n two-dmensonal stagnaton flows around heated surfaces by consderng both cases of an arbtrary wall temperature and arbtrary surface heat flux varatons. They found that a reverse flow developed n the buoyancy opposng flow regon, and dual solutons are found to exst for a certan range of the buoyancy parameter. Ths work was then extended by Dev et al. (1991) for the unsteady case, and by Lok et al. (005) for a vertcal surface mmersed n a mcropolar flud. Dual solutons were found to exst by these authors for a certan range of buoyancy parameter. * Correspondng Author E-mal: anuarshak007@yahoo.co.jp Malaysan Journal of Mathematcal Scences 17

2 Anuar Ishak, Roslnda Nazar, Norhan M. Arfn & Ioan Pop All of the above-mentoned nvestgatons consdered the flow mpngng normally to a vertcal or horzontal surface at rest. The stagnaton pont flows toward a surface whch s moved or stretched, have been consdered for example by Cham (1994, 1996), Mahapatra and Gupta (001, 00), Nazar et al. (004a,b) and more recently by Ishak et al. (006a,b, 007a). The am of ths study was to extend the work by Ishak et al. (006b) for the case of a permeable surface. A thorough revew on the flow and heat transfer over a permeable stretchng surface can be found n Gupta and Gupta (1977), Dutta (1989), Watanabe (1991), and Magyar and Keller (000). In the actual manufacturng process, the stretched surface speed and temperature play an mportant role n the coolng process. Furthermore, durng the manufacture of plastc and rubber sheets, t s often necessary to blow a gaseous medum through the not-yet-soldfed materal. The study of flow feld and heat transfer s necessary for determnng the qualty of the fnal products. PROBLEM FORMULATION AND BASIC EQUATIONS Consder the stagnaton flow of an ncompressble vscous flud normal to a vertcal plate as shown n Fg. 1. It s assumed that the ambent flud s moved wth a velocty u e (x) = ax n the y-drecton towards the stagnaton pont on the plate, where a s a constant and a 0. It s also assumed that the surface s stretched n the x-drecton such that the x-component of the velocty and temperature vary lnearly along t,.e. u w (x) = bx and T w (x) = T + cx, respectvely, where b (>0) and c are arbtrary constants. Under these assumptons along wth the Boussnesq and boundary layer approxmatons, the system of equatons whch model the boundary layer flow are gven by u v + = 0, x y u u due u u + v = ue + v ± gβ T T x y dx y α u T + v T = T x y y, ( ), (1) () (3) where u and v are the velocty components along the x and y axes, respectvely, T s the flud temperature, g s the gravty acceleraton, α, ν and β are the thermal dffusvty, knematc vscosty and thermal expanson coeffcents, respectvely. The last term on the rght-hand sde of Eq. () represents the nfluence of the thermal buoyancy force, wth + and sgns pertanng to the buoyancy assstng and opposng flow regons, respectvely. Fg. 1 llustrates such a flow feld for a vertcal, heated surface wth the upper half of the flow feld beng asssted and the lower half of the flow feld beng opposed by the buoyancy force. The reverse trend wll occur f the plate s cooled below the ambent temperature. The reported results are thus true for both the heated and cooled surface condtons when the 18 Malaysan Journal of Mathematcal Scences

3 Mxed Convecton of the Stagnaton-pont Flow Towards a Stretchng Vertcal Permeable Sheet approprate (assstng and opposng) flow regons are selected (see Ramachandran et al. (1988)). We shall assume that the boundary condtons of Eqs. (1) (3) are v = Vw, u = uw( x), T = Tw( x) at y = 0, u u ( x), T T as y. e (4) In order for smlarty solutons to exst, V w has to be of the form (see Gupta and Gupta (1977), Dutta (1989) and Ishak et al. (007b,c,d,e,f)) Vw = ( bν ) 1/ γ, (5) where γ = f(0), wth γ > (0) s for mass sucton and γ < (0) s for mass njecton. The contnuty equaton can be satsfed by ntroducng a stream functon ψ such that ψ ψ u =, v =. y x (6) The momentum and energy equatons can be transformed to the correspondng ordnary dfferental equatons by the followng substtutons: 1/ b ψ T T η = y, f ( η) =, θ( η) =. ν 1/ ( bν ) x Tw T (7) The transformed ordnary dfferental equatons are: f '" + ff " f ' + ε + λθ = 0, 1 " ' ' 0, Pr θ + fθ f θ = (8) (9) u w T w x,u Buoyancy assstng regon u e O y,v u e T w u w Buoyancy opposng regon Fg. 1: Physcal model and coordnate system Malaysan Journal of Mathematcal Scences 19

4 Anuar Ishak, Roslnda Nazar, Norhan M. Arfn & Ioan Pop subject to the boundary condtons (4) whch become f ( 0 ) = γ, f '( 0) = 1, θ( 0) = 1, f '( ) ε, θ( ) 0, (10) where ε = a/b, prmes denote dfferentaton wth respect to η, Pr s the Prandtl number and the constant λ s the buoyancy or mxed convecton parameter defned as Gr λ =± Re x, 1/ x (11) wth the ± sgn havng the same meanng as n Eq. (). Further, Gr x = gβ (T w T ) x 3 /ν s the local Grashof number and Re x = u w x/ν s the local Reynolds number. It should be notced that λ > 0 and λ < 0 correspond to the assstng and opposng buoyant flows, respectvely. When λ = 0 and ε = 1, the soluton of Eq. (8) subject to the boundary condtons s gven by f ( η) = η+ γ. (1) The physcal quanttes of nterest are the skn frcton coeffcent C f and the local Nusselt number Nu x, whch are defned as C f τ xqw =, Nux =, ρu / k( T T) w w w respectvely, where the skn frcton τ w and the heat transfer from the plate q w are gven by (13) u T τ w = µ, q = k, y w y y= 0 y= 0 (14) wth µ and k beng the dynamc vscosty and thermal conductvty, respectvely. Usng the non-dmensonal varables (7), we get 1 1/ 1/ C Re "(0), / Re '( 0 ) f x = f Nu x x = θ. (15) Equatons (8) and (9) subject to the boundary condtons (10) are solved numercally by an mplct fnte dfference scheme. To support the valdty of the numercal method used, we get approxmate solutons of Eqs. (8) and (9) subject to the boundary condtons (10) vald for small values of ε. By nvestgatng the terms n the governng equaton, we assume that the soluton of Eqs. (8) and (9) near ε = 0 s of the form ( ) = f ( ) ( ) = ( ) f η η ε, θ η θ η ε, = 0 = 0 (16) where f and θ are the perturbatons n f and θ, respectvely. Substtutng expressons (16) nto Eqs. (8) and (9), and comparng the lke terms of ε gves the followng set of equatons: 0 Malaysan Journal of Mathematcal Scences

5 Mxed Convecton of the Stagnaton-pont Flow Towards a Stretchng Vertcal Permeable Sheet and λθ0 f + f f f + = 0, 1 θ0 + f0θ0 f0θ0 = 0, Pr (17) f + f f f f + δ + λθ = 0, j j 0 j= 0 1 θ + f θ f θ = 0, (18) Pr j j j j j= 0 j= 0 for 1, subject to the boundary condtons f ( 0 ) = δ f, f ( 0 ) == δ, θ ( 0) = δ f ( ) δ, θ ( ) 0, 1 (19) for 0, where δ j s the delta Kronecker, whch s defned by δ j 1 f = = 0 f j j The skn frcton coeffcent and the local Nusselt number (15) are approxmately gven now, respectvely, by 1 1/ ( ) ( ) ( ) Cf Rex = f0 0 + f1 0 ε + f 0 ε, 1/ ( ) ( ) ( ) Nu /Re = θ0 0 + θ1 0 ε + θ 0 ε x x. (0) RESULTS AND DISCUSSION Equatons (8) and (9) subject to the boundary condtons (10) were solved numercally usng the Keller-box method, whch s descrbed n Cebec and Bradshaw (1988). The results shown are for a study on the nfluences of several non-dmensonal parameters. The results of the skn frcton coeffcent, local Nusselt number, velocty and temperature dstrbutons are llustrated n graphs, whle the values of the skn frcton coeffcent and the local Nusselt number for some parameters are gven n tables. Tables 1 and show the values of the skn frcton coeffcent and the local Nusselt number, respectvely, for certan values of the related parameters. The values of the skn frcton coeffcent f (0) obtaned n ths study are compared wth those of Mahapatra and Gupta (00) and Nazar et al. (004a,b), whle the values of the local Nusselt number θ (0) whch represent the heat transfer rate at the surface are compared wth those of Al (1995). The comparsons revealed good agreement. Malaysan Journal of Mathematcal Scences 1

6 Anuar Ishak, Roslnda Nazar, Norhan M. Arfn & Ioan Pop TABLE 1 The values of f (0) for varous values of ε when Pr =1, λ = 0 and γ = 0 ε Mahapatra and Nazar et al. Nazar et al. Result of Gupta (00) (004a) (004b) ths study TABLE The values of θ (0) for varous values of γ and Pr when ε = 0 and λ = 0 Results of Al (1995) Results of ths study γ Pr = 0.7 Pr = 1 Pr = 10 Pr = 0.7 Pr = 1 Pr = Therefore, the code that was developed can be used wth hghconfdence to study the problem dscussed n ths paper. Moreover, to support the valdty of the numercal method used, the governng equatons (8)-(10) were solved by usng seres expanson that are vald for small values of the velocty rato parameter ε (= a/b). The values of f (0) and θ (0) obtaned by usng both methods are shown n Tables 3 and 4, respectvely. The results show excellent agreement for small ε. The values of θ (0) as shown n Tables and 4 are always postve. Ths follows from the ntegral relatonshp θ ( 0) = Pr f θd η whch 0 s obtaned from Eqs. (9) and (10). Furthermore, the local Nusselt number as shown n Table ncreases wth Pr. Ths s a result of the decreasng thermal boundary layer thckness whch mples an ncrease n the wall temperature gradent. Fg. a shows the numercal results of the dmensonless skn frcton coeffcent for varous values of the sucton/njecton parameter γ when Pr = 1 and ε = 1, whle the respectve local Nusselt number are presented n Fg. b. It s seen from Fg. a that for Pr = 1 and ε = 1, the values of f (0) are postve for the assstng flow (λ > 0), and negatve for the opposng flow (λ < 0). Physcally, a postve sgn of f (0) mples that the flud exerts a Malaysan Journal of Mathematcal Scences

7 Mxed Convecton of the Stagnaton-pont Flow Towards a Stretchng Vertcal Permeable Sheet TABLE 3 The values of f (0) for varous values of ε and λ when Pr = 1 and γ = 0 Numercal Eq. (15) Small ε Eq. (0) ε λ = 0.1 λ = 1 λ = 10 λ = 0.1 λ = 1 λ = TABLE 4 The values of θ (0) for varous values of ε and λ when Pr = 1 and γ = 0 Numercal Eq. (15) Small ε Eq. (0) ε λ = 0.1 λ = 1 λ = 10 λ = 0.71 λ = 1 λ = drag force on the sheet and a negatve sgn mples the opposte. Moreover, all curves ntersect at a pont where λ = 0,.e. when the buoyancy force s absent. The value of f (0) at ths pont s zero. Ths s not surprsng snce Eqs. (8) and (9) are uncoupled when λ = 0, and the stretchng velocty s equal to the ambent flud velocty when ε = 1, whch mples skn frcton τ w = 0. Ths result s n agreement wth the exact soluton (1), whch mples f (η) = 0, for all η. In contrast, Fg. b shows that there are heat transfers from the sheet to the flud even when the skn frcton s zero. Ths s because the sheet and the flud are of dfferent temperatures. Fgs. a and b show the exstence of dual solutons for a certan range of λ < 0 (opposng flow). The soluton for a partcular value of γ exsts up to a crtcal value of λ (say λ c ). Beyond ths value, the boundary layer separated from the surface, thus we are unable to get the soluton usng the boundary layer approxmatons. To proceed wth the soluton, the full Naver-Stokes equatons have to be solved. It s evdent from Fgs. a Malaysan Journal of Mathematcal Scences 3

8 Anuar Ishak, Roslnda Nazar, Norhan M. Arfn & Ioan Pop and b that sucton delays the boundary layer separaton, whle njecton accelerates t. The curve bfurcates at λ = λ c, and the lower branch soluton contnues further and termnates at a certan value of λ. It should be remarked that the computatons have been performed untl the pont where the soluton dd not converge, and the calculatons were termnated at that pont. Fgs. 3a and 3b depct the velocty and temperature profles for selected values of the parameters, whch support the exstence of dual solutons shown n Fgs. a and b. (a) (b) Fg. : (a) Varatons of f (0); (b) Varatons of θ (0), as a functon of λ at selected values of γ when Pr = 1 and ε = 1 (a) (b) Fg. 3: (a) Velocty profles f (η); (b) Temperature profles θ (η), for varous values of γ when Pr = 1, ε = 1 and λ = 5 Fgs. 4a and 4b show the velocty and temperature profles for selected values of ε when Pr = 1 and λ = 1. Fg. 4a, shows that when ε > 1, the flow has a boundary layer structure and the thckness of the boundary layer decreases wth ncrease n ε. Accordng to Mahapatra and Gupta (00), for a fxed value of b correspondng to the stretchng of the surface, an ncrease n a n relaton to b (such that a/b > 1) mples an ncrease n the stranng moton near the stagnaton regon resultng n ncreased acceleraton of the external stream, and ths leads to the thnnng of the boundary layer wth an ncrease n ε. Further, Fg. 4a shows that when ε < 1, the flow has an nverted boundary layer structure. Ths s a 4 Malaysan Journal of Mathematcal Scences

9 Mxed Convecton of the Stagnaton-pont Flow Towards a Stretchng Vertcal Permeable Sheet (a) (b) Fg. 4: (a) Velocty profles f (η); (b) Temperature profles θ (η), for varous values of ε when Pr = 1 and λ = 1 result from the fact that when ε < 1, the stretchng velocty, bx, of the surface exceeds the velocty, ax, of the external stream. Fgs. 3 and 4 show that the boundary condtons (10) are satsfed, whch supports the valdty of the results of ths study. CONCLUSIONS Ths study concerned the theory of the problem of stagnaton-pont flow and heat transfer towards a stretchng vertcal permeable sheet mmersed n a vscous and ncompressble flud. The governng boundary layer equatons were solved numercally usng an mplct fnte dfference method. Both the skn frcton coeffcent and the local Nusselt number ncrease wth ncreasng buoyancy effects. Sucton ncreases the heat transfer from the surface, whereas njecton causes a decrease. Moreover, sucton delays the boundary layer separaton, whle njecton acts n the opposte. Dual solutons were found to exst for the opposng flow, and Prandtl number enhances the heat transfer rate at the surface. ACKNOWLEDGEMENTS Ths work s supported by a research grant (SAGA fund: STGL ) from the Academy of Scences Malaysa. REFERENCES ALI, M.E On thermal boundary layer on a power-law stretched surface wth sucton or njecton. Int. J. Heat Flud Flow 16: CEBECI, T. and P. BRADSHAW Physcal and Computatonal Aspects of Convectve Heat Transfer. New York: Sprnger. CHIAM. T.C Stagnaton-pont flow towards a stretchng plate. J. Phys. Soc. Japan 63: CHIAM. T.C Heat transfer wth varable conductvty n a stagnaton-pont flow towards a stretchng sheet. Int. Comm. Heat Mass Trans. 3: Malaysan Journal of Mathematcal Scences 5

10 Anuar Ishak, Roslnda Nazar, Norhan M. Arfn & Ioan Pop DEVI, C.D.S., H.S. TAKHAR and G. NATH Unsteady mxed convectn flow n stagnaton regon adjacent to a vertcal surface. Heat Mass Trans. 6: DUTTA, B.K Heat transfer from a stretchng sheet wth unform sucton and blowng. Acta Mech. 78: GUPTA, P.S. and A.S. GUPTA Heat and mass transfer on a stretchng sheet wth sucton or blowng, Can. J. Chem. Engng. 55: HIEMENZ, K De Grenzschcht an enem n den glechformgen Flussgketsstrom engetauchten geraden Kreszylnder. Dngl. Polytech. J. 3: ISHAK, A., R. NAZAR and I. POP. 006a. Magnethydrodynamc stagnaton-pont flow towards a stretchng ventcal sheet. Magnetohydrodynamcs 4: ISHAK, A., R. NAZAR and I. POP. 006b. Mxed convecton boundary layers n the stagnaton-pont flow toward a stretchng vertcal sheet. Meccanca 41: ISHAK, A., R. NAZAR and I. POP. 007a. Mxed convecton on the stagnaton pont flow toward a vertcal, contnously stretchng sheet. ASME J. Heat Trans. 19: ISHAK, A., R. NAZAR, N.M. ARIFIN and I. POP. 007b. Dual solutons n mxed convecton flow near a stagnaton pont on a vertcal porous plate. Int. J. Thermal Scences (n press). ISHAK, A., R. NAZAR and I. POP. 007c. Stagnaton flow of a mcropolar flud towards a vertcal permeable surface. Int. Comm. Heat Mass Trans. (n press). ISHAK, A., R. NAZAR and I. POP. 007d. Boundary-layer flow of a mcropolar flud on a contnuously movng or fxed permeable surface. Int. J. Heat Mass Trans. 50: ISHAK, A., R. NAZAR and I. POP. 007e. Dual solutons n mxed convecton boundary-layer flow wth sucton or njecton. IMA J. Appl. Math. 7: ISHAK, A., J.H. MERKIN, R. NAZAR and I. POP. 007f. Mxed convecton boundary layer flow over a permeable vertcal sucface wth prescrbed wall heat flux. Z. ANGEW. Math. Phys. (onlne frst). LOK, Y.Y., N. AMIN, D. CAMPEAN and I. POP Steady mxed convecton flow of a mcropolar flud near the stagnaton pont on a vertcal surface. Int. J. Num. Methods Heat Flud Flow 15: MAGYARI, E. and B. KELLER Exact soluton for self-smlar boundary-layer flows nduced by permeable stretchng surfaces. Eur. J. Mech. B-Fluds 19: MAHAPATRA, T.R. and A.S. GUPTA Magnetohydrodynamc stagnaton-pont flow towards a stretchng sheet. Acta Mech. 15: MAHAPATRA, T.R. and A.S. GUPTA. 00. Heat transfer n stagnaton-pont flow towards a stretchng sheet. Heat Mass Trans. 38: NAZAR, R., N. AMIN, D. FILIP and I. POP. 004a. Stagnaton pont flow of a mcropolar flud towards a stretchng sheet. Int. J. Nonlnear Mech. 39: NAZAR, R., N. AMIN, D. FILIP and I. POP. 004b. Unsteady boundary layer flow n the regon of the stagnaton pont on a stretchng sheet. Int. J. Engng. Sc. 4: RAMACHANDRAN, N., T.S. CHEN and B.F. ARMALY Mxed convecton n stagnaton flows adjacent to vertcal surface. J. Heat Trans. 110: WATANABE, T Forced and free convecton boundary layer flow wth unform sucton or njecton on a vertcal flat plate. Acta Mech. 89: Malaysan Journal of Mathematcal Scences