Hazem Ali Attia Dept. of Engineering Mathematics and Physics Faculty of Engineering, El-Fayoum University El-Fayoum, Egypt

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1 Article Templte for Publiction in the Stgntion Point Flow nd... Hzem Ali Atti Dept. of Engineering Mthemtics nd Physics Fculty of Engineering, El-Fyoum University El-Fyoum, Egypt Stgntion Point Flow nd Het Trnsfer of Micropolr Fluid with Uniform Suction or Blowing The stedy lminr flow of n incompressible non-newtonin micropolr fluid impinging on permeble flt plte with het trnsfer is investigted. A uniform suction or blowing is pplied norml to the plte, which is mintined t constnt temperture. Numericl solution for the governing nonliner momentum nd energy equtions is obtined. The effect of the uniform suction or blowing nd the chrcteristics of the non-newtonin fluid on both the flow nd het trnsfer is presented nd discussed. Keywords: stgntion point flow, non-newtonin fluid, het trnsfer, suction, numericl solution Introduction The two-dimensionl flow of fluid ner stgntion point is clssicl problem in fluid mechnics. It ws first exmined by Hiemenz (9) who demonstrted tht the Nvier-Stokes equtions governing the flow cn be reduced to n ordinry differentil eqution of third order using similrity trnsformtion. Owing to the nonlinerities in the reduced differentil eqution, no nlyticl solution is vilble nd the nonliner eqution is usully solved numericlly subject to two-point boundry conditions, one of which is prescribed t infinity. Lter the problem of stgntion point flow ws extended in numerous wys to include vrious physicl effects. The xisymmetric three-dimensionl stgntion point flow ws studied by Homnn (936). The results of these studies re of gret technicl importnce, for exmple in the prediction of skin-friction s well s het/mss trnsfer ner stgntion regions of bodies in high speed flows. Either in the two or three-dimensionl cse Nvier-Stokes equtions governing the flow re reduced to n ordinry differentil eqution of third order using similrity trnsformtion. The effect of suction on Hiemenz problem hs been considered in the literture. Schlichting nd Bussmn (943) gve the numericl results first. More detiled solutions were lter presented by Preston (946). An pproximte solution to the problem of uniform suction is given by Ariel (994). The effect of uniform suction on Homnn problem where the flt plte is oscillting in its own plne is considered by Weidmn nd Mhlingm (997). In hydromgnetics, the problem of Hiemenz flow ws chosen by N (979) to illustrte the solution of thirdorder boundry vlue problem using the technique of finite differences. An pproximte solution of the sme problem hs been provided by Ariel (994b). The effect of n externlly pplied uniform mgnetic field on the two or three-dimensionl stgntion point flow ws given, respectively, by Atti (3) nd Atti(3b) in the presence of uniform suction or injection. The study of het trnsfer in boundry lyer flows is of importnce in mny engineering pplictions such s the design of thrust berings nd rdil diffusers, trnspirtion cooling, drg reduction, therml recovery of oil, etc. Mssoudi nd Rmezn (99) used perturbtion technique to solve for the stgntion point flow nd het trnsfer of non-newtonin fluid of second grde. Their nlysis is vlid only for smll vlues of the prmeter tht determines the behvior of the non-newtonin fluid. Lter Pper ccepted My, 7. Technicl Editor: Pulo E. Miygi. Mssoudi nd Rmezn (99) extended the problem to nonisotherml surfce. Grg (994) improved the solution obtined by Mssoudi nd Rmezn (99) by computing numericlly the flow chrcteristics for ny vlue of the non-newtonin prmeter using pseudo-similrity solution. Mny reserchers considered non-newtonin fluids. Thus, mong the non-newtonin fluids, the solution of the stgntion point flow, for viscoelstic fluids, hs been given by Rjeshwri nd Rthn (96), Berd nd Wlters (964), Teipel (986), Aril (99), nd others; for power-lw fluid by Djukic (974); nd for second grde fluids by Teipel (988) nd Ariel (995) in the hydrodynmic cse nd by Atti () in the hydromgnetic cse. Stgntion point flow of non-newtonin micropolr fluid ws studied by Nth (975) nd Nzr et l. (4) with zero verticl velocity t the surfce. The potentil importnce of micropolr fluids in industril pplictions hs motivted these studies. The essence of the theory of micropolr fluid flow lies in the extension of the constitutive equtions for Newtonin fluids so tht more complex fluids such s prticle suspensions, liquid crystls, niml blood, lubriction nd turbulent sher flows cn be described by this theory. The theory of micropolr fluids, first proposed by Eringen (966), is cpble of describing such fluids. In prctice, the theory of micropolr fluids requires tht one must dd trnsport eqution representing the principle of conservtion of locl ngulr momentum to the usul trnsport equtions for the conservtion of mss nd momentum, nd dditionl locl constitutive prmeters re lso introduced (Nzr et l., 4). The key points to note in the development of Eringen's microcontinuum mechnics re the introduction of new kinemtics vribles, e.g. the gyrtion tensor nd microinertil moment tensor, nd the ddition of the concept of body moments, stress moments, nd microstress verges to clssicl continuum mechnics. However, serious difficulty is encountered when this theory is pplied to rel, non-trivil flow problems; even for the liner theory, problem deling with simple microfluids must be formulted in terms of system of nineteen prtil differentil equtions in nineteen unknowns nd the underlying mthemticl problem is not esily menble to solution. These specil fetures of micropolr fluids were discussed in comprehensive review pper of the subject nd ppliction of micropolr fluid mechnics by Arimen et l. (973). The purpose of the present pper is to study the effect of uniform suction or blowing directed norml to the wll on the stedy lminr flow of n incompressible non-newtonin micropolr fluid t two-dimensionl stgntion point with het trnsfer. The wll nd strem tempertures re ssumed to be constnts. A numericl solution is obtined for the governing momentum nd energy equtions using finite difference pproximtions, which tkes into J. of the Brz. Soc. of Mech. Sci. & Eng. Copyright 8 by ABCM Jnury-Mrch 8, Vol. XXX, No. / 5

2 Hzem Ali Atti ccount the symptotic boundry conditions. The numericl solution computes the flow nd het chrcteristics for the whole rnge of the non-newtonin fluid chrcteristics, the suction or blowing prmeter nd the Prndtl number. Formultion of the Problem Consider the two-dimensionl stgntion point flow of n incompressible non-newtonin micropolr fluid impinging perpendiculr on permeble wll nd flows wy long the x-xis. This is n exmple of plne potentil flow tht rrives from the y- xis nd impinges on flt wll plced t y=, divides into two strems on the wll nd leves in both directions. The viscous flow must dhere to the wll, wheres the potentil flow slides long it. (u,v) re the components for the viscous flow of velocity t ny point ( for the viscous flow wheres (U,V) re the velocity components for the potentil flow. A uniform suction or blowing is pplied t the plte with trnspirtion velocity t the boundry of the plte given by -v o, where v o > for suction. The velocity distribution in the frictionless flow in the neighborhood of the stgntion point is given by U(x)= V(=-y where the constnt (>) is proportionl to the free strem velocity fr wy from the surfce. The simplified two-dimensionl equtions governing the flow in the boundry lyer of stedy, lminr nd incompressible micropolr fluid re (Nth, 975) (Nzr et l., 4): u v + =, () x microelements close to the wll surfce re unble to rotte (Nzr et l., 4). This cse is lso known s the strong concentrtion of microelements (Gurm nd Smith, 98). The cse n=/ indictes the vnishing of nti-symmetric prt of the stress tensor nd denotes wek concentrtion (Ahmdi, 976) of microelements which will be considered here. The cse n= is used for the modeling of turbulent boundry lyer flows (Nzr et l., 4). The governing equtions ()-(4) subject to the boundry conditions (5) cn be expressed in simpler form by introducing the following trnsformtion = y, u = xf ( ), v = f ( ), N = x g( ), (6) equtions () nd (3) for the functions f() nd g() tke the form K = (7) + f ff f Kg K + g + fg f g K ( g + f ) = (8) It is worth mentioning tht when n=/, we cn tke (Nzr et l., 4) ] g ( ) = f ( ) (9) then Eqs. (7) nd (8) cn be reduced the single eqution K + f + ff f + = () u u du u N u v U ( h) ρ + = + µ + + h, x y dx y () subject to the boundry conditions f ( ) = A, f () =, f ( ) =, () N N γ N h u ρ u + v = N +, (3) x j j where N is the component of microrottion vector norml to the x-y plne or the ngulr velocity of the microelements whose direction of rottion is in the x-y plne, µ is the viscosity of the fluid, ρ is the density ndj is the microinerti density, γ is the spin-grdient viscosity nd h is the vortex viscosity. We follow the work of mny recent uthors by ssuming tht γ is constnt nd given by (Nth, 975) (Nzr et l., 4) (Rees nd Pop, 998): γ = ( µ + h / ) j (4) nd we tke j=v/ s reference length where v is the kinemtic viscosity. Reltion (4) is invoked to llow Eqs. ()-(3) to predict the correct behviour in the limiting cse when microstructure effects become negligible, nd the microrottion, N, reduces to the ngulr velocity (Nzr et l., 4). The pproprite physicl boundry conditions of Eqs. ()-(3) re u u( ) =, v( ) = vo, N( ) = n, (5) y : u( U ( x) = v(, N(, (5b) where K = h/µ (>) is the mteril prmeter, A = vo / is the suction prmeter nd primes denote differentition with respect to. For micropolr boundry lyer flow, the wll skin friction τ w is given by u τ w = ( µ + h) + hn y= () Using U(x)=x s chrcteristic velocity, the skin friction coefficient C cn be defined s f C f τ w, (3) = ρu Substituting (6) nd () into (3), we get / = ( + K / ) f () (4) where / Re x = xu / is the locl Reynolds number. Using the boundry lyer pproximtions nd neglecting the dissiption, the eqution of energy for temperture T is given by (Mssoudi nd Rmezn, 99) (Mssoudi nd Rmezn, 99), where n is constnt nd n. The cse n=, which indictes N= t the wll, represents concentrted prticle flows in which the 5 / Vol. XXX, No., Jnury-Mrch 8 ABCM

3 Article Templte for Publiction in the Stgntion Point Flow nd... ρ T T T c p u + v = k x y (5) where c is the specific het cpcity t constnt pressure of the p fluid, nd k is the therml conductivity of the fluid. A similrity solution exists if the wll nd strem tempertures, T nd w T re constnts relistic pproximtion in typicl stgntion point het trnsfer problems (Mssoudi nd Rmezn, 99) (Mssoudi nd Rmezn, 99). The boundry conditions for the temperture field re y = : T = Tw, (6) y : T T, (6b) Introducing the non-dimensionl vrible T T =, (7) T w T nd using the similrity trnsformtions given in Eq. (6), we find tht Eqs. () nd (3) reduce to, where + Pr f = (8) ( ) =, ( ) =, (9) Pr = µc p / k is the Prndtl number. The het trnsfer t the wll is computed from Fourier's lw (Mssoudi nd Rmezn, 99) (Mssoudi nd Rmezn, 99) s follows; T qw = k = k( T Tw ) G(Pr) y = where G is the dimensionless het trnsfer rte which is given by (). () G = d exp( Pr fds) The flow Eqs. () nd () re decoupled from the energy Eqs. (8) nd (9), nd need to be solved before the ltter cn be solved. The flow Eq. () constitutes non-liner, non-homogeneous boundry vlue problem (BVP). In the bsence of n nlyticl solution of problem, numericl solution is indeed n obvious nd nturl choice. The boundry vlue problem given by Eqs. () nd () my be viewed s prototype for numerous other situtions which re similrly chrcterized by boundry vlue problem hving third order differentil eqution with n symptotic boundry condition t infinity. Therefore, its numericl solution merits ttention from prcticl point of view. The flow Eqs. () nd () re solved numericlly using finite difference pproximtions. A qusi-lineriztion technique is first pplied to replce the non-liner terms t liner stge, with the corrections incorported in subsequent itertive steps until convergence. The qusi-linerized form of Eq. () is, ( + K / ) fn+ + fn fn + + fn fn+ fn fn fn fn + + fn + = () where the subscript n or n+ represents the nth or (n+)th pproximtion to the solution. Then, Crnk-Nicolson method is used to replce the different terms by their second order centrl difference pproximtions. An itertive scheme is used to solve the qusi-linerized system of difference equtions. The solution for the Newtonin cse is chosen s n initil guess nd the itertions re continued till convergence within prescribed ccurcy. Finlly, the resulting block tri-digonl system ws solved using generlized Thoms' lgorithm. The energy Eq. (8) is liner second order ordinry differentil eqution with vrible coefficient, f(), which is known from the solution of the flow Eqs. () nd () nd the Prndtl number Pr is ssumed constnt. Eqution (8) is solved numericlly under the boundry condition (9) using centrl differences for the derivtives nd Thoms' lgorithm for the solution of the set of discretized equtions. The resulting system of equtions hs to be solved in the infinite domin <<. A finite domin in the -direction cn be used insted with chosen lrge enough to ensure tht the solutions re not ffected by imposing the symptotic conditions t finite distnce. Grid-independence studies show tht the computtionl domin << cn be divided into intervls ech is of uniform step size which equls.. This reduces the number of points between << without scrificing ccurcy. The vlue = ws found to be dequte for ll the rnges of prmeters studied here. Convergence is ssumed when the rtio of every one of f, f, f, or f for the lst two pproximtions differed from unity by less thn -5 t ll vlues of in <<. Results nd Discussion Figures nd present the profiles of f nd f, respectively, for vrious vlues of K nd A. The figures show tht incresing the prmeter K decreses both f nd f due to the increse in the dmping effect of the viscous forces. On the other hnd, incresing A increses them which is expected since incresing suction opens n esier pth for the incoming flow towrds the wll nd, in turn, increses both f nd f. The figures indicte lso tht the effect of K on f nd f is more pronounced for higher vlues of A (cse of suction). However, the effect of A on f nd f becomes more pronounced for smller vlues of K. Also, incresing K increses the velocity boundry lyer thickness while incresing A decreses it. Figure 3 presents the profile of temperture for vrious vlues of K nd A nd Pr=.5. It is cler tht incresing K increses nd the thickness of the therml boundry lyer. Incresing A decreses for ll K nd its influence becomes more pprent for smller K. This emphsizes the influence of the injected flow in the cooling process. The ction of fluid injection (A<) is to fill the spce immeditely djcent to the disk with fluid hving nerly the sme temperture s tht of the disk. As the injection becomes stronger, so tht does the blnket extends to greter distnces from the surfce. As shown in Fig. 3, the progressive flttening of the temperture profile djcent to the disk mnifests these effects. Thus, the injected flow forms n effective insulting lyer, decresing the het trnsfer from the disk. Suction, on the other hnd, serves the function of bringing lrge quntities of mbient fluid into the immedite neighborhood of the disk surfce. As consequence of the incresed het-consuming bility of this ugment flow, the temperture drops quickly s we proceed wy from the disk. The presence of fluid t ner-mbient temperture close to the surfce increses the het trnsfer. Figures 4 nd 5 present the temperture profiles for vrious vlues of K nd Pr nd for A=-.5 nd.5, respectively. The figures bring out clerly the effect of the Prndtl number on the therml boundry lyer thickness. As shown in Figs. 4 nd 5, incresing Pr J. of the Brz. Soc. of Mech. Sci. & Eng. Copyright 8 by ABCM Jnury-Mrch 8, Vol. XXX, No. / 53

4 Hzem Ali Atti decreses the therml boundry lyer thickness for ll K nd A. It is shown in Fig. 4 the influence of blowing in flttening of the temperture profiles djcent to the disk for higher Pr. The effect of K on is more pronounced for higher vlues of Pr for the blowing cse (see Fig. 4). f K=,A=- K=,A= K=,A= K=,A=- K=,A= K=,A= Figure. Effect of the prmeters K nd A on the profile of f. f` K=,A=- K=,A= K=,A= K=,A=- K=,A= K=,A= Figure. Effect of the prmeters K nd A on the profile of f K=,A=- K=,A= K=,A= K=,A=- K=,A= K=,A= Figure 3. Effect of the prmeters K nd A on the profile of (Pr=.5) K=,Pr=. K=,Pr=.5 K=,Pr= K=,Pr=. K=,Pr=.5 K=,Pr= Figure 4. Effect of the prmeters K nd Pr on the profile of (A=-.5) K=,Pr=. K=,Pr=.5 K=,Pr= K=,Pr=. K=,Pr=.5 K=,Pr= Figure 5. Effect of the prmeters K nd Pr on the profile of (A=.5) Tbles nd present the vrition of the wll sher stress / nd the het trnsfer rte t the wll G(Pr), respectively, for vrious vlues of K nd A nd for Pr=.5. Tble shows tht, for A<, incresing K increses / stedily. However, for A, incresing K increses / nd then incresing K more decreses /. Incresing A increses / for ll K nd its effect is more pprent for smller K. Tble shows tht incresing K decreses G(Pr) due to its dmping ffect for the coming flow towrds the wll. Incresing A increses G(Pr) for ll K s incresing suction helps bringing fluid t ner-mbient towrds the surfce of the wll which increses the het trnsfer. Tble Vrition of the wll sher stress / with K nd A. A K= K=.5 K= K=.5 K= K= Tble Vrition of the wll het trnsfer G(Pr) with K nd A (Pr=.5). A K= K=.5 K= K=.5 K= Tble 3 presents the effect of K on G(Pr) for vrious vlues of Pr nd for A=. Incresing K decreses G(Pr) for ll Pr nd its effect is more for higher Pr. Incresing Pr increses G(Pr) for ll K. Tble 4 shows the vrition of G(Pr) for vrious vlues of Pr nd A 54 / Vol. XXX, No., Jnury-Mrch 8 ABCM

5 Article Templte for Publiction in the Stgntion Point Flow nd... nd for K=. Incresing A increses G(Pr) nd its effect is more pprent for higher Pr. For the suction cse (A ), incresing Pr increses G stedily. On the other hnd, for the lrge blowing cse, incresing Pr decreses G stedily. But for moderte blowing velocity (A=-), incresing Pr incresing G nd incresing Pr more decreses G. Tble 3 Vrition of the wll het trnsfer G(Pr) with K nd Pr (A=-.5). K Pr=.5 Pr=. Pr=.5 Pr= Pr= Tble 4 Vrition of the wll het trnsfer G(Pr) with A nd Pr (K=). A Pr=.5 Pr=. Pr=.5 Pr= Pr= Conclusions The two-dimensionl stgntion point flow of n incompressible non-newtonin micropolr fluid with het trnsfer is studied in the presence of uniform suction or blowing. A numericl solution for the governing equtions is obtined which llows the computtion of the flow nd het trnsfer chrcteristics for vrious vlues of the non-newtonin prmeter K, the suction prmeter A, nd the Prndtl number Pr. The results indicte tht incresing the prmeter K increses both the velocity nd therml boundry lyer thickness while incresing A decreses the thickness of both lyers. The effect of the prmeter K on the velocity is more pprent for suction thn blowing. The influence of the prmeter K on the temperture is more pprent for higher vlues of Prndtl number. The effect of the suction velocity on the sher stress t the wll depends on the vlue of the non-newtonin prmeter K. On the other hnd, the influence of the blowing velocity on the het trnsfer rte t the wll depends on the vlue of the non-newtonin prmeter K. References Ahmdi, G. (976), Self-similr solution of incompressible micropolr boundry lyer flow over semi-infinite plte, Interntionl Journl of Engineering Science, Vol. 4, No. 7, pp Ariel, P.D. (99), Hybrid method for computing the flow of viscoelstic fluids, Interntionl J. Numericl Methods Fluids, Vol. 4, No. 7, pp Ariel, P.D. (994), Hiemenz flow in hydromgnetics, Act Mech., Vol. 3, No. -4, pp Ariel, P.D. (994), Stgntion point flow with suction: An pproximte solution, J. Appl. Mech., Vol. 6, No. 4, pp Ariel, P.D. (995), A numericl lgorithm for computing the stgntion point flow of second grde fluid with/without suction, Journl of Computtionl nd Applied Mthemtics,Vol. 59, No., pp Arimen, T.; Turk, M.A. nd Sylvester, N.D. (973), Microcontinuum fluid mechnics review, Interntionl Journl of Engineering Science, Vol., No. 8, pp Atti, H.A. (), Hiemenz mgnetic flow of non-newtonin fluid of second grde with het trnsfer, Cndin Journl of Physics, Vol. 78, No. 9, pp Atti, H.A. (3), Homnn mgnetic flow nd het trnsfer with uniform suction or injection, Cn. J. Phys., Vol. 8, No., pp Atti, H.A. (3), Hydromgnetic stgntion point flow with het trnsfer over permeble surfce, Arb. J. Sci. Engg. Vol. 8, No.B, pp. 7-. Berd, D.W. nd Wlters, K. (964), Elstico-viscous boundry-lyer flows. I. Two-dimensionl flow ner stgntion point, Proc. Cmbridge Philos. Soc. Vol. 6, pp Djukic, D.S. (974), Hiemenz mgnetic flow of power lw fluids, Journl of Applied Mechnics, Trnsctions ASME, Vol. 4, Ser E, No.3, pp Eringen, A.C. (966), Theory of micropolr fluids, J. Mth. Mech., Vol. 6, pp. -8. Grg, V.K. (994), Het trnsfer due to stgntion point flow of non- Newtonin fluid, Act Mech., Vol. 4, No. 3-4, pp Gurm, G.S. nd Smith, A.C. (98), Stgntion flows of micropolr fluids with strong nd wek interctions, Computers & Mthemtics with Applictions, Vol. 6, No., pp Hiemenz, K. (9), Die Grenzschicht n einem in den gleichförmigen Flüssigkeitsstrom eingetuchten gerden Kreiszylinder, Dingler s Polytech Journl, Vol.36, pp.3-4. Homnn, F. (936), Der Einfluss grosset Zähigkeit bei der Strömung um den Zylinder nd um die Kugel, Z. Angew. Mth. Mech., Vol. 6, pp Mssoudi, M. nd M. Rmezn, M. (99), Boundry lyer het trnsfer nlysis of viscoelstic fluid t stgntion point, ASME Het Trnsfer Division, Vol. 3, pp Mssoudi, M. nd M. Rmezn, M. (99), Het trnsfer nlysis of viscoelstic fluid t stgntion point, Mech. Res. Commun., Vol. 9, No., pp N, T.Y. (979), Computtionl methods in engineering boundry vlue problem. Acdemic Press, New York. Nth, G. (975), Similr solutions for the incompressible lminr boundry lyer with pressure grdient in micropolr fluids, Rheologic Act, Vol. 4, No. 9, pp Nzr, R.;Amin, N.; Filip, D. nd Pop, I. (4), Stgntion point flow of micropolr fluid towrds stretching sheet, Interntionl Journl of Non-Liner Mechnics, Vol. 39, No. 7, pp Preston, J. H. (946), The Boundry-lyer Flow over Permeble Surfce through which Suction is Applied, Aeronuticl Reserch Council - Reports nd Memornd, London, No. 44. Rjeshwri, G.K. nd S.L. Rthn, S.L. (96), Flow of prticulr clss of non-newtonin visco-elstic nd visco-inelstic fluids ner stgntion point, Z. Angew. Mth. Phys., Vol. 3, No., pp Rees, D.A.S. nd Pop, I. (998), Free convection boundry-lyer flow of micropolr fluid from verticl flt plte, IMA Journl of Applied Mthemtics (Institute of Mthemtics nd its Applictions), Vol. 6, No., pp Schlichting, H. nd Bussmnn, K. (943), Exkte Losungen fur die Lminre Grenzchicht mit Absugung und Ausblsen, Schri. Dtsch. Akd. Luftfhrtforschung, Ser. B, Vol. 7, No., pp. 5. Teipel, I. (986), Die rumliche Stupunktstromung fur ein viskoelstisches fluid, Rheologic Act, Vol. 5, No., pp Teipel, I. (988), Stgntion point flow of non-newtonin second order fluid, Trnsctions of the Cndin Society for Mechnicl Engineering, Vol., No., pp Weidmn, P.D. nd Mhlingm, S. (997), Axisymmetric stgntionpoint flow impinging on trnsversely oscillting plte with suction, J. Engg. Mth. 3, No. 4, pp J. of the Brz. Soc. of Mech. Sci. & Eng. Copyright 8 by ABCM Jnury-Mrch 8, Vol. XXX, No. / 55

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