(1) continuity equation: 0. momentum equation: u v g (2) u x. 1 a

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1 Comment on The effect of vible viscosity on mied convection het tnsfe long veticl moving sufce by M. Ali [Intentionl Jounl of Theml Sciences, 006, Vol. 45, pp ] Asteios Pntoktos Associte Pofesso of Fluid Mechnics School of Engineeing, Democitus Univesity of Thce, Xnthi Geece 1. INTRODUCTION The poblem of foced convection long n isotheml moving plte is clssicl poblem of fluid mechnics tht hs been solved fo the fist time in 1961 by Skidis (1961). It ppes tht the fist ok concening mied convection long moving plte is tht of nd Chen (1980). Theefte, mny solutions hve been obtined fo diffeent spects of this clss of boundy lye poblems. In the pevious oks the fluid popeties hve been ssumed constnt. Ali (006) in ecent ppe teted, fo the fist time, the mied convection poblem ith vible viscosity. He used the locl simility method to solve this poblem but thee e doubts bout the vlidity of his esults. Fo tht eson e esolved the bove poblem ith the diect numeicl solution of the boundy lye equtions ithout ny tnsfomtion.. THE MATHEMATICAL MODEL Conside the flo long veticl flt plte ith u nd v denoting espectively the velocity components in the nd y diection, hee is the coodinte long the plte nd y is the coodinte pependicul to. Fo stedy, to-dimensionl flo the boundy lye equtions including vible viscosity nd buoyncy foces e u v y continuity eqution: 0 u momentum eqution: u v g ( T T ) y y y v 1 u (1) ()

INTRODUCTION The poblem of foced convection long n isotheml moving plte is clssicl poblem of fluid mechnics tht hs been solved fo the fist time in 1961 by Skidis (1961).

2 T T y enegy eqution: u v (3) y hee T is the fluid tempetue, μ is the dynmic viscosity, α is the theml diffusivity, nd ρ is the mbient fluid density. The folloing boundy conditions hve been pplied: T t y = 0 u=u, v=0, T=T (4) s y u =0,T = T (5) hee T is the plte tempetue, T is the mbient fluid tempetue nd U is the constnt velocity of the moving plte. The viscosity is ssumed to be n invese line function of tempetue given by the folloing eqution (Ali 006) 1 1 [1 ( T T )] (6) hee μ is the mbient fluid dynmic viscosity nd γ is theml popety of the fluid. The equtions (1)-(3) epesent to-dimensionl pbolic flo. Such flo hs pedominnt velocity in the stemise coodinte (unidiectionl flo) hich in ou cse is the diection long the plte. The equtions ee solved diectly, ithout ny tnsfomtion, using the finite diffeence method of Ptnk (1980). The solution pocedue stts ith knon distibution of velocity nd tempetue t the plte edge (=0) nd mches long the plte. At the leding edge the tempetue s tken unifom nd equl to mbient one nd the velocity s lso unifom ith vey smll vlue. At ech donstem position the discetized equtions () nd (3) e solved using the tidigonl mti lgoithm (TDMA). The coss-stem velocities v ee obtined fom the continuity eqution. The fod step size Δ s mm nd e used nonunifom ltel gid ith 500 points hee Δy inceses long y. In the numeicl solution of the boundy lye poblems the clcultion domin must lys be t lest equl o ide thn the boundy lye thickness. Hoeve, it is knon tht the boundy lye thickness inceses ith. Theefoe, it ould be desible to hve gid hich confoms to the ctul shpe of the boundy lye. Fo tht eson n epnding gid hs been used in the pesent ok. The esults e gid independent. The pbolic solution pocedue is ell knon solution method nd hs been used etensively in the litetue. It ppeed fo the fist time in 1970 (Ptnk nd Splding, 1970) nd hs been included in clssicl fluid mechnics tetbooks (see pge 75 in White, 1991). Andeson et l. (1984) mention 7 numeicl methods fo the solution of the boundy lye equtions (pge 364) nd mong them is the ell knon Ptnk Splding method. The method is fully implicit nd cn be pplied to both simil

plte. The viscosity is ssumed to be n invese line function of tempetue given by the folloing eqution (Ali 006) 1 1 [1 ( T T )] (6) hee μ is the mbient fluid dynmic viscosity nd γ is theml popety of

3 nd nonsimil poblems. The dynmic viscosity μ nd the Pndtl numbe, hich is function of viscosity, hve been consideed vible duing the solution pocedue. A detiled desciption of the solution pocedue, ith vible themophysicl popeties, my be found in Pntoktos (00). 3. RESULTS AND DISCUSSION The locl Nusselt numbe nd the locl Reynolds numbe hve been defined s follos by Ali (006) h Nu k U Re (8) (7) thus the tem Nu Re - is h k T T Nu Re Re Re Re ( ) (9) k k T T y T T y y0 y0 The quntity C f hs not been defined by Ali (006) nd e used the folloing eqution fo this quntity (Bejn 1995, pge 51) C f U (10) hee is the ll she stess given by u (11) y y0 Consequently the tem C f Re is C f Re U Re u y y0 (1) Ali (006) tnsfomed equtions (1)-(3) into the folloing equtions

RESULTS AND DISCUSSION The locl Nusselt numbe nd the locl Reynolds numbe hve been defined s follos by Ali (006) h Nu k U Re (8) (7) thus the tem Nu Re - is h k T T Nu Re Re Re Re ( ) (9) k

4 ''' 1 ( ) ff '' ' f '' 0 f (13) '' P f ' 0 (14) hee f nd θ e the dimensionless velocity nd dimensionless tempetue defined s f ' u U (15) T T T T (16) λ=g /Re is the buoyncy pmete nd G is the Gshof numbe defined s G =gβ(t -T ) 3 /ν (17) θ is the viscosity pmete defined by 1 ( T T (18) ) It should be mentioned hee tht hen θ the fluid viscosity becomes equl to mbient viscosity. In equtions (13) nd (14) the pime epesents diffeentition ith espect to simility vible η defined s (Ali, 006) y Re 1/ (19) Ali (006) solved equtions (13) nd (14) using the fouth ode Runge-Kutt method. Loclly simility solutions ee obtined fo incesing vlues of λ t ech constnt θ. At ech ne θ the pocedue stts fom knon solution hich coesponds to pue foced convection (λ=0). The Pndtl numbe included in the tnsfomed enegy eqution (15) s ssumed constnt nd equl to mbient Pndtl numbe P (0) Hoeve, the Pndtl numbe is function of viscosity nd s viscosity vies coss the boundy lye, the Pndtl numbe vies, too. In tble 1 the skin fiction coefficient C f Re nd the Nusselt numbe Nu Re e given fo mbient Pndtl numbe 0.7. In this tble the esults by Ali (006)

In equtions (13) nd (14) the pime epesents diffeentition ith espect to simility vible η defined s (Ali, 006) y Re 1/ (19) Ali (006) solved equtions (13) nd (14) using the fouth ode Runge-Kutt method.

5 hve been lso included fo compison. The esults by Ali hve been tken fom his figues 4 nd 8. It s difficult to etct vlues fo θ ne 0 nd 1 nd fo tht eson e took vlues fo -10 θ -1.0 nd 1.5θ 10. In the lst column of the tble the Pndtl numbes t the plte (P ) e included. P (1) Tble 1. Vlues of C f Re nd Nu Re fo P =0.7 θ constnt viscosity constnt viscosity Pesent Wok ( fom nd Chen ( fom nd Chen C f Re Ali (006) Pesent Wok = < (0.349 fom nd Chen = ( fom nd Chen Nu Re Ali. (005) Diffeence % Diffeence % P 0.35 < < < < < < < < < < < <1 0.80

8875 fom nd Chen 0.3886 (0.3885 fom nd Chen C f Re Ali (006) Pesent Wok =0-0.88 <1 0.3555 (0.349 fom nd Chen =1 0.88 16 0.4559 (0.4550 fom nd Chen Nu Re Ali. (005) Diffeence % Diffeence % P 0.35 <1 0.

6 = <1 0.7 constnt viscosity (4.798 fom nd Chen (909 fom nd Chen < < < < < < < < < < λ=10 constnt viscosity (8.9 fom Chen ( fom Chen 0.68 <1 0.7 λ= Fom tble 1 it is seen tht the skin fiction coefficient C f Re nd the Nusselt numbe Nu Re clculted by the pesent method e in vey good geement ith those clculted by Ali (006) nd nd Chen (1980) fo the cse λ=0 (pue foced convection) nd constnt viscosity. Ecept tht the bove quntities clculted by the pesent method e in vey good geement ith those clculted by nd Chen (1980) fo the cses λ=1, 5 (mied convection) nd constnt viscosity. Ou esults compe lso vey ell ith those of Chen (000) fo λ=10 nd constnt viscosity. In ddition ou method hs been used ecently successfully to to simil poblems (Pntoktos, 004, 005). The Nusselt numbes given by Ali

98 36 965 9 <1 0.80 λ=10 constnt viscosity 8.504 (8.9 fom Chen 10.87 3 0.6884 (0.6800 fom Chen 0.68 <1 0.7 λ=0-10 14.8304 18.94 8 0.8054 0.79 0.65-7.5 14.719 18.87 8 0.8085 0.79 0.64-5.0 14.304 18.6 30 0.

7 (006) e in good geement ith ou esults fo ll cses of the buoyncy pmete λ. Fo the skin fiction coefficient C f Re things e diffeent. Fo λ=0 thee is vey good geement but fo λ=1 lge diffeences ppe. The divegence eist lso fo highe vlues of λ t smlle te. It is seen tht ou C f Re vlues e lys loe thn those of Ali (006) nd this is in ccodnce ith the velocity pofiles included in figues 1 nd hee e see tht the velocity pofiles clculted by the pesent method ly loe thn those of Ali. It is dvocted hee tht the esults of the skin fiction coefficient given by Ali (006) fo λ1 e ong. The eo is cused pobbly by the locl simility method tht hs been used fo the solution of the equtions. Minkoycz nd Spo (1974) mention tht n unothodo vesion of the locl simility method yields esults of uncetin ccucy. It should be noted hee tht Ali (006) tested the ccucy of his method comping the esults only ith those of the pue foced convection cse (λ=0). If the compison hd been etended to eisting esults fo the mied convection poblem ith constnt viscosity ( nd Chen, 1980, Chen, 000) the eo ould ppe.

It is seen tht ou C f Re vlues e lys loe thn those of Ali (006) nd this is in ccodnce ith the velocity pofiles included in figues 1 nd hee e see tht the velocity pofiles clculted by the pesent method

8 REFERENCES 1. Ali, M. (006). The effect of vible viscosity on mied convection het tnsfe long veticl moving sufce, Intentionl Jounl of Theml Sciences, Vol. 45, pp Andeson, D., Tnnehill, J. nd Pletche, R. (1984). Computtionl Fluid Mechnics nd Het Tnsfe, McG-Hill Book Compny, Ne Yok. 3. Bejn A. (1995). Convection Het Tnsfe, John Wiley & Sons, Ne Yok. 4. Chen, C. H. (000). Mied convection cooling of heted continuously stetching sufce, Het nd Mss Tnsfe, Vol. 36, pp Minkoycz, W. J. nd Spo, E. M. (1974). Locl nonsimil solutions fo ntul convection on veticl cylinde, ASME Jounl of Het Tnsfe, Vol. 96, pp , A. nd Chen, T. S. (1980). Buoyncy effects in boundy lyes on inclined, continuous moving sheets, ASME Jounl of Het Tnsfe, Vol. 10, pp Pntoktos, A. (00). Lmin fee-convection ove veticl isotheml plte ith unifom bloing o suction in te ith vible physicl popeties, Intentionl Jounl of Het nd Mss Tnsfe, Vol. 45, pp Pntoktos, A. (004). Futhe esults on the vible viscosity on flo nd het tnsfe to continuous moving flt plte, Intentionl Jounl of Engineeing Science, Vol. 4, pp, Pntoktos, A. (005). Foced nd mied convection boundy lye flo long flt plte ith vible viscosity nd vible Pndtl numbe: ne esults, Het nd Mss Tnsfe, vol. 41, pp Ptnk, S. V. nd Splding D. B. (1970). Het nd Mss Tnsfe in Boundy Lyes, Intetet, London. 11.Ptnk, S.V. (1980). Numeicl Het Tnsfe nd Fluid Flo, McG- Hill Book Compny, Ne Yok. 1.Skidis, B.C. (1961). Boundy lye behvio on continuous solid sufces: I. Boundy lye equtions fo to-dimensionl nd isymmetic flo. II.

Mied convection cooling of heted continuously stetching sufce, Het nd Mss Tnsfe, Vol. 36, pp. 79-86. 5. Minkoycz, W. J. nd Spo, E. M. (1974).

9 Boundy lye on continuous flt sufce. III. Boundy lye on continuous cylindicl sufce, AIChE Jounl, Vol. 7, pp. 6-8, 1-5, White, F. (1991). Viscous Fluid Flo, McG-Hill, Ne Yok.

AIChE Jounl, Vol. 7, pp. 6-8, 1-5, 467-47. 13.

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