Numerical solutions for unsteady flow past a semi-infinite inclined plate with temperature oscillations

Transcription

1 Joural of Mechaical Sciece ad Techology 3 (009) 1710~1717 Joural of Mechaical Sciece ad Techology DOI /s Numerical solutios for usteady flow past a semi-ifiite iclied plate with temperature oscillatios G.Palai ad Kwag-Yog Kim * Departmet of Mechaical Egieerig, Iha Uiversity Icheo, , Korea (Mauscript Received October 16, 008; Revised March 18, 009; Accepted March 3, 009) Abstract A study of the velocity ad thermal boudary layers o a semi-ifiite iclied plate with temperature oscillatios is preseted i this work. The o-liear, coupled parabolic itegro-partial differetial equatios goverig flow ad heat trasfer have bee solved umerically usig a implicit fiite differece scheme of Crak-Nicolso type. The umerical values for the flow field, temperature, shearig stress, ad heat trasfer coefficiets are preseted i a graphical form. It is observed that the velocity ad temperature profiles decrease as the frequecy parameter icreases. Keywords: Fiite differece; Oscillatios; Trasiet flow; Grashof umber; Ski frictio Itroductio This paper was recommeded for publicatio i revised form by Associate Editor Yag Na * Correspodig author. Tel.: , Fax.: address: kykim@iha.ac.kr KSME & Spriger 009 Natural covectio heat trasfer plays a importat role i our eviromet ad i may egieerig devices. The buoyacy force iduced by the desity differeces i a fluid causes atural covectio. The applicatios iclude, for example, the coolig of the core of a uclear reactor i the case of a pump, or power failures ad the warmig ad coolig of electroic compoets. Fujii ad Imura [1] studied experimetally the atural covectio heat trasfer from a plate with arbitrary icliatio usig two plates with differet dimesios. Pera ad Gebhart [] studied the lamiar atural covectio boudary layer flow above horizotal ad slightly iclied surfaces with uiform temperature ad heat flux. Perturbatio aalysis was employed to solve the problem. Che et al. [3] studied the flow ad heat trasfer characteristics of lamiar free covectio i boudary layer flows from horizotal, iclied, ad vertical plates with variable wall temperature ad heat flux. Ekambavaa ad Gaesa [4] preseted a fiitedifferece solutio of usteady atural covectio boudary layer flow over a iclied plate with variable surface temperature. These problems are of basic importace, thus other physical situatios also eed ivestigatio. Oe such case is whe the plate temperature is oscillatig; how is the flow affected? Such issue is particularly importat ad useful i the fields of space sciece, uclear egieerig, structural egieerig, ad so o. Keller ad Yag [5] studied the heat trasfer resposes of a lamiar free covectio boudary layer alog a vertical heated plate to surface temperature oscillatios, whe the mea surface temperature is proportioal to x, where x is the distace measured from the leadig edge of the plate. A exact aalysis of Stoke's problem for a ifiite vertical plate, whose temperature varies liearly with time, was studied by Soudalgekar ad Patil [6]. The motio of a semiifiite icompressible viscous fluid caused by the oscillatios of a vertical plate was studied by Soudalgekar [7]. They derived closed-form solutios to velocity, temperature, ad peetratio distace through which the leadig edge effect propagates. The two-dimesioal usteady flow of a viscous i-

2 G. Palai ad K.-Y. Kim / Joural of Mechaical Sciece ad Techology 3 (009) 1710~ compressible fluid past a ifiite vertical plate has bee studied by Soudalgekar [8] uder the followig coditios: (i) costat suctio at the plate, (ii) wall temperature oscillatig at a costat o-zero mea, ad (iii) a costat free stream. The approximate solutios to the coupled o-liear equatios goverig the flow have bee carried out for trasiet velocity, trasiet temperature, the amplitude ad phase of ski frictio, ad the rate of heat trasfer. Verma [9] aalyzed the effect of the oscillatio of surface temperature o usteady free covectio from a horizotal plate. Soudalgekar et al. [10] studied the flow of a viscous icompressible fluid past a impulsively started ifiite vertical plate i the presece of foreig mass uder the coditio of variable plate temperature ad costat heat flux. Houssai [11] solved the problem of simultaeous heat ad mass trasfer i two-dimesioal free covectio from a semi-ifiite vertical flat plate. A itegral method is used to fid a solutio for zero wall velocity with small-amplitude oscillatory wall temperature. Das et al. [1] addressed the trasiet free covectio flow past a ifiite vertical plate with the periodic oscillatio of surface temperature. They simplified the problem by assumig small values of the Grashof umber Gr (<<1). I this case, the temperature is idepedet of the flow, ad the heat is trasferred by coductio oly. They used the Laplace trasform techique to solve the simplified equatios, ad the results show that the trasiet velocity profile ad the peetratio distace decrease with a icrease i the frequecy of plate temperature oscillatio. Hossai et al. [13] studied the heat-trasfer respose of a lamiar free covectio boudary layer flow of a viscous icompressible ad electrically coductig fluid alog a vertical plate to surface temperature oscillatios. Revakar [14] ivestigated the free covectio effects o flow field for two cases: (1) whe a submerged ifiite plate is set ito motio impulsively with oscillatig plate temperature, ad () whe the submerged ifiite plate is set ito motio with liear harmoic oscillatio parallel to itself with oscillatig temperature. The exact solutios for temperature ad flow field are preseted. The steady ad usteady free covectio from a vertical wall with stream-wise surface temperature oscillatio was ivestigated by Li et al. [15]. For small values of the Grashof umber, they obtaied a asymptotic formula for the average Nusselt umber usig a perturbatio method. Lorezo ad Padet [16] coducted a parametric study of the free covectio alog the vertical wall whe a periodic heat flux desity cotaiig adiabatic period is applied. Their results show the importace of the adiabatic period to let the surroudig fluid refresh itself before applyig a ew heatig period, leadig to a icrease i free covectio heat trasfer. Rai ad Devaraj [17] preseted a umerical solutio for free covectio flow past a vertical cylider with temperature oscillatios. They solved o-dimesioal goverig equatios by usig a implicit fiite differece scheme of Crak- Nicolso type. The effect of the periodic oscillatio of the surface temperature o the trasiet free covectio from a vertical plate was ivestigated by Saeid [18]. They observed that icreasig the amplitude ad the frequecy of the oscillatig surface temperature will decrease the free covectio heat trasfer from the plate to both air ad water. The usteady atural covectio flow past a semiifiite iclied plate with temperature oscillatios has bee give very scat attetio i the literature. Hece, we propose to study the problem of heat trasfer effects o a semi-ifiite iclied plate with temperature oscillatios. The goverig boudary layer equatios are first cast ito a dimesioless form, ad the resultig system of equatios is the solved by a implicit fiite differece scheme. I the preset aalysis, a relatively higher Grashof umber is cosidered (10 4 < Gr <10 9 ), where the lamiar boudary layer is applicable for studyig the effect of oscillatig plate temperature o the free covectio effects o the semi-ifiite iclied plate.. Mathematical aalysis We cosider a two-dimesioal usteady flow of a viscous icompressible fluid flow past a semi-ifiite iclied plate with temperature oscillatios. The aalysis of the preset paper is based o the followig assumptios: (1) The plate makes a icliatio agle φ to the horizotal. () The x-axis is measured alog the plate, ad the y-axis is take alog upward ormal to the plate. (3) Iitially at time t 0, it is assumed that the plate ad the fluid are at the same temperature, ad at time t > 0, the temperature of the plate is maitaied at a oscillatig temperature. (4) The effect of viscous dissipatio is ot cosidered i the eergy equatio. (5) All the fluid properties are assumed to be co-

3 171 G. Palai ad K.-Y. Kim / Joural of Mechaical Sciece ad Techology 3 (009) 1710~1717 Fig. 1. Schematic diagram. stat, except for the ifluece of desity variatio with temperature, which is cosidered oly i the body force term. (6) There is o chemical reactio betwee the diffusig species ad the fluid. By assumig Boussiesq's approximatio, the usteady two-dimesioal boudary layer flow ca be show to be govered by the followig equatios: u v = 0 x y u u u u v = g cos ( T T ) dy t x y x β φ y u gβsi φ( T T ) υ T T T T u v = α t x y y The iitial ad boudary coditios are as follows: t 0 : u = 0, v = 0, T = T, t > 0 : u = 0, v = 0, T = ( T w T )cosω t T w at y = 0 u = 0, T = T, at x = 0 u 0, T T, as y y (1) () (3) (4) where u ad v are the velocity compoets i the x ad y directios, respectively; α is the thermal diffusivity; g is the acceleratio due to gravity; t is the time; T is the temperature of the fluid i the boudary layer; β is the volumetric coefficiet of thermal expasio; is the kiematic viscosity; ω is the frequecy parameter; ad φ is the icliatio agle to the horizotal. The followig o-dimesioal quatities are itroduced: x y 1/4 ul 1/ X =, Y= Gr, U = Gr, L L vl t T T V Gr t Gr T 1/4 1/ =, =, = L Tw T gβ L 3 ( T w T ), Gr = L ω 1/ ω = Gr Pr = α (5) Eqs. (1)-(4) are reduced to the followig odimesioal form: U V = 0 (6) X Y U U U U V t X Y (7) 1/4 U = Gr cosφ TdY T siφ X Y Y T T T 1 T U V = (8) t X Y Pr Y The correspodig iitial ad boudary coditios i dimesioless form are as follows: t 0 : U = 0, V = 0,T = 0, t > 0 : U = 0, V = 0,T = 1 cosωt, at Y = 0 (9) U = 0, T = 0, at X = 0 U 0,T 0, asy The o-dimesioal forms of local ad average ski frictio, Nusselt umber, are as follows: 3/4 U τ X = Gr Y Y = 0 (10) 1 3/ 4 U τ= Gr dx Y Y = 0 0 (11) 1/ 4 T Nu X = X Gr Y TY = 0 (1) Y = 0 1 1/ 4 T Nu = Gr TY = 0 dx 0 Y Y = 0 3. Numerical techiques (13) The two-dimesioal, o-liear, usteady, coupled, ad itegro-partial differetial Eqs. (6)-(8) uder

4 G. Palai ad K.-Y. Kim / Joural of Mechaical Sciece ad Techology 3 (009) 1710~ the iitial ad boudary coditios (9) are solved by usig a implicit fiite differece scheme of the Crak-Nicolso type Ui,j 1 Ui 1,j 1 Ui,j Ui 1,j Ui,j 1 Ui 1,j 1 Ui,j Ui 1,j 4 X 1 1 V i, j V i, j 1 V i, j V i, j 1 (14) = 0 Y Ui,j Ui,j ( Ui,j U U U i 1,j i,j i 1,j) Ui,j X (15) 1 1 ( Ui,j 1 Ui,j 1 Ui,j 1 Ui,j 1) 1/ 4 Vi,j = Gr cosφ TdY 4 Y X Y [ Ti,j Ti,j] ( Ui,j 1 Ui,j Ui,j 1 Ui,j 1 U i,j Ui,j 1) siφ ( Y) Ti, j T i, j (T i, j T i 1, j T i, j T i 1, j ) U i, j t X (16) 1 1 (T i, j 1 T i, j 1 T i, j 1 T i, j 1 ) V i, j 4 Y (T i, j 1 Ti, j T i, j 1 Ti, j 1 Ti, j Ti, j 1 ) = Pr ( Y ) Now, we cosider the solutio of the two-dimesioal equatio usig the Crak-Nicolso method. The fiite differece grid advaces the solutio from the time level () to the time level (1) as illustrated i Fig.. For the Dirchilet boudary coditios (i.e., the value of the fuctio is specified at the boudaries), the fiite differece equatios must be applied at the iterior poits. The regio of itegratio is cosidered as a rectagle with sides X max (=1) ad Y max (=14), where Y max correspods to Y=,which lies very well outside the mometum ad eergy boudary layers. The maximum of Y was chose as 14 after some prelimiary ivestigatios so that the last two of the boudary coditios (9) are satisfied. Here, the subscript i- desigates the grid poit alog the X-directio, the j- alog the Y directio, ad the superscript alog the t directio. At ay oe time step, the coefficiets U i,j ad Vi,j appearig i the differece equatios are treated as costats. The values of U, V, ad T are kow at all grid poits at t = 0 from the iitial coditios. The computatios of U, V, ad T at the time level (1) usig the values at the previous time level () are carried out as follows: The fiite differece Eq. (16) at every iteral odal poit o a particular i-level costitutes a tridiagoal system of equatios. Such systems of equatios are solved usig the Thomas algorithm as de- Fig.. The grid system. scribed i the work of Caraha et al. [19].Thus, the values of T are foud at every odal poit for a particular i at the (1) th time level. Usig the values of T at the (1) th time level i Eq. (15), the values of U at the (1) th time level are foud i a similar maer. Thus, the values of T ad U are kow o a particular i-level. Fially, the values of V are calculated explicitly usig Eq. (14) at every odal poit o a particular i-level at the (1) th time level. This process is repeated for various i-levels. Thus, the values of T, U, ad V are kow at all grid poits i the rectagular regio at the (1) th time level. Computatios are carried out util the steady state is reached. The steady-state solutio is assumed to have bee reached whe the absolute differece betwee the values of U, as well as temperature T, at two cosecutive time steps is less tha 10-5 at all grid poits. After experimetig with a few sets of mesh sizes, they have bee fixed at the level X=0.05, Y=0.5, ad the time step =0.01. I this case, the spatial mesh sizes are reduced by 50% i oe directio, the i both directios; the results are compared. It is observed that whe the mesh size is reduced by 50% i the X ad Y directios, the results differ i the fourth decimal place. Hece, the above-metioed sizes have bee cosidered as the appropriate mesh sizes for calculatio. The derivatives ivolved i Eqs. (10)-(13) are evaluated usig a five-poit approximatio formula, ad the the itegrals are evaluated usig the Newto-Cotes closed itegratio formula. 4. Stability of the scheme The stability criterio of the fiite differece scheme for costat mesh sizes is examied usig the Vo-Neuma techique as explaied by Caraha et al. [19]. The geeral term of the Fourier expasio for U ad T at a time arbitrarily called t=0 is assumed to

5 1714 G. Palai ad K.-Y. Kim / Joural of Mechaical Sciece ad Techology 3 (009) 1710~1717 i X i X be of the form e α β e (here i = 1 ). At a later time t, these terms will become U = F(t)e T = G(t)e iαx iβy e iαx iβy e (17) The we substitute Eq. (17) i Eqs. (15) ad (16) uder the assumptio that the coefficiets U ad T are costats over ay oe time step ad deotig the values after oe time step by F ad G. After simplificatio, we obtai iα x (F F) U (F F)(1 e ) V (F F)isi β Y X Y 1/ 4 α iβy i Y i Y (G G) β β = Gr (cosφ) G(e e )e (siφ) β (F F)(cosβ Y 1) (18) ( Y) iα x ( G G) U ( G G)(1 e ) X V ( G G) isi β Y (19) Y 1 (G G )(cos β Y 1) = Pr ( Y ) Eqs. (18) ad (19) ca be rewritte as (1 AF ) = (1 AF ) α φ β where 1/4 [ (cos ) ( i β Y i β Gr G e e Y ) e i β Y (G G) (si φ)] (0) ( 1 B)G = (1 B)G (1) U iα X A= (1 e ) X V isi( β Y) (cos β Y 1) Y ( Y) U iα X V B = (1 e ) isi( β Y) X Y (cos β Y 1) Pr ( Y ) After elimiatig G i Eq. (0) usig Eq. (1), the resultat equatio is give by α = φ β 1/ 4 iβy iβy iβy (1 AF ) (1 AF ) GGr [ (cos ) ( e e ) e (si φ) ] (1 B) () Eqs. () ad (1) ca be writte i matrix form: 1 A = 1 A G 0 where F 1 D F 1 B G 1 B α D = Gr φ e (1 A) β e e (si φ) ] (1 B) 1/4 iβy iβy iβy 1 [ (cos ) ( ) (3) Now, for the stability of the fiite differece scheme, the modulus of each eige value of the amplificatio matrix does ot exceed uity. Sice matrix Eq. (3) is triagular, the eige values are its diagoal elemets. The eige values of the amplificatio matrix are (1-A)/ (1A) ad (1-B)/ (1B). Assumig that U is everywhere o-egative, ad V is everywhere o-positive, we obtai α X β Y A= asi csi ia ( si( α X) bsi( β Y)) where U a =, X b = V, Y c = ( Y) Because the real part of A is greater tha or equal to zero, (1-A)/ (1A) 1 always. Similarly, (1-B)/ (1B) 1. Hece, the fiite differece scheme is ucoditioally stable. The local trucatio error is O( t Y X), ad it teds to be zero as, X, ad Y also ted to be zero. Hece, the scheme is compatible. Stability ad compatibility esure covergece. 5. Results ad discussio To assess the accuracy of the umerical results, the preset result is compared with previous studies available i the literature. The velocity profiles for φ = , Gr = 10 6, ad Pr = 0.7 are compared with the theoretical results of the work of Che et al. [3] as show i Fig. 3, which are foud to be i good agreemet.

6 G. Palai ad K.-Y. Kim / Joural of Mechaical Sciece ad Techology 3 (009) 1710~ Fig. 3. Compariso of velocity profiles at X=1.0 The trasiet velocity profiles at X=1.0 for differet values of the Grashof umber ad the icliatio agle φ to the horizotal are preseted i Fig. 4. Velocity icreases steadily as time advaces, ad it reaches a temporal maximum ad cosequetly the steady state because the tagetial compoet of the buoyacy force icreases with φ ad domiates i the dowstream. A higher velocity is experieced throughout the trasiet period as well as i the steady-state level for a system havig larger agles of icliatio to the horizotal. The differece betwee the temporal maximum ad the steady state decreases as φ icreases. The velocity of air decreases as Gr icreases. I Fig.5, the trasiet velocity profiles are plotted for differet values of the Pradtl umber of the fluid Pr ad the frequecy parameter ωt. It is observed that there is a decrease i the velocity of the fluid whe Pr icreases (i.e., for Pr=0.71 (air), 7.0 (water)). The higher the values of Pr, the higher is the rate of heat trasfer; hece, the time take to reach the steady state is high for higher values of Pr. The differece betwee the temporal maximum ad steady state decreases with a icrease i the value of the Pradtl umber of the fluid. For the frequecy parameter ωt, the velocity is observed to decrease for a icrease i the value of ωt. The time take to reach the steady state icreases as ωt icreases. It is also observed that the temporal maximum for velocity icreases with a icrease i the value of ωt. The profiles of trasiet temperature for various values of the Grashof umber ad the icliatio agle φ are show i Fig. 6. The temperature profiles preseted are those at the leadig edge of the plate, that is, at X=1.0. We observe from this figure that a lower temperature is experieced for a system havig a higher value of φ. The effect of Grashof umber o the temperature profile is il. The effects of Pradtl umber Pr ad the frequecy parameter ωt o temperature are show i Fig. 7. The Fig. 4. Trasiet velocity profiles at X=1.0 for differet Gr ad φ (*-steady state). Fig. 5. Trasiet Velocity profiles at X=1.0 for differet Pr ad ω t(*-steady state). larger value of Pr gives rise to the thier thermal boudary layer because icreasig the value of Pr gives rise to a higher heat trasfer. The temporal maximum is attaied at a early stage for lower values of the frequecy parameter ωt. It is also observed that the temperature profiles decrease with a icrease i the value of ωt. I Fig. 8, the values of local shear stress are plotted for various values of Pr, φ, ad ωt. The local wall shear stress decreases as φ decreases because the velocity decreases with a decrease i the value of φ as show i Fig. 4. The icreasig value of Pr gives rise to a lower shear stress, which is also observed i Fig. 5. This is quite expected because icreasig Pr gives a thicker velocity profile, which i tur results i a lower shear stress value. It is also observed that shear stress decreases as ωt icreases. The steady-state local Nusselt umber, that is, the local heat trasfer rate, is plotted i Fig. 9. It icreases as X icreases. The local heat trasfer is stroger o Pr tha o the other parameters because a lower Pr results i thicker temperature profiles. It is also observed that the local Nusselt umber icreases with a icrease i the value of φ. But local Nusselt umber decreases as ω t icreases. The average values of ski frictio ad the Nusselt umber are show i Figs. 10 ad 11, respectively. I

7 1716 G. Palai ad K.-Y. Kim / Joural of Mechaical Sciece ad Techology 3 (009) 1710~1717 Fig. 6. Trasiet temperature profiles at X=1.0 for differet Gr ad φ (*-steady state). Fig. 10. Average ski frictio. Fig. 7. Trasiet temperature profiles at X=1.0 for differet Pr ad ω t(*-steady state). Fig. 11. Average Nusselt umber. creases as φ decreases. Furthermore, it is oticed that the average Nusselt umber decreases with a icrease i the value of ωt. Fig. 8. Local ski frictio. Fig. 9. Local Nusselt umber. Fig. 10, it is observed that the average ski frictio icreases with time ad reaches the steady state after some time. Owig to icliatio with the horizotal, the average ski frictio is foud to decrease. A lower value of ski frictio is observed for a higher value of Pr ad ωt. The average Nusselt umber is the same at a particular time level i the iitial period for a fixed Pr. This shows that there is oly heat coductio i the iitial time level. The average Nusselt umber de- 6. Coclusio Usteady atural covectio flow past a semiifiite iclied plate with oscillatig temperature is cosidered i this paper. The goverig partial differetial equatios are trasformed ito a set of dimesioless goverig equatios, which are solved umerically by usig a implicit fiite differece method. The coclusios of the study are as follows: (1) The differece betwee the temporal maximum ad the steady-state value is reduced whe φ icreases. () The velocity is observed to be maximum ear the upstream ad decreases i the flow directio. (3) Velocity ad temperature are foud to decrease whe the frequecy parameter ωt icreases. (4) The local shear stress decreases as ωt icreases. (5) The local Nusselt umber icreases with a icrease i the value of the icliatio agle φ. Ackowledgmet This work was supported by the BK1 Research Program, Iha Uiversity, Korea. Refereces [1] T. Fujii ad H. Imura, Natural Covectio heat

8 G. Palai ad K.-Y. Kim / Joural of Mechaical Sciece ad Techology 3 (009) 1710~1717 trasfer from a plate with arbitrary icliatio, It.J. Heat Mass Tr. 15 ( 197) [] L. Pera ad B. Gebhart Natural Covectio boudary layer flow over horizotal ad slightly iclied surfaces, It. J. Heat Mass Tr, 16 (1973) [3] T. S. Che H. C. Tie ad B. F. Armaly, Natural Covectio o horizotal, iclied ad vertical plates with variable surface temperature or heat flux, It. J. Heat Mass Trasfer. 9 (1986) [4] K. Ekambavaa ad P. Gaesa, Fiite differece solutio of usteady atural covectio boudary layer flow over a iclied plate with variable surface temperature, Wärme ad Stoffübertragug, 30 (1994), [5] M. D. Kelleher ad K. T. Yag, Heat trasfer resposes of lamiar free covectio boudary layers alog a vertical heated plate to surface temperature oscillatios, J. Appl. Math. Phys. 19 (1968) [6] V. M. Soudalgekar ad M. R., Patil, Stoke s problem for a vertical ifiite plate with variable temperature, Astrophysics ad Space Sci. 59 (1978) [7] V. M. Soudalgekar, Free covectio effects o the flow past a ifiite vertical oscillatig plate, Astrophysics ad Space Sci. 64 (1979) [8] V. M. Soudalgekar, Usteady forced ad free covective flow past a ifiite vertical porous plate with oscillatig wall temperature ad costat suctio, Astrophysics ad Space Sci,66 (1979) 333. [9] R. L. Verma, Free covectio fluctuatig boudary layer o a horizotal plate, J.Appl. Math. Mech, 63 (198) [10] V. M. Soudalgekar, N. S. Birajdar ad V. K. Darwhekar, Mass-Trasfer effects o the flow past a impulsively started ifiite vertical plate with variable temperature or costat heat flux, Astrophysics ad Space Sci. 100 (1984) [11] M. A. Hossai, Simultaeous heat ad mass trasfer o oscillatory free covectio boudary layer flow, It. J. Eergy Research, 1 (1998) [1] U. N. Das R. K. Deka ad V. M. Soudalgekar, Trasiet free covectio flow past a ifiite vertical plate with periodic temperature variatio, Tras. ASME, J. Heat Tr. 11 (1999) [13] M. A. Hossai, S. K. Das ad I. Pop, Heat trasfer respose of MHD free covectio alog a vertical plate to surface temperature oscillatios, It. J. Noliear Mechaics. 33 (1998) [14] Shripad T. Revakar, Free covectio effect o a flow past a impulsively started or oscillatig ifiite vertical plate, Mechaics Research Commuicatios. 7 () (000) [15] J. Li, D. Igham ad I. Pop, Natural covectio from a vertical plate with a surface temperature oscillatio, It. J. Heat Mass Tr, 44 (001) [16] T. Lorezo ad J. Padet, Parametric study of trasiet free covectio heat trasfer, It. J. Heat Mass Tr. 45 ( 00) [17] H. P. Rai ad R. Devaraj, Numerical solutio of usteady flow past a vertical cylider with temperature oscillatios, Forschug im Igeieurwese. 68 (003) [18] Nawaf H. Saeid., Periodic free covectio from vertical plate subjected to periodic surface temperature oscillatig, It.J thermal sci.43 (004) [19] B. Caraha, H. A. Luther ad J. O. Wilkes, Applied Numerical Methods. Joh Wiley ad Sos, G. Palai received his B.Sc. ad M.Sc. degrees from Madras Uiversity, Idia, i 1991 ad 1993, respectively, ad his Ph.D. degree from Aa Uiversity, Idia i 001. Dr. G. Palai is curretly a Post Doctoral Research Fellow at the School of Mechaical Egieerig of Iha Uiversity i Icheo, Korea. Kwag-Yog Kim received his B.S. degree from Seoul Natioal Uiversity i 1978, ad his M.S. ad Ph.D. degrees from the Korea Advaced Istitute of Sciece ad Techology (KAIST), Korea, i 1981 ad 1987, respectively. He is curretly a professor ad the chairma of the School of Mechaical Egieerig of Iha Uiversity, Icheo, Korea. Professor Kim is also the curret editor-ichief of the Trasactios of Korea Society of Mechaical Egieers (KSME), the editor-i-chief of the Iteratioal Joural of Fluid Machiery ad Systems (IJFMS), ad the chief vice presidet of the Korea Fluid Machiery Associatio (KFMA). He is likewise a fellow of the America Society of Mechaical Egieers (ASME).