Yu, Y.-S. et al.: Homotopy Perturbatio Method for Viscous Heatig THERMAL SCIENCE, Year 13, Vol. 17, No. 5, pp. 1355-136 1355 HOMOTOPY PERTURBATION METHOD FOR VISCOUS HEATING IN PLANE COUETTE FLOW by Yi-Sha YUN a* ad Chaolu TEMUER b a College of Scieces, Ier Mogolia Uiversity of Techology, Hohhot, Chia b College of Arts ad Scieces, Shaghai Maritime Uiversity, Shaghai, Chia Origial scietific paper DOI: 1.98/TSCI135355Y I this paper, the problem of viscous heatig i plae Couette flow is cosidered by the homotopy perturbatio method. The o-liear terms are expaded to Taylor series of the homotopy parameter. The obtaied solutios are show graphically ad are compared with the exact solutios. The obtaied results illustrate the efficiecy ad coveiece of the method. Key words: homotopy perturbatio method, homotopy equatio, Taylor series, plae Couette flow, viscous dissipatio Itroductio We cosider the stea flow of a icompressible Newtoia fluid betwee two ifiite parallel plates, oe of which is movig. The goverig equatios of the problem are [1]: d dv d dt dv µ =, k + µ = dx dx dx dx dx with the boudary coditios: x = : v=, T = T, x= b: v= V, T = T () where v is the axial velocity, T the temperature, µ the viscosity, k the thermal coductivity, b the distace of the two ifiite parallel plates, T the temperature aroud the two plates, ad V the velocity of the movig plate. Here, oe cosider costat thermal coductivity k = k ad viscosity with expoetial temperature depedece, µ = µ exp[ α(t T )], where α is a costat. Through the followig dimesioless trasformatios: x v µ, y, u, T, T T V θ = = = β = α ε = (3) T b V k T the goverig eqs. (1) ad the boudary coditios () are rewritte as: (1) d d d * Correspodig author; e-mail: yyisha@sia.com d du d θ du exp( βθ) =, + ε exp( βθ) = y y y (4)
Yu, Y.-S. et al.: Homotopy Perturbatio Method for Viscous Heatig 1356 THERMAL SCIENCE, Year 13, Vol. 17, No. 5, pp. 1355-136 with y = : u =, θ =, y = : u = 1, θ = (5) I this paper, a powerful aalysis techique, the homotopy perturbatio method (HPM), is employed to solve the problem. The HPM was proposed by He [, 3] ad was further developed ad improved by himself [4-9]. For the effectiveess ad coveiece of HPM, may other mathematicias ad egieers are attracted to stu its improvemet, covergece ad applicatios i may areas. A ew modificatio of HPM was proposed by addig ad subtractig a liear term i the cosidered origial equatios i [1]. Two algorithms for costructig the homotopy equatio were give ad applied to solve quadratic Riccati differetial equatio of fractioal order i [11]. Combiatio of the Laplace trasform ad HPM was studied i [1]. Combiatio of the variatioal iteratio method ad HPM was studied i [13]. HPM was improved by trucatig the ifiite series correspodig to the first-order approximate solutio before itroducig this solutio i the secod-order liear differetial equatio i [14]. The multistage homotopy-perturbatio method was proposed ad used to solve the Lorez system i [15]. A ew exteded homotopy perturbatio method was proposed i [16]. The covergece of HPM was ivestigated i [17-19]. The applicatios of the HPM o may areas were also ivestigated such as the high-order boudary value problems [-3], Static pull-i istability [4], Lae-Emde type sigular IVPs problem [5], itegrodifferetial equatios [6], iverse problem [7], Hamilto-Jacobi-Bellma equatio [8], heat trasfer problem [9, 3] et al. I [9], this problem was cosidered by the same techique. I order to improve those results of [9], we costruct the homotopy equatio oly based o oe of equatios istead of all equatios as [9]. Meawhile, the o-liear terms are expaded to Taylor series of the homotopy parameter p. The obtaied solutios are show graphically ad are compared with the exact solutios. The computatioal results show that the approximate solutio obtaied i this paper has higher accuracy tha that i [9]. Solutio by HPM Through deotig the o-liear terms as: du du N1[ θ, u] = exp( βθ ), N[, u] exp( ) θ = βθ the eqs. (4) are rewritte as: d θ 1 d N [ θ, u] =, + εn [ θ, u] = (6) Followig the HPM, the expressios of u(y) ad θ(y) are assumed: 1 1 θ( y) = θ ( y) + pθ ( y) + + p θ ( y) +, u( y) = u ( y) + pu ( y) + + p u ( y) + (7) The o-liear terms are expaded to Taylor series with respect to p o p = : 1 θ = θ = = = N [, u] A' p, N [, u] A'' p
Yu, Y.-S. et al.: Homotopy Perturbatio Method for Viscous Heatig THERMAL SCIENCE, Year 13, Vol. 17, No. 5, pp. 1355-136 1357 where A' = N p, u p, =,1,,, 1 d 1 θ! dp = = p = A'' = N p, u p, =,1,,. 1 d θ! dp = = p = (8) Thus, the eqs. (6) are rewritte: d A' p = (9) = d θ + ε A'' p = (1) = The homotopy equatio is costructed oly based o the eq. (1): d θ + p A'' p = (11) ε = Substitutig the expressios of solutios (7) to the eq. (9) ad the homotopy eq. (11), the settig the coefficiets of p i (i =, 1,,, ) are zero, we obtai a series of liear boudary problems: θ ''( y) = p : θ() =, θ(1) = p u''( y) βu'( y) θ'( y) = : u() =, u(1) = 1 βθ ( ) y 1 εe [ u'( y)] + θ1''( y) = p : θ1() =, θ1(1) = β θ1( yu ) '( y) θ'( y) βu1'( y) θ'( y) βu'( y) θ1'( y) 1 p : βu''( y) θ1( y) + u1''( y) = u1() =, u1(1) = Solvig these liear problems, the solutios are obtaied: θ = ε y y ε θ 1 =
Yu, Y.-S. et al.: Homotopy Perturbatio Method for Viscous Heatig 1358 THERMAL SCIENCE, Year 13, Vol. 17, No. 5, pp. 1355-136 1 [ ( ) ] θ = 4 yyy y+ βε βε u = y 1 3 1 yβε u1 = y βε + y βε 6 4 1 1 5 1 β ε u = (1 y) β ε yβ ε + 96 48 96 Thus the -order approximate solutios of θ(y) ad u(y) are obtaied by let p = 1: θ( y) Ξ ( y) = θ( y) + θ1( y) + + θ 1( y), u( y) U( y) = u( y) + u1( y) + + u 1( y) These approximate solutios may be compared with the exact solutio of [1] which is give by: εβ εβ exp [ βθ( y) ] = 1 cosh (y 1) arcsih +, 8 8 1 8 εβ u = 1+ tah (y 1)arcsih + 1 εβ 8 This compariso of θ is give i fig.1 whe β = 1 ad a rage of values of ε. The graphs of the deviatio of the preset results from the exact solutio are depicted i fig. whe β = 1 ad a rage of values of ε..15 Θ.1.5 ε = 1.5 ε = 1 ε =.5..4.6.8 1. y Figure 1. Temperature distributio i a plae Couette flow, β = 1. (Dots: the 3-order approximate solutios i the preset work, Solid lie: the exact solutio) Deviatio.15.1.5 ε = 1.5 ε = 1 ε =.5..4.6.8 1. y Figure. Deviatio of the 3-order approximate solutio from the exact solutio, β = 1 Coclusios I this paper the homotopy equatio is costructed based o oe of two goverig equatios. The HPM is also combied with Taylor series. The approximate solutios of the problem of viscous heatig i plae Couette flow are obtaied with high accuracy. The efficiecy ad coveiece of the HPM have cofirmed agai oce.
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