A Decomposition Method for Volume Flux. and Average Velocity of Thin Film Flow. of a Third Grade Fluid Down an Inclined Plane

Transcription

1 Adv. Theor. Appl. Mech., Vol. 1, 8, no. 1, 9 A Decomposition Method for Volme Flx and Average Velocit of Thin Film Flow of a Third Grade Flid Down an Inclined Plane A. Sadighi, D.D. Ganji,. Sabzehmeidani Noshirvani Universit of Technolog Department of Mechanical Engineering P. O. Box 8, Babol, Iran Abstract Pertrbation methods depend on a small parameter which is difficlt to be fond for real-life nonlinear problems. To overcome this shortcoming, a powerfl analtical method is introdced to solve the thin film flow problem with a third grade flid on an inclined plane. Here, Adomian Decomposition method is applied to solve nonlinear eqation of the velocit field. The reslts obtained b this method are then compared with the traditional pertrbation method to illstrate the effectiveness of this method. Finall volme flx and average film velocit is given graphicall. Corresponding athor: Amin Sadighi Amiri address: am_sadighi@ahoo.com

2 A. Sadighi, D.D. Ganji,. Sabzehmeidani Kewords: Decomposition Method; Pertrbation method; Thin film flow; Third grade flid; Volme flx 1. Introdction Most scientific problems and phenomena in different fields of science and engineering occr nonlinearl. Except in a limited nmber of these problems, we enconter difficlties in finding their exact analtical soltions. Pertrbation method provides the most versatile tools available in nonlinear analsis of engineering problems, bt its limitations hamper its application: 1. Pertrbation method is based on assming a small parameter. An overwhelming majorit of nonlinear problems, especiall those having strong nonlinearit, have no small parameters at all.. The approximate soltions obtained b the pertrbation methods, in most cases, are valid onl for the small vales of the small parameter. The pertrbation soltions are generall niforml valid as long as a specific sstem parameter is small. However, we cannot rel fll on the approximations, becase there is no criterion on which the small parameter shold exist. Ths, it is essential to check the validit of the approximations nmericall and/or experimentall. To overcome these difficlties, approximate analtical soltions, sch as the tanh method [1,], the sine-cosine method [,], the homogeneos balance method [,], the variational iteration method [-8], the homotop-pertrbation method [9-1] and the Adomian decomposition method [1-1] are introdced, among which Adomian decomposition method [1-1] is the most effective and convenient one for both weakl and strongl nonlinear problems. This method has been shown to effectivel and accratel solve a large class of nonlinear problems with approximations converging rapidl to accrate soltions. There are few exact soltions of the Navier-Stokes eqations becase of their highl nonlinearit and these become rare when non-newtonian flids eqations are sed. Pertrbation techniqes [1,17] are widel applied for obtaining approximate soltions to these eqations involving a small parameter ε. Bt the have limitations that mentioned above.

3 Decomposition method 7 In this paper, we appl Adomian Decomposition method to std the thin film flow problem with a third grade flid on an inclined plane. The capabilit and effectiveness of this method are revealed b obtaining the analtical soltions of the model and comparing with pertrbation method.. Fndamentals of Adomian decomposition method Let s discss a brief otline of the Adomian Decomposition method. For this, we consider a general nonlinear eqation in the form [18] L R N = g (1) where L is the highest order derivative which is assmed to be easil invertible, R the linear differential operator of less order than L, N presents the nonlinear terms 1 and g is the sorce term. Appling the inverse operator L to the both sides of Eq. (1), and sing the given conditions we obtain: 1 1 = f ( x) L ( R) L ( N) () where the fnction f ( x) represents the terms arising from integration the sorce term g ( x), sing given conditions. For nonlinear differential eqations, the nonlinear operator N = F( ) is represented b an infinite series of the so-called Adomian polnomials ( ) = = F () m A m The polnomials A m are generated for all kind of nonlinearit so that A depends onl on, A 1 depends on and 1, and so on. The Adomian polnomials introdced above show that the sm of sbscripts of the components of for each term of A m is eqal to n [19]. The Adomian method defines the soltion ( x) b the series = m () m=

4 8 A. Sadighi, D.D. Ganji,. Sabzehmeidani In the case of F ( ), the infinite series is a Talor expansion abot, as follows: ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) F = F F F F ()!! B rewriting Eq. () as =..., sbstitting it into Eq. () and 1 then eqating two expressions for F ( ) fond in Eq. () and Eq. (), defines formlas for the Adomian polnomials in the form of [18] ( ) ( ) ( )( ) ( ) ( ) 1... F = A1 A... = F F 1... F... ()! B eqating terms in Eq. (), the first few Adomian s polnomials A, A 1, A, A and A are given: ( ) F ( ) A = F (7) A = (8) A = F ( ) 1 F ( ) (9)! 1 A = F ( ) 1 F ( ) 1 F ( ) (1)! ( iv) A = F ( ) 1 F ( ) 1 F ( ) 1 F ( ) (11)!!! M Now that the A m are known, Eq. () can be sbstitted in Eq.() to specif the terms in the expansion for the soltion of Eq. ().. Mathematical modeling of the problem The basic eqations governing the motion of an incompressible flid, neglecting the thermal effects, are []: V = (1) DV ρ Dt = p ρf divτ (1)

5 Decomposition method 9 where ρ the constant densit, V the velocit vector, p the pressre, τ the stress tensor and D / Dt denoting the material derivative. The stress tensor defining a third grade flid is given b [] S i i= 1 τ = (1) where S S S = (1) 1 A 1 1A A1 = (1) ( A 1 A A A 1 ) ( tra ) 1 1 A A = (17) Here is the coefficient of viscosit and 1,, 1, and, are material constants []. The Rivilin-Ericksen tensors, A n are defined b A = I, the identit tensor, and DAn1 T An = An1 ( V ) ( V ) An1, n 1 (18) Dt We consider a thin film of an incompressible flid of third grade flow down an inclined plane. The ambient air is assmed stationar so that the flow is de to gravit alone. We assme that the srface tension of the flid is negligible and the film is of niform thickness. We seek a velocit field of the form ( ( ),,) v = (19) Sbstitting for v and τ in Eqs. (1) and (1) and assming the absence of pressre gradient we obtain d d d d d d Sbject to the bondar conditions of ( ) =, ( ) ρg sin = at = (1) d d =, at () = ()

6 A. Sadighi, D.D. Ganji,. Sabzehmeidani. Soltion of the problem In this section, we std the velocit field and find expressions for ( ) b traditional pertrbation method and Adomian decomposition method..1. Soltion b pertrbation method In Eq. (), we takeε =, where =. So Eq. () transforms to []: d d d ρg ε = () d d d Let s assme ε as a small parameter. In order to solve Eq. () b traditional pertrbation method, we expand ( ) in the form of (, ) = ( ) ε ( ) ε ( ) ε 1 () Sbstitting Eq. () into Eq. () and rearranging based on powers of ε -terms, we can obtain: ε d : d ρg = ( ) d ( ) =, d = 1 d : 1 d d ε = (7) d d d ( ) d1 1 ( ) =, = d d d d 1 d d1 d ε : 1 = (9) d d d d d d ( ) d ( ) =, = d Solving Eq. ()-(), we obtain: ρg ( ) = () () (8) () (1)

7 Decomposition method 1 1 ρ g sin ( ) = 1 ρ g sin ( ) = And approximate soltion obtained b pertrbation method will be as follows: ρg ( ) = ε 1 ρ g sin () () ρ g sin 1 ε () Soltion b Adomian Decomposition method Following the Adomian decomposition analsis, the linear operator is defined as: d L = () d Conseqentl, Eq. () can be written as follows: d d ρg ρg L= = N () d d where =. The nonlinear term is d d = N = A m (7) d d m= A Hence, sing Eqs. (7) - (11) gives: d d = d d d d1 d d1 d 1 = A A d d d d d (8) (9) d d d d1 d 1 d1 d = () d d d d d d d

8 A. Sadighi, D.D. Ganji,. Sabzehmeidani d d1 d d d d 1 d1 d 1 d1 d d = (1) A d d d d d d d d d d d d1 d d1 d d 1 d d = () A d d d d d d d d d d M 1 Appling the inverse operator L to both sides of Eq. (), we obtain: ρg L L = L N L () 1 If L is a second-order operator, L is a twofold indefinite integral. Performing the indicated operations we obtain: d ( ) ( ) 1 1 ρg sin = L N () d ths, d ( ) ( ) ( ) 1 ρg = () d Appling bondar conditions given in Eqs. (1) and (), we obtain: ρg ( ) = The next iterates are determined recrsivel b Using above iteration formla, we obtain: 1 m 1 = L A m 1 ρ g sin ( ) = 1 ρ g sin ( ) = 9 18 and so on. In the same manner the rest of the components of the iteration formla can be obtained. Upon smming above iterations, the second order approximation is expressed as 1 d d d () (7) (8) (9)

9 Decomposition method ( ) = 1 sin sin g g ρ ρ sin g ρ ε () which is the same as that obtained b pertrbation method. Now let s do the following change of parameters: ν =, =, ρ ν =, sin ν g m = (1) where ν is kinematic viscosit, is dimensionless velocit and is non-newton parameter. After parameter change, Eq. () transforms into ( ) = m m m () It is worth pointing ot that if we set the non-newton parameter eqal to zero, i.e. =, non-newtonian soltion, i.e. Eq. (), transforms into Newtonian soltion.. Flow rate and average velocit After finding expression for velocit profile, the flow rate per nit width is given b [] ( )d W Q = () where W is width of the film. Using Eq. () we obtain the following expression for the flow rate: 7 sin 7 1 sin sin ρ ρ ρ = g g g W Q () Introdcing

10 A. Sadighi, D.D. Ganji,. Sabzehmeidani ρg ϕ =, ψ = () transforms Eq. () into Q W = ϕ ψϕ ψ ϕ 7 The average velocit over the cross section of the film is Q 1 1 ( ) = = ϕ ψϕ ψ ϕ (7) W 7 (). Reslts and discssions Fig. 1 shows the dimensionless velocit profile with different vales of, as non-newton parameter, and given m. It is apparent that the velocit profile converges to the Newtonian flid velocit profile as the non-newton parameter decreases. The velocit profile for different vales of m and given is depicted in Fig. where it is obvios that when we decrease m, the velocit profile meets the Newtonian flid velocit profile. Fig. represents the average velocit over the cross section of the film. As it is seen when we eqal ψ to zero, the flid shows the Newtonian flid behavior. 7. Conclsion In this paper Adomian decomposition method has been sccessfll sed to obtain the velocit profile of thin film flow of a third grade flid down an inclined plane. The reslts obtained b decomposition method are in excellent agreement with pertrbation method. Bt sing the common pertrbation method is based pon the existence of a small parameter, so developing the method for different applications is not eas and finding this small parameter is also difficlt. Bt Adomian

11 Decomposition method decomposition method does not need an small parameter and can be applied to wide class of nonlinear problems, whether or not with small parameter. In conclsion, Adomian decomposition method provides highl accrate nmerical soltions for nonlinear problems. It also has man merits in comparison with other methods, sch as: 1. Adomian decomposition method does not reqire small parameters which are needed b pertrbation method.. Adomian decomposition method avoids linearization and phsicall nrealistic assmptions. References [1] Engi Fan, Extended tanh-fnction method and its applications to nonlinear eqations, Phsics Letters A, 77 (), [] A.-M. Wazwaz, A sine-cosine method for handlingnonlinear wave eqations, Mathematical and Compter Modeling, (), [] A.-M. Wazwaz, The tanh and the sine-cosine methods for the complex modified K-dV and the generalized K-dV eqations, Compters & Mathematics with Applications, 9 (), [] Mingliang Wang, bin Zho, Zhibin Li, Application of a homogeneos balance method to exact soltions of nonlinear eqations in mathematical phsics, Phsics Letters A, 1 (199), 7-7. [] Engi Fan, Two new applications of the homogeneos balance method, Phsics Letters A, (), -7. [] J.H. He, Variational iteration method for atonomos ordinar differential sstems, Applied Mathematics and Comptation, 11 (), 11-1.

12 A. Sadighi, D.D. Ganji,. Sabzehmeidani [7] J.H. He, Variational iteration method - a kind of non-linear analtical techniqe: Some examples, International Jornal of Non-linear Mechanics, (1999), [8] J.H. He, Approximate analtical soltion for seepage flow with fractional derivatives in poros media, Compter Methods in Applied Mechanics and Engineering, 17 (1998), 7-8. [9] J.H. He, Homotop pertrbation method for bifrcation of nonlinear problems, International Jornal of Nonlinear Science and Nmerical Simlation, (), 7-8. [1] J.H. He, A copling method of a homotop techniqe and a pertrbation techniqe for non-linear problems, International Jornal of Non-linear Mechanics, (), 7-. [11] J.H. He, Homotop pertrbation method: a new nonlinear analtical techniqe, Applied Mathematics and Comptation, 1 (), [1] J.H. He, Homotop pertrbation techniqe, Compter Methods in Applied Mechanics and Engineering, 178 (1999), 7-. [1] N. Shawagfeh, D. Kaa, Comparing nmerical methods for the soltions of sstems of ordinar differential eqations, Applied Mathematics Letters, 17 (), -8. [1] G. Adomian, Soltion of phsical problems b decomposition, Compters & Mathematics with Applications, 7 (199), 1-1. [1] A. M. Wazwaz, Constrction of solitar wave soltions and rational soltions for the KdV eqation b Adomian decomposition method, Chaos, Solitons, & Fractals, 1 (1), 8-9.

13 Decomposition method 7 [1] A.H. Nafeh, Introdction to Pertrbation techniqes, wile, [17] A. Aziz, T.. Na, Pertrbation methods in heat transfer, Springer-Verlag, 198, Berlin. [18] C. Arslantrk, A decomposition method for fin efficienc of convective straight fins with temperatre-dependent thermal condctivit, International commnications in heat and mass transfer, (), [19] A.M. Wazwaz, A new algorithm for calclating Adomian polnomials for nonlinear operators, Applied Mathematics and comptation, 111 (), -9. [] A.M. Siddiqi, R. Mahmood, Q.K. Ghori, Homotop-pertrbation method for thin film flow of a third grade flid down an inclined plane, Choas, Solitons and Fractals, (), Article in press. Received: December 7, 7

14 8 A. Sadighi, D.D. Ganji,. Sabzehmeidani. m =. =... =. = Fig. 1. The dimensionless velocit profile with different vales of and given m. =. m=1.. m =.7. m= Fig.. The dimensionless velocit profile for different vales of m and given

15 Decomposition method ϕ=1 ψ=.8 ψ=. ψ=. ψ= _ Fig.. The average velocit over the cross section of the film for given amonts of ψ.