ADM Solution of Flow Field and Convective Heat Transfer over a Parallel Flat Plate and Comparison with the Forth Order Runge Kutta Method

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1 Australia Joural of Basic ad Applied Scieces, 5(: -8, ISSN ADM Solutio of Flow Field ad Covective Heat Trasfer over a Parallel Flat Plate ad Compariso with the Forth Order Ruge Kutta Method Arma Hasapour, Davod Domiri Gaji, Mohammad Soleimai Lashkeari Youg Researchers Club, Gorga Brach, Islamic Azad Uiversity, Gorga, Ira. Babol Noshirvai Uiversity of Techology, Faculty of Mechaical Egieerig, Babol, Ira. Youg Youg Researchers Club, Chalus Brach, Islamic Azad Uiversity-Chalus Brach, Chalus, Ira. Abstract: I this paper, the case of forced covectio over a horizotal flat plate is preseted ad the Adomia decompositio method (ADM is employed to calculate a approximatio to the solutio of the system of oliear differetial equatios goverig o the problem. It has bee attempted to show the potetials ad wide-rage applicatios of the Adomia decompositio method i compariso with the previous oes i solvig heat trasfer problems. The results are compared by the umerical techique so-called Forth Order Ruge Kutta method ad Homotopy perturbatio method (HPM. A clear coclusio ca be draw from the umerical results that the ADM provides highly precise umerical solutios for oliear differetial equatios. The effect of Adomia polyomials terms is cosidered o accuracy of the results. Also isotherm cotours have bee obtaied i various coditio of Re ad Pr. Boudary layer thickess icreases with Pr but decreases with Re ad X velocity layer thickess decreases with Re too. Keywords: Adomia Decompositio Method, Covectio heat trasfer, Flow field, Numerical Solutio INTRODUCTION Most scietific problems such as heat trasfer problems are iheretly of oliearity. There are few pheomea i differet fields of sciece occurrig liearly. Therefore, these oliear equatios should be solved usig other methods. Some of them are solved usig umerical techiques (Hasapour et al. ad some of them are solved usig the aalytical method of such as perturbatio techique, HPM, HAM ad etc. (Gaji et al. 8, Gaji et al. 9, Gaji et al., Gaji et al.. The other well-kow method is Adomias decompositio method (ADM (Adomia 994, Adomia 998, Gaji et al.. The ADM was used to solve a wide rage of physical problems. This method provides a direct scheme for solvig liear ad oliear determiistic ad stochastic equatios without the eed for liearizatio ad yields coverget series solutios rapidly. A advatage of this method is that, it ca provide aalytical or a approximated solutio to a rather wide class of oliear (ad stochastic equatios without liearizatio, perturbatio, closure approximatio, or discretizatio methods. Ulike the commo methods are oly applicable to systems with weak oliearity ad small perturbatio ad may chage the physics of the problem due to simplificatio. ADM gives the approximated solutio of the problem without ay simplificatio. Thus, its results are more pragmatic. Durig recet years, several researchers have tried to modify the ADM. Wazwaz ( developed a fast ad accurate algorithm for solutio of sixth-order boudary value problems. Jafari ad Daftardar-Gejji (6 modified ADM to solve a system of oliear equatios. They obtaied a series solutio with faster accelerated covergece tha the series obtaied by the stadard ADM. Luo (5 proposed a efficiet modificatio to ADM, amely two-step Adomia Decompositio Method (TSADM that facilitated the calculatios. Luo et al. (6 revised ADM for cases ivolvig ihomogeeous boudary coditios, usig a suitable trasformatio. Zhag (5 preseted a modified ADM to solve a class of oliear sigular boudary value problems, which start as oliear ormal model equatios i oliear coservative vibratory systems. Also several researchers have used the ADM to solve a wide rage of physical problems i various egieerig fields such as fluid flow ad heat ad mass trasfer (Gaji et al. 7. I this paper, the case of forced covectio over a horizotal flat plate is preseted ad the Adomia decompositio method (ADM is employed to calculate a approximatio to the solutio of the system of oliear differetial equatios goverig o the problem. The goal of this study ca be achieved by implemetig the ADM to determie distributio of velocity ad temperature boudary layer. Limit literature works o cotours of temperature ad velocity i aalytical solutios. I this article we ivestigated these cotours for showig results ad effect of dimesioless umber (Re ad Pr o problem clearly. Correspodig Author: Arma Hasapour, Youg Researchers Club, Gorga Brach, Islamic Azad Uiversity, Gorga, Ira.

2 Aust. J. Basic & Appl. Sci., 5(: -8, MATERIALS AND METHODS Adomia Decompositio Method (ADM: Cosider equatio Fu( t g( t, where F represets a geeral oliear ordiary or partial differetial operator icludig both liear ad oliear terms. The liear terms are decomposed ito L+R, where L is easily ivertible (usually the highest order derivative ad R is the remais of the liear operator. Thus, the equatio ca be writte as: Lu Nu Ru g ( Where, Nu idicates the oliear terms. By solvig this equatio for Lu, sice L is ivertible, we ca write: L Lu L g L Nu L Ru If L is a secod-order operator, L - is a twofold idefiite itegral, by solvig Eq. ( for u, we get: u A Bt L g L Nu L Ru ( ( Where A ad B are costats of itegratio ad ca be foud from the boudary or iitial coditios. ADM assumes the solutio u ca be expaded ito ifiite series as: u u Also, the oliear term Nu will be writte as: Nu A (4 (5 Where A are the special Adomia polyomials. By substitutig Eqs. (4 ad (5 i Eq. (, the solutio ca be writte as: u u L Ru L A Where u is idetified as: A + Bt + L g []. I Eq. (6, the Adomia polyomials ca be geerated by several meas. Here we used the followig recursive formulatio: A d [! d [ N i ui ]],,,,,... (7 Sice the method does t resort to liearizatio or assumptio of weak oliearity, the solutio is geerated i the form of geeral solutio ad it is more realistic compared to the method of simplifyig the physical problems. Basic Equatios: Boudary layer flow over a horizotal flat plate is govered by the cotiuity ad the Navier Stokes equatios. Uder the boudary layer assumptios ad a costat property assumptio, the cotiuity ad Navier Stokes equatios ad eergy trasport equatio become: (6 u v x y (8 u u dp u u v v g ( T T x y dx y (9

3 Aust. J. Basic & Appl. Sci., 5(: -8, T T T u v x x y ( From Eqs. (9 ad (, the solutios of the eergy ad mometum equatios are coupled. However, the buoyacy force may be eglected if there is a pressure gradiet at right agles to the gravitatioal force. Thus, i the case of the forced covectio over a horizotal flat plate, the solutio to the mometum equatio is decoupled from the eergy solutio. The followig dimesioless variables are itroduced i the trasformatio: y.5 Re x x ( T T ( Tw T ( Where θ is o-dimesioal form of the temperature ad the Reyolds umber is defied as: ux Rex ( By usig Eqs. (8-(, the goverig equatios ca be reduced to two equatios where F is a fuctio of the similarity variable (η: F FF (4 Pr( F (5 The referece velocity is the free stream velocity of forced covectio. The boudary coditios for are obtaied from the similarity variables. For the forced covectio case (995: (, (, ( F F F (, ( If ADM applies o the Eqs. (-(6 i operator form as: LF FF (6 (7 (8 Pr( F L (9 d Where the differetial operator L is give by L = ad have bee assumed the iverse of the operator L d exists ad it ca be itegrated from to, i.e. L If L Operates o Eqs. (8 ad (9, yields L LF = we have: F( F( F ( F ( L ( FF (. ddd. FF L ( ad L L = Pr( F. The, ( ( ( ( ( L ( Pr( F (

4 Aust. J. Basic & Appl. Sci., 5(: -8, From the boudary coditios (Eqs. (6 ad (7 ad takig ca be obtaied by: F( L ( FF ( L ( Pr( F ADM is itroduced i the followig expressio: F( ( F ( ( F ( ad (, ADM solutio ( ( (4 (5 The ADM is defied as the oliear fuctio G (F( by a ifiite series of polyomials: G( F( ( ( A B (6 (7 Adomia polyomials A represet the oliear term G (F( ad ca be calculated from Eq. (7. By substitutig Eqs. (4 - (7 ito Eqs.( ad ( yields: F ( L ( A (8 ( L ( B (9 To determie the compoets of A ad F (, F ( was defied from the boudary coditio of Eqs.(6 ad (7 at = : F ( ( For determiatio of the other compoets of F ( ad (, we have: F ( L ( A,,,... ( L ( B,,,... ( ( ( ( By usig Eq (7, we obtai the followig terms of Adomia polyomials A : 4

5 Aust. J. Basic & Appl. Sci., 5(: -8, A FF A ( FF FF A ( FF FF FF A ( FF FF FF FF A4 ( FF 4 FF FF FF FF 4 ad this calculatio ca be cotiued util A. Ad from Eq(8, for determiig of F (, we have: F 5 F.46 8 F.68 4 F. F (4 (5 We use: F F F F F F F 4... (6 Ad similar to above: Pr B ( Pr B ( Pr B ( (7 Ad from Eqs. (9, for determiig of (, we have: Pr( Pr( We use:... (9 (8 5

6 Aust. J. Basic & Appl. Sci., 5(: -8, Clarificatio of Numerical Method: The type of curret problem is Boudary Value Problem ad Maple software like appropriate software is used as a umerical solver. I this study the midpoit method has bee used. There are two major cosideratios whe choosig a method for a problem. The trapezoid method is geerally efficiet for typical problems but the midpoit method is so capable of hadlig harmless ed-poit sigularities that the trapezoid method caot. The midpoit method, also kow as the Ruga-Kutta method, improves the Euler method by addig a midpoit i the step which icreases the accuracy. RESULTS AND DISCUSSIONS This study cosiders the steady state boudary layer flow. ADM have bee used for calculatio the mometum ad eergy equatios of the forced covectio over a horizotal flat plate. The series for F, F (the velocity profile ad temperature field,, evaluated ad are preseted i Figs. ( (. Figures ( - ( show the effect of umber of series term i correctess of ADM solutio, For F ad F (velocity field. It is obvious from this figure that whe the terms of series is expad, the covergece for velocity distributio is better. I Fig. ( The variatios of the Pradtl umber (Pr o the temperature ( are show..5 F Terms 5Terms F Terms 5Terms 4 4 Fig. : Compariso F (η ad F (η betwee terms ad 5 terms i ADM series, Pr= Temp Pr =.7 Pr = Pr =.. 4 Fig. : Compariso of temperature betwee various Pr umber i temperature distributio It ca be see that the temperature ( decrease with Pr. Because velocity profile is idepedet of Pr umber, therefore the same curve is ot plotted for F. I order to verify the accuracy of the curret study illustrates the compariso betwee ADM method ad homotopy perturbatio method (7 ad umerical solutio are show i Table (. The results prove that the aswers for velocity ad temperature distributios have a good agreemet with aother method ad particularly with umerical solutio. Also the results for aalytical solutios drow i -D graph for comparig effect of Re ad Pr umber. Effect of Re umber o isotherm cotour are show i Figs. ( ad 4. 6

7 Aust. J. Basic & Appl. Sci., 5(: -8, Table : The results of ADM, HPM [4] ad Numerical methods for F(η, F (η ad θ(η F(η F(η θ(η η ADM HPM NUM ADM HPM NUM ADM HPM NUM y x Fig. : Cotour isotherms for Re/x=., Pr= Boudary layer thickess decreases with Re. Boudary layer thickess icreases with Pr, i smaller isotherm lie ( this differece have bee bigger tha larger isotherm lie. These results are show i Fig. 4. y x Fig. : 4 Effect of Pr o Cotour isotherms for Re/x=., Pr=.7 (dashed lies,.5 (solid lies Coclusios: The mai aim of this paper is the Applicatio of Adomia Decompositio Method to boudary layer flow ad covectio heat trasfer over a horizotal flat plate. Graphical results are preseted to examie the effects of the Pradtl umber (Pr, the extesio of polyomial series of Adomia Decompositio method o the F, F (velocity profile ad temperature profiles (. We obtaied isotherm cotours i various coditio of Re ad Pr ad are show Boudary layer thickess icreases with Pr but decreases with Re. accordig to the aalytical solutio X velocity layer thickess decreases with Re. REFERENCES Adomia, G., 994. Solvig Frotier problems of physics: the decompositio method. Kluwer Academic Publicatio. Adomia, G., 998. A review of the decompositio method i applied mathematics. J Math. Aal. Appl., 5:

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