Variational Approach of Marangoni Mixed Convection Boundary Layer Flow with Pade Technique

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1 Joural of Applied Fluid Mechaics, Vol. 9, No. 6, pp , 6. Available olie at ISSN 7-7, EISSN Variatioal Approach of Maragoi Mied Covectio Boudary Layer Flow with Pade Techique M. Gubes ad G. Oturac Departmet of Mathematics, Karamaoglu Mehmetbey Uiversity, Karama, 7, Turkey Departmet of Mathematics, Selcuk Uiversity, Koya, 4, Turkey Correspodig Author (Received September 9, ; accepted February, 6) ABSTRACT I this paper, Variatioal Iteratio method with combiig Pade approimatio (Modified Variatioal Iteratio Method-MVIM) is performed to Maragoi covectio flow over the surface with buoyacy effects which is occurred gravity ad eteral pressure. After the appropriate trasformatio of equatios, we get the dimesioless form to solve umerically with modified variatioal iteratio method. We compare the our results with well-kow asymptotic epasio method used by Zhag Ya ad Zheg Liacu ad also compare with Fourth order Ruge Kutta solutio which are preseted i tables. Very efficiet ad accurate results are obtaied with preseted method. Keywords: Variatioal iteratio method; Maragoi covectio; Pade approimatio; Boudary layer flow. NOMENCLATURE f velocity fuctio g gravitatioal acceleratio Pr Pradtl umber S iterface T temperature T positive Temp. icremet alog the iterface compoet of velocity ue ( ) velocity outside boudary layer y compoet of velocity y Cartesia coordiates ormal to S Cartesia coordiate alog Marao mied covectio parameter similarity variable buoyacy forces parameter surface tesio temperature fuctio Lagrage multiplier stream fuctio. INTRODUCTION Maragoi covectio flow is stimulated by variatios of surface tesio throughout liquidliquid or liquid-gas surfaces ad it is importat i may fields of ature ad egieerig. Fudametal treatmet of Maragoi flow has bee aalyzed by (Gelles 978; Napolitao 98, 979, 978; Okao et al. 989). Some of umerical studies deped o the Maragoi covectio i various geometries have bee preseted by (Golia ad Viviai 98, 986), (Cristopher ad Wag ), (Pop et al. ), (Chamkha et al. 6), (Arafue ad Hirata 999), (Magyari ad Chamkha 8), (Aii et al. ), (Hamid ad Arifi 4), (Remeli et al. ), (Zhag ad Zheg ), (Che 7) ad (Al-Mudhaf ad Chamkha ). I real applicatios, mathematical problems are usually modeled by oliear ordiary ad partial differetial equatios such as physical ad egieerig applicatios. I geerally oliear models may ot be a eact solutios. Therefore, we try to fid approimate or umerical solutios of these models as see i refereces (Freidooimehr ad Rashidi ; Jhakal ). Oe of the most popular techique is variatioal iteratio method that is very powerful. I this cotet, we will cosider to eted the work studied by (Zhag ad Zheg 4) i order to fid aalytical solutios by usig variatioal iteratio method with combiig Pade approimatio called modified variatioal iteratio method (MVIM). Also, we will give compariso betwee MVIM ad

2 M. Gubes ad G. Oturac / JAFM, Vol. 9, No. 6, pp , 6. asymptotic epasio method (Zhag ad Zheg 4). Additioally, ukow parameters of velocity gradiet f () a ad temperature gradiet () b which obtaied by MVIM is compared with Ruge-Kutta method as show i tables. The effects ad variatios of Pr, ad o velocity ad temperature profiles are preseted graphically. All the calculatios for solutios are provided by oly oe or two step iteratios. Thus, it is foud that the preset results are i very good agreemet with other kow result as preseted i Table, ad.. MATHEMATICAL DESCRIPTION OF PROBLEMS We will cosider two-dimesioal Maragoi boudary layer flow with buoyacy effects due to eteral pressure gradiet ad gravity. It's occurs alog a iterface S betwee two fluids as i Fig.. if y, u u ( ) u, T () e where u ad v are the velocity compoets correspodig to ad y ais respectively. ue ( ) is eteral velocity, T is fluid temperature, is Maragoi mied covectio parameter. Also, if, the buoyacy force is available ad if, the buoyacy force is ot available. Additioally, variatio of surface tesio as d T, mtt Tm. () dt By usig these facts ad boudary coditios, we ca write the trasform variables as 4 T y, T T ( ) ( y, ) 4 T f( ) u r, 4 T k T (4) Combiig ()-(4) ad cosiderig the literature (Golia ad Viviai 98, 986; Chamkha et al 6; Al-Mudhaf ad Chamkha ) we obtai the mai ordiary differetial system which is the reduced form of () as Fig.. Schematic of the problem. f f ff r Pr f f ad boudary coditios tur ito f (), f( ) r, f() k, () (6) Gravity g occurs throughout o iterface S, the surface tesio chages with temperature. Viscous dissipatio ad iterface tortuosity are egligible. Also the flow fields for two iterfacig fluids are idepedet (Golia ad Viviai 98, 986; Zhag ad Zheg 4). Cosiderig these iformatio, we ca write the goverig equatios for the Maragoi boudary layer with water based fluid as u v y u u du e u u v u T y e d y u T v T T y Pr y Also boudary coditios are give as u T if y,, v, T T y () (), ( ) Here, prime deotes the derivatives with respect to.. MODIFIED VARIATIONAL ITERATION METHOD (MVIM) Variatioal iteratio method (VIM) is oe of the powerful mathematical tool to solve various kids of liear ad oliear problems as show some of i ref. (He 7, 999, 997; He ad Hog 7). I order to basic defiitio of VIM, we cosider the followig geeral oliear problem (He 7, 999, 997; He ad Hog 7)?? EMBED Equatio.DSMT4 Lu( ) Ru( ) N u( ) g( ) (7) d m where L, m ad R are liear operators, d m N is a oliear operator ad g is give cotiuous fuctio. Accordig to the origially VIM, we costruct the correctio fuctioal as 9

3 M. Gubes ad G. Oturac / JAFM, Vol. 9, No. 6, pp , 6. u ( ) u ( ) Lu ( s) Ru ( s) ( s) ds Nu ( s) g( s) (8) Here,, is a Lagrage multiplier (Iokuti et al. 978), u is cosidered as a restricted variatio, i.e. u. If we apply the variatio to correctio fuctioal (8) by usig variatioal aalysis, the we write dow as u ( ) u ( ) Lu ( s) Ru ( s) (, s) ds Nu ( s) g( s) Ku ( ) ( ) (, ) s u s ds g( s) (9) () From solutio of Euler-Lagrage problem show i (), we determie the Lagrage multiplier ad successive iteratios u ( ), are obtaied by usig Lagrage multiplier ad iitial approimatio u that satisfy, at least, the iitial ad boudary coditios with possible ukows. Cosequetly the eact solutio of (7) ca be obtaied by usig (He 7, 999, 997; He ad Hog 7) as u ( ) lim u( ) (). Pade Techic Some techiques eist to icrease the covergece of a give series. Amog them so-called Pade Techique is widely applied (Baker ad Morris 98). Supposed that a fuctio y ( ) is preseted by a power series c LM, Pade approimat is () L g g gl LM M O () hh hm which agree with () as far as possible. Here there are L idepedet umerator coefficiets ad M idepedet deomiator coefficiets, so makig LM ukow coefficiet i all. This is suggested that ormally LM, ought to fit the power series () amely L gg gl c M hh hm (4) L M O If the equatios equate with respect to L L LM,,...,, we write dow h c h c M LM M LM hc L hmclm hm clm hc L h c h c h c M L M L LM () If, c for cosistecy. Sice h, () become a set of M liear equatios for M ukow deomiator coefficiets ad also the umerator coefficiets g, g,..., g L follow immediately from (4) by equatig the coefficiets L M of,,,..., as g c, g cch, g c hc h c, mi LM, g c h c L L L (6) Thus, (6) ormally determie the Pade umerator ad deomiator. The LM, Pade approimat is costructed which agrees with c, through L M order. I order to fid the ifiite boudary coditios i (6) ad icrease covergece ad efficiecy of the series solutio (), we apply the Pade approimatio techic to (). Therefore, we combie the variatioal iteratio method ad pade techic so called modified variatioal iteratio method (MVIM). 4. SOLUTION PROCEDURE OF PROBLEMS Now, we will apply our proposed method MVIM to eqs. ()-(6) to obtai aalytic solutios. ad Let, assumed that f a b for the boudary coditios (6). By usig these cases, the iitial approimatios f ad which provided boudary coditios (6) are cosidered as 9

4 M. Gubes ad G. Oturac / JAFM, Vol. 9, No. 6, pp , 6. f a ad b (7) where ab, are ukow coefficiets that will be obtaied by applyig boudary coditios (6). By usig the variatios theory (9)-(), Lagrage multipliers are foud as follow respectively ( ) f ad ( ) s s (8) Thus, Lagrage multipliers (8) put ito (8), the successive iteratio equatios are writte as f ( ) f ( ) f s( f s) ( s ) f ( s) f s r ds s ad s ( s ) f( s) s ds Pr f ss (9) () By applyig (7) to (9)-(), we obtai the solutios of () respect to boudary coditios (6) as follow 6 f a a 4 a b 6 4 Pr a a a a 4 6 Pr a Prba 4 ( ) b Pra Pr abpr 6 ( a Pr Pr) 4 4 Pr( a ) Prb 8 () () (Pr a bpr b) Pr a b 4 Prb Pra Pr a 6 PrPr Pr a 6 7 4aPrb Prb 6 4 Pr a Pr a 4 For () ad (), iteratio process cotiues sufficietly (as see () ). I the series solutios () ad (), the ukow costat ab, which are deoted to velocity gradiet ad temperature gradiet respectively, are foud by applyig very efficiet approimatio called Pade Techique (-6) ad ifiite boudary coditios (6). For umerical values of, Pr ad (opposig buoyacy force), we fid ukow costats as f a.4746 b.886 () From () ad put, Pr ad ito (- ), the umerical solutio of () with MVIM is foud as follow respectively, f RESULTS Ad DISCUSSION The dimesioless form () ad (6) of Maragoi boudary layer flow equatios () ad () are cosidered. These equatios have bee solved by modified variatioal iteratio method (MVIM) ad very efficiet ad accurate results are obtaied by MVIM. For both opposig ad favorable buoyacy effected Maragoi flow with various values of Pradtl umber (Pr) ad Maragoi mied covectio parameter, our results are compared with well kow asymptotic iteratio method (Zhag ad Zheg 4) ad fourth order Ruge-Kutta method i Table,,. 9

5 M. Gubes ad G. Oturac / JAFM, Vol. 9, No. 6, pp , 6. Table Compariso of velocity ad temperature gradiets values for buoyacy force effects MVIM Ref (Zhag & Zheg) Numerical Pr f () () f () () f () () MVIM Ref (Zhag & Zheg) Numerical Pr f () () f () () f () () Fourth order Ruge Kutta Table Compariso of Temperature ad Velocity gradiets for various values of Pr ad for MVIM Numerical Pr f () () f () () Fourth order Ruge Kutta Table Compariso of Temperature ad Velocity gradiets for various values of Pr ad for MVIM Numerical Pr f () () f () () Fourth order Ruge Kutta It is evidet from Table that our results better tha the results i ref. (Zhag ad Zheg 4) with compared umerical method. Also from Table ad, it tells us that our preseted method (MVIM) is efficiet ad powerful mathematical tool. Fig. - demostrate the variatios of velocity profiles for both opposig ad favorable buoyacy forces with differet values of Pradtl Number (Pr). Also, Fig. 4-7 show the variatios of temperature profiles as the same meaig of Fig. - with various values of Pr ad Maragoi mied covectio parameter respectively. 6. CONCLUSION I this paper, we cosider the oliear ordiary differetial equatios which correspods to Maragoi boudary layer flow with buoyacy effects. These equatios are solved by Modified variatioal iteratio method aalytically. The velocity gradiet f, temperature gradiet as well as the temperature ad velocity profiles are aalyzed ad compared for buoyacy 9

6 M. Gubes ad G. Oturac / JAFM, Vol. 9, No. 6, pp , 6. opposed ad buoyacy favorable cases. Cosequetly, results show that MVIM is very powerful ad coveiet method for aalytical ad umerical solutios for oliear flow equatios. Fig.. Pradtl Number effects o temperature profile for eq. () with assistig buoyacy forces. Fig.. Pradtl Number effects o velocity profile for eq. () with opposig buoyacy forces. Fig. 6. Maragoi covectio parameter effects o temperature profile for eq. () with opposig buoyacy forces. Fig.. Pradtl Number effects o velocity profile for eq. () with assistig buoyacy forces Fig. 7. Maragoi covectio parameter effects o temperature profile for eq. () with assistig buoyacy forces. Fig. 4. Pradtl Number effects o temperature profile for eq. () with opposig buoyacy forces. ACKNOWLEDGEMENTS The authors would like to thak editor ad all referees for their cotributios. 94

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