Research Article Two-Phase Flow in Wire Coating with Heat Transfer Analysis of an Elastic-Viscous Fluid

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1 Advances in Mathematical Physics Volume 016, Aticle ID , 19 pages Reseach Aticle Two-Phase Flow in Wie Coating with Heat Tansfe Analysis of an Elastic-Viscous Fluid Zeeshan Khan, 1, Rehan Ali Shah, 3 Saeed Islam, 1 and Bilal Jan 1 Depatment of Mathematics, Abdul Wali Khan Univesity, Madan, Khybe Pakhtunkhwa, Pakistan Sahad Univesity of Science and Infomation Technology, Peshawa, Pakistan 3 Depatment of Mathematics, Univesity of Engineeing and Technology, Peshawa, Khybe Pakhtunkhwa, Pakistan Coespondence should be adessed to Zeeshan Khan; zeeshansuit@gmail.com Received 4 Apil 016; Revised 1 May 016; Accepted 7 June 016 Academic Edito: Alexande Iomin Copyight 016 Zeeshan Khan et al. This is an open access aticle distibuted unde the Ceative Commons Attibution License, which pemits unesticted use, distibution, and epoduction in any medium, povided the oiginal wok is popely cited. This wok consides two-phase flow of an elastic-viscous fluid fo double-laye coating of wie. The wet-on-wet (WOW) coating pocess is used in this study. The analytical solution of the theoetical model is obtained by Optimal Homotopy Asymptotic Method (OHAM). The expession fo the velocity field and tempeatue distibution fo both layes is obtained. The convegence of the obtained seies solution is established. The analytical esults ae veified by Adomian Decomposition Method (ADM). The obtained velocity field is compaed with the existing exact solution of the same flow poblem of second-gade fluid and with analytical solution of a thid-gade fluid. Also, emeging paametes on the solutions ae discussed and appopiate conclusions ae awn. 1. Intoduction The study of non-newtonian fluids has gained deep attention fom eseaches due to its vaious applications in industies like oil, polyme, plastic, and so foth. Vaious models, both analytical and numeical, have been discussed in the study of non-newtonian fluids. Fluids models ae chaacteized by the undelying fluid gades like second gade, thid gade, and so foth genealizing to n-gade fluids. In this study non-newtonian thid-gade fluids have been studied fo thei applicability in optical fibe coating. Thid-gade fluids have been studied by many eseaches. Siddiqui et al. [1] studied the tosion flow of such fluid. The same autho studied heat flux of such fluids in two paallel plates which is discussed in []. Islam et al. [3] studied thid-gade fluid with heat tansfe. Aksoy and Pakdemili [4] investigated the thid-gade fluid flow in paallel plates with a poous medium. Thesubjectoftwoimmisciblefluidsflowswithheattansfe has been biefly studied due to its impotance in nuclea and chemical industies. It can be classified into thee goups, namely, segegated flows, tansitional o mixed flows, and dispesed flows [5]. Siddiqui et al. [6] studied two immiscible fluids in poous media. Batchelo [7] studied two immiscible fluidsinananalogousplate.twoimmisciblefluidshavebeen extensively studied theoetically and expeimentally [8, 9]. Diffeent types of fluids ae used fo wie and fibe optics coating, which depends upon the geomety of die, fluid viscosity, tempeatue of the wie o fibe optics, and the molten polyme. Most elevant woks on the wie and fibe optics coating aethussummaizedinthefollowing. Shah et al. [10] studied the wie coating analysis with linealy vaying tempeatue. Unsteady second-gade fluid with oscillating bounday condition inside the wie coating die was investigated by Shah et al. [11]. Exact solution was obtained fo unsteady second-gade fluid in wie coating analysis [1]. Shah et al. [13] studied thid-gade fluid with heat tansfe in the wie coating analysis. All these attempts wee elated to single laye coating flow. Immiscible fluid flow is used fo many industial and manufactuing pocesses such as oil industy o polyme poduction. Kim et al. [14] examined the theoetical pediction on the double-laye coating in wet-on-wet optical fibe coating pocess. Double-laye coating liquid flows wee used by Kim and Kwak [15] in optical fibe manufactuing. Fo this pupose powe-law fluid model was used. Recently Zeeshan et al. [16] used Phan Thien Tanne fluid in doublelaye optical fibe coating. The same autho [17] investigated

2 Advances in Mathematical Physics Q 1 Pimay coating applicato Seconday Q coating applicato Die cup Die cup Wie in Double coated wie out Figue 1: Manufactuing pocess of wie. Pimay coating esin Seconday coating esin double-laye esin coating of optical fibe glass using wet-onwet coating pocess with constant pessue gadient. Twophase flow of an Oloyd 8-constant fluid was used fo optical fibe coating by Zeeshan et al. [18]. Flow and heat tansfe in double-laye optical fibe coating using wet-on-wet coating pocess wee investigated by Khan et al. [19]. Keeping in view the wide ange of applications, an attempt is made to analyze the flow and heat tansfe in two-phase flow of an elastic-viscous fluid in a pessue type die. In this pape, the task is to find the analytical solutions fo the govening nonlinea equation aising in a coating metallic wie pocess inside a cylinical oll die, to study the fluid flow behavio in paticula, and to examine the effects of the non-newtonian fluid paametes and axial distance fom the cente of the metallic wie. This is ou fist attempt to investigate the double-laye coating flow of an elastic-viscous fluid on the wie using wet-on-wet coating pocess. Apat fom this, no one investigated the double-laye wie coating in wet-on-wet coating pocess using two immiscible elasticviscous fluids in a pessuized coating die. To the best of ouknowledge,nosuchanalysisofthedouble-layecoating flows of two immiscible elastic-viscous fluids on the wie is available in the liteatue. The pesent pape is stuctued as follows. Section isesevedfomodelingofthepoblem.solutionofthe poblem is given in Section 3. Section 4 is eseved fo analysis of esults. Section 5 contains the concluding emaks.. Modeling of the Poblem The geomety of the poblem is shown in Figue 1. The die and wie ae concentic. The coodinate system is taken at the cente of the wie, in which is taken pependicula to the flow diection z. The coating pocess is pefomed in two phases. In the fist phase the uncoveed wie of adius R w is agged with constant velocity V 1 into the pimay coating liquid. In the second phase the wet coating passes though the seconday coating die of adius R d and length L. Inthisway,thewie leaves the system with two layes of coating. The wet layes ae ied up by ultaviolet (UV) lamps. The liquid paametes at each phase ae genealized by coesponding phase numbe denoted by j (j = 1, ). The liquids ae paameteized by tempeatue Θ (j), the fluid density ρ (j),theviscosityμ (j), themal conductivity K (j), themal expansion coefficient β (j), and the specific heat c (j) p.thegasvelocitysuoundingthe polymeatthesufaceofcoatedwieisepesentedbyv as shown in Figue. The govening equations fo the two fluids ae the continuity, momentum, and enegy equations given as follows: u (j) =0, (1) ρ (j) D u (j) Dt = T (j), () c (j) DΘ (j) p =k Dt j Θ (j) + t (T (j) L), (3) whee D/Dt = / t + u (j) isthe substantive acceleation. No-slip bounday conditions ae taken at velocity. The tempeatue conditions Θ (1) and Θ () aetakenatthefibe opticsanddiewall,espectively.atthefluidinteface,we utilize the assumptions that the velocity, the shea stess, the pessue gadient, the tempeatue, and the heat flux ae continuous. The Cauchy stess T (j) given in () is T (j) = pi + S (j). (4) In the above equation p is the pessue and I and S (j) ae the identity and exta stess tenso, espectively. Fo thid-gade fluid S (j) is defined as [1 4, 13] S (j) =μ (j) A (j) 1 +α (j) 1 A(j) +α (j) A(j) 1 +τ (j) 1 A(j) (5) +τ (j) (A (j) 1 A(j) + A (j) A(j) 1 )+τ(j) 3 (t A (j) ) A(j) 1. In which α (j) 1, α(j), τ(j) 1, τ(j),andτ(j) 3 ae constants and A (j) 1, A (j), A(j) 3 ae kinematic tensos defined by A (j) 1 = L (j)t + L (j), A (j) n = A (j) n 1 LT + LA (j) n 1 + DA(j) n 1 Dt, n =,3, whee T stands fo tanspose of a matix. Fo steady and unidiectional flow velocity, tempeatue and stess fields ae defined as u (j) =[0,0,w (j) ()], Θ (j) =Θ (j) (), S (j) =S (j) (). (6) (7)

3 Advances in Mathematical Physics 3 Die =R d =R Fist phase flow Second phase flow =R w z Wie V 1 Figue : Two-phase flow model in wie coating die. In view of (7), (1) is satisfied identically and fom () (6) we have S (j) 3 =(α(j) 1 +α (j) )(dw(j) ), S (j) zz =α(j) ( dw(j) ), S (j) dw (j) 3 z =μ(j) +(τ (j) +τ (j) 3 )(dw(j) ). (8) The petinent bounday conditions on the velocity ae [14 19] w (1) =V 1 at =R w, w () =0 at =R d, w (1) =w (), S (1) z =S() z at =R. (10) (11) The petinent bounday conditions on the tempeatue ae [16 19] Fom (8), () and (3) educe to Θ (1) = Θ (1) at =R w, Θ () = Θ () at =R d, (1) (τ (j) +τ (j) 3 ) d 3 ( (dw(j) ) ) + μ(j) d (dw(j) )=0, K (j) ( d + 1 d )Θ(j) +μ (j) ( dw(j) ) 4 +(τ (j) +τ (j) 3 )(dw(j) ) =0. (9) K (1) dθ (1) Θ (1) =Θ (), =K () dθ () at =R, (13) whee (11) and (13) ae the inteface conditions on velocity and tempeatue, espectively. The volume flow ate at some contol suface is Q (j) =πv 1 (h (j) R w ), (14) whee h (j) is the adius of the coated wie.

4 4 Advances in Mathematical Physics The volume flow ate is [14 19] Q (j) R d = w (j) (). (15) R w Thickness of the coating wie is [14 19] Q(1) h (1) =R w ((1 + πv 1 Rw ) 1), h () =R w ([(1 + h(1) ) + Q() ] R w πv 1 Rw [ ] +(1+ h(1) R w )). 1/ (16) In view of (17), (9) (16) can be educed to the following set of nondimensional equations, espectively: d w j d θ j + dw j +β j (3 d w j (dw j ) +( dw 3 j ) )=0, =0, + 1 w 1 (1) =1, w (δ) =U, w 1 =w, S (1) z =S() z θ 1 (1) =0, θ (δ) =1, θ 1 =θ, dθ j +B j ( dw j ) +B j β j ( dw 4 j ) at =Ω, (18) (19) (0) (1) () = R w, w j = w(j) V 1, θ j = Θ(j) Θ () Θ () Θ (1), τ (j) 0 =τ (j) +τ (j) 3, R d =δ>1, R w R =Ω, R w (17) dθ 1 = K dθ δ Q (j) Q j =Q 1 +Q = πrw V 1 δ + w ()(), Ω h j =h 1 +h = h(j) R w Ω = w 1 ()() 1 at =Ω, Ω 1/ =([1+ w 1 ()() ] 1 δ 1/ 1)+([(1+h 1 ) + w ()() ] Ω (3) (4) (5) V V 1 =U, +(1+h 1 )). λ= μ() μ (1), K= K() K (1), μ (j) V 1 B j =, K (j) ( Θ () Θ (1) ) β j = τ (j) 0 μ (j) (R w /V 1 ). 3. Solution of the Poblem The OHAM is a steadfast method which has been boadly used by the eseaches to solve nonlinea poblems. One special aea of application of this method is to solve equations aising when non-newtonian fluids [0 5] ae studied. To solve (18) coesponding to the bounday conditions given in (0) and (1), we apply OHAM and ADM [6 8]. The details of OHAM and ADM ae given in Appendix. Hee, only the OHAM solution is given.

5 Advances in Mathematical Physics 5 By using OHAM, the zeoth-, fist-, and second-ode solutions fo both layes ae given below: Collectingtheesults,wewitethevelocityfieldobtainedby OHAM up to second-ode appoximation as w (1)0 =σ 1 +σ ln, w ()0 =σ 3 +σ 4 ln, w (1)1 =σ 5 +σ 6 ln k C 1 k ln + 1 k β C 1 k β 1, w ()1 =σ 7 +σ 8 ln k C 3 k ln + 1 k β C 3 k β, w (1) =σ 9 +σ 10 ln ln C 1 k 1 ln C 1 k 1 +k w 1 () =w (1)0 () +w (1)1 (, C 1 )+w (1) (, C 1,C ) +, w () =w ()0 () +w ()1 (, C 1 )+w () (, C 3,C 4 ) +. (7) The seies solution up to second-ode appoximation fo both layes is w 1 =Φ 11 +Φ 1 ln +Φ Φ Γ Φ Φ Φ 18, (8) C 1k 1 C 1 k C 1 k + 1 C 1 k β C 1 k β C 1k β C 1 k β C 1 k β C 1 k β C 1 k β 1 4 C k β 1 + 3C 1 k 1k β 1 + 6C 1k 3 β 1 σ 4 C 1 k4 β 1 C 1 k4 β C 1 k4 β 1 4 3C 1k β 1σ 4 σ 4 ln +C 1 σ 4 ln, w () =σ 11 +σ 1 ln ln C 3 k ln C 3 k 1 +k (6) w =Γ 11 +Γ 1 ln +Γ Γ Γ Γ Γ Γ 18. Inviewof(4)thevolumeflowateineachphaseisobtained as follows: Q 1 = 1 4 +Φ 14 (Ω 1) +Φ 11 (Ω 1) Φ 1 4 Ω + Φ 13 3 (Ω3 1)+( Φ 1 3 Ω +Φ 15 ) ln Ω Φ 16 ( 1 Ω 1) Φ 17 ( 1 Ω 1) C 3k 1 C 1 k C 3 k + 1 C 1 k β C 3 k β 3 3 C 3k β 4 + C 3 k β 10 5 C 3 k β 3 4 C 3 k β C 3 k β 4 C 4k β + 3C 3 k 1k β + 6C 3k 3 β σ 6 C 3 k4 β C 3 k4 β 3 6 3C 3 k4 β 4 3C 3k β σ 6 σ 6 ln +C 3 σ 6 ln. Q Φ 18 3 ( 1 Ω 3 1), = (Ω δ) (( Γ 11 (Ω+δ) + Γ 13 (Ω+δ) +Γ 14 )) +Γ 15 (ln ln ( Ω δ )) ( 1 Ω 1 δ )Γ Γ ( 1 Ω + 1 δ ) + Γ 86 ( 1 Ω + 1 δ ). (9)

6 6 Advances in Mathematical Physics 4. Solution fo the Tempeatue Distibution Now inseting w 1 () and w () fom (8) into (19) and solving with bounday conditions given in () and (3), we obtain the tempeatue fields fo both layes as θ 1 =c 1 +c ln + Ψ 1 + Ψ 1 + Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ 1 + Ψ +Ψ 3 + Ψ 4 +Ψ 5 (ln ), θ =c 3 +c 4 ln + Λ 1 + Λ 1 + Λ Λ Λ Λ Λ Λ Λ Λ Λ Λ Λ Λ Λ Λ Λ Λ Λ Λ Λ 1 + Λ +Λ 3 + Λ 4 +Λ 5 (ln ), (30) whee c 1, c, c 3, c 4, σ 1, σ, σ 3, σ 4, σ 5, σ 6, σ 7, σ 8, σ 9, σ 10, σ 11, σ 1, Ψ 11, Ψ 1, Ψ 13, Ψ 14, Ψ 15, Ψ 16, Ψ 17, Ψ 18, Γ 11, Γ 1, Γ 13, Γ 14, Γ 15, Γ 16, Γ 17, Γ 18, Λ 1, Λ, Λ 3, Λ 4, Λ 5, Λ 6, Λ 7, Λ 8, Λ 9, Λ 10, Λ 11, Λ 1, Λ 13, Λ 14, Λ 15, Λ 16, Λ 17, Λ 18, Λ 19, Λ 0, Λ 1, Λ, Λ 3, and Λ 4 ae all constants containing the convegence-contol paametes C 1,...,C 4, which ae optimally detemined by themethodofleastsquaesandaegiveninappendix. 5. Analysis of the Results Two-phase flow of an elastic-viscous thid-gade fluid is used fo wie coating. Actually, we ae making a platfom fo coating of polyme on the wie using theoetical appoach. The wie needs flexibility; that is why we need two-laye coating of the polyme. The inne coating o pimay coating potectsthewiefombending,whiletheoutecoating o seconday coating potects the pimay coating fom mechanical damage. Fo coating of the double-laye wie the wet-on-wet coating pocess is applied. Axial velocity distibution, flow ate, thickness of the coated wie, and tempeatuedistibutionsfoeachphaseaeobtainedby OHAM. The convegence of the method is also necessay to check the eliability of the methodology. The convegence of the obtained seies is shown in Figues 3 and 4. The cuent computed esults obtained ae also compaed with ADM, and an outstanding coespondence is seen to exist between the two sets of data as evealed in Figues 5 and 6 and Table 1. Futhemoe, the obtained esults ae also compaed with Table 1: Numeical compaison of OHAM and ADM when β 1 = 0.1, β = 0.5, λ = 0.3, δ =, C 1 = , C = , C 3 = , andc 4 = OHAM ADM Absolute eo Table : Compaison of the pesent wok with published wok [1] when β 1 = 0.1, β = 0.5, λ = 0.3, δ=, C 1 = , C = , C 3 = , andc 4 = Pesent wok Published wok [1] Absolute eo Table 3: Compaison of the pesent wok with published wok [13] when β 1 = 1 = β, λ = 0.3, δ =, C 1 = , C = , C 3 = , andc 4 = Pesent wok Published wok [13] Absolute eo peceding published elated liteatue, as a special case of the poblem and admiable ageement is obseved in this case also, as shown in Tables and 3.

7 Advances in Mathematical Physics 7 1.0E E E E E 01.00E 01 w() E Figue 3: Eo gaph (showing convegence of OHAM): OHAM up to second-ode solution when U = 0., β 1 = 0.3, β =1, λ = 0.3, and δ=. 3.50E E E E E E OHAM ADM Figue 5: Velocity compaison between OHAM and ADM esults when U = 0., β 1 = 0.1, β = 0.5, λ = 0.3, C 1 = , C = , C 3 = ,andC 4 = E E θ() Figue 4: Eo gaph (showing convegence of OHAM): OHAM up to second-ode solution when U = 0., β 1 = 0.1, β = 0.3, λ = 0.3, and δ= The vaiation of the non-newtonian paamete β 1,the Binkman numbe B 1, the conductivity atio K,andtheadii atio δ onthevelocity,tempeatue,andthicknessofcoated wie ae elaboated numeically in Tables 4 8. Table 4 shows the effects of the non-newtonian paamete β 1 on the velocity pofile. Hee, we vaied β 1 = 0., 0.3, 0.5, 0.8 and fixed the values of U = 0., λ = 0.1, β = 0.5,andδ=.Itistobenoted that the ise of the non-newtonian paamete deceases the speedoftheflow.theeffectsofthebinkmannumbeb 1 and the conductivity atio K on the tempeatue distibution ae givenintables5and6.intable5wevaiedb 1 = 5, 10, 13, 18 and fixed the values of λ = 0.1, β 1 = 0.1, β = 0.3, and δ=. Table 6 gives the numeical values of the tempeatue distibution by taking diffeent values of the conductivity atio K. We obseve fom these tables that the tempeatue inside the fluid inceases by inceasing the values of the OHAM ADM Figue 6: Tempeatue compaison between OHAM and ADM esults when β 1 = 0.1, β = 0.5, λ = 0.3, B 1 =3, B =5, δ=, C 1 = , C = , C 3 = , and C 4 = conductivity atio. The incease in the Binkman numbe significantly affects the tempeatue distibution as shown in Table 5. Futhemoe, the tempeatue pofile attains its maximum values at the cente of the annula die fo diffeent values of B 1 ; then it deceases to meet the fa field bounday conditions fo fixed paametes. Tables 7 and 8 pesent the impact of enlaging the adii atio δ and the viscosity atio λ on the thickness of the coated wie, espectively. Fom these

8 8 Advances in Mathematical Physics Table 4: Velocity pofile fo vaious values of β 1,whenβ = 0., λ= 0.3, δ =, C 1 = , C = , C 3 = , andc 4 = β 1 = 0. β 1 = 0.3 β 1 = 0.5 β 1 = Table 5: Tempeatue distibutions fo seveal values of B 1,when β 1 = 0.1, β = 0.3, B = 10, K = 0.3, λ = 0.3, δ =, C 1 = , C = , C 3 = , andc 4 = B 1 =5 B 1 =10 B 1 =13 B 1 = Table 6: Tempeatue distibutions fo seveal values of conductivity atio K, whenβ 1 = 0.1, β = 0.3, B 1 = 10, B = 15, λ = 0.3, δ =, C 1 = , C = , C 3 = , andc 4 = K = 0.1 K = 0.3 K = 0.7 K = tables it is obseved that the thickness of coated wie inceases with incease of δ and λ,espectively. Table 7: Thickness of coated wie fo vaious values of δ when β 1 = 0.1, β = 0.3, λ = 0.3, C 1 = , C = , C 3 = , andc 4 = δ = 1. δ = 1.4 δ = 1.8 δ = Table 8: Thickness of coated wie fo vaious values of λ when β 1 = 0.1, β = 0.3, δ =, C 1 = , C = , C 3 = , andc 4 = λ = 0. λ = 0.4 λ = 0.5 λ = Additionally, the thickness of the coated wie can be maintained at a equied level by adjusting these paametes. 6. Conclusions Two-phase flow of an elastic-viscous thid-gade fluid and wet-on-wet coating pocess is applied fo wie coating analysis. The obtained nonlinea equations ae solved fo velocity fields and tempeatue distibution by OHAM. ADM is also used fo claity. The effect of vaious emeging paametes on the velocity pofile, thickness of coated wie, and tempeatue distibution is discussed numeically. It is obseved that, with inceasing β 1,thevelocityofthefluiddeceases. The tempeatue inside the fluid is found to be inceased with inceasing the Binkman numbe B 1 and conductivity atio K. Futhemoe, the thickness of the coated wie also inceases with δ and λ. Appendix A. Analysis of ADM ADM is an analytical technique fo decomposing an unknown function into infinitely many components. Fo bette undestanding we conside the following: w (i) (, t) = n=0 w (i)n (, t), i = 1,. (A.1) To find the components w (i)0, w (i)1, w (i),..., sepaately, decomposition method is used.

9 Advances in Mathematical Physics 9 Conside the following nonlinea diffeential equation: L t w (i) (, t) +L w (i) (, t) +R (i) w (i) (, t) +N (i) w (i) (, t) =g (i) (, t), L w (j) (, t) =g (j) (, t) L t w (j) (, t) R (j) w (j) (, t) N (j) w (j) (, t). (A.) (A.3) Hee L = / and L t = / taelinea opeatos, g (i) (, t) is a souce tem, R (i) w (i) (, t) is a emainde linea opeato, and N (i) u (i) (, t) is a nonlinea tem. Applying L 1 on (A.3) to both side we have L 1 L w (i) (, t) =L 1 g (i) (, t) L 1 L tw (i) (, t) L 1 R (i)w (i) (, t) L 1 N (i)w (i) (, t), w (i) (, t) =f (i) (, t) L 1 L tw (i) (, t) L 1 R (i)w (i) (, t) L 1 N (i)w (i) (, t). (A.4) The function f (j) (, t) is obtained by utilizing the bounday conditions given in (0) and (1). The opeato L 1 = ( ) is used fo second-ode diffeential equations. With the seies solution of w (i) using ADM, we have n=0 w (i) (, t) = n=0 w (i)n (, t), w (i)n (, t) =f (i) (, t) L 1 L 1 N (i) n=0 R (i) n=0 w (i)n (, t) w (i)n (, t) (A.5) In view of Adomian polynomials the nonlinea tem N (i) n=0 w (i)n(, t) can be expessed as N (i) w (i)n (, t) = n=0 n=0 A (i)n, (A.6) whee the components w (i)0, w (i)1, w (i), w (i)3,... ae detemined as w (i)0 +w (i)1 +w (i) +w (i)3 +w (i)4 + Theecusiveelationisdefinedas w (i)0 (, t) =f (i)0 (, t) w (i)1 (, t) = L 1 R (i) [w (i)0 (, t)] L 1 (A (i)0 ), w (i) (, t) = L 1 R (i) [w (i)1 (, t)] L 1 (A (i)1 ), w (i)3 (, t) = L 1 R (i) [w (, t)] L 1 (A (i) ), and so on. B. Analysis of OHAM (A.8) To study the basic idea of OHAM, we conside the following nonlinea diffeential equation: A(w (i) ())+G (i) () =0, Λ, B(w (i), dw (i) )=0, Δ, (B.1) whee A is the diffeential opeato and B is the bounday opeato, w (i) is the unknown function, is the spatial independent vaiable, Δ is the bounday of the domain Λ, and G (i) () is the unknown analytical function. The opeato A can be witten as A=L (i) +N (i), (B.) whee L (i) and N (i) ae the linea and nonlinea opeatos, espectively. Fom (9), we have L (j) [φ (j) (, p)]= d w (j) N (j) [φ (j) (, p)] =β (j) (3 d w (j) G (j) =0, + dw (j), ( dw (j) ) +( dw 3 (j) ) ), (B.3) whee L (j), N (j), and G (j) ae linea opeato, nonlinea opeato, and souce tem, espectively, and p [0,1]is an embedding paamete. We conside a homotopy φ (j) (,p) : Λ [0,1] Rthat satisfies =f (i) (, t) L 1 R (i) (w (i)0 +w (i)1 +w (i) +w (i)3 + ) L 1 N (i) (A (i)0 + A (i)1 + ). (A.7) To detemine the seies components w (i)0, w (i)1, w (i), w (i)3,...,itshouldbenotedthatadmsuggestthatf (j) (, t), in fact descibe the zeoth component w (i)0. [1 p] [L (j) [φ (j) (, p)]+ G (j) ()] H (j) (p) [L (j) [φ (j) (, p)]+ N (j) [φ (j) (, p)]+ G j ()] =0, B(φ (j) (, p), φ (j) (, p) )=0, (B.4)

10 10 Advances in Mathematical Physics with bounday conditions φ 1 (1, p) = 1, φ (δ, p) = U, φ 1 (Ω, p) = φ (Ω, p), S z(1) (Ω, p) = S z() (Ω, p), (B.5) whee H (j) (p) is a nonzeo auxiliay function and φ (j) (, p) is an unknown function. Fo p=0, (11) only ecupeate the linea pat of solution; that is, φ (j) (, p) = w (j)0 (), Fist-ode poblem with bounday conditions is shown as follows: p 1 : d w (1)1 + dw (1)1 dw (1)0 dw (1)0 C 1 β 1 C 1 ( dw 3 (1)0 ) d w (1)0 d w (1)0 C 1 d w (1)0 6β 1 C 1 ( dw (1)0 ) =0, L (j) [φ (j) (, 0)]+G() =0, B(w (j)0, w (j)0 ), (B.6) Fo p = 1, we ecupeate the nonlinea bounday value poblem and the solution conveges to exact solution; that is, φ (j) (, 1) = w (j) (). Thesolutionφ (j) (, p) appoaches fom w (j)0 to w (j) () as p vaies fom 0 to 1. In ode to impove the accuacy of the esults and also in ode to ensue a faste convegence to the exact solution, we use the following genealized auxiliay function involving an inceased numbe of convegence-contol paametes even in the fist ode of appoximation, including also a physical paamete o a function of the physical paamete p 1 : d w ()1 + dw ()1 dw (1)0 dw ()0 C 3 β C 3 ( dw 3 ()0 ) d w ()0 d w ()0 C 3 d w ()0 6β C 3 ( dw ()0 ) =0, w (1)1 (1) =0, w ()1 (δ) =0, w (1)1 (Ω) =w ()1 (Ω), (B.10) H (t; p, C i )=ph 1 (t, C i )+p H (t, C i )+, (B.7) whee H i (t; C j ), i = 1,,..., ae auxiliay functions depending on it (o anothe physical paamete) and unknown convegence-contol paametes C j, j=1,,... Fo appoximate solution, φ (j) (, p) is expanding with espect to p by using Taylo seies φ (j) (, p, C i )=w (j)0 () + k=1 w (j)k (, C i )p k, i=1,,... (B.8) Substituting both (B.7) and (B.8) into (B.4), and equating each coefficient like powe of p and p, we have diffeent ode poblems. Zeoth-ode poblem with bounday conditions is shown as follows: S z(1)1 (Ω) =S z()1 (Ω). Second-ode poblem with bounday conditions is shown as follows: p : d w (1) + dw (1) dw (1)0 C β 1 C ( dw 3 (1)0 ) dw (1)1 d w (1)0 d w (1)0 dw (1)1 C C 1 6β 1 C 1 dw (1)1 ( dw (1)0 ) p 0 : d w (1)0 + dw (1)0 =0, p 0 : d w ()0 + dw ()0 =0, w (1)0 (1) =1, w ()0 (δ) =U, w (1)0 (Ω) =w ()0 (Ω), S z(1)0 (Ω) =S z()0 (Ω). (B.9) d w (1)0 6β 1 C ( dw (1)0 ) d w (1)0 dw (1)0 dw (1)1 1β 1 C 1 d w (1)1 C 1 d w (1)1 6β 1 C 1 d w (1)1 p : d w () + dw () dw ()0 C 4 ( dw (1)0 ) =0,

11 Advances in Mathematical Physics 11 β C 4 ( dw 3 ()0 ) dw ()1 d w ()0 d w ()0 dw ()1 C 4 C 3 6β C 3 dw ()1 ( dw ()0 ) d w ()0 6β C 4 ( dw ()0 ) d w ()0 dw ()0 dw ()1 1β C 3 d w ()1 C 3 d w ()1 6β C 3 d w ()1 w (1) (1) =0, w () (δ) =0, w (1) (Ω) =w () (Ω), S z(1) (Ω) =S z(1) (Ω). ( dw ()0 ) =0, (B.11) The convegence of (B.8) depends upon the auxiliay constant and ode of the poblem. If it conveges at p=1,onehas w (j) (, C 1,C,...,C n ) =w (j)0 () + w (j)n (, C 1,C,...,C n ). n=1 (B.1) Using (B.1) in (B.1), the expession fo the esidual in the following is obtained as R (j) (, C n )=L (j) w(,c n )+G (j) () +N (j) (w (, C n )), n=1,,...,m. (B.13) Many methods such as Ritz Method, Galekin s Method, collocationmethod,andleastsquaemethodaeusedtofind the auxiliay constants. Hee we use the least squae method to find the auxiliay constant: b J(C 1,C,...,C m )= R (j) (, C 1,C,...,C m ), a J = J (B.14) = =0, C 1 C whee a and b (taking fom domain) ae constant that locate the auxiliay constants which minimize the esidual. The above sequence of poblems given in (B.9) (B.11) ae solved using Mathematica softwae (see (6) in Solution of the Poblem): σ 1 =1, σ 3 =U σ 4 ln δ, σ = U 1 ln Ω σ ln δ 4 ( ln Ω 1), σ 4 = p 3p 3 + 1/3 ( p +3p 1p 3 ) 3p 3 ( p 3 +9p 1p p 3 7p3 p 4 + 4( p +3p 1p 3 ) 3 +( p 3 +9p 1p p 3 7p3 p 4) 1/3 ) ( p 3 +9p 1p p 3 7p 3 p 4 + 4( p +3p 1p 3 ) 3 +( p 3 +9p 1p p 3 7p3 p 4) 1/3 ), 3 1/3 p 3 ln δ p 1 =1+( ln Ω 1)(1 6γ 1 Ω (U 1 ln Ω ) ), p = 6γ 1 Ω (U 1 δ )(ln ln Ω ln Ω 1), p 3 = 3 Ω (γ ln δ γ 1 ( ln Ω 1) ), p 4 = U 1 ln Ω,

12 1 Advances in Mathematical Physics σ 5 =k β 1C 1 k 8 + β 1C k, σ 6 = 1 ln Ω Ωk +C 1 k β 1 + λc 1k β 1 Ω +C 3 k ln δ Ωk C 3 k ln Ω σ 8 (ln (Ωδ)), σ 7 = σ 8 ln δ+δk +C 3 k ln δ λc 4k β 8δ 4 + λc 3k β δ, σ 8 = λ (ζ/ω + ln (δω)) [C 1k ln Ω C 1 k β 0 + C 1k β 1 Ω +C 3 k ln δ C 4k β 8δ 4 + C 3k β δ C 3 k ln Ω+ C 4k β 8Ω 4 ] C 3k β Ω Ω ln Ω (ζ/ω + ln (δω)) [k + C 1k Ω + C 1k β 1 Ω 5 C 1k β 1 Ω 3 C 1k β 1 Ω 3 ζ(k + C 3k Ω + C 1k β 1 Ω 5 C 3k β 1 Ω 3 )], σ 9 = k (1 + C 1 ) λc 1k β C 1 k β 1 +C k β 1 3C 1 k β 1 6C 1 k 3 β 1 3 C 1 k3 β C 1 k4 β 1 +σ 4 (1 +k 1 +3C 1 k β 1), σ 10 = λ ln Ω [ σ 9 k (C 1 (1 + C 1 ) C 3 (1 + C 3 ) 3 (σ 4 +σ 6 )) ln Ω k Ω (C3 1 C3 3 +C 1 C 3 )+Ω(C 1 +C 3 ) +( k 10Ω 5 k 3Ω 4 )(β 1C 1 +β C 3 )+( k 3Ω 3 + k 4Ω )(β 1C 1 +β C 3 )+( k 10Ω 5 + k 3Ω 4 )(β 1C 1 +β C 3 ) + k 3Ω 3 (β 1C 1 +β C 3 ) k 4Ω 4 ((β 1C 1 β C 3 )+(β 1C +β C 4 )) ( 3k 1k Ω 3k Ω )(β 1C 1 β C 3 ) + k4 3Ω 6 (β 1C 1 +β C 3 ) k 4Ω (β 1C 1 β C 3 )] + k4 Ω 4 (β 1C 1 +β C 3 )+ 1 Ω (σ 4 +σ 6 )+ 1 Ω (C 1σ 4 +C 3 σ 6 ) + 3k4 Ω (β 1C 1 σ 4 +β C 3 σ 6 )+σ 11 +σ 1 ln Ω, σ 11 = [σ 1 ln Ω + λk 1 (C 3 C 3 ) ln Ω δk + λk δ C 3 δk C 3 + k δ C 3 k 10δ 5 β C 3 + λk 3Ω 4 β C 3 k 3δ 3 β C 3 k 10δ 5 β C 3 k 3δ 4 β C 3 + k 3δ 3 β C 3 + k 4δ β C 3 k δ β C 4 + 3k 1k δ β C 3 + 6k3 δ β C 3 + 3k3 δ 3 β C 3 + k4 3δ 6 β C 3 σ 4 δ σ 4 ln δ+c 1 σ 4 ln δ 3k δ β C 3 σ 4], σ 1 =( λ Ω (ln δ 1))[ k 1 (C 1 C 3 ) ln Ω k 1 (C 3 1 C3 3 ) ln Ω+ k Ω (C 1 +C 3 ) k (C 1 C 3 )+ k Ω (C 1 +C 3 ) + k Ω 6 (β 1C 1 +β C 3 )+ k 8Ω 5 (β 1C 1 +β C 3 )+ k Ω 4 (β 1C 1 +β C 3 )+ k Ω 3 (β 1C 1 +β C 3 )+ k Ω 6 (β 1C 1 +β C 3 ) + k 8Ω 5 (β 1C 1 +β C 3 )+ k Ω 4 (β 1C 1 +β C 3 )+ k Ω 3 (β 1C 1 +β C 3 )+k Ω 3 (β 1C +β C 4 )+3k 1k Ω 3 (β 1 C 1 +β C 3 ) + 6k Ω (β 1C 1 +β C 3 )+ 3k3 Ω 3 (β 1C 1 +β C 3 ) k4 Ω 7 (β 1C 1 +β C 3 ) 6k4 Ω 5 (β 1 C 1 β C 3 ) 1 Ω (σ 4 σ 6 )

13 Advances in Mathematical Physics 13 +(σ 4 +σ 6 ) ln Ω 1 Ω (C 1σ 4 C 3 σ 6 ) (C 1 σ 4 C 3 σ 6 ) ln Ω 6k Ω 3 (β 1C 1 σ 4 β C 3 σ 6 )+ 1 ln Ω σ 9], Φ 11 =1 β 1 C 1k +β 1 σ 5, Γ 11 =1 β C 3k +β σ 7, Φ 1 = U 1 ln δ +β 1σ 6 β 1 C 1 k +β 1 σ 10, Γ 1 = U 1 ln δ +β σ 8 β C 3 k +β σ 1, Φ 13 =β 1 k β 1 k, Γ 13 =β k β k, Φ 14 =β 1 C 1 k +6β 1 C 1 k 3 σ 4 (1 + C 1 )+β 1 C 1k, Γ 14 =β C 3 k +6β C 3 k 3 σ 6 (1 + C 3 )+β C 3k, Φ 15 = 1 C 1k β C 1k β3 1 +3C 1k β 1 σ C 1k β C 1 k β 1 +3C 1 k β3 1 k 1 C k β3 1, Φ 16 = 1 3 C 1k β C 1 k β3 1, Γ 16 = 1 3 C 3k β3 1 3 C 3 k β3, Φ 17 = 1 8 C 1k β C 1k β C 1 k β3 1 1 C 1 k4 β3 1, Φ 18 = 1 10 C 1k β C 1 k β 1, Γ 18 = 1 10 C 3k β C 3 k β, Γ 15 = 1 C 3k β 1 4 C 3k β3 +3C 3k β σ C 3k β + 1 C 3 k β +3C 3 k β3 k 1 C 4 k β3, Γ 17 = 1 8 C 3k β 1 3 C 3k β3 1 3 C 3 k β3 1 C 3 k4 β3, Ψ 1 = 65 4 B 1β 1 Φ 4 18, Ψ = B 1β 1 Φ 17 Φ 3 18, Ψ 3 = 1B 1 β 1 Φ 17 Φ B 1β 1 Φ 16 Φ 3 18, Ψ 4 = 1B 1 β 1 Φ 17 Φ B 1β 1 Φ 16 Φ 3 18, Ψ 5 = B 1β 1 Φ 3 17 Φ B 1β 1 Φ 16 Φ 17 Φ B 1β 1 Φ 15 Φ 3 18, Ψ 6 = B 1β 1 Φ B 1β 1 Φ 16 Φ 17 Φ B 1β 1 Φ 16 Φ B 1β 1 Φ 15 Φ 17 Φ B 1β 1 Φ 14 Φ 3 18, Ψ 8 = 7 4 B 1β 1 Φ 16 Φ 17 4B 1β 1 Φ 15 Φ B 1β 1 Φ 3 16 Φ B 1β 1 Φ 15 Φ 16 Φ 17 Φ B 1β 1 Φ 14 Φ 17 Φ B 1 β 1 Φ 15 Φ B 1β 1 Φ 14 Φ 16 Φ B 1β 1 Φ 1 Φ 17 Φ B 1β 1 Φ 13 Φ 3 18,

14 14 Advances in Mathematical Physics Ψ 9 = 96 5 B 1β 1 Φ 3 16 Φ B 1β 1 Φ 15 Φ 16 Φ B 1β 1 Φ 14 Φ B 1β 1 Φ 15 Φ 16 Φ B 1β 1 Φ 15 Φ 17Φ B 1β 1 Φ 14 Φ 16 Φ 17 Φ B 1β 1 Φ 1 Φ 17 Φ B 1β 1 Φ 14 Φ 15 Φ 18 +8B 1β 1 Φ 1 Φ 16 Φ B 1 β 1 Φ 13 Φ 17 Φ 18, Ψ 10 = B 1β 1 Φ B 1β 1 Φ 15 Φ 16 Φ B 1β 1 Φ 15 Φ B 1β 1 Φ 14 Φ 16 Φ B 1β 1 Φ 1 Φ B 1 β 1 Φ 15 Φ 16Φ B 1β 1 Φ 14 Φ 16 Φ B 1β 1 Φ 14 Φ 15 Φ 17 Φ B 1β 1 Φ 1 Φ 16 Φ 17 Φ B 1 β 1 Φ 13 Φ 17 Φ B 1β 1 Φ 14 Φ B 1β 1 Φ 1 Φ 15 Φ B 1β 1 Φ 13 Φ 16 Φ 18, Ψ 11 = B 1β 1 Φ 15 Φ B 1β 1 Φ 15 Φ 16Φ B 1β 1 Φ 14 Φ 16 Φ B 1β 1 Φ 14 Φ 15 Φ B 1 β 1 Φ 1 Φ 16 Φ B 1β 1 Φ 13 Φ B 1β 1 Φ 3 15 Φ B 1β 1 Φ 14 Φ 15 Φ 16 Φ B 1β 1 Φ 1 Φ 16 Φ B 1β 1 Φ 14 Φ 17Φ B 1β 1 Φ 1 Φ 15 Φ 17 Φ B 1β 1 Φ 13 Φ 16 Φ 17 Φ B 1β 1 Φ 1 Φ 14 Φ B 1 β 1 Φ 13 Φ 15 Φ 18, Ψ 1 = 3B 1 β 1 Φ 15 Φ 16 3 B 1β 1 Φ 14 Φ B 1β 1 Φ 3 15 Φ 17 8B 1 β 1 Φ 14 Φ 15 Φ 16 Φ 17 +6B 1 β 1 Φ 1 Φ 16 Φ B 1 β 1 Φ 14 Φ B 1β 1 Φ 1 Φ 15 Φ 17 +8B 1β 1 Φ 13 Φ 16 Φ B 1β 1 Φ 14 Φ 15 Φ 18 5 B 1β 1 Φ 14 Φ 16Φ B 1 β 1 Φ 1 Φ 15 Φ 16 Φ B 1β 1 Φ 13 Φ 16 Φ B 1β 1 Φ 1 Φ 14 Φ 17 Φ B 1β 1 Φ 13 Φ 15 Φ 17 Φ B 1 β 1 Φ 1 Φ B 1β 1 Φ 13 Φ 14 Φ 18, Ψ 13 = B 1β 1 Φ 3 15 Φ B 1β 1 Φ 14 Φ 15 Φ B 1β 1 Φ 1 Φ B 1β 1 Φ 14 Φ 15 Φ B 1 β 1 Φ 14 Φ 16Φ B 1β 1 Φ 1 Φ 15 Φ 16 Φ B 1β 1 Φ 13 Φ 16 Φ B 1β 1 Φ 1 Φ 14 Φ B 1 β 1 Φ 13 Φ 15 Φ B 1β 1 Φ 14 Φ 15Φ B 1β 1 Φ 1 Φ 15 Φ B 1β 1 Φ 1 Φ 14 Φ 16 Φ B 1 β 1 Φ 13 Φ 15 Φ 16 Φ B 1β 1 Φ 1 Φ 17Φ B 1β 1 Φ 13 Φ 14 Φ 17 Φ B 1β 1 Φ 1 Φ 13 Φ 18, Ψ 14 = 8 5 B 1β 1 Φ B 1β 1 Φ 14 Φ 15 Φ B 1β 1 Φ 14 Φ B 1β 1 Φ 1 Φ 15 Φ B 1β 1 Φ 13 Φ B 1 β 1 Φ 14 Φ 15Φ B 1β 1 Φ 1 Φ 15 Φ B 1β 1 Φ 1 Φ 14 Φ 16 Φ B 1β 1 Φ 13 Φ 15 Φ 16 Φ B 1 β 1 Φ 1 Φ B 1β 1 Φ 13 Φ 14 Φ 17 5 B 1β 1 Φ 3 14 Φ B 1β 1 Φ 1 Φ 14 Φ 15 Φ B 1β 1 Φ 13 Φ 15 Φ B 1 β 1 Φ 1 Φ 16Φ B 1β 1 Φ 13 Φ 14 Φ 16 Φ B 1β 1 Φ 1 Φ 13 Φ 17 Φ B 1Φ 18 3B 1β 1 Φ 13 Φ 18, Ψ 15 = B 1β 1 Φ 14 Φ B 1β 1 Φ 14 Φ 15Φ B 1β 1 Φ 1 Φ 15 Φ B 1β 1 Φ 1 Φ 14 Φ B 1β 1 Φ 13 Φ 15 Φ B 1β 1 Φ 3 14 Φ B 1β 1 Φ 1 Φ 14 Φ 15 Φ B 1β 1 Φ 13 Φ 15 Φ B 1β 1 Φ 1 Φ 16Φ

15 Advances in Mathematical Physics 15 B 1 β 1 Φ 13 Φ 14 Φ 16 Φ B 1β 1 Φ 1 Φ 13 Φ B 1β 1 Φ 1 Φ 14 Φ B 1β 1 Φ 1 Φ 15Φ B 1 β 1 Φ 13 Φ 14 Φ 15 Φ B 1β 1 Φ 1 Φ 13 Φ 16 Φ B 1Φ 17 Φ B 1β 1 Φ 13 Φ 17Φ 18, Ψ 16 = 3 4 B 1β 1 Φ 14 Φ 15 +B 1β 1 Φ 1 Φ B 1β 1 Φ 3 14 Φ B 1β 1 Φ 1 Φ 14 Φ 15 Φ B 1β 1 Φ 13 Φ 15 Φ B 1 β 1 Φ 1 Φ B 1β 1 Φ 13 Φ 14 Φ B 1β 1 Φ 1 Φ 14 Φ 17 3B 1 β 1 Φ 1 Φ 15Φ 17 +6B 1 β 1 Φ 13 Φ 14 Φ 15 Φ 17 9B 1 β 1 Φ 1 Φ 13 Φ 16 Φ B 1Φ 17 3B 1β 1 Φ 13 Φ B 1β 1 Φ 1 Φ 14Φ B 1β 1 Φ 13 Φ 14 Φ B 1 β 1 Φ 1 Φ 13 Φ 15 Φ B 1Φ 16 Φ B 1β 1 Φ 13 Φ 16Φ 18, Ψ 17 = B 1β 1 Φ 3 14 Φ B 1β 1 Φ 1 Φ 14 Φ B 1β 1 Φ 13 Φ B 1β 1 Φ 1 Φ 14 Φ B 1β 1 Φ 1 Φ 15Φ B 1β 1 Φ 13 Φ 14 Φ 15 Φ B 1β 1 Φ 1 Φ 13 Φ B 1β 1 Φ 1 Φ 14Φ B 1β 1 Φ 13 Φ 14 Φ B 1 β 1 Φ 1 Φ 13 Φ 15 Φ B 1Φ 16 Φ B 1β 1 Φ 13 Φ 16Φ B 1β 1 Φ 3 1 Φ B 1β 1 Φ 1 Φ 13 Φ 14 Φ B 1Φ 15 Φ B 1β 1 Φ 13 Φ 15Φ 18, Ψ 18 = 1 18 B 1β 1 Φ B 1β 1 Φ 1 Φ 14 Φ B 1β 1 Φ 1 Φ B 1β 1 Φ 13 Φ 14 Φ 15 B 1β 1 Φ 1 Φ 14Φ 16 +B 1 β 1 Φ 13 Φ 14 Φ 16 8B 1 β 1 Φ 1 Φ 13 Φ 15 Φ B 1Φ 16 3B 1β 1 Φ 13 Φ B 1β 1 Φ 3 1 Φ B 1 β 1 Φ 1 Φ 13 Φ 14 Φ B 1Φ 15 Φ B 1β 1 Φ 13 Φ 15Φ B 1β 1 Φ 1 Φ 13Φ B 1Φ 14 Φ B 1 β 1 Φ 13 Φ 14Φ 18, Ψ 19 = 8 5 B 1β 1 Φ 1 Φ B 1β 1 Φ 1 Φ 14Φ B 1β 1 Φ 13 Φ 14 Φ B 1β 1 Φ 1 Φ 13 Φ B 1β 1 Φ 3 1 Φ B 1β 1 Φ 1 Φ 13 Φ 14 Φ B 1Φ 15 Φ B 1β 1 Φ 13 Φ 15Φ B 1β 1 Φ 1 Φ 13Φ B 1Φ 14 Φ B 1 β 1 Φ 13 Φ 14Φ B 1Φ 1 Φ B 1β 1 Φ 1 Φ 13 Φ 18, Ψ 0 = 3 4 B 1β 1 Φ 1 Φ B 1β 1 Φ 13 Φ B 1β 1 Φ 3 1 Φ 15 6B 1 β 1 Φ 1 Φ 13 Φ 14 Φ B 1Φ 15 3B 1β 1 Φ 13 Φ B 1 β 1 Φ 1 Φ 13Φ B 1Φ 14 Φ 16 9 B 1β 1 Φ 13 Φ 14Φ B 1Φ 1 Φ 17 +6B 1 β 1 Φ 1 Φ 13 Φ B 1Φ 13 Φ B 1 β 1 Φ 3 13 Φ 18, Ψ 1 = 8 9 B 1β 1 Φ 3 1 Φ B 1β 1 Φ 1 Φ 13 Φ B 1β 1 Φ 1 Φ 13Φ B 1Φ 14 Φ B 1β 1 Φ 13 Φ 14Φ B 1 Φ 1 Φ 16 +8B 1 β 1 Φ 1 Φ 13 Φ B 1Φ 13 Φ B 1β 1 Φ 3 13 Φ 17, Ψ = 1 B 1β 1 Φ B 1β 1 Φ 1 Φ 13Φ B 1Φ 14 3B 1β 1 Φ 13 Φ 14 +B 1Φ 1 Φ 15 +1B 1 β 1 Φ 1 Φ 13 Φ B 1 Φ 13 Φ 16 +6B 1 β 1 Φ 3 13 Φ 16,

16 16 Advances in Mathematical Physics Ψ 3 = 8B 1 β 1 Φ 3 1 Φ 13 +B 1 Φ 1 Φ B 1 β 1 Φ 1 Φ 13 Φ 14 +4B 1 Φ 13 Φ B 1 β 1 Φ 3 13 Φ 15, Ψ 4 = B 1 Φ 1 Φ 13 8B 1 β 1 Φ 1 Φ 3 13, Ψ 5 = 1 4 B 1Φ 13 1 B 1β 1 Φ 4 13, Ψ 6 = 1 B 1Φ 1 6B 1β 1 Φ 1 Φ 13 +B 1Φ 13 Φ 14 +4B 1 β 1 Φ 3 13 Φ 14, Λ 1 = 65 4 B β Γ 4 18, Λ = B β Γ 17 Γ 3 18, Λ 3 = 1B β Γ 17 Γ B β Γ 16 Γ 3 18, Λ 4 = 1B β Γ 17 Γ B β Γ 16 Γ 3 18, Λ 5 = B β Γ 3 17 Γ B β Γ 16 Γ 17 Γ B β Γ 15 Γ 3 18, Λ 6 = B β Γ B β Γ 16 Γ 17 Γ B β Γ 16 Γ B β Γ 15 Γ 17 Γ B β Γ 14 Γ 3 18, Λ 7 = B β Γ 16 Γ B β Γ 16 Γ 17Γ B β Γ 15 Γ 17 Γ B β Γ 15 Γ 16 Γ B β Γ 14 Γ 17 Γ B β Γ 1 Γ 3 18, Λ 8 = 7 4 B β Γ 16 Γ 17 4B β Γ 15 Γ B β Γ 3 16 Γ B β Γ 15 Γ 16 Γ 17 Γ B β Γ 14 Γ 17 Γ B β Γ 15 Γ B β Γ 14 Γ 16 Γ B β Γ 1 Γ 17 Γ B β Γ 13 Γ 3 18, Λ 9 = 96 5 B β Γ 3 16 Γ B β Γ 15 Γ 16 Γ B β Γ 14 Γ B β Γ 15 Γ 16 Γ B β Γ 15 Γ 17Γ B β Γ 14 Γ 16 Γ 17 Γ B β Γ 1 Γ 17 Γ B β Γ 14 Γ 15 Γ 18 +8B β Γ 1 Γ 16 Γ B β Γ 13 Γ 17 Γ 18, Λ 10 = B β Γ B β Γ 15 Γ 16 Γ B β Γ 15 Γ B β Γ 14 Γ 16 Γ B β Γ 1 Γ B β Γ 15 Γ 16Γ B β Γ 14 Γ 16 Γ B β Γ 14 Γ 15 Γ 17 Γ B β Γ 1 Γ 16 Γ 17 Γ B β Γ 13 Γ 17 Γ B β Γ 14 Γ B β Γ 1 Γ 15 Γ B β Γ 13 Γ 16 Γ 18, Λ 11 = B β Γ 15 Γ B β Γ 15 Γ 16Γ B β Γ 14 Γ 16 Γ B β Γ 14 Γ 15 Γ B β Γ 1 Γ 16 Γ B β Γ 13 Γ B β Γ 3 15 Γ B β Γ 14 Γ 15 Γ 16 Γ B β Γ 1 Γ 16 Γ B β Γ 14 Γ 17Γ B β Γ 1 Γ 15 Γ 17 Γ B β Γ 13 Γ 16 Γ 17 Γ B β Γ 1 Γ 14 Γ B β Γ 13 Γ 15 Γ 18, Λ 1 = 3B β Γ 15 Γ 16 3 B β Γ 14 Γ B β Γ 3 15 Γ 17 8B β Γ 14 Γ 15 Γ 16 Γ 17 +6B β Γ 1 Γ 16 Γ B β Γ 14 Γ B β Γ 1 Γ 15 Γ 17 +8B β Γ 13 Γ 16 Γ B β Γ 14 Γ 15 Γ 18 5 B β Γ 14 Γ 16Γ B β Γ 1 Γ 15 Γ 16 Γ

17 Advances in Mathematical Physics 17 B β Γ 13 Γ 16 Γ B β Γ 1 Γ 14 Γ 17 Γ B β Γ 13 Γ 15 Γ 17 Γ B β Γ 1 Γ B β Γ 13 Γ 14 Γ 18, Λ 13 = B β Γ 3 15 Γ B β Γ 14 Γ 15 Γ B β Γ 1 Γ B β Γ 14 Γ 15 Γ B β Γ 14 Γ 16Γ B β Γ 1 Γ 15 Γ 16 Γ B β Γ 13 Γ 16 Γ B β Γ 1 Γ 14 Γ B β Γ 13 Γ 15 Γ B β Γ 14 Γ 15Γ B β Γ 1 Γ 15 Γ B β Γ 1 Γ 14 Γ 16 Γ B β Γ 13 Γ 15 Γ 16 Γ B β Γ 1 Γ 17Γ B β Γ 13 Γ 14 Γ 17 Γ B β Γ 1 Γ 13 Γ 18, Λ 14 = 8 5 B β Γ B β Γ 14 Γ 15 Γ B β Γ 14 Γ B β Γ 1 Γ 15 Γ B β Γ 13 Γ B β Γ 14 Γ 15Γ B β Γ 1 Γ 15 Γ B β Γ 1 Γ 14 Γ 16 Γ B β Γ 13 Γ 15 Γ 16 Γ B β Γ 1 Γ B β Γ 13 Γ 14 Γ 17 5 B β Γ 3 14 Γ B β Γ 1 Γ 14 Γ 15 Γ B β Γ 13 Γ 15 Γ B β Γ 1 Γ 16Γ B β Γ 13 Γ 14 Γ 16 Γ B β Γ 1 Γ 13 Γ 17 Γ B Γ 18 3B β Γ 13 Γ 18, Λ 15 = B β Γ 14 Γ B β Γ 14 Γ 15Γ B β Γ 1 Γ 15 Γ B β Γ 1 Γ 14 Γ B β Γ 13 Γ 15 Γ B β Γ 3 14 Γ B β Γ 1 Γ 14 Γ 15 Γ B β Γ 13 Γ 15 Γ B β Γ 1 Γ 16Γ B β Γ 13 Γ 14 Γ 16 Γ B β Γ 1 Γ 13 Γ B β Γ 1 Γ 14 Γ B β Γ 1 Γ 15Γ B β Γ 13 Γ 14 Γ 15 Γ B β Γ 1 Γ 13 Γ 16 Γ B Γ 17 Γ B β Γ 13 Γ 17Γ 18, Λ 16 = 3 4 B β Γ 14 Γ 15 +B β Γ 1 Γ B β Γ 3 14 Γ B β Γ 1 Γ 14 Γ 15 Γ B β Γ 13 Γ 15 Γ B β Γ 1 Γ B β Γ 13 Γ 14 Γ B β Γ 1 Γ 14 Γ 17 3B β Γ 1 Γ 15Γ 17 +6B β Γ 13 Γ 14 Γ 15 Γ 17 9B β Γ 1 Γ 13 Γ 16 Γ B Γ 17 3B β Γ 13 Γ B β Γ 1 Γ 14Γ B β Γ 13 Γ 14 Γ B β Γ 1 Γ 13 Γ 15 Γ B Γ 16 Γ B β Γ 13 Γ 16Γ 18, Λ 17 = B β Γ 3 14 Γ B β Γ 1 Γ 14 Γ B β Γ 13 Γ B β Γ 1 Γ 14 Γ B β Γ 1 Γ 15Γ B β Γ 13 Γ 14 Γ 15 Γ B β Γ 1 Γ 13 Γ B β Γ 1 Γ 14Γ B β Γ 13 Γ 14 Γ B β Γ 1 Γ 13 Γ 15 Γ B Γ 16 Γ B β Γ 13 Γ 16Γ B β Γ 3 1 Γ B β Γ 1 Γ 13 Γ 14 Γ B Γ 15 Γ B β Γ 13 Γ 15Γ 18, Λ 18 = 1 18 B β Γ B β Γ 1 Γ 14 Γ B β Γ 1 Γ B β Γ 13 Γ 14 Γ 15 B β Γ 1 Γ 14Γ 16 +B β Γ 13 Γ 14 Γ 16 8B β Γ 1 Γ 13 Γ 15 Γ B Γ 16 3B β Γ 13 Γ B β Γ 3 1 Γ B β Γ 1 Γ 13 Γ 14 Γ B Γ 15 Γ B β Γ 13 Γ 15Γ B β Γ 1 Γ 13Γ B Γ 14 Γ B β Γ 13 Γ 14Γ 18, Λ 19 = 8 5 B β Γ 1 Γ B β Γ 1 Γ 14Γ B β Γ 13 Γ 14 Γ B β Γ 1 Γ 13 Γ B β Γ 3 1 Γ

18 18 Advances in Mathematical Physics B β Γ 1 Γ 13 Γ 14 Γ B Γ 15 Γ B β Γ 13 Γ 15Γ B β Γ 1 Γ 13Γ B Γ 14 Γ B β Γ 13 Γ 14Γ B Γ 1 Γ B β Γ 1 Γ 13 Γ 18, Λ 0 = 3 4 B β Γ 1 Γ B β Γ 13 Γ B β Γ 3 1 Γ 15 6B β Γ 1 Γ 13 Γ 14 Γ B Γ 15 3B β Γ 13 Γ B β Γ 1 Γ 13Γ B Γ 14 Γ 16 9 B β Γ 13 Γ 14Γ B Γ 1 Γ 17 +6B β Γ 1 Γ 13 Γ B Γ 13 Γ B β Γ 3 13 Γ 18, Λ 1 = 8 9 B β Γ 3 1 Γ B β Γ 1 Γ 13 Γ B β Γ 1 Γ 13Γ B Γ 14 Γ B β Γ 13 Γ 14Γ B Γ 1 Γ 16 +8B β Γ 1 Γ 13 Γ B Γ 13 Γ B β Γ 3 13 Γ 17, Λ = 1 B β Γ B β Γ 1 Γ 13Γ B Γ 14 3B β Γ 13 Γ 14 +B Γ 1 Γ 15 +1B β Γ 1 Γ 13 Γ B Γ 13 Γ 16 +6B β Γ 3 13 Γ 16, Λ 3 = 8B β Γ 3 1 Γ 13 +B Γ 1 Γ 14 +4B β Γ 1 Γ 13 Γ 14 +4B Γ 13 Γ B β Γ 3 13 Γ 15, Λ 4 = B Γ 1 Γ 13 8B β Γ 1 Γ 3 13, Λ 5 = 1 4 B Γ 13 1 B β Γ 4 13, Λ 6 = 1 B Γ 1 6B β Γ 1 Γ 13 +B Γ 13 Γ 14 +4B β Γ 3 13 Γ 14, k 1 = U 1 ln Ω ( p 3p 3 1/3 ( p +3p 1p 3 ) 3p 3 ( p 3 +9p 1p p 3 7p3 p 4 + 4( p +3p 1p 3 ) 3 +( p 3 +9p 1p p 3 7p3 p 4) 1/3 ) ( p 3 +9p 1p p 3 7p 3 p 4 + 4( p +3p 1p 3 ) 3 +( p 3 +9p 1p p 3 7p3 p 4) 1/3 ) ln δ + )( 3 1/3 p 3 ln Ω 1), k = p 3p 3 1/3 ( p +3p 1p 3 ) 3p 3 ( p 3 +9p 1p p 3 7p3 p 4 + 4( p +3p 1p 3 ) 3 +( p 3 +9p 1p p 3 7p3 p 4) 1/3 ) + ( p 3 +9p 1p p 3 7p 3 p 4 + 4( p +3p 1p 3 ) 3 +( p 3 +9p 1p p 3 7p3 p 4) 1/3 ). 3 1/3 p 3 (B.15) Competing Inteests The authos declae that they have no competing inteests. Refeences [1] A. M. Siddiqui, T. Haoon, and S. Ium, Tosional flow of thid gade fluid using modified homotopy petubation method, Computes & Mathematics with Applications, vol.58,no.11-1, pp , 009. [] A. M. Siddiqui, A. Zeb, Q. K. Ghoi, and A. Benhabit, Homotopy petubation method fo heat tansfe flow of a thid gade fluid between paallel plates, Chaos, Solitons & Factals,vol.36, no. 1, pp , 008. [3] S.Islam,R.A.Shah,andI.Ali, Optimalhomotopyasymptotic solutions of couette and poiseuille flows of a thid gade fluid

19 Advances in Mathematical Physics 19 with heat tansfe analysis, Intenatioal Jounal of Non-Linea Sciences and Numeical Simulation,vol.11,pp ,010. [4] Y. Aksoy and M. Pakdemili, Appoximate analytical solutions fo flow of a thid-gade fluid though a paallel-plate channel filled with a poous medium, Tanspot in Poous Media, vol. 83,no.,pp ,010. [5] A. M. Siddiqui, R. Mahmood, and Q. K. Ghoi, Thin film flow of a thid gade fluid on a moving belt by He s homotopy petubation method, Intenational Jounal of Nonlinea Sciences and Numeical Simulation,vol.7,no.1,pp.7 14,006. [6] A. M. Siddiqui, R. Mahmood, and Q. K. Ghoi, Homotopy petubation method fo thin film flow of a thid gade fluid down an inclined plane, Chaos, Solitons & Factals,vol.35,no. 1,pp ,008. [7] G. K. Batchelo, An Intoduction to Fluid Dynamics, Cambidge Univesity Pess, Cambidge, UK, [8] M. Ishii, Themo-Fluid Dynamic Theoy of Two-Phase Flow, Eyolles, Pais, Fance, [9] M. Akai, A. Inoue, and S. Aoki, The pediction of statified two-phase flow with a two-equation model of tubulence, Intenational Jounal of Multiphase Flow,vol.7,no.1,pp.1 39, [10] R. A. Shah, S. Islam, A. M. Siddiqui, and T. Haoon, Heat tansfe by lamina flow of an elastico-viscous fluid in postteatment analysis of wie coating with linealy vaying tempeatue along the coated wie, Heat and Mass Tansfe,vol.48,no.6,pp , 01. [11] R. A. Shah, S. Islam, A. M. Siddiqui, and T. Haoon, Optimal homotopy asymptotic method solution of unsteady second gade fluid in wie coating analysis, JounaloftheKoean Society fo Industial and Applied Mathematics,vol.15,no.3,pp. 01, 011. [1] R. A. Shah, S. Islam, and A. M. Siddiqui, Exact solution of a diffeential equation aising in the wie coating analysis of an unsteady second gade fluid, Mathematical and Compute Modelling,vol.57,no.5-6,pp ,013. [13] R. A. Shah, S. Islam, M. Ellahi, T. Haoon, and A. M. Siddiqui, Analytical solutions fo heat tansfe flows of a thid Gade fluid in case of post-teatment of wie coating, Intenational Jounal of the Physical Sciences, vol. 6, pp , 011. [14] K. Kim, H. S. Kwak, S. H. Pak, and Y. S. Lee, Theoetical pediction on double-laye coating in wet-on-wet optical fibe coating pocess, Jounal of Coatings Technology Reseach, vol. 8, no. 1, pp , 011. [15] K. J. Kim and H. S. Kwak, Analytic study of non-newtonian double laye coating liquid flows in optical fibe manufactuing, AppliedMechanicsandMateials, vol. 4, pp , 01. [16] Zeeshan, R. A. Shah, S. Islam, and A. M. Siddique, Doublelaye optical fibe coating using viscoelastic phan thien tanne fluid, New Yok Science Jounal, vol. 6, no.10, pp.66 73, 013. [17] Zeeshan, S. Islam, R. A. Shah, I. Khan, and T. Gul, Exact solution of PTT fluid in optical fibe coating analysis using two-laye coating flow, Jounal of Applied Envionmental and Biological Sciences,vol.5,pp ,015. [18] Zeeshan, R. A. Shah, I. Khan, T. Gul, and P. Gaskel, Doublelaye optical fibe coating analysis by withawal fom a bath of oloyd 8-constant fluid, Jounal of Applied Envionmental and Biological Sciences, vol. 5, pp , 015. [19] Z. Khan, S. Islam, R. A. Shah, and I. Khan, Flow and heat tansfe of two immiscible fluids in double laye optical fibe coating, Jounal of Coatings Technology and Reseach,016. [0]T.Gul,R.A.Shah,S.Islam,andM.Aif, MHDthinfilm flows of a thid gade fluid on a vetical belt with slip bounday conditions, Jounal of Applied Mathematics, vol. 013, Aticle ID70786,14pages,013. [1] V. Mainca, N. Heişanu, and I. Nemeş, Optimal homotopy asymptotic method with application to thin film flow, Cental Euopean Jounal of Physics,vol.6,no.3,pp ,008. [] V. Mainca and N. Heişanu, Application of Optimal Homotopy Asymptotic Method fo solving nonlinea equations aising in heat tansfe, Intenational Communications in Heat and Mass Tansfe,vol.35,no.6,pp ,008. [3] V. Mainca, N. Heisanu, C. Bota, and B. Mainca, An optimal homotopy asymptotic method applied to the steady flow of a fouth-gade fluid past a poous plate, Applied Mathematics Lettes,vol.,no.,pp.45 51,009. [4] N. Heisanu, V. Mainca, and G. Madescu, An analytical appoach to non-linea dynamical model of a pemanent magnet synchonous geneato, Wind Enegy,vol.18,no.9,pp , 015. [5] N. Heişanu and V.Mainca, Optimal homotopy petubation method fo a non-consevative dynamical system of a otating electical machine, Zeitschift fu Natufoschung A,vol.67,no. 8-9, pp , 01. [6] G. Adomian, A eview of the decomposition method and some ecent esults fo nonlinea equations, Mathematical and Compute Modelling,vol.13,no.7,pp.17 43,1990. [7] A.-M. Wazwaz, Adomian decomposition method fo a eliable teatment of the Batu-type equations, Applied Mathematics and Computation,vol.166,no.3,pp ,005. [8] A.-M. Wazwaz, Adomian decomposition method fo a eliable teatment of the Emden-Fowle equation, Applied Mathematics and Computation,vol.161,no.,pp ,005.

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