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Open Physics

formerly Central European Journal of Physics

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Volume 15, Issue 1

Issues

Volume 13 (2015)

Optimal homotopy perturbation method for nonlinear differential equations governing MHD Jeffery-Hamel flow with heat transfer problem

Vasile Marinca
  • University Politehnica Timisoara, Department of Mechanics and Vibration, Timisoara, 300222, Romania
  • Department of Electromechanics and Vibration, Center for Advanced and Fundamental Technical Research, Romania Academy, Timisoara, 300223, Romania
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Remus-Daniel Ene
Published Online: 2017-03-15 | DOI: https://doi.org/10.1515/phys-2017-0006

Abstract

In this paper, the Optimal Homotopy Perturbation Method (OHPM) is employed to determine an analytic approximate solution for the nonlinear MHD Jeffery-Hamel flow and heat transfer problem. The Navier-Stokes equations, taking into account Maxwell’s electromagnetism and heat transfer, lead to two nonlinear ordinary differential equations. The results obtained by means of OHPM show very good agreement with numerical results and with Homotopy Perturbation Method (HPM) results.

Keywords: optimal homotopy perturbation method; Jeffery-Hamel; nonlinear ordinary differential equations

PACS: 02.60.-x; 47.11.-j; 47.50.-d

1 Introduction

Incompressible fluid flow with heat transfer is one of the most applicable cases in various fields of engineering due to its industrial applications. The problem of a viscous fluid between two nonparallel walls meeting at a vertex and with a source or sink at the vertex was pioneered by Jeffery [1] and Hamel [2]. Since then, the Jeffery-Hamel problem has been studied by several researchers and discussed in many textbooks and articles. A stationary problem with a finite number of “outlets” to infinity in the form of infinite sectors is considered by Rivkind and Solonnikov [3]. The problem of steady viscous flow in a convergent channel is analyzed analytically and numerically for small, moderately large and asymptotically large Reynolds numbers over the entire range of allowed convergence angles by Akulenko, et al. [4]. The MHD Jeffery-Hamel problem is solved by Makinde and Mhone [5] using a special type of Hermite-Padé approximation semi-numerical approach and by Esmaili et al. [6] by applying the Adomian decomposition method. The classical Jeffery-Hamel flow problem is solved by Ganji et al. [7] by means of the variational iteration and homotopy perturbation methods, and by Joneidi et al. [8] by the differential transformation method, Homotopy Perturbation Method and Homotopy Analysis Method. The classical Jeffery-Hamel problem was extended in [9] to include the effects of external magnetic field in a conducting fluid. The optimal homotopy asymptotic method is applied by Marinca and Herişanu [10] and by Esmaeilpour and Ganji [11]. The effect of magnetic field and nanoparticles on the Jeffery-Hamel flow are studied in [12, 13]. A numerical treatment using stochastic algorithms is applied by Raja and Samar [14].

During recent years, several methods have been used for solving different problems such as the traveling-wave transformation method [15], the Cole-Hopf transformation method [16], the optimal homotopy asymptotic method [17] and the generalized boundary element approach [18].

In general, the problems of Jeffery-Hamel flows and other fluid mechanics problems are inherently nonlinear. Excepting a limited number of these problems, most do not have analytical solutions. The aim of this paper is to propose an accurate approach to the MHD Jeffery-Hamel flow with heat transfer problem using an analytical technique, namely the Optimal Homotopy Perturbation Method (OHPM) [1921]. Our approach does not require a small or large parameter in the governing equations, and is based on the construction and determination of some auxiliary functions combined with a convenient way to optimally control the convergence of the solution.

2 Problem statement and governing equations

We consider a system of cylindrical coordinates with a steady flow of an incompressible conducting viscous fluid from a source or sink at channel walls lying in planes, with angle 2 α, taking into account the effect of electromagnetic induction, as shown in Fig.1, and heat transfer.

Geometry of the MHD Jeffery-Hamel problem.
Figure 1

Geometry of the MHD Jeffery-Hamel problem.

The continuity equation, the Navier-Stokes equations and energy equation in cylindrical coordinates can be written as [2225]: 1rrrur+1rrruφ=0,(1) ururr+uφrurφuφ2r=1ρPr+ν1r(rεrr)r+1rεrφrεrφrσB02ρr2ur,(2) uruφr+uφruφφuφurr=1ρrPφ+ν1r2(rεrφ)r+1rεφφφεrφrσB02ρr2uφ,(3) urTr=kρcp2T+νcp2urr2+urr2+1rurr2+σB02ρr2ur2,(4)

where ρ is the fluid density, P is the pressure, ν is the kinematic viscosity, T is the temperature, k is the thermal conductivity, cp is the specific heat at constant pressure, σ is the electrical conductivity, B0 is the induced magnetic field and the stress components are defined as εrr=2urr23divu¯,(5) εφφ=2ruφφ+2urr23divu¯,(6) εrφ=2rurφ+2ruφr.(7)

By considering the velocity field as only along the radial direction, i.e., uφ = 0 and substituting Eqs. (5)(7) into Eqs. (2) and (3), the continuity, Navier-Stokes and energy equations become: 1rrrur=0,(8) ururr=1ρPr+ν2ururr2σB02ρr2ur,(9) 1ρrPφ+2νr2urφ=0.(10)

The relevant boundary conditions, due to the symmetry assumption at the channel centerline, are as follows: urφ=Tφ=0,ur=ucratφ=0,(11)

and at the plates making the body of the channel: ur=0,T=Tcr2atφ=α,(12)

where uc and Tc are the centerline rate of movement and the constant wall temperature, respectively.

From the continuity equation (8), one can get rur=f(φ),(13)

where f(φ) is an arbitrary function of φ only.

By integrating Eq. (10), it holds that P(r,φ)=2ρνr2f(φ)+ρg(r),(14)

in which g(r) is an arbitrary function of r only.

Now, we define the dimensionless parameters: η=φα,F(η)=f(φ)uc,θ(η)=r2TTc,(15)

where Tc is the ambient temperature, and substituting these into Eqs. (4) and (9) and then eliminating the pressure term, one can put F+2αReFF+(4H)α2F=0,(16) θ+2α2α+RePrFθ+βPr(H+4α2)F2+F2=0,(17)

subject to the boundary conditions F(0)=1,F(0)=0,F(1)=0,(18) θ(1)=0,θ(0)=0,(19)

where Re=αucν is the Reynolds number, H=σB02ρν is the Hartmann number, Pr=νcpkρ,β=uccpand prime denotes derivative with respect to η.

3 Basic ideas of the optimal homotopy perturbation method

To explain the ideas of the optimal homotopy perturbation method, consider the non-linear differential equation Lu,u,u,u,η+g(η)+Nu,u,u,u,η=0,(20)

that is subject to the initial / boundary condition Bu,uη=0,ηΓ,(21)

where L is a linear operator, g is a known function, N is a nonlinear operator, B is a boundary operator and Γ is the boundary of the domain of interest [1921]. We construct the homotopy [26] Hu,p=Lu,u,u,u,η+g(η)+pNu,u,u,u,η=0,(22)

for Eq. (20), where p is the homotopy parameter, p ∈ [0, 1]. From Eq. (22), one gets Hu,0=Lu,u,u,u,η+g(η)=0Hu,1=Lu,u,u,u,η+g(η)+Nu,u,u,u,η=0.(23)

Assuming that the approximate analytical solution of the second-order can be expressed in the form u¯(η)=u0+pu1+p2u2,(24)

and expanding the nonlinear operator N in series with respect to the parameter p, we have: Nu¯,u¯,u¯,u¯,η=Nu0,u0,u0,u0,η+pu1Nu¯u0,u0,u0,u0,η++u1Nu¯u0,u0,u0,u0,η+u1Nu¯u0,u0,u0,u0,η++u1Nu¯u0,u0,u0,u0,η+p2u2Nu¯+u2Nu¯+,(25)

where Fu¯=Fu¯. By introducing a number of unknown auxiliary functions Hi(η, Ck), i = 0, 1, 2, … that depend on the variable η and some parameters Ck, k = 1, 2, …, s, we can construct a new homotopy: Hu¯,p=Lu¯,u¯,u¯,u¯,η+g(η)+pH0(η,Ck)Nu0,u0,u0,u0,η+p2H1(η,Ck)u1Nu¯u0+H2(η,Ck)u1Nu¯u0+H3(η,Ck)u1Nu¯u0+H4(η,Ck)u1Nu¯u0+p2H5(η,Ck)u2Nu¯u0+H6(η,Ck)u2Nu¯u0+.(26)

Equating the coefficients of like powers of p yields the linear equations: Lu0,u0,u0,u0,η+g(η)=0,Bu0,u0η=0,(27) Lu1+H0(η,Ck)Nu0,u0,u0,u0,η=0,Bu1,u1η=0,(28) Lu2+H1(η,Ck)u1Nu¯u0,u0,u0,u0,η+H2(η,Ck)u1Nu¯u0+H3(η,Ck)u1Nu¯u0+H4(η,Ck)u1Nu¯u0=0,Bu2,u2η=0.(29)

The functions Hi(η, Ck), i = 0, 1, 2, … are not unique and can be chosen such that the products HiujNu and ujNu are of the same form. In this way, a maximum of only two iterations are required to achieve accurate solutions.

The unknown parameters Ck, k = 1, 2, …, s which appear in the functions Hi(η, Ck) can be determined optimally by means of the least-square method, collocation method, the weighted residuals, the Galerkin method, and so on.

In this way, the solution of Eq. (20) subject to the initial/boundary condition (21) can be readily determined. It follows that the basic ideas of our procedure are the construction of a new homotopy (26), the auxiliary functions Hi with parameters Ck that can be determined optimally leading to the conclusion that the convergence of the approximate solutions can be easily controlled.

4 Application of OHPM to the MHD Jeffery-Hamel flow and heat transfer problem

Let us present the approximate analytic expressions of f(η) and θ(η) from Eqs. (16)(19) by means of OHPM.

For Eqs. (16) and (18), the linear operator is chosen as L(F) = F‴, while the nonlinear operator is defined as N(F) = 2αFF′ + (4 − H)α2 F′, g(η) = 0. The initial approximation F0 is obtained from Eq. (27) F0=0,F0(0)=1,F0(0)=0,F0(1)=0.(30)

The solution of Eq. (30) is hence F0(η)=1η2.(31)

On the other hand, from Eq. (16), one obtains NFF=2αReF,NFF=2αReF+(4H)α2.(32)

By substituting Eq. (31) into the nonlinear operator N and into Eq. (32), one retrieves NF0=2Aη22(A+B)η,NFF0=2Aη,NFF0=Aη2+A+B,(33)

where A = 2αRe, B = (4 − H)α2.

Eq. (28) becomes F1+H0(η,Ck)[2Aη22(A+B)η]=0,F1(0)=F1(0)=F1(1)=0.(34)

We choose H0(η, Ck) = −60C1 where C1 is an unknown parameter and from Eq. (34) we obtain F1(η)=2AC1η55(A+B)C1η4+(3A+5B)C1η2.(35)

Eq. (29) can be written in the form F2+H1(η,Ck)(2Aη)F1+H2(η,Ck)(Aη2+A+B)F1=0F2(0)=F2(0)=F2(1)=0.(36)

In this case we choose H1(η,Ck)=12AC2η2+C3η+C4+C5η,H2(η,Ck)=C62+C7η,

such that the solution of Eq. (36) is given by F2(η)=AC1C2495η11+2AC1C35(A+B)C1C2720η10+2AC1C45(A+B)C1C3+5A2C1C6504η9+(3A+5B)C1C25(A+B)C1C4+2AC1C510(A2+AB)C1C6336η8+(3A+5B)C1C35(A+B)C1C55(A2+AB)C1C6+2AC1C7210η7+(3A+5B)C1C4+(13A2+25AB+10B2)C1C65(A+B)C1C7120η6+(3A+5B)C1C560η5(3A2+8AB+5B2)C1C6+(3A+5B)C1C724η4+Mη2,(37)

where M=C1C214951144+1112A+53361144BC1C313605504+170A+1425504BC1C713A140+B6C1C412525336+140A+1245336BC1C59A280+5B84C1C6550451685210160A25168+5210+18AB18B2.

For p = 1 in Eq. (24), we obtain the second-order approximate solution, using Eqs. (31), (35) and (37): F¯(η)=F0(η)+F1(η)+F2(η)=AC1C2495η11+2AC1C35(A+B)C1C2720η10+2AC1C45(A+B)C1C3+5A2C1C6504η9+(3A+5B)C1C25(A+B)C1C4+2AC1C510(A2+AB)C1C6336η8+(3A+5B)C1C35(A+B)C1C55(A2+AB)C1C6+2AC1C7210η7+(3A+5B)C1C4+(13A2+25AB+10B2)C1C65(A+B)C1C7120η6+2AC1+(3A+5B)60C1C5η5(3A2+8AB+5B2)C1C6+(3A+5B)C1C724+(5A+5B)C1η4+3A+5B+M1η2.(38)

Now, we present the approximate analytic solution for Eqs. (17) and (19). The linear and nonlinear operators and the function g are, respectively, L(θ)=θ,g(η)=1,N(θ)=1+4α2θ+2αRePrFθ+βPr(H+4α2)F2+F2.(39)

Eq. (27) becomes θ01=0,θ0(1)=0,θ0(0)=0.(40)

Eq. (40) has the solution θ0(η)=12(1η2).(41)

From Eq. (39) it follows that Nθ(θ)=4α2+2αRePrF,Nθ(θ)=0.(42)

By substituting Eq. (41) into Eqs. (39) and (42), one gets respectively N(θ0)=C2Dη2+Eη4,Nθ(θ0)=L+Kη2,(43)

where C=1+2α2+αRePr+4βα2Pr+PrHβD=α2+2αRePr+2βPr(2α21)+PrHβE=αRePr+4βα2Pr+PrHβL=4α2+2αRePr,K=2αRePr.(44)

Eq. (28) can be written θ1+h0(η,C8)(C2Dη2+Eη4)=0,θ1(1)=θ1(0)=0.(45)

Choosing h0(η, C8) = −30C8 in Eq. (45), one obtains θ1(η)=C815C(η21)5D(η41)+E(η61).(46)

Eq. (29) can be written in the form θ2+h1(η,Ck)(L+Kη2)θ1=0,θ2(1)=θ2(0)=0,(47)

and therefore it is natural to choose the auxiliary function h1 as h1(η,Ck)=1C8C9+C10η+C11η2+C12η3+C13η4.

From Eq. (47), it can shown that θ2(η)=L(15C5D+E)12C9(η21)+16C10(η31)+15LCC9(15C5D+E)(KC9+LC11)η4112+15LCC10(15C5D+E)(KC10+LC12)η5120+(15CK5DK)C9+15LCC11(15C5D+E)(KC11+LC13)η6130+(15CK5DL)C10+15LCC12K(15C5D+E)C12η7142+(LE5DK)C9+(15CK5DL)C11+15LCC13K(15C5D+E)C13η8156+(LE5DK)C10+(15CK5DL)C12η9172+EKC9+(LE5DK)C11+(15CK5DL)C13η10190+EKC11+(LE5DK)C13η121132+EKC12η131156+EKC13η141182.(48)

The second-order approximate solution of Eqs. (17) and (19) is θ¯(η)=θ0(η)+θ1(η)+θ2(η),(49)

where θ0, θ1 and θ2 are given by Eqs. (41), (46) and (48), respectively.

5 Numerical results

In order to show the efficiency and accuracy of OHPM, we consider some cases for different values of the parameters α and H. In all cases we consider Re = 50, Pr = 1, β = 3.492161428 ⋅ 10−13.

Case 5.1 Consider α=π24 and H = 0. By means of the least-square method, the values of the parameters Ci, i = 1, 2, …, 13 are C1=0.000025737857,C2=128165.4247848388,C3=360860.8730122449,C4=315974.73981422884,C5=20823.768289134576,C6=319.2089575339067,C7=62701.61063351235,C8=8.9044474496021012,C9=2.97992391317725371012,C10=0.4418087268641012,C11=1.8087786626281012,C12=0.1116990016091012,C13=5.2934509552141012.

One gets approximate solutions from Eqs. (38) and (49), respectively: F¯(η)=12.3104494668η2+2.4868857696η4+0.3531718162η53.4153413394η6+1.7515474936η7+1.2031805064η81.6209066880η9+0.6391440832η100.0872321749η11,(50) θ¯(η)=59.560673288998(η21)9.116379402803(η31)142.753455187914(η41)+44.843714524611(η51)+168.833421156648(η61)18.843109698135(η71)183.399486372212(η81)+2.145923952284(η91)+122.114218185962(η101)36.681425691368(η121)0.061343981363(η131)+2.491809125293(η141)1012.(51)

Case 5.2 For α=π24,H=250, the parameters Ci are: C1=0.011565849071,C2=0.707159448177,C3=290.324617452708,C4=435.512207098156,C5=98.780455208880,C6=0.371954495218,C7=82.314218258861,C8=37.2855014590401012,C9=15.2534046781601012,C10=12.5728094369481012,C11=22.0832717731951012,C12=1.0903391984001012,C13=14.2411690610941012,

and therefore the approximate solutions (38) and (49) may be written as: F¯(η)=11.638305627622η2+1.206009669620η4+0.043648374556η51.078984595104η6+0.156296141929η7+0.738925954400η80.549972600570η9+0.122598968736η100.000216285946η11,(52) θ¯(η)=1249.857282929460(η21)9.116379418809(η31)956.378989402938(η41)+1351.297888810558(η51)1018.834483075130(η61)646.214253910126(η71)+1232.314945834321(η81)+129.244515099455(η91)589.041009734213(η101)+116.176788092689(η121)0.598803450274(η131)6.703807283538(η141)1012.(53)

Case 5.3 For α=π24,H=500 we obtain: F¯(η)=11.1724778890η2+0.4686048815η40.098961423266η50.3550702077η6+0.788918793543η71.8412450049η8+2.166714191416η91.2119558765η10+0.255472535060η11,(54) θ¯(η)=3025.399926380127(η21)9.116379434815(η31)3566.696711595512(η41)+4417.102370290691(η51)2124.393833887592(η61)2172.379321553169(η71)+2808.629798261231(η81)+471.928890384626(η91)1267.595946519067(η101)+212.551207395542(η121)1.000321320275(η131)11.126037049812(η141)1012.(55)

Case 5.4 For α=π24,H=1000, it holds that: F¯(η)=10.6223664982η20.1660109481η40.2259232591η5+0.415889430872η60.5758423027η7+0.198748794050η8+0.0038449003η9+0.029311285675η100.0576514026η11,(56) θ¯(η)=8164.566371333924(η21)9.116379466826(η31)12276.720695370957(η41)+13537.207073598438(η51)5217.969886112186(η61)6768.312414390118(η71)+8081.157802168993(η81)+1537.402367705613(η91)3743.086450834657(η101)+613.584536626787(η121)1.303403201992(η131)31.505440255694(η141)1012.(57)

Case 5.5 For α=π36,H=0, we obtain F¯(η)=11.7695466647η2+1.2754140854η4+0.1103023597η51.1550256736η6+0.4434051824η7+0.3358564399η80.3325192185η9+0.1045556849η100.0124421957η11,(58) θ¯(η)=111.208165621670(η21)6.895054111348(η31)219.101666428999(η41)+112.000412353392(η51)+120.895605108821(η61)47.637714830801(η71)104.355375516421(η81)+6.984672282609(η91)+75.737792150272(η101)24.982664253556(η121)0.097447216002(η131)+1.768576986049(η141)1012.(59)

Case 5.6 If α=π36,H=250 then F¯(η)=11.5111014276η2+2876996η4+0.0128407190η568759509η6+0.0367598458η7+198667899η80.1656253347η9+019977968η10+0.0108498620η11,(60) θ¯(η)=5094.867741624686(η21)6.895054147361(η31)4500.773385702808(η41)+6249.12145418685(η51)4283.444052831423(η61)2834.31757828637(η71)+5102.615992457279(η81)+528.701361603855(η91)2382.96513866767(η101)+460.279590044557(η121)2.850139405026(η131)26.598784857626(η141)1012.(61)

Case 5.7 For α=π36,H=500 the approximate solutions are F¯(η)=11.2942214724η2+334630869η4+0.0019708504η50.3330602404η60.0197377497η7+0.2108709627η80.1151093836η9+0.0147650020η10+0.0010589440η11,(62) θ¯(η)=10479.441564298402(η21)6.895054183374(η31)9629.602717192312(η41)+12532.057010176612(η51)7708.77471180501(η61)5687.792565325205(η71)+9313.682156269786(η81)+1063.274966246270(η91)4206.734533154069(η101)+761.164805945711(η121)5.659649234867(η131)42.382058179415(η141)1012.(63)

Case 5.8 If α=π36,H=1000 then F¯(η)=10.9581900583η2+0.0913634769η4+0.0030073285η50.1611067985η6+0.0668486226η70.0696973759η8+0.0454374262η90.0111066789η100.0065559425η11,(64) θ¯(η)=22457.262743604428(η21)6.895054255400(η31)22247.17539598977(η41)+26605.166448464304(η51)14220.534699997283(η61)12131.493178637833(η71)+17921.197074275697(η81)+2301.819882082563(η91)7836.564793794578(η101)+1301.948445444968(η121)11.186604132870(η131)68.576887703688(η141)1012.(65)

From Tables 116, it is obvious that the second-order approximate solutions obtained by OHPM are of a high accuracy in comparison with the homotopy perturbation method and with numerical solution obtained by means of a fourth-order Runge-Kutta method in combination with the shooting method using Wolfram Mathematica 6.0 software.

Table 1

Comparison between HPM results [9], OHPM results (50) and numerical results for the velocity F(η) for α=π24andH=0

Table 2

Comparison between OHPM results (51) and numerical results for the temperature θ(η) for α=π24andH=0

Table 3

Comparison between HPM results [9], OHPM results (52) and numerical results for the velocity F(η) for α=π24andH=250

Table 4

Comparison between OHPM results (53) and numerical results for the temperature θ(η) for α=π24andH=250

Table 5

Comparison between HPM results [9], OHPM results (54) and numerical results for the velocity F(η) for α=π24andH=500

Table 6

Comparison between OHPM results (55) and numerical results for the temperature θ(η) for α=π24andH=500

Table 7

Comparison between HPM results [9], OHPM results (56) and numerical results for the velocity F(η) for α=π24andH=1000

Table 8

Comparison between OHPM results (57) and numerical results for the temperature θ(η) for α=π24andH=1000

Table 9

Comparison between OHPM results (58) and numerical results for the velocity F(η) for α=π36andH=0

Table 10

Comparison between OHPM results (59) and numerical results for the temperature θ(η) for α=π36andH=0

Table 11

Comparison between OHPM results (60) and numerical results for the velocity F(η) for α=π36andH=250

Table 12

Comparison between OHPM results (61) and numerical results for the temperature θ(η) for α=π36andH=250

Table 13

Comparison between OHPM results (62) and numerical results for the velocity F(η) for α=π36andH=500

Table 14

Comparison between OHPM results (63) and numerical results for the temperature θ(η) for α=π36andH=500

Table 15

Comparison between OHPM results (64) and numerical results for the velocity F(η) for α=π36andH=1000

Table 16

Comparison between OHPM results (65) and numerical results for the temperature θ(η) for α=π36andH=1000

In Figs.2 and 3 are presented the effect of the Hartmann number on the velocity profile for Re = 50 and α=π24andα=π36, respectively. It is observed that velocity increases with increase of the Hartmann number for any value of α. The same effect of Hartmann number on the thermal profile are presented in Figs. 4 and 5 for α=π24andα=π36, respectively. In this case, the temperature decreases with increase of the Hartmann number in both cases. The effect of the half angle α on the velocity profile is presented in Figs. 69. With increasing value of α, velocity decreases for H = 0 and H = 250, but increases for H = 500 and H =1000. From Figs. 1013, it is interesting to remark that the temperature increases as the half angle α increases. In all cases, the maximum temperature occurs near the walls for H = 0 and precisely at the wall for H ≠ 0, while the minimum occurs near the channel axis.

Effect of the Hartmann number on the velocity profile for α = π/24, Re = 50: —— numerical solution, ……‥ OHPM solution
Figure 2

Effect of the Hartmann number on the velocity profile for α = π/24, Re = 50: —— numerical solution, ……‥ OHPM solution

Effect of the Hartmann number on the velocity profile for α = π/36, Re = 50: —— numerical solution, ……‥ OHPM solution
Figure 3

Effect of the Hartmann number on the velocity profile for α = π/36, Re = 50: —— numerical solution, ……‥ OHPM solution

Effect of the Hartmann number on the thermal profile for α = π/24, Re = 50: —— numerical solution, ……‥ OHPM solution
Figure 4

Effect of the Hartmann number on the thermal profile for α = π/24, Re = 50: —— numerical solution, ……‥ OHPM solution

Effect of the Hartmann number on the thermal profile for α = π/36, Re = 50: —— numerical solution, ……‥ OHPM solution
Figure 5

Effect of the Hartmann number on the thermal profile for α = π/36, Re = 50: —— numerical solution, ……‥ OHPM solution

Velocity profile for α = π/36 and α = π/24, H = 0, Re = 50, Eqs. (50) and (58): —— numerical solution, ……‥ OHPM solution
Figure 6

Velocity profile for α = π/36 and α = π/24, H = 0, Re = 50, Eqs. (50) and (58): —— numerical solution, ……‥ OHPM solution

Velocity profile for α = π/36 and α = π/24, H = 250, Re = 50, Eqs. (52) and (60): —— numerical solution, ……‥ OHPM solution
Figure 7

Velocity profile for α = π/36 and α = π/24, H = 250, Re = 50, Eqs. (52) and (60): —— numerical solution, ……‥ OHPM solution

Velocity profile for α = π/36 and α = π/24, H = 500, Re = 50, Eqs. (54) and (62): —— numerical solution, ……‥ OHPM solution
Figure 8

Velocity profile for α = π/36 and α = π/24, H = 500, Re = 50, Eqs. (54) and (62): —— numerical solution, ……‥ OHPM solution

Velocity profile for α = π/36 and α = π/24, H = 1000, Re = 50, Eqs. (56) and (64): —— numerical solution, ……‥ OHPM solution
Figure 9

Velocity profile for α = π/36 and α = π/24, H = 1000, Re = 50, Eqs. (56) and (64): —— numerical solution, ……‥ OHPM solution

Thermal profile for α = π/36 and α = π/24, H = 0, Re = 50, Eqs. (51) and (59): —— numerical solution, ……‥ OHPM solution
Figure 10

Thermal profile for α = π/36 and α = π/24, H = 0, Re = 50, Eqs. (51) and (59): —— numerical solution, ……‥ OHPM solution

Thermal profile for α = π/36 and α = π/24, H = 250, Re = 50, Eqs. (53) and (61): —— numerical solution, ……‥ OHPM solution
Figure 11

Thermal profile for α = π/36 and α = π/24, H = 250, Re = 50, Eqs. (53) and (61): —— numerical solution, ……‥ OHPM solution

Thermal profile for α = π/36 and α = π/24, H = 500, Re = 50, Eqs. (55) and (63): —— numerical solution, ……‥ OHPM solution
Figure 12

Thermal profile for α = π/36 and α = π/24, H = 500, Re = 50, Eqs. (55) and (63): —— numerical solution, ……‥ OHPM solution

Thermal profile for α = π/36 and α = π/24, H = 1000, Re = 50, Eqs. (57) and (65): —— numerical solution, ……‥ OHPM solution
Figure 13

Thermal profile for α = π/36 and α = π/24, H = 1000, Re = 50, Eqs. (57) and (65): —— numerical solution, ……‥ OHPM solution

6 Conclusions

In this paper, the Optimal Homotopy Perturbation Method (OHPM) is employed to propose a new analytic approximate solution for the MHD Jeffery-Hamel flow with heat transfer problem. Our procedure does not need restrictive hypotheses, is very rapidly convergent after only two iterations with the convergence of the solutions ensured in a rigorous way. The cornerstone of the validity and flexibility of our procedure is the choice of the linear operator and the optimal auxiliary functions which contribute to very accurate solutions. The parameters which are involved in the composition of the optimal auxiliary functions are optimally identified via various methods in a rigorous way. Our technique is very effective, explicit, and easy to apply—which proves that this method is very efficient in practice.

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About the article

Received: 2016-08-29

Accepted: 2016-12-06

Published Online: 2017-03-15


Conflict of interestConflict of Interests: The authors declare that there is no conflict of interests regarding the publication of this paper.


Citation Information: Open Physics, Volume 15, Issue 1, Pages 42–57, ISSN (Online) 2391-5471, DOI: https://doi.org/10.1515/phys-2017-0006.

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© 2017 V. Marinca and R.-D. Ene. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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