Show Summary Details
More options …

# Open Physics

### formerly Central European Journal of Physics

Editor-in-Chief: Seidel, Sally

Managing Editor: Lesna-Szreter, Paulina

IMPACT FACTOR 2018: 1.005

CiteScore 2018: 1.01

SCImago Journal Rank (SJR) 2018: 0.237
Source Normalized Impact per Paper (SNIP) 2018: 0.541

ICV 2017: 162.45

Open Access
Online
ISSN
2391-5471
See all formats and pricing
More options …
Volume 15, Issue 1

# 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:
/ 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.

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.

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]: $1r∂∂rrur+1r∂∂rruφ=0,$(1) $ur∂ur∂r+uφr∂ur∂φ−uφ2r=−1ρ∂P∂r+ν1r∂(rεrr)∂r+1r∂εrφ∂r−εrφr−σB02ρr2ur,$(2) $ur∂uφ∂r+uφr∂uφ∂φ−uφurr=−1ρr∂P∂φ+ν1r2∂(rεrφ)∂r+1r∂εφφ∂φ−εrφr−σB02ρr2uφ,$(3) $ur∂T∂r=kρcp∇2T+νcp2∂ur∂r2+urr2+1r∂ur∂r2+σB02ρr2ur2,$(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 $εrr=2∂ur∂r−23divu¯,$(5) $εφφ=2r∂uφ∂φ+2urr−23divu¯,$(6) $εrφ=2r∂ur∂φ+2∂∂ruφr.$(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: $1r∂∂rrur=0,$(8) $ur∂ur∂r=−1ρ∂P∂r+ν∇2ur−urr2−σB02ρr2ur,$(9) $−1ρr∂P∂φ+2νr2∂ur∂φ=0.$(10)

The relevant boundary conditions, due to the symmetry assumption at the channel centerline, are as follows: $∂ur∂φ=∂T∂φ=0,ur=ucratφ=0,$(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 $F‴+2αReFF′+(4−H)α2F′=0,$(16) $θ″+2α2α+RePrFθ+βPr(H+4α2)F2+F′2=0,$(17)

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

where $Re=\frac{\alpha {u}_{c}}{\nu }$ is the Reynolds number, $H=\sqrt{\frac{\sigma {B}_{0}^{2}}{\rho \nu }}$ is the Hartmann number, $Pr=\frac{\nu {c}_{p}}{k\rho },\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\beta =\frac{{u}_{c}}{{c}_{p}}$and 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 $Bu,∂u∂η=0,η∈Γ,$(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: $Nu¯,u¯′,u¯″,u¯‴,η=Nu0,u0′,u0″,u0‴,η+pu1Nu¯u0,u0′,u0″,u0‴,η++u1′Nu¯′u0,u0′,u0″,u0‴,η+u1″Nu¯″u0,u0′,u0″,u0‴,η++u1‴Nu¯‴u0,u0′,u0″,u0‴,η+p2u2Nu¯+u2′Nu¯′+…,$(25)

where ${F}_{\overline{u}}=\frac{\mathrm{\partial }F}{\mathrm{\partial }\overline{u}}.$ 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: $Hu¯,p=Lu¯,u¯′,u¯″,u¯‴,η+g(η)+pH0(η,Ck)Nu0,u0′,u0″,u0‴,η+p2H1(η,Ck)u1Nu¯u0+H2(η,Ck)u1′Nu¯′u0+H3(η,Ck)u1″Nu¯″u0+H4(η,Ck)u1‴Nu¯‴u0+p2H5(η,Ck)u2Nu¯u0+H6(η,Ck)u2′Nu¯′u0+….$(26)

Equating the coefficients of like powers of p yields the linear equations: $Lu0,u0′,u0″,u0‴,η+g(η)=0,Bu0,∂u0∂η=0,$(27) $Lu1+H0(η,Ck)Nu0,u0′,u0″,u0‴,η=0,Bu1,∂u1∂η=0,$(28) $Lu2+H1(η,Ck)u1Nu¯u0,u0′,u0″,u0‴,η+H2(η,Ck)u1′Nu¯′u0+H3(η,Ck)u1″Nu¯″u0+H4(η,Ck)u1‴Nu¯‴u0=0,Bu2,∂u2∂η=0.$(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 $F0(η)=1−η2.$(31)

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

By substituting Eq. (31) into the nonlinear operator N and into Eq. (32), one retrieves $NF0=2Aη2−2(A+B)η,NFF0=−2Aη,NF′F0=−Aη2+A+B,$(33)

where A = 2αRe, B = (4 − H)α2.

Eq. (28) becomes $F1‴+H0(η,Ck)[2Aη2−2(A+B)η]=0,F1(0)=F1′(0)=F1(1)=0.$(34)

We choose H0(η, Ck) = −60C1 where C1 is an unknown parameter and from Eq. (34) we obtain $F1(η)=2AC1η5−5(A+B)C1η4+(3A+5B)C1η2.$(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 $F2(η)=AC1C2495η11+2AC1C3−5(A+B)C1C2720η10+2AC1C4−5(A+B)C1C3+5A2C1C6504η9+(3A+5B)C1C2−5(A+B)C1C4+2AC1C5−10(A2+AB)C1C6336η8+(3A+5B)C1C3−5(A+B)C1C5−5(A2+AB)C1C6+2AC1C7210η7+(3A+5B)C1C4+(13A2+25AB+10B2)C1C6−5(A+B)C1C7120η6+(3A+5B)C1C560η5−(3A2+8AB+5B2)C1C6+(3A+5B)C1C724η4+Mη2,$(37)

where $M=−C1C21495−1144+1112A+5336−1144B−C1C31360−5504+170A+142−5504B−C1C713A140+B6−C1C41252−5336+140A+124−5336B−C1C59A280+5B84−C1C65504−5168−5210−160A2−5168+5210+18AB−18B2.$

For p = 1 in Eq. (24), we obtain the second-order approximate solution, using Eqs. (31), (35) and (37): $F¯(η)=F0(η)+F1(η)+F2(η)=AC1C2495η11+2AC1C3−5(A+B)C1C2720η10+2AC1C4−5(A+B)C1C3+5A2C1C6504η9+(3A+5B)C1C2−5(A+B)C1C4+2AC1C5−10(A2+AB)C1C6336η8+(3A+5B)C1C3−5(A+B)C1C5−5(A2+AB)C1C6+2AC1C7210η7+(3A+5B)C1C4+(13A2+25AB+10B2)C1C6−5(A+B)C1C7120η6+2AC1+(3A+5B)60C1C5η5−(3A2+8AB+5B2)C1C6+(3A+5B)C1C724+(5A+5B)C1η4+3A+5B+M−1η2.$(38)

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

Eq. (27) becomes $θ0″−1=0,θ0(1)=0,θ0′(0)=0.$(40)

Eq. (40) has the solution $θ0(η)=12(1−η2).$(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 $N(θ0)=C−2Dη2+Eη4,Nθ(θ0)=L+Kη2,$(43)

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

Eq. (28) can be written $θ1″+h0(η,C8)(C−2Dη2+Eη4)=0,θ1(1)=θ1′(0)=0.$(45)

Choosing h0(η, C8) = −30C8 in Eq. (45), one obtains $θ1(η)=C815C(η2−1)−5D(η4−1)+E(η6−1).$(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 $θ2(η)=−L(15C−5D+E)⋅12C9(η2−1)+16C10(η3−1)+15LCC9−(15C−5D+E)(KC9+LC11)η4−112+15LCC10−(15C−5D+E)(KC10+LC12)η5−120+(15CK−5DK)C9+15LCC11−(15C−5D+E)(KC11+LC13)η6−130+(15CK−5DL)C10+15LCC12−K(15C−5D+E)C12⋅η7−142+(LE−5DK)C9+(15CK−5DL)C11+15LCC13−−K(15C−5D+E)C13η8−156+(LE−5DK)C10+(15CK−5DL)C12η9−172+EKC9+(LE−5DK)C11+(15CK−5DL)C13η10−190+EKC11+(LE−5DK)C13η12−1132+EKC12η13−1156+EKC13η14−1182.$(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 $\alpha =\frac{\pi }{24}$ and H = 0. By means of the least-square method, the values of the parameters Ci, i = 1, 2, …, 13 are $C1=−0.000025737857,C2=128165.4247848388,C3=−360860.8730122449,C4=315974.73981422884,C5=−20823.768289134576,C6=−319.2089575339067,C7=−62701.61063351235,C8=−8.904447449602⋅10−12,C9=−2.9799239131772537⋅10−12,C10=0.441808726864⋅10−12,C11=−1.808778662628⋅10−12,C12=0.111699001609⋅10−12,C13=−5.293450955214⋅10−12.$

One gets approximate solutions from Eqs. (38) and (49), respectively: $F¯(η)=1−2.3104494668η2+2.4868857696η4+0.3531718162η5−3.4153413394η6+1.7515474936η7+1.2031805064η8−1.6209066880η9+0.6391440832η10−0.0872321749η11,$(50) $θ¯(η)=−59.560673288998(η2−1)−9.116379402803(η3−1)−142.753455187914(η4−1)+44.843714524611(η5−1)+168.833421156648(η6−1)−18.843109698135(η7−1)−183.399486372212(η8−1)+2.145923952284(η9−1)+122.114218185962(η10−1)−36.681425691368(η12−1)−0.061343981363(η13−1)+2.491809125293(η14−1)⋅10−12.$(51)

Case 5.2 For $\alpha =\frac{\pi }{24},H=250,$ the parameters Ci are: $C1=−0.011565849071,C2=0.707159448177,C3=−290.324617452708,C4=435.512207098156,C5=−98.780455208880,C6=−0.371954495218,C7=−82.314218258861,C8=−37.285501459040⋅10−12,C9=−15.253404678160⋅10−12,C10=12.572809436948⋅10−12,C11=−22.083271773195⋅10−12,C12=1.090339198400⋅10−12,C13=14.241169061094⋅10−12,$

and therefore the approximate solutions (38) and (49) may be written as: $F¯(η)=1−1.638305627622η2+1.206009669620η4+0.043648374556η5−1.078984595104η6+0.156296141929η7+0.738925954400η8−0.549972600570η9+0.122598968736η10−0.000216285946η11,$(52) $θ¯(η)=−1249.857282929460(η2−1)−9.116379418809(η3−1)−956.378989402938(η4−1)+1351.297888810558(η5−1)−1018.834483075130(η6−1)−646.214253910126(η7−1)+1232.314945834321(η8−1)+129.244515099455(η9−1)−589.041009734213(η10−1)+116.176788092689(η12−1)−0.598803450274(η13−1)−6.703807283538(η14−1)⋅10−12.$(53)

Case 5.3 For $\alpha =\frac{\pi }{24},H=500$ we obtain: $F¯(η)=1−1.1724778890η2+0.4686048815η4−0.098961423266η5−0.3550702077η6+0.788918793543η7−1.8412450049η8+2.166714191416η9−1.2119558765η10+0.255472535060η11,$(54) $θ¯(η)=−3025.399926380127(η2−1)−9.116379434815(η3−1)−3566.696711595512(η4−1)+4417.102370290691(η5−1)−2124.393833887592(η6−1)−2172.379321553169(η7−1)+2808.629798261231(η8−1)+471.928890384626(η9−1)−1267.595946519067(η10−1)+212.551207395542(η12−1)−1.000321320275(η13−1)−11.126037049812(η14−1)⋅10−12.$(55)

Case 5.4 For $\alpha =\frac{\pi }{24},H=1000,$ it holds that: $F¯(η)=1−0.6223664982η2−0.1660109481η4−0.2259232591η5+0.415889430872η6−0.5758423027η7+0.198748794050η8+0.0038449003η9+0.029311285675η10−0.0576514026η11,$(56) $θ¯(η)=−8164.566371333924(η2−1)−9.116379466826(η3−1)−12276.720695370957(η4−1)+13537.207073598438(η5−1)−5217.969886112186(η6−1)−6768.312414390118(η7−1)+8081.157802168993(η8−1)+1537.402367705613(η9−1)−3743.086450834657(η10−1)+613.584536626787(η12−1)−1.303403201992(η13−1)−31.505440255694(η14−1)⋅10−12.$(57)

Case 5.5 For $\alpha =\frac{\pi }{36},H=0,$ we obtain $F¯(η)=1−1.7695466647η2+1.2754140854η4+0.1103023597η5−1.1550256736η6+0.4434051824η7+0.3358564399η8−0.3325192185η9+0.1045556849η10−0.0124421957η11,$(58) $θ¯(η)=−111.208165621670(η2−1)−6.895054111348(η3−1)−219.101666428999(η4−1)+112.000412353392(η5−1)+120.895605108821(η6−1)−47.637714830801(η7−1)−104.355375516421(η8−1)+6.984672282609(η9−1)+75.737792150272(η10−1)−24.982664253556(η12−1)−0.097447216002(η13−1)+1.768576986049(η14−1)⋅10−12.$(59)

Case 5.6 If $\alpha =\frac{\pi }{36},H=250$ then $F¯(η)=1−1.5111014276η2+2876996η4+0.0128407190η5−68759509η6+0.0367598458η7+198667899η8−0.1656253347η9+019977968η10+0.0108498620η11,$(60) $θ¯(η)=−5094.867741624686(η2−1)−6.895054147361(η3−1)−4500.773385702808(η4−1)+6249.12145418685(η5−1)−4283.444052831423(η6−1)−2834.31757828637(η7−1)+5102.615992457279(η8−1)+528.701361603855(η9−1)−2382.96513866767(η10−1)+460.279590044557(η12−1)−2.850139405026(η13−1)−26.598784857626(η14−1)⋅10−12.$(61)

Case 5.7 For $\alpha =\frac{\pi }{36},H=500$ the approximate solutions are $F¯(η)=1−1.2942214724η2+334630869η4+0.0019708504η5−0.3330602404η6−0.0197377497η7+0.2108709627η8−0.1151093836η9+0.0147650020η10+0.0010589440η11,$(62) $θ¯(η)=−10479.441564298402(η2−1)−6.895054183374(η3−1)−9629.602717192312(η4−1)+12532.057010176612(η5−1)−7708.77471180501(η6−1)−5687.792565325205(η7−1)+9313.682156269786(η8−1)+1063.274966246270(η9−1)−4206.734533154069(η10−1)+761.164805945711(η12−1)−5.659649234867(η13−1)−42.382058179415(η14−1)⋅10−12.$(63)

Case 5.8 If $\alpha =\frac{\pi }{36},H=1000$ then $F¯(η)=1−0.9581900583η2+0.0913634769η4+0.0030073285η5−0.1611067985η6+0.0668486226η7−0.0696973759η8+0.0454374262η9−0.0111066789η10−0.0065559425η11,$(64) $θ¯(η)=−22457.262743604428(η2−1)−6.895054255400(η3−1)−22247.17539598977(η4−1)+26605.166448464304(η5−1)−14220.534699997283(η6−1)−12131.493178637833(η7−1)+17921.197074275697(η8−1)+2301.819882082563(η9−1)−7836.564793794578(η10−1)+1301.948445444968(η12−1)−11.186604132870(η13−1)−68.576887703688(η14−1)⋅10−12.$(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 $\alpha =\frac{\pi }{24}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}H=0$

Table 2

Comparison between OHPM results (51) and numerical results for the temperature θ(η) for $\alpha =\frac{\pi }{24}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}H=0$

Table 3

Comparison between HPM results [9], OHPM results (52) and numerical results for the velocity F(η) for $\alpha =\frac{\pi }{24}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}H=250$

Table 4

Comparison between OHPM results (53) and numerical results for the temperature θ(η) for $\alpha =\frac{\pi }{24}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}H=250$

Table 5

Comparison between HPM results [9], OHPM results (54) and numerical results for the velocity F(η) for $\alpha =\frac{\pi }{24}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}H=500$

Table 6

Comparison between OHPM results (55) and numerical results for the temperature θ(η) for $\alpha =\frac{\pi }{24}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}H=500$

Table 7

Comparison between HPM results [9], OHPM results (56) and numerical results for the velocity F(η) for $\alpha =\frac{\pi }{24}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}H=1000$

Table 8

Comparison between OHPM results (57) and numerical results for the temperature θ(η) for $\alpha =\frac{\pi }{24}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}H=1000$

Table 9

Comparison between OHPM results (58) and numerical results for the velocity F(η) for $\alpha =\frac{\pi }{36}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}H=0$

Table 10

Comparison between OHPM results (59) and numerical results for the temperature θ(η) for $\alpha =\frac{\pi }{36}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}H=0$

Table 11

Comparison between OHPM results (60) and numerical results for the velocity F(η) for $\alpha =\frac{\pi }{36}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}H=250$

Table 12

Comparison between OHPM results (61) and numerical results for the temperature θ(η) for $\alpha =\frac{\pi }{36}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}H=250$

Table 13

Comparison between OHPM results (62) and numerical results for the velocity F(η) for $\alpha =\frac{\pi }{36}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}H=500$

Table 14

Comparison between OHPM results (63) and numerical results for the temperature θ(η) for $\alpha =\frac{\pi }{36}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}H=500$

Table 15

Comparison between OHPM results (64) and numerical results for the velocity F(η) for $\alpha =\frac{\pi }{36}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}H=1000$

Table 16

Comparison between OHPM results (65) and numerical results for the temperature θ(η) for $\alpha =\frac{\pi }{36}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}H=1000$

In Figs.2 and 3 are presented the effect of the Hartmann number on the velocity profile for Re = 50 and $\alpha =\frac{\pi }{24}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\alpha =\frac{\pi }{36},$ 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 $\alpha =\frac{\pi }{24}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\alpha =\frac{\pi }{36},$ 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.

Figure 2

Effect of the Hartmann number on the velocity profile for α = π/24, 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

Figure 4

Effect of the Hartmann number on the thermal profile for α = π/24, 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

Figure 6

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

Figure 8

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

Figure 10

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

Figure 12

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

## References

• [1]

Jeffery G. B., The two-dimensional steady motion of a viscous fluid, Philos. Mag. A, 1915, 6(20), 455–465. Google Scholar

• [2]

Hamel G., Spiralförmige bewegungen zäher flussigkeiten, Jahresbericht der Deutschen Mathematiker-Vereinigung, 1916, 25, 34–60. Google Scholar

• [3]

Rivkind L., Solonnikov V. A., Jeffery-Hamel asymptotics for steady state Navier-Stokes flow in domains with sector-like outlets to infinity, J. Math. Fluid Mech., 2000, 2, 324–352.

• [4]

Akulenko L. D., Georgevskii D. V., Kumakshev S. A., Solutions of the Jeffery-Hamel problem regularly extendable in the Reynolds number, Fluid Dyn., 2004, 39(1), 12–28.

• [5]

Makinde O. D., Mhone P. Y., Hermite-Padé approximation approach to MHD Jeffery-Hamel flows, Appl.Math. Comput., 2006, 181, 966–972. Google Scholar

• [6]

Esmaili Q., Ramiar A., Alizadeh E., Ganji D. D., An approximation of the analytical solution of the Jeffery-Hamel flow by decomposition method, Phys. Lett., 2008, 372, 3434–3439.

• [7]

Ganji Z. Z., Ganji D. D., Esmaeilpour M., Study of nonlinear Jeffery-Hamel flow by He’s semi-analytical methods and comparison with numerical results, Comput. Math. Appl., 2009, 58, 2107–2116.

• [8]

Joneidi A. A., Domairry G., Babaelahi M., Three analytical methods applied to Jeffery-Hamel flow, Commun. Nonlinear Sci. Numer. Simul., 2010, 15, 3423–3434.

• [9]

Moghimi S. M., Ganji D. D., Bararnia H., Hosseini M., Jalbal M., Homotopy perturbation method for nonlinear MHD Jeffery-Hamel problem, Comput. Math. Appl., 2011, 61, 2213–2216.

• [10]

Marinca V., Herisanu N., An optimal homotopy asymptotic approach applied to nonlinear MHD Jeffery-Hamel flow, Math. Probl. Eng., 2011, Article ID 169056, 16 pages.

• [11]

Esmaeilpour M., Ganji D. D., Solution of the Jeffery-Hamel flow problem by optimal homotopy asymptotic method, Comput. Math. Appl., 2010, 59, 3405–3411.

• [12]

Sheikholeshami M., Ganji D. D., Ashorynejad H. R., Rokni H. B., Analytical investigation of Jeffery-Hamel flow with high magnetic fiwld and nanoparticle by Adomian decomposition method, Appl. Math. Mech.-Engl., 2012, 33(1), 25–36.

• [13]

Rostami A. K., Akbari M. R., Ganji D. D., Heydari S., Investigating Jeffery-Hamel flow with high-magnetic field and nanoparticles by HPM and AGM, Cent. Eur. J. Eng., 2014, 4(4), 357–370. Google Scholar

• [14]

Raja M. A. Z., Samar R., Numerical treatment of nonlinear MHD Jeffery-Hamel problems using stochastic algorithms, Comput. Fluids, 2014, 91, 111–115.

• [15]

Yang X.-J., Tenreiro Machado J. A., Baleanu D., Cattani C., On exact traveling-wave solutions for local fractional Korteweg-de Vries equation, Chaos, 2016, 26, Article ID 084312.

• [16]

Yang X.-J., Tenreiro Machado J. A., Hristov J., Nonlinear dynamics for local fractional Burgers’ equation arising in frctal flow, Nonlinear Dynam., 2016, 84, 3–7.

• [17]

Jafari H., Ghorbani M., Ebadattalab E., Moallem R., Baleanu D., Optimal homotopy asymptotic method - a tool for solving fuzzy differential equations, J. Comput. Complex. Appl., 2016, 2(4), 112–123. Google Scholar

• [18]

Kakuda K., Tosaka N., The generalized boundary element approach to Burgers’ equation, Int. J. Numer. Meth. Eng., 1990, 29(2), 245–261.

• [19]

Marinca V., Herişanu N., Nonlinear Dynamical Systems in Engineering - Some Approximate Approaches, Springer Verlag, Heidelberg, 2011. Google Scholar

• [20]

Marinca V., Herişanu N., Nonlinear dynamic analysis of an electrical machine rotor-bearing system by the optimal homotopy perturbation method, Comput. Math. Appl., 2011, 61, 2019’ 2024.

• [21]

Herişanu N., Marinca V., Optimal homotopy perturbation method for a non-conservative dynamical system of a rotating electrical machine, Z. Naturforsch A, 2012, 67a, 509–516.

• [22]

Schlichting H., Boundary-layer theory, Mc Graw-Hill Book Company, 1979. Google Scholar

• [23]

Ali F. M., Nazar R., Arifin N. M., Pop I., MHD stagnation-point flow and heat transfer towards a stretching sheet with induced magnetic field, Appl. Math. Mech.-Engl., 2011, 32, 409–418.

• [24]

Choi S. H., Wilhelm H.-E., Incompressible magnetohydrodynamic flow with heat transfer between inclined walls, Phys. Fluids, 1979, 22, 1073–1078.

• [25]

Turkylmazoglu M., Extending the traditional Jeffery-Hamel flow to strechable convergent / divergent channels, Comput. Fluids, 2014, 100, 196–203.

• [26]

He J. H., A coupling method of homotopy technique and perturbation technique for nonlinear problems, Int. J. Nonlin. Mech., 2000, 35, 37–43.

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,

Export Citation