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

  • Vasile Marinca and Remus-Daniel Ene EMAIL logo
Published/Copyright: March 15, 2017
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Open Physics
From the journal Open Physics Volume 15 Issue 1

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) [19–21]. 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.
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 [22–25]:

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=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,∂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 [19–21]. 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 Fu¯=∂F∂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 Hi ⋅ ujNu 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)α2F′, 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 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=−8.904447449602⋅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 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=−37.285501459040⋅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 α=π24,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 α=π24,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 α=π36,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 α=π36,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 α=π36,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 α=π36,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 1–16, 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

ηFHPM[9]FnumericF¯OHPM,Eq. (50)relative error = |Fnumeric−F¯OHPM|
01110
0.10.97707110.97714260470.97714449591.8 ⋅ 10−6
0.20.91120200.91147922780.91148020549.7 ⋅ 10−7
0.30.81041150.81100524030.81100546572.2 ⋅ 10−7
0.40.68592300.68691482200.68691630341.4 ⋅ 10−6
0.50.54984270.55128832120.55128953021.2 ⋅ 10−6
0.60.41316980.41510889470.41510917402.7 ⋅ 10−7
0.70.28460240.28705463200.28705549988.6 ⋅ 10−7
0.80.17027910.17312213900.17312315211.01 ⋅ 10−6
0.90.07442320.07687157560.07687210585.3 ⋅ 10−7
10000
Table 2

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

ηθnumericθ¯OHPM from Eq. (51)relative error = |θnumeric−θ¯OHPM|
0-9.134405103300 ⋅ 10−12-9.134559900000 ⋅ 10−121.5 ⋅ 10−16
0.1-8.553981690585 ⋅ 10−12-8.561731325450 ⋅ 10−127.7 ⋅ 10−15
0.2-7.029011445513 ⋅ 10−12-7.029011445531 ⋅ 10−121.7 ⋅ 10−23
0.3-4.968059593695 ⋅ 10−12-4.959904647105 ⋅ 10−128.1 ⋅ 10−15
0.4-2.830354806908 ⋅ 10−12-2.830354806921 ⋅ 10−121.3 ⋅ 10−23
0.5-1.003581673783 ⋅ 10−12-1.015625108634 ⋅ 10−121.2 ⋅ 10−14
0.62.755715234490 ⋅ 10−132.755715234410 ⋅ 10−137.9 ⋅ 10−24
0.79.398183452650 ⋅ 10−139.682947296376 ⋅ 10−132.8 ⋅ 10−14
0.81.062337244632 ⋅ 10−121.062337244629 ⋅ 10−123.3 ⋅ 10−24
0.97.684282928672 ⋅ 10−136.524090716991 ⋅ 10−131.1 ⋅ 10−13
1000
Table 3

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

ηFHPM[9]FnumericF¯OHPM,Eq. (52)relative error = |Fnumeric−F¯OHPM|
01110
0.10.98373400.98373677910.98373692461.4 ⋅ 10−7
0.20.93633500.93634599480.93634592606.8 ⋅ 10−8
0.30.86168940.86171335810.86171321891.3 ⋅ 10−7
0.40.76534050.76538147530.76538139807.7 ⋅ 10−8
0.50.65339610.65345730870.65345702262.8 ⋅ 10−7
0.60.53146210.53154666090.53154629753.6 ⋅ 10−7
0.70.40381300.40392273540.40392242053.1 ⋅ 10−7
0.80.27287080.27299803170.27299751305.1 ⋅ 10−7
0.90.13894330.13904160790.13904110705.0 ⋅ 10−7
10000
Table 4

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

ηθnumericθ¯OHPM from Eq. (53)relative error = |θnumeric−θ¯OHPM|
0-8.520036945014 ⋅ 10−10-8.520036944914 ⋅ 10−109.9 ⋅ 10−21
0.1-8.394493067234 ⋅ 10−10-8.395974340239 ⋅ 10−101.4 ⋅ 10−13
0.2-8.032504411694 ⋅ 10−10-8.032504411616 ⋅ 10−107.8 ⋅ 10−21
0.3-7.450795118612 ⋅ 10−10-7.450297754118 ⋅ 10−104.9 ⋅ 10−14
0.4-6.677062404515 ⋅ 10−10-6.677062404494 ⋅ 10−102.1 ⋅ 10−21
0.5-5.746573136109 ⋅ 10−10-5.746735798562 ⋅ 10−101.6 ⋅ 10−14
0.6-4.698328124476 ⋅ 10−10-4.698328124489 ⋅ 10−101.3 ⋅ 10−21
0.7-3.570882806332 ⋅ 10−10-3.571240799348 ⋅ 10−103.5 ⋅ 10−14
0.8-2.398013591946 ⋅ 10−10-2.398013591937 ⋅ 10−109.8 ⋅ 10−22
0.9-1.206024561793 ⋅ 10−10-1.200670094842 ⋅ 10−105.3 ⋅ 10−13
1000
Table 5

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

ηFHPM[9]FnumericF¯OHPM,Eq. (54)relative error = |Fnumeric−F¯OHPM|
01110
0.10.98831970.98831966680.98832079941.1 ⋅ 10−6
0.20.95379550.95379524790.95380263517.3 ⋅ 10−6
0.30.89785150.89785104380.89786104309.9 ⋅ 10−6
0.40.82246990.82246904390.82246960005.5 ⋅ 10−7
0.50.72968170.72968034250.72967193658.4 ⋅ 10−6
0.60.62097480.62097285970.62097068812.1 ⋅ 10−6
0.70.49666440.49666178660.49667001568.2 ⋅ 10−6
0.80.35521150.35520791130.35520972611.8 ⋅ 10−6
0.90.19238210.19237752150.19236836429.1 ⋅ 10−6
10000
Table 6

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

ηθnumericθ¯OHPM from Eq. (55)relative error = |θnumeric−θ¯OHPM|
0-1.783303641361 ⋅ 10−9-1.783303641351 ⋅ 10−99.9 ⋅ 10−21
0.1-1.753016328789 ⋅ 10−9-1.753373570315 ⋅ 10−93.5 ⋅ 10−13
0.2-1.666810282194 ⋅ 10−9-1.666810282186 ⋅ 10−97.8 ⋅ 10−21
0.3-1.531366088677 ⋅ 10−9-1.531258066872 ⋅ 10−91.08 ⋅ 10−13
0.4-1.356325163065 ⋅ 10−9-1.356325163062 ⋅ 10−92.1 ⋅ 10−21
0.5-1.152391353408 ⋅ 10−9-1.152436518900 ⋅ 10−94.5 ⋅ 10−14
0.6-9.301091200084 ⋅ 10−10-9.301091200095 ⋅ 10−101.1 ⋅ 10−21
0.7-6.989817680976 ⋅ 10−10-6.987973346248 ⋅ 10−101.8 ⋅ 10−13
0.8-4.652100850091 ⋅ 10−10-4.652100850080 ⋅ 10−101.1 ⋅ 10−21
0.9-2.319273360260 ⋅ 10−10-2.321569863571 ⋅ 10−102.2 ⋅ 10−13
1000
Table 7

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

ηFHPM[9]FnumericF¯OHPM,Eq. (56)relative error = |Fnumeric−F¯OHPM|
01110
0.10.99376070.99376114130.99375783493.3 ⋅ 10−6
0.20.97478860.97479053770.97478718583.3 ⋅ 10−6
0.30.94227940.94228386430.94228376619.8 ⋅ 10−8
0.40.89474310.89475135410.89474996421.3 ⋅ 10−6
0.50.82974710.82976056180.82975643994.1 ⋅ 10−6
0.60.74347560.74349599280.74349412811.8 ⋅ 10−6
0.70.62998660.63001641900.63001679363.7 ⋅ 10−7
0.80.47993380.47997572860.47997248423.2 ⋅ 10−6
0.90.27827890.27832978890.27832805911.7 ⋅ 10−6
10000
Table 8

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

ηθnumericθ¯OHPM from Eq. (57)relative error = |θnumeric−θ¯OHPM|
0-3.885903481818 ⋅ 10−9-3.885903481801 ⋅ 10−91.7 ⋅ 10−20
0.1-3.804390715058 ⋅ 10−9-3.805365047292 ⋅ 10−99.7 ⋅ 10−13
0.2-3.575104095175 ⋅ 10−9-3.575104095175 ⋅ 10−95.7 ⋅ 10−23
0.3-3.222962704714 ⋅ 10−9-3.222630117770 ⋅ 10−93.3 ⋅ 10−13
0.4-2.782964569511 ⋅ 10−9-2.782964569511 ⋅ 10−94.4 ⋅ 10−23
0.5-2.293213461698 ⋅ 10−9-2.293504856351 ⋅ 10−92.9 ⋅ 10−13
0.6-1.790055410191 ⋅ 10−9-1.790055410191 ⋅ 10−93.5 ⋅ 10−23
0.7-1.302936492610 ⋅ 10−9-1.301687731203 ⋅ 10−91.2 ⋅ 10−12
0.8-8.442618285524 ⋅ 10−10-8.442618285525 ⋅ 10−101.4 ⋅ 10−23
0.9-4.107432220830 ⋅ 10−10-4.158108458702 ⋅ 10−105.06 ⋅ 10−12
1000
Table 9

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

ηFnumericF¯OHPM,Eq. (58)relative error = |Fnumeric−F¯OHPM|
0110
0.10.98243123640.98243207018.3 ⋅ 10−7
0.20.93122595770.93122654665.8 ⋅ 10−7
0.30.85061061610.85061073391.1 ⋅ 10−7
0.40.74679080180.74679150086.9 ⋅ 10−7
0.50.62694816820.62694900728.3 ⋅ 10−7
0.60.49823444640.49823474632.9 ⋅ 10−7
0.70.36696633860.36696680374.6 ⋅ 10−7
0.80.23812374630.23812456868.2 ⋅ 10−7
0.90.11515193120.11515239534.6 ⋅ 10−7
1000
Table 10

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

ηθnumericθ¯OHPM from Eq. (59)relative error = |θnumeric−θ¯OHPM|
0-2.552530215568 ⋅ 10−11-2.552530214568 ⋅ 10−119.9 ⋅ 10−21
0.1-2.442980325655 ⋅ 10−11-2.444079060328 ⋅ 10−111.09 ⋅ 10−14
0.2-2.143998685224 ⋅ 10−11-2.143998685224 ⋅ 10−111.9 ⋅ 10−25
0.3-1.714518106214 ⋅ 10−11-1.713385692946 ⋅ 10−111.1 ⋅ 10−14
0.4-1.227729149275 ⋅ 10−11-1.227729149275 ⋅ 10−111.3 ⋅ 10−25
0.5-7.579058130819 ⋅ 10−12-7.588196380463 ⋅ 10−129.1 ⋅ 10−15
0.6-3.636709774083 ⋅ 10−12-3.636709774083 ⋅ 10−121.6 ⋅ 10−25
0.7-8.316369531201 ⋅ 10−13-8.099883012752 ⋅ 10−132.1 ⋅ 10−14
0.86.930669117891 ⋅ 10−136.930669117890 ⋅ 10−137.8 ⋅ 10−26
0.99.721365514984 ⋅ 10−138.898023093848 ⋅ 10−138.2 ⋅ 10−14
1000
Table 11

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

ηFnumericF¯OHPM,Eq. (60)relative error = |Fnumeric−F¯OHPM|
0110
0.10.98497463470.98497468274.8 ⋅ 10−8
0.20.94090305960.94090303712.2 ⋅ 10−8
0.30.87062109850.87062104914.9 ⋅ 10−8
0.40.77830943040.77830940382.6 ⋅ 10−8
0.50.66881948540.66881938779.7 ⋅ 10−8
0.60.54696072780.54696059711.3 ⋅ 10−7
0.70.41687574530.41687563361.1 ⋅ 10−7
0.80.28157372220.28157353801.8 ⋅ 10−7
0.90.14262910450.14262892441.8 ⋅ 10−7
1000
Table 12

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

ηθnumericθ¯OHPM from Eq. (61)relative error = |θnumeric−θ¯OHPM|
0-3.397742006028 ⋅ 10−9-3.397742006018 ⋅ 10−99.9 ⋅ 10−21
0.1-3.346638363267 ⋅ 10−9-3.347192325339 ⋅ 10−95.5 ⋅ 10−13
0.2-3.199501303768 ⋅ 10−9-3.199501303768 ⋅ 10−91.9 ⋅ 10−22
0.3-2.964278501171 ⋅ 10−9-2.964072110215 ⋅ 10−92.06 ⋅ 10−13
0.4-2.653181572584 ⋅ 10−9-2.653181572584 ⋅ 10−96.7 ⋅ 10−23
0.5-2.281170448213 ⋅ 10−9-2.281224162207 ⋅ 10−95.3 ⋅ 10−14
0.6-1.864037686164 ⋅ 10−9-1.864037686164 ⋅ 10−91.4 ⋅ 10−23
0.7-1.416766717489 ⋅ 10−9-1.416988166987 ⋅ 10−92.2 ⋅ 10−13
0.8-9.521674889800 ⋅ 10−10-9.521674889800 ⋅ 10−102.8 ⋅ 10−24
0.9-4.798620033623 ⋅ 10−10-4.772452449206 ⋅ 10−10 2.6 ⋅ 10−12
1000
Table 13

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

ηFnumericF¯OHPM,Eq. (62)relative error = |Fnumeric−F¯OHPM|
0110
0.10.98711080490.98711081821.3 ⋅ 10−8
0.20.94906423160.94906422665.0 ⋅ 10−9
0.30.88761045400.88761044865.3 ⋅ 10−9
0.40.80531445120.80531446161.03 ⋅ 10−8
0.50.70510322970.70510322415.5 ⋅ 10−9
0.60.58975345970.58975345524.4 ⋅ 10−9
0.70.46138672870.46138673981.1 ⋅ 10−8
0.80.32100640880.32100639611.2 ⋅ 10−8
0.90.16806496520.16806498141.6 ⋅ 10−8
1000
Table 14

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

ηθnumericθ¯OHPM from Eq. (63)relative error = |θnumeric−θ¯OHPM|
0-6.861779213872 ⋅ 10−9-6.861779213862 ⋅ 10−99.9 ⋅ 10−21
0.1-6.756689795935 ⋅ 10−9-6.757837516749 ⋅ 10−91.1 ⋅ 10−12
0.2-6.454596023395 ⋅ 10−9-6.454596023395 ⋅ 10−91.4 ⋅ 10−23
0.3-5.973076729675 ⋅ 10−9-5.972618579646 ⋅ 10−94.5 ⋅ 10−13
0.4-5.338639377096 ⋅ 10−9-5.338639377096 ⋅ 10−95.7 ⋅ 10−24
0.5-4.583265280198 ⋅ 10−9-4.583356882876 ⋅ 10−99.1 ⋅ 10−14
0.6-3.739726496008 ⋅ 10−9-3.739726496008 ⋅ 10−96.6 ⋅ 10−24
0.7-2.838920429095 ⋅ 10−9-2.839180849233 ⋅ 10−92.6 ⋅ 10−13
0.8-1.906149341683 ⋅ 10−9-1.906149341683 ⋅ 10−95.7 ⋅ 10−24
0.9-9.597066242185 ⋅ 10−10-9.553857275377 ⋅ 10−104.3 ⋅ 10−12
1000
Table 15

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

ηFnumericF¯OHPM,Eq. (64)relative error = |Fnumeric−F¯OHPM|
0110
0.10.99042721100.99042711071.003 ⋅ 10−7
0.20.96181006770.96180992991.3 ⋅ 10−7
0.30.91440363560.91440366392.8 ⋅ 10−8
0.40.84847368910.84847371672.7 ⋅ 10−8
0.50.76406422110.76406408491.3 ⋅ 10−7
0.60.66067723560.66067718395.1 ⋅ 10−8
0.70.53685205780.53685218211.2 ⋅ 10−7
0.80.38960184410.38960178296.1 ⋅ 10−8
0.90.21361113250.21361112943.1 ⋅ 10−9
1000
Table 16

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

ηθnumericθ¯OHPM from Eq. (65)relative error = |θnumeric−θ¯OHPM|
0-1.406496797937 ⋅ 10−8-1.406496797936 ⋅ 10−89.9 ⋅ 10−21
0.1-1.383991714524 ⋅ 10−8-1.384237616580 ⋅ 10−82.4 ⋅ 10−12
0.2-1.319483359510 ⋅ 10−8-1.319483359510 ⋅ 10−88.2 ⋅ 10−24
0.3-1.217244871718 ⋅ 10−8-1.217139646331 ⋅ 10−81.05 ⋅ 10−12
0.4-1.083591345359 ⋅ 10−8-1.083591345359 ⋅ 10−81.1 ⋅ 10−23
0.5-9.260353523538 ⋅ 10−9-9.260364501045 ⋅ 10−91.09 ⋅ 10−14
0.6-7.519751014022 ⋅ 10−9-7.519751014022 ⋅ 10−94.9 ⋅ 10−24
0.7-5.683228753360 ⋅ 10−9-5.683311365697 ⋅ 10−98.2 ⋅ 10−14
0.8-3.802282135607 ⋅ 10−9-3.802282135607 ⋅ 10−93.3 ⋅ 10−24
0.9-1.908111552553 ⋅ 10−9-1.902733734257 ⋅ 10−95.3 ⋅ 10−12
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. 6–9. With increasing value of α, velocity decreases for H = 0 and H = 250, but increases for H = 500 and H =1000. From Figs. 10–13, 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 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 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 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 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 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 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 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 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 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 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 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
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.

  1. Conflict of interest

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

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Received: 2016-8-29
Accepted: 2016-12-6
Published Online: 2017-3-15

© 2017 V. Marinca and R.-D. Ene

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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https://doi.org/10.1515/phys-2017-0006
Keywords for this article
optimal homotopy perturbation method; Jeffery-Hamel; nonlinear ordinary differential equations
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  71. Regular Articles
  72. Numerical solution for fractional Bratu’s initial value problem
  73. Regular Articles
  74. Structure and optical properties of TiO2 thin films deposited by ALD method
  75. Regular Articles
  76. Quadruple multi-wavelength conversion for access network scalability based on cross-phase modulation in an SOA-MZI
  77. Regular Articles
  78. Application of ANNs approach for wave-like and heat-like equations
  79. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  80. Study on node importance evaluation of the high-speed passenger traffic complex network based on the Structural Hole Theory
  81. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  82. A mathematical/physics model to measure the role of information and communication technology in some economies: the Chinese case
  83. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  84. Numerical modeling of the thermoelectric cooler with a complementary equation for heat circulation in air gaps
  85. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  86. On the libration collinear points in the restricted three – body problem
  87. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  88. Research on Critical Nodes Algorithm in Social Complex Networks
  89. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  90. A simulation based research on chance constrained programming in robust facility location problem
  91. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  92. A mathematical/physics carbon emission reduction strategy for building supply chain network based on carbon tax policy
  93. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  94. Mathematical analysis of the impact mechanism of information platform on agro-product supply chain and agro-product competitiveness
  95. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  96. A real negative selection algorithm with evolutionary preference for anomaly detection
  97. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  98. A privacy-preserving parallel and homomorphic encryption scheme
  99. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  100. Random walk-based similarity measure method for patterns in complex object
  101. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  102. A Mathematical Study of Accessibility and Cohesion Degree in a High-Speed Rail Station Connected to an Urban Bus Transport Network
  103. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  104. Design and Simulation of the Integrated Navigation System based on Extended Kalman Filter
  105. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  106. Oil exploration oriented multi-sensor image fusion algorithm
  107. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  108. Analysis of Product Distribution Strategy in Digital Publishing Industry Based on Game-Theory
  109. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  110. Expanded Study on the accumulation effect of tourism under the constraint of structure
  111. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  112. Unstructured P2P Network Load Balance Strategy Based on Multilevel Partitioning of Hypergraph
  113. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  114. Research on the method of information system risk state estimation based on clustering particle filter
  115. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  116. Demand forecasting and information platform in tourism
  117. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  118. Physical-chemical properties studying of molecular structures via topological index calculating
  119. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  120. Local kernel nonparametric discriminant analysis for adaptive extraction of complex structures
  121. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  122. City traffic flow breakdown prediction based on fuzzy rough set
  123. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  124. Conservation laws for a strongly damped wave equation
  125. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  126. Blending type approximation by Stancu-Kantorovich operators based on Pólya-Eggenberger distribution
  127. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  128. Computing the Ediz eccentric connectivity index of discrete dynamic structures
  129. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  130. A discrete epidemic model for bovine Babesiosis disease and tick populations
  131. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  132. Study on maintaining formations during satellite formation flying based on SDRE and LQR
  133. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  134. Relationship between solitary pulmonary nodule lung cancer and CT image features based on gradual clustering
  135. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  136. A novel fast target tracking method for UAV aerial image
  137. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  138. Fuzzy comprehensive evaluation model of interuniversity collaborative learning based on network
  139. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  140. Conservation laws, classical symmetries and exact solutions of the generalized KdV-Burgers-Kuramoto equation
  141. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  142. After notes on self-similarity exponent for fractal structures
  143. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  144. Excitation probability and effective temperature in the stationary regime of conductivity for Coulomb Glasses
  145. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  146. Comparisons of feature extraction algorithm based on unmanned aerial vehicle image
  147. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  148. Research on identification method of heavy vehicle rollover based on hidden Markov model
  149. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  150. Classifying BCI signals from novice users with extreme learning machine
  151. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  152. Topics on data transmission problem in software definition network
  153. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  154. Statistical inferences with jointly type-II censored samples from two Pareto distributions
  155. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  156. Estimation for coefficient of variation of an extension of the exponential distribution under type-II censoring scheme
  157. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  158. Analysis on trust influencing factors and trust model from multiple perspectives of online Auction
  159. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  160. Coupling of two-phase flow in fractured-vuggy reservoir with filling medium
  161. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  162. Production decline type curves analysis of a finite conductivity fractured well in coalbed methane reservoirs
  163. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  164. Flow Characteristic and Heat Transfer for Non-Newtonian Nanofluid in Rectangular Microchannels with Teardrop Dimples/Protrusions
  165. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  166. The size prediction of potential inclusions embedded in the sub-surface of fused silica by damage morphology
  167. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  168. Research on carbonate reservoir interwell connectivity based on a modified diffusivity filter model
  169. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  170. The method of the spatial locating of macroscopic throats based-on the inversion of dynamic interwell connectivity
  171. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  172. Unsteady mixed convection flow through a permeable stretching flat surface with partial slip effects through MHD nanofluid using spectral relaxation method
  173. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  174. A volumetric ablation model of EPDM considering complex physicochemical process in porous structure of char layer
  175. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  176. Numerical simulation on ferrofluid flow in fractured porous media based on discrete-fracture model
  177. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  178. Macroscopic lattice Boltzmann model for heat and moisture transfer process with phase transformation in unsaturated porous media during freezing process
  179. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  180. Modelling of intermittent microwave convective drying: parameter sensitivity
  181. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  182. Simulating gas-water relative permeabilities for nanoscale porous media with interfacial effects
  183. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  184. Simulation of counter-current imbibition in water-wet fractured reservoirs based on discrete-fracture model
  185. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  186. Investigation effect of wettability and heterogeneity in water flooding and on microscopic residual oil distribution in tight sandstone cores with NMR technique
  187. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  188. Analytical modeling of coupled flow and geomechanics for vertical fractured well in tight gas reservoirs
  189. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  190. Special Issue: Ever New "Loopholes" in Bell’s Argument and Experimental Tests
  191. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  192. The ultimate loophole in Bell’s theorem: The inequality is identically satisfied by data sets composed of ±1′s assuming merely that they exist
  193. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  194. Erratum to: The ultimate loophole in Bell’s theorem: The inequality is identically satisfied by data sets composed of ±1′s assuming merely that they exist
  195. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  196. Rhetoric, logic, and experiment in the quantum nonlocality debate
  197. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  198. What If Quantum Theory Violates All Mathematics?
  199. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  200. Relativity, anomalies and objectivity loophole in recent tests of local realism
  201. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  202. The photon identification loophole in EPRB experiments: computer models with single-wing selection
  203. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  204. Bohr against Bell: complementarity versus nonlocality
  205. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  206. Is Einsteinian no-signalling violated in Bell tests?
  207. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  208. Bell’s “Theorem”: loopholes vs. conceptual flaws
  209. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  210. Nonrecurrence and Bell-like inequalities
  211. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  212. Three-dimensional computer models of electrospinning systems
  213. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  214. Electric field computation and measurements in the electroporation of inhomogeneous samples
  215. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  216. Modelling of magnetostriction of transformer magnetic core for vibration analysis
  217. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  218. Comparison of the fractional power motor with cores made of various magnetic materials
  219. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  220. Dynamics of the line-start reluctance motor with rotor made of SMC material
  221. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  222. Inhomogeneous dielectrics: conformal mapping and finite-element models
  223. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  224. Topology optimization of induction heating model using sequential linear programming based on move limit with adaptive relaxation
  225. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  226. Detection of inter-turn short-circuit at start-up of induction machine based on torque analysis
  227. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  228. Current superimposition variable flux reluctance motor with 8 salient poles
  229. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  230. Modelling axial vibration in windings of power transformers
  231. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  232. Field analysis & eddy current losses calculation in five-phase tubular actuator
  233. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  234. Hybrid excited claw pole generator with skewed and non-skewed permanent magnets
  235. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  236. Electromagnetic phenomena analysis in brushless DC motor with speed control using PWM method
  237. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  238. Field-circuit analysis and measurements of a single-phase self-excited induction generator
  239. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  240. A comparative analysis between classical and modified approach of description of the electrical machine windings by means of T0 method
  241. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  242. Field-based optimal-design of an electric motor: a new sensitivity formulation
  243. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  244. Application of the parametric proper generalized decomposition to the frequency-dependent calculation of the impedance of an AC line with rectangular conductors
  245. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  246. Virtual reality as a new trend in mechanical and electrical engineering education
  247. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  248. Holonomicity analysis of electromechanical systems
  249. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  250. An accurate reactive power control study in virtual flux droop control
  251. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  252. Localized probability of improvement for kriging based multi-objective optimization
  253. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  254. Research of influence of open-winding faults on properties of brushless permanent magnets motor
  255. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  256. Optimal design of the rotor geometry of line-start permanent magnet synchronous motor using the bat algorithm
  257. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  258. Model of depositing layer on cylindrical surface produced by induction-assisted laser cladding process
  259. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  260. Detection of inter-turn faults in transformer winding using the capacitor discharge method
  261. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  262. A novel hybrid genetic algorithm for optimal design of IPM machines for electric vehicle
  263. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  264. Lamination effects on a 3D model of the magnetic core of power transformers
  265. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  266. Detection of vertical disparity in three-dimensional visualizations
  267. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  268. Calculations of magnetic field in dynamo sheets taking into account their texture
  269. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  270. 3-dimensional computer model of electrospinning multicapillary unit used for electrostatic field analysis
  271. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  272. Optimization of wearable microwave antenna with simplified electromagnetic model of the human body
  273. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  274. Induction heating process of ferromagnetic filled carbon nanotubes based on 3-D model
  275. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  276. Speed control of an induction motor by 6-switched 3-level inverter
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