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  and Hamel . 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 . 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. . The MHD Jeffery-Hamel problem is solved by Makinde and Mhone  using a special type of Hermite-Padé approximation semi-numerical approach and by Esmaili et al.  by applying the Adomian decomposition method. The classical Jeffery-Hamel flow problem is solved by Ganji et al.  by means of the variational iteration and homotopy perturbation methods, and by Joneidi et al.  by the differential transformation method, Homotopy Perturbation Method and Homotopy Analysis Method. The classical Jeffery-Hamel problem was extended in  to include the effects of external magnetic field in a conducting fluid. The optimal homotopy asymptotic method is applied by Marinca and Herişanu  and by Esmaeilpour and Ganji . 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 .
During recent years, several methods have been used for solving different problems such as the traveling-wave transformation method , the Cole-Hopf transformation method , the optimal homotopy asymptotic method  and the generalized boundary element approach .
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.
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 (5) (6) (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: (8) (9) (10)
The relevant boundary conditions, due to the symmetry assumption at the channel centerline, are as follows: (11)
and at the plates making the body of the channel: (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 (13)
where f(φ) is an arbitrary function of φ only.
By integrating Eq. (10), it holds that (14)
in which g(r) is an arbitrary function of r only.
Now, we define the dimensionless parameters: (15)
subject to the boundary conditions (18) (19)
where is the Reynolds number, is the Hartmann number, 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 (20)
that is subject to the initial / boundary condition (21)
Assuming that the approximate analytical solution of the second-order can be expressed in the form (24)
and expanding the nonlinear operator N in series with respect to the parameter p, we have: (25)
where 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: (26)
Equating the coefficients of like powers of p yields the linear equations: (27) (28) (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
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) (30)
The solution of Eq. (30) is hence (31)
On the other hand, from Eq. (16), one obtains (32)
where A = 2αRe, B = (4 − H)α2.
Eq. (28) becomes (34)
We choose H0(η, Ck) = −60C1 where C1 is an unknown parameter and from Eq. (34) we obtain (35)
Eq. (29) can be written in the form (36)
In this case we choose
such that the solution of Eq. (36) is given by (37)
Eq. (27) becomes (40)
Eq. (40) has the solution (41)
From Eq. (39) it follows that (42)
Eq. (28) can be written (45)
Choosing h0(η, C8) = −30C8 in Eq. (45), one obtains (46)
Eq. (29) can be written in the form (47)
and therefore it is natural to choose the auxiliary function h1 as
From Eq. (47), it can shown that (48)
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 and H = 0. By means of the least-square method, the values of the parameters Ci, i = 1, 2, …, 13 are
Case 5.2 For the parameters Ci are:
Case 5.3 For we obtain: (54) (55)
Case 5.4 For it holds that: (56) (57)
Case 5.5 For we obtain (58) (59)
Case 5.6 If then (60) (61)
Case 5.7 For the approximate solutions are (62) (63)
Case 5.8 If then (64) (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.
In Figs.2 and 3 are presented the effect of the Hartmann number on the velocity profile for Re = 50 and 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 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.
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
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.
© 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