1. Introduction

The forced convection heat transfer over a permeable stretching plate has relevance in applications such as solar receivers exposed to wind currents, electronic devices cooled by fans, nuclear reactors cooled during emergency shutdown, heat exchanges placed in a low-velocity environment, extrusion processes, cooling of reactors, glass fiber production and crystal growing ^{[1, 2, 3]}. During the last decade, nanofluid heat transfer problems have been given considerable attention by researchers. Most of the nonlinear differential equations do not have an analytical solution. However, so far there have been many researchers that attempted to solve the nonlinear differential equations by using numeric methods. Using the numeric methods, a tremendous amount of CPU time as well as huge memory is required. Semi analytical methods which are more suitable than the numerical methods are applied for the solution of nonlinear nonhomogeneous partial differential equations [4-10]^[4]. Comparing with other methods, the Semi analytical methods have the advantage of simplicity when applying to solve complicated nonlinear problems. The HPM, ADM, and VIM methods are used to solve the nonhomogeneous variable coefficient partial differential equations with accurate approximation. Consequently, to extend the validity of the solution to a broader range, one needs to handle huge amount of computational effort. The most powerful Semi analytical method to the solution of nonhomogeneous variable coefficient partial differential equations is the homotopy perturbation method (HPM).

He [11-15]^[11] developed the homotopy perturbation method for solving linear, nonlinear, and initial and boundary value problems by combining the standard homotopy and the perturbation methods. The homotopy perturbation method was formulated by taking the full advantage of the standard homotopy and perturbation methods and has been modified later by some scientists to obtain more accurate results, rapid convergence, and to reduce the amount of computation ^{[16, 17, 18, 19]}.

Recently, some of researchers have solved many problems in different fields of engineering. Singh et al. ^[20] solved space-time fractional solidification in a finite slab with HPM. Ajadi and Zuilino ^[21] applied HPM to reaction-diffusion equations with source term. They concluded that rapid convergence is obtained to the exact solution by HPM. Slota ^[22] applied the HPM to Stefan solidification heat equation problem, and his results show that HPM is a capable method for solving the problems under consideration.

The basic motivation of this paper is to solve a two-dimensional forced-convection boundary-layer MHD problem formed by a magneto hydrodynamic (MHD) incompressible nanofluid flow in the presence of variable magnetic field over a horizontal flat plate including the viscous dissipation term using the VIM, HPM and ADM. The two-dimensional forced-convection boundary-layer MHD problem is also exact solutions, and the results of simulation are compared with the results obtained by solving the problem using the VIM, HPM and ADM. In the present problem, a nano incompressible fluid in the presence of a variable magnetic field and the viscous dissipation effect over a horizontal stretching flat plate are considered. The results are compared with the previous results of exact simulation. To our knowledge, there have been no results reported so far for the boundary layer flow of nanofluid, using the VIM, HPM and ADM methods, including the MHD with variable magnetic field, and viscous dissipation effect.

2. Decribe Analytical Methods

Consider the following function:

(1)

with the boundary condition of:

(2)

Where A (u) is defined as follows:

(3)

homotopy-perturbation structure is shown as:

(4)

or

(5)

where

(6)

Obviously, considering Eqs. (4) and (5), we have:

(7)

Where p ∈ [0, 1] is an embedding parameter and is the first approximation that satisfies the boundary condition. The process of the changes in p from zero to unity is that of v(r, p) changing from to . We consider v as:

(8)

and the best approximation is:

(9)

The above convergence is discussed in ^{[23, 24]}.

To clarify the basic ideas of VIM, we consider the following differential equation:

(10)

where L is a linear operator, F is a nonlinear operator and g (t) is a heterogeneous term.

According to VIM, we can write down a correction functional as follows:

(11)

Where λ is a general Lagrangian multiplier ^{[25, 26]} which can be identified optimally via the variational theory. The subscript n indicates the nth approximation and is considered as a restricted variation ^{[25, 26]}, i.e., .

To clarify the basic ideas of ADM, we consider the following differential equation:

(12)

Where L is the highest order derivative which is assumed to be easily invertible, R the linear differential operator of less order than L, Nu presents the nonlinear terms and g is the source term. Applying the invers operator L^-1 to the both sides of Eq. (12), and using the given conditions we obtain:

(13)

where the function f(x) represents the terms arising from integration the source term g(x), using given conditions. For nonlinear differential equations, the nonlinear operator is represented by an infinite series of the so-called Adomian polynomials.

(14)

The polynomials are generated for all kind of nonlinearity so that depends only on , depends on and , and so on. The Adomian method defines the solution by the series:

(15)

In the case of , the infinite series is a Taylor expansion about . In other words

(16)

By rewriting Eq. (15) as , substituting it into Eq. (16) and then equating two expressions for F(u) found in Eq. (16) and Eq. (14) defines formulas for the Adomian polynomials:

(17)

By equating terms in Eq. (17), the first few Adomian’s polynomials and are given:

(18)

(19)

(20)

(21)

Now that the are known, Eq. (15) can be substituted in Eq. (14) to specify the terms in the expansion for the solution of Eq. (16).

3. Mathematical Formulation

The governing two-dimensional forced-convection boundary-layer flow over a horizontal stretching flat plate including the viscous dissipation term is written as

(22)

(23)

(24)

Figure 1. Schematic of the physical model and coordinate system

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Eq (22) describes the continuity equation, where u and v are the velocity components in the x and y directions, respectively, (see Figure 1). Eq (23) describes the two dimensional momentum equation in the presence of a variable magnetic field, where u and v are the x and y components of velocity, respectively, and are the dynamic viscosity and the density of the nanofluid, respectively, σ is the electrical conductivity, and B(x) is the variable magnetic field acting in the perpendicular direction to the horizontal flat plate. Eq (24) describes the two-dimensional energy equation including the viscous dissipation term, where, u, v, and T are the x and y components of velocity and temperature, respectively, is the thermal diffusivity, and is the heat capacitance of the nanofluid.

The boundary conditions are defined as

(25)

where is the x-component of velocity on the horizontal flat plate, b and m are constants, and and are the plate and ambient temperatures, respectively. The nanofluid properties such as the density, , the dynamic viscosity, , the heat capacitance, , and the thermal conductivity, , are defined in terms of fluid and nanoparticles properties as in ^[27],

(26)

where is the density of fluid, is the density of nanoparticles, φ is defined as the volume fraction of the nanoparticles, is the dynamic viscosity of fluid, is the thermal capacitance of fluid, is the thermal capacitance of nanoparticles, and are the thermal conductivities of fluid and nanoparticles, respectively.

The variable magnetic field is defined as ^{[28, 29]}

(27)

The following dimensionless similarity variable is used to transform the governing equations into the ordinary differential equations

(28)

The dimensionless stream function and dimensionless temperature are defined as

(29)

where the stream function is defined as

(30)

By applying the similarity transformation parameters, the momentum Eq (23) and the energy Eq (24) can be rewritten as

(31)

Therefore, the transformed boundary conditions are

(32)

The dimensionless parameters of Mn, Pr, Ec, and Re_x are the magnetic parameter, Prandtl, Eckert, and Reynolds numbers, respectively. They are defined as

(33)

Eq (31) is rewritten as

(34)

(35)

where coefficients, A, B, C, D, and E are written as

(36)

4. Analytical Methods Applied to the Problem

A homotopy perturbation method can be constructed as follows:

(37)

(38)

One can now try to obtain a solution of Eqs. (37, 38) in the form of:

(39)

(40)

where are functions yet to be determined. According to Eqs. (37,38) the initial approximation to satisfy initial condition is:

(41)

(42)

Substituting Eqs. (39, 40) into Eqs. (37, 38) yields:

(43)

(44)

The solutions of Eqs. (43, 44) may be written as follow:

(45)

(46)

In the same manner, the rest of components were obtained by using the Maple package. According to the HPM, we can conclude:

(47)

(48)

In order to solve Eqs. (34, 35) with boundary conditions (33) using VIM, we construct a correction functional, as follows:

(49)

(50)

Its stationary conditions can be obtained as follows:

(51)

(52)

The Lagrangians multiplier can therefore be identified as:

(53)

(54)

As results, we obtain the following iteration formula:

(55)

(56)

Now we start with an arbitrary initial approximation that satisfies the initial conditions:

(57)

(58)

Using the above Variational formula (49, 50), we have:

(59)

(60)

Substituting Eq. (57, 58) in to Eq. (59, 60) and after simplification, we have:

(61)

(62)

and so on. In the same manner the rest of the components of the iteration formula can be obtained.

In order to apply ADM to nonlinear equation in fluids problem, we rewrite Eqs (34, 35) in the following operator form:

(63)

(64)

where the notation:

(65)

is the linear operator. By using the inverse operator, we can write Eqs (63, 64) in the following form:

(66)

(67)

where the inverse operator is defined by:

(68)

where

(69)

The nonlinear operators are defined by the following infinite series:

(70)

where is called Adomian polynomials and defined by:

(71)

Hence we obtain the components series solution by the following recursive relations:

(72)

(73)

Where . Adomian’s polynomials formula, Eq. (71), is easy to set computer cod to get as many polynomials as we need in the calculation. We can give the first few Adomian’s polynomials of the as:

(74)

(75)

and so on, the rest of the polynomials can be constructed in a similar manner. Using the recursive relation, Eqs (72, 73) and Adomian’s polynomials formula, Eq. (71), with the initial conditions, Eq. (32), gives:

(76)

(77)

where

(78)

(79)

and so on. In the same manner the rest of the components of the iteration formula can be obtained.

Table 1. Thermophysical properties of water and nanoparticles

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These functions, f and θ, are calculated for the case where, , , , , and . The physical properties of the fluid, water, and the nanoparticles, aluminum oxide , are given in Table 1.

5. Results and Discussion

Comparison between the results of VIM, HPM and ADM methods is shown in (Figure 2). It can be seen that the HPM method of this analytic methods is closer to numeric method, also Table 2, Table 3 display the numerical magnitude of velocity and temperature profiles. Based on results of these tables discrepancy between the obtained result of HPM and exact solution is less than discrepancy the result of VIM and ADM with exact solution. This is obvious that HPM method is appropriate method for this problem. With observation of results we are starting to clear study about HPM with more iteration to approach to better answer.

Figure 2. Comparison of dimensionless velocity & temperature profiles versus the numeric method with the results obtained by HPM, VIM & ADM at , , , , and .

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Table 2. The results of VIM, HPM and ADM methods and their errors for f’

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Table 3. The results of VIM, HPM and ADM methods and their errors for θ

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Figure 3 shows the comparison of dimensionless velocity profiles versus the numeric method with the results obtained by the HPM at VIM & ADM at , , , and . The results obtained from the HPM are reported for four different sums of terms, S=2,4,8, and 12, in the HPM series solution. It is obvious from Figure 3 that as the number of sums of terms in the HPM series solution increases, the results approach towards the profile obtained from the exact solution. The mean discrepancies between the results of velocity obtained from the HPM for S=12 and the results obtained from the exact solution are at most 1%.

Figure 3. Comparison of dimensionless velocity profiles versus the numeric method with the results obtained by HPM at VIM & ADM at , , , , and .

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Figure 4 shows the comparison of dimensionless temperature profiles versus the numeric method with the results obtained by the HPM at VIM & ADM at , , , and . The results obtained from the HPM are reported for four different sums of terms, S=2,4,8, and 12, in the HPM series solution. It is obvious from Figure 4 that as the number of sums of terms in the HPM series solution increases, the results approach towards the profile obtained from the exact solution. The mean discrepancies between the results of velocity obtained from the HPM for S=12 and the results obtained from the exact solution are at most 3%.

Figure 4. Comparison of dimensionless temperature profiles versus the numeric method with the results obtained by HPM at VIM & ADM at , , , , and

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Figure 5. The influence of volume fraction of the nanofluid (ϕ) on (a) velocity, (b) temperature

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Figure 6. The effect of different value of m on (a) velocity, (b) temperature

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The fluctuation of temperature and velocity with different values of nanoparticles volume is shown in Figure 5. Considering this figure it is obvious that with increasing of nanoparticles volume, the temperature value decreases and with increasing of nanoparticles volume, the velocity value increases. Figure 6 indicates the influence of m on the temperature and velocity profiles. According to this figure it is obtained that the values of velocity decrease with increasing of parameter m, but for temperature the reverse trend is observed and with increasing of m the value of temperature profile increases. Finally Figure 7 show the effect of Eckert number on temperature profile. Considering this figure it is obvious that with increasing of Ec, the temperature value increases.

Figure 7. The effect of Eckert number on temperature profile

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6. Conclusion

In this work, the nonlinear two-dimensional forced-convection boundary-layer magneto hydrodynamic (MHD) incompressible flow of nanofluid over a horizontal stretching flat plate with variable magnetic field including the viscous dissipation effect is solved using the homotopy perturbation method (HPM), Variational iteration method (VIM) and Adomian decomposition method (ADM). The results are justified and compared with the results obtained from the numeric method. Our results obtained from the HPM, when the number of sums of terms in the HPM series solution increases, showed a monotonic convergence towards the results using the exact solution. The results obtained from the HPM show at most less than 5% mean deviations when compared with the results obtained from the exact solution. For the nonlinear MHD problem, this is encouraging because these results are only achieved by including at most S=12 number of sums of terms in the HPM series solution. Also the influence of physical factors such as m, Eckert number (Ec) and the percentage of nanoparticles (ϕ) on the velocity and temperature profiles have been investigated. The results show with increasing of the parameter m, velocity decreases but for temperature the reverse trend is observed and also with increasing of nanoparticles volume, the temperature value decreases and the velocity value increases. Also with increasing of Eckert number, the temperature value increases.

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