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Journal of Applied Mathematics
Volume 2011, Article ID 876437, 17 pages
http://dx.doi.org/10.1155/2011/876437
Research Article

The HPM Applied to MHD Nanofluid Flow over a Horizontal Stretching Plate

Mechanical Engineering Department, Amirkabir University of Technology, Tehran, Iran

Received 28 July 2011; Accepted 31 August 2011

Academic Editor: Yansheng Liu

Copyright © 2011 S. S. Nourazar et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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). In the present work, our results of the HPM are compared with the results of simulation using the finite difference method, Keller's box-scheme. The comparisons of the results show that the HPM has the capability of solving the nonlinear boundary layer MHD flow of nanofluid with sufficient accuracy.

1. Introduction

Recently, lots of attention are devoted toward the semianalytical solution of real-life mathematical modeling that is inherently nonlinear differential equations with variable coefficients. 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. Semianalytical methods which are more suitable than the numerical methods are applied for the solution of nonlinear nonhomogeneous partial differential equations [17]. Comparing with other methods, the Semianalytical 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 Semianalytical method to the solution of nonhomogeneous variable coefficient partial differential equations is the homotopy perturbation method (HPM).

He [812] 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 [1316].

Recently, some of researchers have solved many problems in different fields of engineering. Singh et al. [17] solved space-time fractional solidification in a finite slab with HPM. Ajadi and Zuilino [18] applied HPM to reaction-diffusion equations with source term. They concluded that rapid convergence is obtained to the exact solution by HPM. Slota [19] 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 HPM. The two-dimensional forced-convection boundary-layer MHD problem is also simulated with the numerical Keller’s box-scheme [20], and the results of simulation are compared with the results obtained by solving the problem using the HPM. In the present problem, a nanoincompressible 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 numerical simulation. To our knowledge, there have been no results reported so far for the boundary layer flow of nanofluid, using the HPM method, including the MHD with variable magnetic field, and viscous dissipation effect.

2. Basic Idea of Homotopy Perturbation Method

The homotopy perturbation method (HPM) is originally initiated by He [19]. This is a combination of the classical perturbation technique and homotopy technique. The basic idea of the HPM for solving nonlinear differential equations is as follows: consider the following nonlinear differential equation: (𝑢)=0(2.1) subject to boundary condition 𝐵𝑢,𝜕𝑢𝜕𝑛=0,(2.2) where is a general nonlinear differential operator and𝐵 is a boundary operator.

Usually the main differential equation does not include the small parameter; however, to construct a homotopy, the nonlinear operator is divided into two parts, the first part includes the linear operator, 𝐿, and the second part includes the nonlinear operator, 𝑁. Therefore, (2.1) is rewritten as 𝐿(𝑢)+𝑁(𝑢)=0.(2.3) We now write the homotopy that constructed by He [19] as follows:𝐻𝑢(𝑣,𝑝)=𝐿(𝑣)+𝑝𝑁(𝑣)(1𝑝)𝐿0=0,(2.4) where 𝑝 is called the homotopy parameter which is usually assumed to vary between [0,1]. In (2.4), when 𝑝 is equal to 1 it converts back to the main differential equation (2.1), and in case where 𝑝 is equal to zero, (2.4) gives the zero-order approximation of the main differential equation (2.1). According to the perturbation method, the approximate solution to (2.4) is expressed as a series of the power of the homotopy parameter 𝑝 as 𝑣=𝑣0+𝑝𝑣1+𝑝2𝑣2+𝑝3𝑣3+,(2.5) where in the limit when 𝑝 approaches 1, (2.5) becomes 𝑢=lim𝑝1𝑣=𝑣0+𝑣1+𝑣2+𝑣3+.(2.6)

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 𝜕𝑢+𝜕𝑥𝜕𝑣𝑢𝜕𝑦=0,(3.1)𝜕𝑢𝜕𝑥+𝑣𝜕𝑢=1𝜕𝑦𝜌nf𝜇nf𝜕2𝑢𝜕𝑦2𝜎𝐵(𝑥)2𝑢𝑢,(3.2)𝜕𝑇𝜕𝑥+𝑣𝜕𝑇𝜕𝑦=𝛼nf𝜕2𝑇𝜕𝑦2+𝜇nf𝜌𝐶𝑝nf𝜕𝑢𝜕𝑦2.(3.3) Equation (3.1) describes the continuity equation, where 𝑢 and 𝑣are the velocity components in the 𝑥 and 𝑦 directions, respectively, (see Figure 1). Equation (3.2) describes the two-dimensional momentum equation in the presence of a variable magnetic field, where 𝑢and 𝑣are the 𝑥 and 𝑦 components of velocity, respectively, 𝜇nf and 𝜌nfare 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. Equation (3.3) 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, 𝛼nf is the thermal diffusivity, and (𝜌𝐶𝑝)nfis the heat capacitance of the nanofluid.

876437.fig.001
Figure 1: Schematic of the physical model and coordinate system.

The boundary conditions are defined as 𝑢=𝑢𝑤=𝑏𝑥𝑚,𝑣=0,𝑇=𝑇𝑤,at𝑦=0,𝑢0,𝑇𝑇,as𝑦,(3.4) where 𝑢𝑤 is the 𝑥-component of velocity on the horizontal flat plate, 𝑏 and 𝑚 are constants, and 𝑇𝑤and 𝑇 are the plate and ambient temperatures, respectively. The nanofluid properties such as the density,𝜌nf, the dynamic viscosity, 𝜇nf, the heat capacitance, (𝜌𝐶𝑝)nf, and the thermal conductivity, 𝑘nf, are defined in terms of fluid and nanoparticles properties as in [21], 𝜌nf=(1𝜙)𝜌𝑓+𝜙𝜌𝑠,𝜇nf=𝜇𝑓(1𝜙)2.5,𝑘nf𝑘𝑓=𝑘𝑠+2𝑘𝑓𝑘2𝜙𝑓𝑘𝑠𝑘𝑠+2𝑘𝑓𝑘+2𝜙𝑓𝑘𝑠,𝜌𝐶𝑝nf=(1𝜙)𝜌𝐶𝑝𝑓+𝜙𝜌𝐶𝑝𝑠,𝛼nf=𝑘nf𝜌𝐶𝑝nf,(3.5) where 𝜌𝑓 is the density of fluid, 𝜌𝑠 is the density of nanoparticles, 𝜙 is defined as the volume fraction of the nanoparticles,𝜇𝑓 is the dynamic viscosity of fluid, (𝜌𝐶𝑝)𝑓 is the thermal capacitance of fluid, (𝜌𝐶𝑝)𝑠 is the thermal capacitance of nanoparticles, and 𝑘𝑓 and 𝑘𝑠 are the thermal conductivities of fluid and nanoparticles, respectively.

The variable magnetic field is defined as [22, 23] 𝐵(𝑥)=𝐵0𝑥𝑚1,(3.6) where 𝐵0 and 𝑚 are constant.

The following dimensionless similarity variable is used to transform the governing equations into the ordinary differential equations 𝑦𝜂=𝑥Re𝑥1/2,Re𝑥=𝜌𝑓𝑢𝑤(𝑥)𝜇𝑓𝑥.(3.7) The dimensionless stream function and dimensionless temperature are defined as 𝑓(𝜂)=𝜓(𝑥,𝑦)Re𝑥1/2𝑢𝑤,(𝑥)𝜃(𝜂)=𝑇𝑇𝑇𝑤𝑇,(3.8) where the stream function 𝜓(𝑥,𝑦) is defined as 𝑢=𝜕𝜓𝜕𝑦,𝑣=𝜕𝜓𝜕𝑥.(3.9) By applying the similarity transformation parameters, the momentum equation (3.1) and the energy equation (3.2) can be rewritten as 𝑓+𝜌(1𝜙)+𝜙𝑠𝜌𝑓(1𝜙)2.5𝑚+122𝑓𝑓(𝜌1𝜙)+𝜙𝑠𝜌𝑓(1𝜙)2.5(𝑚)𝑓2(1𝜙)2.5𝑓Mn𝜃=0,+(1𝜙)+𝜙𝜌𝐶𝑝𝑠𝜌𝐶𝑝𝑓Pr𝑓𝜃+EcPr(1𝜙)2.5=0.(3.10) Therefore, the transformed boundary conditions are𝑓(0)=1,𝑓(0)=0,𝜃(0)=1,𝑓()=0,𝜃()=0.(3.11) The dimensionless parameters of Mn,Pr,Ec, and Re𝑥are the magnetic parameter, Prandtl, Eckert, and Reynolds numbers, respectively. They are defined as Mn=𝜎𝐵20𝜌𝑓𝑏,Pr=𝜌𝐶𝑝𝑓𝑘𝑒𝑓𝜐𝑓𝑢,Ec=𝑤(𝑥)2𝐶𝑝Δ𝑇,Re𝑥=𝜌𝑓𝑢𝑤(𝑥)𝜇𝑓𝑥.(3.12) Equation (3.10) is rewritten as𝑓+𝐴𝑓𝑓𝐵𝑓2𝐶𝑓𝜃=0,(3.13)+𝐷𝑓𝜃+𝐸𝑓2=0.(3.14) The boundary conditions for 𝑓 and 𝜃 in (3.13) and (3.14) are as follows: 𝑓(0)=1,𝑓(0)=0,𝜃(0)=1,𝑓()=0,𝜃()=0,(3.15) where coefficients, 𝐴, 𝐵, 𝐶, 𝐷, and 𝐸 are written as𝜌𝐴=(1𝜙)+𝜙𝑠𝜌𝑓(1𝜙)2.5𝑚+122,(𝜌𝐵=1𝜙)+𝜙𝑠𝜌𝑓(1𝜙)2.5(𝑚),𝐶=(1𝜙)2.5,Mn𝐷=(1𝜙)+𝜙𝜌𝐶𝑝𝑠𝜌𝐶𝑝𝑓Pr,𝐸=EcPr(1𝜙)2.5.(3.16)

4. The HPM Applied to the Problem

We are ready now to apply the HPM to solve the similarity nonlinear ordinary differential equations (3.13) and (3.14) with boundary conditions defined as in (3.11). First we construct a homotopy for each of (3.13) and (3.14) as follows: 𝑓(1𝑝)𝑓𝟎𝑓+𝑝+𝐴𝑓𝑓𝐵𝑓2𝐶𝑓𝜃=0,(4.1)(1𝑝)𝜃𝟎𝜃+𝑝+𝐷𝑓𝜃+𝐸𝑓2=0.(4.2) The approximation for each of 𝑓and 𝜃 in terms of the power series of homotopy parameter 𝑝 is written as 𝑓=𝑓0+𝑝𝑓1+𝑝2𝑓2+𝑝3𝑓3+=𝑛𝑖=1𝑝𝑖𝑓𝑖,(4.3)𝜃=𝜃0+𝑝𝜃1+𝑝2𝜃2+𝑝3𝜃3+=𝑛𝑖=1𝑝𝑖𝜃𝑖.(4.4) Substituting (4.3) and (4.4) into (4.1) and (4.2), respectively, and after manipulations, the coefficients of terms of different powers for 𝑝 are written as follows: 𝑝0𝑓𝟎=0,𝜃𝟎𝑓=0,0(0)=1,𝑓0𝜂=0,𝑓0(0)=0,𝜃0(0)=1,𝜃0𝜂𝑝=0,1𝐴𝑓0𝑓0𝐴𝐵𝑓02+𝑓1+𝑓0𝐾𝑓0=0,𝜃0+𝐷𝑓0𝜃0+𝜃1+𝐸𝑓02𝑓=0,1(0)=0,𝑓1𝜂=0,𝑓1(0)=0,𝜃1(0)=0,𝜃1𝜂𝑝=0,2𝐾𝑓12𝐴𝐵𝑓1𝑓0+𝑓2+𝐴𝑓0𝑓1+𝐴𝑓1𝑓0=0,𝐷𝑓1𝜃0+2𝐸𝑓0𝑓1+𝜃2+𝐷𝑓0𝜃0𝑓=0,2(0)=0,𝑓2𝜂=0,𝑓2(0)=0,𝜃2(0)=0,𝜃2𝜂𝑝=0,3𝐾𝑓22𝐴𝐵𝑓0𝑓2+𝑓3+𝐴𝑓1𝑓1+𝐴𝑓2𝑓0=0,𝐷𝑓2𝜃0+𝐷𝑓1𝜃1+𝐸𝑓12+𝐷𝑓0𝜃2+𝜃3+2𝐸𝑓0𝑓2𝑓=0,3(0)=0,𝑓3𝜂=0,𝑓3(0)=0,𝜃3(0)=0,𝜃3𝜂=0.(4.5) The above sets of recursive ordinary differential equations along with their boundary conditions are solved using the MAPLE software. Some samples of these functions obtained by the MAPLE software are brought to the reader’s attention as follows:𝑓01(𝜂)=𝜂𝜂102,𝑓1(𝜂)=7.333333333𝜂1000005+1.833333333𝜂100046.875000000𝜂1002,𝑓2(𝜂)=1.056349206𝜂100000008+4.225396825𝜂100000074.03333333𝜂10000061.008333333𝜂100005+1.260416667𝜂10004+4.501488101𝜂10002,𝑓3(𝜂)=1.600529100𝜂1000000000011+8.802910053𝜂1000000000101.514100529𝜂100000009+5.743898810𝜂100000008+5.809920634𝜂100000072.772916667𝜂10000062.805927579𝜂10000058.252728185𝜂1000004+3.197932176𝜂10002,𝑓4(𝜂)=2.398776866𝜂1000000000000014+1.679143807𝜂100000000000134.131499118𝜂1000000000012+3.693701059𝜂100000000011+4.599520498𝜂1000000000102.081888227𝜂100000009+4.091584733𝜂100000008+1.616748748𝜂10000007+1.815600201𝜂10000006+9.229301027𝜂100000055.862875656𝜂10000042.794411931𝜂10002,𝑓5(𝜂)=3.539668656𝜂1000000000000000017+3.008718359𝜂100000000000000169.621663665𝜂1000000000000015+1.336240956𝜂100000000000144.703056270𝜂100000000000136.487744707𝜂1000000000012+5.258958021𝜂100000000011+1.973384501𝜂1000000000105.793349681𝜂10000000094.024040085𝜂10000000085.317835354𝜂100000007+1.289832644𝜂100000061.022487452𝜂10000005+5.123088540𝜂1000004+1.227506044𝜂10002,𝜃01(𝜂)=15𝜃𝜂,1(𝜂)=197𝜂2500004+197𝜂125003611𝜂25000023541𝜃12500𝜂,2(𝜂)=243689𝜂656250000007+381234399997𝜂3000000000000000620923179307𝜂2000000000000052571375199𝜂12000000000004+4353487𝜂18750000031568233333𝜂2000000000002+18522049449𝜃448000000000𝜂,3(𝜂)=1348247347𝜂9000000000000000010+128997197𝜂1800000000000009600744859𝜂560000000000008+68494603𝜂21000000000007+1127538653𝜂300000000000063252643783𝜂200000000000052291723651𝜂40000000000004546104693𝜂3000000000003926887907𝜂2500000000002+11162966633099𝜃201600000000000𝜂,4(𝜂)=4078298911𝜂7800000000000000000013+1051026059𝜂33000000000000000011+2184149803𝜂450000000000000010+309004361𝜂18000000000000091846846597𝜂560000000000008+2550925297𝜂420000000000007+219441079𝜂12000000000006582975921𝜂200000000000005+186565583𝜂15000000000042901970381𝜂6000000000003+239346723𝜂800000000002698438049321211𝜃9225216000000000𝜂,5(𝜂)=1944523579𝜂1200000000000000000000016+833864223𝜂7000000000000000000015368935767𝜂113750000000000000014+184656149𝜂52000000000000000132568874313𝜂6600000000000000000123176798263𝜂1100000000000000011+1479712693𝜂90000000000000010+128033843𝜂900000000000009536161247𝜂280000000000008+1526910869𝜂14000000000000072319614257𝜂150000000000006+6768377991𝜂2000000000000052079006273𝜂40000000000004+308834747𝜂5000000000034211857587𝜂20000000000002+1919667324454739173690086400000000000𝜂.(4.6)

These functions, 𝑓 and𝜃, are calculated for the case where, Ec=0.1, 𝑚=0, Pr=6.2, 𝜑=0.2, and Mn=0.2. The physical properties of the fluid, water, and the nanoparticles, aluminum oxide (Al2O3), are given in Table 1.

tab1
Table 1: Thermophysical properties of water and nanoparticles.

5. Numerical Method

The differential equations, (3.13) and (3.14), along with the boundary conditions, (3.15), are split into five first-order differential equations by introducing new dependent variables. The five split first-order differential equations are discretized using the first-order backward finite difference scheme, the so-called Keller’s box method [20]. The discretized form of the five split differential equations are linearized using the Newton’s method [2426]. The discretized and linearized equations form a system of block-tridiagonal equations which are solved using the block-tridiagonal-elimination technique [26]. A step size of Δ𝜂=0.005 is selected to satisfy the convergence criterion of 104 in all cases. In our simulation, 𝜂 is chosen to be equal to 5 in order to suffice for taking into account the full effect of boundary layer growth. Then the differential equations, (3.13) and (3.14), along with the boundary conditions, (3.15), are solved using the HPM. The recursive differential equations with the relevant boundary conditions resulting from the HPM are solved using the MAPLE software.

6. Results and Discussions

Table 2 shows the comparison between the results obtained from HPM and the results obtained from the numerical method (NM) at Ec=0.1, 𝑚=0, Pr=6.2, 𝜑=0.2, and Mn=0.2. The comparison of the results obtained from the HPM and the results obtained from the NM shows excellent agreements at different values of the similarity parameter. Figure 2 shows the comparison of dimensionless velocity profiles versus the normalized coordinates using the Keller’s box numerical method with the results obtained by the HPM at Ec=0.1, 𝑚=0, Pr=6.2, 𝜑=0.2, and Mn=0.2. The results obtained from the HPM are reported for three different sums of terms, 𝑆=4, 8, and 12, in the HPM series solution. It is obvious from Figure 2 that as the number of sums of terms in the HPM series solution increases, the results approach towards the profile obtained from the NM. The mean discrepancies between the results of velocity obtained from the HPM for 𝑆=12 and the results obtained from the NM are at most 2%. Figure 3 shows the comparison of dimensionless temperature profiles versus the normalized coordinates using the Keller’s box numerical method with the results obtained by the HPM at Ec=0.1, 𝑚=0, Pr=6.2, 𝜑=0.2, and Mn=0.2. The results obtained from the HPM are reported for three different sums of terms, 𝑆=4, 8, and 12, in the HPM series solution. As the number of sums of terms in the HPM series solution increases the agreement between the results obtained from the HPM and the results obtained from the NM is more pronounced. The mean discrepancies between the results of temperature obtained from the HPM for 𝑆=12 and the results obtained from the NM are less than 5%. Figure 4 shows the comparison of dimensionless velocity profiles versus the normalized coordinates using the Keller’s box numerical method with the results obtained by the HPM at Ec=0.1, 𝑚=0, Pr=6.2, 𝜑=0.2, and Mn=0. The results obtained from the HPM are reported for three different number of sums of terms, 𝑆=4, 8, and 12, in the HPM series solution. As the number of sums of terms in the HPM series solution increases the agreement between the results of dimensionless velocity obtained from the HPM and the results obtained from the NM is more pronounced. The results of velocity obtained from the HPM for 𝑆=8 and 12 and the results obtained from the NM are almost the same. Figure 5 shows the comparison of dimensionless temperature profiles versus the normalized coordinates using the Keller’s box numerical method with the results obtained by the HPM at Ec=0.1, 𝑚=0, Pr=6.2, 𝜑=0.2, and Mn=0. The results obtained from the HPM are reported for three different sums of terms, 𝑆=4, 8, and 12, in the HPM series solution. However, as the number of sums of terms in the HPM series solution increases, the agreement between the results obtained from the HPM and the results obtained from the NM is more apparent. For the temperature profiles, the mean discrepancies between the results obtained from the HPM when 𝑆=12 and the results obtained from the NM are at most 8%, whereas the discrepancies between the results obtained for velocity from HPM and NM are negligible at the same conditions. The reason of this behavior is due to the complex nonlinearity that exists in the nature of the governing equations which makes it so difficult to exactly realize the obsessive interaction existing in the problem. Figure 6 shows the comparison between dimensionless velocity profiles versus the normalized coordinates using the Keller’s box numerical method and the results obtained by the HPM at Ec=0, 𝑚=0, Pr=6.2, 𝜑=0.2, and Mn=0.2. The results obtained from the HPM are reported for three different sums of terms 𝑆=4, 8, and 12 in the HPM series solution. Figure 7 shows the comparison of dimensionless temperature profiles versus the normalized coordinates using the Keller’s box numerical method with the results obtained by the HPM at Ec=0.1, 𝑚=0, Pr=6.2, 𝜑=0.2, and Mn=0.2. The results obtained from the HPM are reported for three different sums of terms, 𝑆=4, 8, and 12, in the HPM series solution. It is obvious that as the number of sums of terms in the HPM series solution increases, the results approach toward the profile obtained from the NM. The mean discrepancies between the results of velocity obtained from the HPM for 𝑆=12 and the results obtained from the NM are at most 5%. Figure 8 shows the comparison between dimensionless velocity profiles versus the normalized coordinates using the Keller’s box numerical method with the results obtained by the HPM at Ec=0.1, 𝑚=0.1, Pr=6.2, 𝜑=0.2, and Mn=0.2. The results obtained from the HPM are reported for three different numbers of sums of terms, 𝑆=4, 8, and 12, in the HPM series solution. One can realize from Figure 8, as the number of sums of terms in the HPM series solution increases, the results approach toward the profile obtained from the NM. The mean discrepancies between the results of velocity obtained from the HPM for 𝑆=12 and the results obtained from the NM are at most 5%. Figure 9 shows the comparison of dimensionless temperature profiles versus the normalized coordinates using the Keller’s box numerical method with the results obtained by the HPM atEc=0.1, 𝑚=0.1, Pr=6.2, 𝜑=0.2, and Mn=0.2. The results obtained from the HPM are reported for three different numbers of sums of terms, 𝑆=4, 8, and 12, in the HPM series solution. As the number of sums of terms in the HPM series solution increases, the results approach towards the profile obtained from the NM. The mean discrepancies between the results of velocity obtained from the HPM for 𝑆=12 and the results obtained from the NM are at most 4%.

tab2
Table 2: Comparison between HPM and NM at Ec=0.1, 𝑚=0, Pr=6.2, 𝜑=0.2, and Mn=0.2.
876437.fig.002
Figure 2: Comparison of dimensionless velocity profiles versus the normalized coordinates using the Keller’s box numerical method with the results obtained by HPM at Ec=0.1, 𝑚=0, Pr=6.2, 𝜑=0.2, and Mn=0.2.
876437.fig.003
Figure 3: Comparison of dimensionless temperature profiles versus the normalized coordinates using the Keller’s box numerical method with the results obtained by HPM at Ec=0.1, 𝑚=0, Pr=6.2, 𝜑=0.2, and Mn=0.2.
876437.fig.004
Figure 4: Comparison of dimensionless velocity profiles versus the normalized coordinates using the Keller’s box numerical method with the results obtained by HPM at Ec=0.1, 𝑚=0, Pr=6.2, 𝜑=0.2, and Mn=0.
876437.fig.005
Figure 5: Comparison of dimensionless temperature profiles versus the normalized coordinates using the Keller’s box numerical method with the results obtained by HPM at Ec=0.1, 𝑚=0, Pr=6.2, 𝜑=0.2, and Mn=0.
876437.fig.006
Figure 6: Comparison of dimensionless velocity profiles versus the normalized coordinates using the Keller’s box numerical method with the results obtained by HPM at Ec=0, 𝑚=0, Pr=6.2, 𝜑=0.2, and Mn=0.2.
876437.fig.007
Figure 7: Comparison of dimensionless temperature profiles versus the normalized coordinates using the Keller’s box numerical method with the results obtained by HPM at Ec=0, 𝑚=0, Pr=6.2, 𝜑=0.2, and Mn=0.2.
876437.fig.008
Figure 8: Comparison of dimensionless velocity profiles versus the normalized coordinates using the Keller’s box numerical method with the results obtained by HPM at Ec=0.1, 𝑚=0.1, Pr=6.2, 𝜑=0.2, and Mn=0.2.
876437.fig.009
Figure 9: Comparison of dimensionless temperature profiles versus the normalized coordinates using the Keller’s box numerical method with the results obtained by HPM at Ec=0.1, 𝑚=0.1, Pr=6.2, 𝜑=0.2, and Mn=0.2.

7. Conclusions

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). The results are justified and compared with the results obtained from the numerical method (NM). 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 NM. The results obtained from the HPM show at most less than 8% mean deviations when compared with the results obtained from the NM. 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.

Nomenclature

𝐵(𝑥):Magnetic field
𝑏:Constant parameter
Ec:Eckert number
(𝐶𝑝)𝑠:Thermal capacitance of solid
(𝐶𝑝)𝑓:Thermal capacitance of fluid
𝑓:Dimensionless velocity variable
𝑘𝑠:Thermal conductivity of nanoparticles
𝑘𝑓:Thermal conductivity of fluid
𝑚:Index of power law velocity
Mn:Magnetic parameter
Pr:Prandtl number
Re:Reynolds number
𝑆:No. of terms in the HPM
𝑇:Absolute temperature
𝑇:Constant temperature of the fluid far away from the plate
𝑇𝑤:Given temperature at the plate
𝑢:Velocity in x-direction
𝑣:Velocity in y-direction
𝑢𝑤:Velocity of the plate
𝑥:Horizontal coordinate
𝑦:Vertical coordinate.
Greek Symbols
𝜎:Electrical conductivity
𝜃:Dimensionless temperature
𝜓:Stream function
𝜇𝑓:Fluid viscosity
𝜑:Nanoparticles volume fraction
𝜌𝑠:nanoparticles density
𝜌𝑓:Fluid density.

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