Document Type: Full Lenght Research Article
Authors
^{1} Assistant Professor Department of Mathematics Sri Ramakrishna Mission Vidyalaya College of Arts and Science, Coimbatore, Tamil Nadu
^{2} Department of Mathematics,Providence College for Women, Coonoor  643 104, INDIA
^{3} Department of Mathematics, Jamal Mohamed College, Trichy  6420 020, INDIA
^{4} of Mathematics, Sri Ramakrishna Mission Vidyalaya College of Arts & Science, Coimbatore  641 020, INDIA.
Abstract
Keywords
Main Subjects
History:
Submitted 25 September 2015 Received 6 January 2015 Accepted 6 August 2016
Keywords:
Homotopy analysis method Magnetohydrodynamics Nanofluid Runge–Kutta method Vertical plate 
A B S T R A C T
The magnetohydrodynamic boundarylayer flow of twophase model nanofluid over a vertical plate is investigated both analytically and numerically. A system of governing nonlinear partial differential equations is converted into a set of nonlinear ordinary differential equations by suitable similarity transformations, and the transformed equations are solved analytically using the homotopy analysis method and numerically by the fourthorder Runge–Kutta method along with a shooting technique. The effects of pertinent physical parameters on velocity, temperature, and concentration profiles of the nanofluid are discussed graphically. Furthermore, comparisons have been made with benchmark solutions for a special case and have obtained very good agreement.
© 2016 Published by Semnan University Press. All rights reserved. 
Analytical and Numerical Studies on Hydromagnetic Flow of Boungiorno Model Nanofluid over a Vertical Plate
B. Ganga, S. Mohamed Yusuff Ansari, N.Vishnu Ganesh, A.K. Abdul Hakeem^{* } ^{ }^{ } Department of Mathematics,Providence College for Women, Coonoor  643 104, INDIA. Department of Mathematics, Jamal Mohamed College, Trichy  620 020, INDIA. Department of Mathematics, Sri Ramakrishna Mission Vidyalaya College of Arts & Science, Coimbatore  641 020, INDIA.


Journal of Heat and Mass Transfer Research
Journal homepage: http://jhmtr.journals.semnan.ac.ir 

Corresponding Author: A.K. Abdul Hakeem Department of Mathematics, Sri Ramakrishna Mission Vidyalaya College of Arts & Science, Coimbatore  641 020, INDIA Email: abdulhakeem6@gmail.com

Kuznetsov and Nield [16] studied the classical problem of free convection boundarylayer flow of viscous and incompressible fluids passed through a vertical flat plate in the case of twophase model nanofluids. The same authors [17] extended their investigation to study the doublediffusive natural convection boundarylayer flow of nanofluid over a vertical plate. Khan and Pop [18] analyzed the development of steady boundarylayer flow, heat transfer, and nanoparticle volume fraction over a linear stretching surface in a nanofluid. Khan and Aziz [19] investigated the boundarylayer flow of nanofluid past a vertical surface with a constant heat flux. Gorla and Chamkha [20] analyzed the natural convection flow of a nanofluid past an isothermal horizontal plate in a porous medium saturated by a nanofluid. Aziz and Khan [21] studied the natural convection flow of a nanofluid over a convectively heated vertical plate.
The applied magnetic field may play an important role in controlling momentum and heat transfers in the boundarylayer flow of different fluids. Ibrahim et al. [22] investigated the magneticfield effects on free convection and masstransfer flow past a semiinfinite vertical flat plate. Muthucumaraswamy and Janakiraman [23] studied the thermalradiation effects on flow past an impulsively started infinite vertical plate with uniform temperature and variable mass diffusion in the presence of a transverseapplied magnetic field. Yahyazadeh et al. [24] inspected the effect of a magnetic field on the free convection flow of a nanofluid over a linear stretching sheet.
Nonlinear phenomena play a crucial role in applied mathematics and physics. The theory of nonlinear problems recently has been the focus of many studies. In most cases, it is difficult to solve nonlinear problems, especially to obtain the analytical solutions. An analytical method for strongly nonlinear problems, namely the homotopy analysis method (HAM), was proposed by Liao in 1992. The HAM is a general analytic approach to derive a series of solutions for various types of nonlinear equations, including algebraic equations, ordinary differential equations, partial differential equations, differentialintegral equations, differentialdifference equations, and coupled equations. Unlike perturbation methods, the HAM is independent of small or large physical parameters and, thus, is valid whether or not a nonlinear problem contains small or large physical parameters. More importantly, different from all perturbation and traditional non perturbation methods, the HAM provides a simple way to ensure the convergence of a solution series, and therefore, the HAM is valid even for strongly nonlinear problems. In addition, the HAM provides great freedom to choose proper base functions to approximate a nonlinear problem [2532]. Motivated by the advantages of the HAM, we have applied it to investigate the boundarylayer flow of an incompressible, viscous nanofluid over a vertical plate in the presence of the magneticfield effect. To verify the HAM solutions, we have compared the present solutions with the numerical solutions obtained by the fourthorder Runge–Kutta method along with the shooting iteration technique.
2. Formulation of the problem
Consider the steady twodimensional boundarylayer flow of a nanofluid over a vertical plate in the presence of a magnetic field. We select a coordinate frame in which the x axis is aligned vertically upwards. We consider a vertical plate at y = 0. At this boundary, the temperature T and the nanoparticle volume fraction f take constant values T_{w} and f_{w}, respectively. The temperature T and the nanoparticle volume fraction of the nanofluid f take values T_{¥} and f_{¥},respectively, as y®¥. We also consider the influence of a constant magneticfield strength B_{0}, which is applied normally to the plate. It is assumed further that the induced magnetic field is negligible in comparison to the applied magnetic field. Under the above assumptions, the boundarylayer equations governing the flow and thermal and concentration fields can be written in dimensional form as follows [16]:
, (1)
(2)
, (3)
, (4)
where u and v are the velocity components along the x and y directions, respectively, p is the fluid pressure, r_{f }is the density of the base fluid, r_{p }is the nanoparticle density, m is the absolute viscosity of the base fluid, is the thermal diffusivity of the base fluid, is the ratio of nanoparticle heat capacity and the base fluid heat capacity, f is the local solid volume fraction of the nanofluid, b is the volumetric thermal expansion coefficient of the base fluid, D_{B} is the Brownian diffusion coefficient, D_{T}is the thermophoretic diffusion coefficient, T is the local temperature, g is the acceleration due to gravity, and B_{0 }is the constant magnetic field.
The boundary conditions are taken to be
at , (5)
as (6)
3. Similarity transformations
The following quantities are introduced to transform Eqs. (2)–(4) into ordinary differential equations:
(10)
with the local Rayleigh number, which is defined as
, (11)
and the stream function y( x,y) is defined such that
. (12)
Therefore, the continuity equation from Eq. (1) is satisfied identically. After some algebraic manipulation, the momentum, energy, and solidvolume fraction equations are obtained as follows:
, (13)
, (14)
, (15)
where primes denote differentiation with respect to h and the nondimensional parameters—the Prandtl number (Pr), buoyancyratio parameter (Nr), Brownian motion parameter (Nb), thermophoresis parameter (Nt), Lewis number (Le), and magnetic parameter(M)—are defined as follows:
and
The corresponding boundary conditions are as follows:
at , (16)
as . (17)
The reduced local Nusselt number Nur and the reduced local Sherwood number Shr can be introduced and represented as follows:
.
4. Analytical solution by the HAM
Eqs. (13)–(15) are solved under corresponding boundary conditions (16) and (17) by using the HAM. For the HAM solutions, we choose the initial guesses and auxiliary linear operators in the following form:
, (20)
, (21)
and
(22)
c_{1}, c_{2},and c_{3} are constants, pÎ[0,1] denotes the embedding parameter, and h_{1}, h_{2}, and h_{3} indicate the nonzero auxiliary parameters. We then construct the following problems:
Zerothorder deformation problems,
(23)
(24)
(25)
(26)
(27)
(28)
and
(29)
(30)
(31)
For p = 0 and p = 1, we have
(32)
(33)
(34)
Due to the Taylor series with respect to p,the following is provided:
(35)
(36)
(37)
(38)
(39)
(40)
and thus, the m^{th}order deformation problems are as follows:
(41)
(42)
(43)
and
,(44)
, (45)
, (46)
where
(47)
(48)
(49)
and
, (50)
for which h is chosen in such a way that these three series are convergent at p = 1.Therefore,we have
, (51)
, (52)
. (53)
4.1 Convergence of the HAM
As Liao [2426] has pointed out, the convergence rate of approximation for the HAM solution strongly depends on the value of auxiliary parameter h. Fig. 1 clearly depicts that the range for the admissible value of h is ‑1.1 £ h £ 0.1. Our calculation for this case indicates that h= 0.3 for the whole region h.
Fig.1 s¢¢(0), θ ¢(0), and f ¢(0) plots for determining the optimum h coefficient. 
5. Numerical method for the solution
Nonlinear coupled differential Eqs.(13)–(15), along with boundary conditions (16) and (17), form a twopoint boundary value problem thatis solved using a shooting technique along with the fourthorder Runge–Kutta integration scheme by converting it into an initial value problem. In this method, we have to choose a suitable finite value of , say .We set the following firstorder system:
, (54)
with the boundary conditions
and . (55)
To solve (54) with (55) as an initial value problem, we need the values for y_{3}(0) i.e. s''(0), y_{5}(0) i.e. q¢(0), and y_{7}(0) i.e. f¢(0), but no such values are given. The initial guess values for s''(0),q¢(0), and f¢(0) are chosen, and the fourthorder Runge–Kutta integration scheme is applied to obtain the solution. Then, we compare the calculated values of s'(h),q(h),and f(h)at h_{¥} with the given boundary conditions s'(h_{¥}) = 0, q (h_{¥}) = 0, and f(h_{¥}) = 0and adjust the values of s''(0), q¢(0), and f'(0) using the shooting technique to give a better approximation for the solution. The process is repeated until we get the correct results up to the desired accuracy level of10^{8}, which fulfils the convergence criterion.
6. Results and discussion
The analytical solutions for the hydromagnetic flow of an incompressible viscous nanofluid over a vertical plate were obtained using the HAM method. The obtained analytical solutions were verified by the numerical solutions obtained using the fourthorder Runge–Kutta method. The results are displayed with the help of graphical illustrations. To test the accuracy of these analytical and numerical solutions, we compared the resulting calculated values with those reported in Bejan [34]. The comparison, shown in Table 1,was found to be in good agreement.
The effects of the Prandtl number and magnetic parameter on the dimensionless velocity profile along the vertical plate are shown in Fig. 2. A comparison of the velocity profiles for the two different Prandtl numbers shows that the velocity profile rises as Pr increases, unlike for the flow of a regular fluid over a vertical plate [16]. It also shows that the presence of the magnetic parameter reduces the dimensionless velocity. This is due to the fact that, when a transverse magnetic field is applied to an electrically conducting fluid, it gives rise to a resistive force known as the Lorentz force. This force causes the fluid to experience resistance by increasing the friction between its layers, which decreases velocity. Thus, the increasing values of the magnetic parameter decrease the velocity of the nanofluid.
Fig. 3 shows the effects of the buoyancy ratio on the dimensionless velocity profile in the presence of a magnetic field. The velocity profile decreases as the buoyancy ratio increases. Fig. 4 shows the effects of the Lewis number on the dimensionless velocity profile in the presence of a magnetic field for the selected parameters. It is clear that the velocity is enhanced as the Lewis number increases. Fig. 5 shows the effects of Brownian motion and thermophoresis parameters on the dimensionless velocity profile. It can be seen that the increasing values of Nb and Nt enhance the velocity profile of the nanofluid.
The effects of the magnetic parameter on the dimensionless temperature profile along the vertical plate are shown in Fig. 6. It can be seen that the increasing values of the magnetic parameter causes the nanofluid to become warmer and, therefore, increase in temperature, which is similar to the temperature profile in the natural convective boundary layer of a regular fluid. Fig. 7 displays the effects of the Prandtl number on the temperature profile. It can be noted that the temperature decreases with the increasing values of the Prandtl number. This is due to the fact that the thermal boundarylayer thickness decreases with an increase in the Prandtl number.
Fig. 8 demonstrates the effects of the buoyancy ratio on the dimensionless temperature profile in the presence of magnetic field. From this figure, it is apparent that the temperature increases as the buoyancy ratio increases. Fig. 9 depicts the effects of the Lewis number on the dimensionless temperature profile in the presence of a magnetic field. It can be observed that the increasing values of Le decrease the temperature profile. Fig. 10 illustrates the dimensionless temperature profile in the presence of magnetic field and nanofluid parameters for different values of the Brownian motion and thermophoresis parameters. It can be noted that the Brownian motion and thermophoresis parameters augment the temperature profile. This is because increasing Nt means a higher thermophoresis force, which tends to move nanoparticles from hot to cold areas and consequently increases the fluid temperature.
The magnetic parameter’s effect on the dimensionless concentration profile along the vertical plate is shown in Fig. 11. From the figure, it can be seen that the dimensionless nanosolid fraction increases as the magnetic parameter increases. Fig. 12 shows the effects of the Prandtl number on the dimensionless nanoparticle volume fraction. It can be observed that the nanoparticle volume fraction decreases as the Prandtl number increases.
Fig. 13 depicts the effects of the buoyancy ratio on the dimensionless nanoparticle volume fraction profile. The nanoparticle volume fraction increases as the buoyancy ratio increases.
Fig.14 shows the effect of the Lewis number on the dimensionless nanoparticle volume fraction profile in the presence of a magnetic field. It is interesting to note that the increasing values of Le decrease the nanoparticle volume fraction. Close observation of Figs. 9 and 14 revealed that the increasing values of the Lewis number arrest the concentration boundarylayer thickness.
Fig. 15 exhibits the effects of the Brownian motion and thermophoresis parameters on the dimensionless nanoparticle volume fraction profile in the presence of a magnetic field. It is clear that an increase in the values of both nanofluid parameters, thermophoresis and Brownian motion, causes a decrease of the nanosolid fraction. Due to the presence of nanoparticles in a nanofluid system, Brownian motion of the particles takes place, and by increasing it, the heattransfer characteristics of the fluid changes, which results in increased temperature. Moreover, a slight decrease in the nanoparticle volume boundarylayer thickness is observed with increasing Nb.
Figs. 16 and 17 show the effects of the Brownian motion parameter, thermophoresis parameter, magnetic parameter, and Lewis number on the reduced Nusselt number and local Sherwood number, respectively. It can be noted that the reduced Nusselt number decreases with magnetic parameter, Brownian motion parameter, thermophoresis parameter, and Lewis number (Fig. 16).
The local Sherwood number decreases with magnetic parameter and increases with Brownian motion parameter, thermophoresis parameter, and Lewis number (Fig. 17).
Table 2 shows the values of the reduced Nusselt and Sherwood numbers. The reduced Nusselt and Sherwood numbers decrease with the increasing values of Nr and M. The increasing values of Pr increase both the reduced Nusselt and Sherwood numbers. The parameters Nb and Le decrease the values of the Nusselt number and increase the Sherwood number. Table 2 provides the information about the heat and masstransfer characteristics of the flow in a convenient form for research and engineering calculations.
Fig.3 Effects of the buoyancyratio on the dimensionless velocity profiles for Pr = 1, Le = 10, M = 0.5, and Nb = Nt = 0.1. 
Fig.2 Effects of the magnetic parameter and Prandtl number on the dimensionless velocity for Le = 1 and Nb = Nt = Nr = 0.1.

Fig.4 Effects of the Lewis number on the dimensionless velocity profiles for Pr = 1, M = 0.5, and Nb = Nt = Nr = 0.1. 
Fig.7 Effects of the Prandtl number on the temperature profiles for M = 0.5, Le = 1, and Nb = Nt = Nr = 0.1.

Fig.6 Effects of the magnetic parameter on the dimensionless temperature profiles for Pr = 1, Le = 1, and Nb = Nt = Nr = 0.1.

Fig.5 Effects of Brownian motion and thermophoresis parameters on dimensionless velocity profiles for Pr = 1, Le = 10, M = 0.5, and Nr = 0.1. 
Fig.12 Effects of the Prandtl number on the dimensionless solid volume fraction of the nanofluid profiles for Pr = 1, Le = 1, and Nb=Nt=Nr=0.1.

Fig.9 Effects of the Lewis number on the dimensionless temperature profiles for Pr = 1, Nb = Nt = Nr = 0.1, and M = 0.5.

Fig.11 Effects of themagnetic parameter on the dimensionless solid volume fraction of the nanofluid profiles for Pr = 1, Le = 1, and Nb = Nt = Nr = 0.1.

Fig.8 Effects of the buoyancy ratio on the dimensionless temperature profiles for Pr = 1, Le = 10, Nb = Nt = 0.1, and M = 0.5.

Fig.10 Effects of Brownian motion and thermophoresis parameters on the dimensionless temperature profiles for Pr = 1, Le = 10, M = 0.5, and Nr = 0.1. 
Fig.13 Effects of buoyancyratio on the dimensionless solid volume fraction of the nanofluid profiles for Pr = 1, Le = 10, Nb = Nt = 0.1, and M = 0.5.

Fig.14 Effects of the Lewis number on the dimensionless solid volume fraction of the nanofluid profiles for Pr=1, Nb = Nt = Nr = 0.1, and M = 0.5.

Fig.15 Effects of the Brownian motion parameters on the dimensionless temperature profiles for Pr = 1, Le = 10, M = 0.5, and Nt = Nr = 0.1. 
Figure 17. Effects of the Brownian motion parameters, Lewis number, and the magnetic parameter on the local Sherwood number for Pr = 7 and Nr = 0.1. 
Fig.16 Effects of Brownian motion parameters, the Lewis number and magnetic parameter on the reduced Nusselt number for Pr = 7 and Nr = 0.1. 
Table 1. Comparison test results. Values of the reduced Nusselt number Nur = Ra_{x}^{1/4} Nu in the limiting case of a regular fluid. The present results are with Le = 10 and Nr = Nb = Nt = 10^{5}.

Pr 
Bejan [34] 
Present results (absence of magnetic parameter) 

Analytical 
Numerical 

1 
0.401 
0.40103 
0.40102817 
10 
0.465 
0.46496 
0.46496287 
100 
0.490 
0.49000 
0.49000028 
Table 2. Variation of Nur and Shr with Pr, Nb, and Nr for Nt = 0.1 and Le = 10.
Le 
M 
Nb 
Nr 
Pr=1 
Pr=5 

Nur 
Shr 
Nur 
Shr 

Analytical 
Numerical 
Analytical 
Numerical 
Analytical 
Numerical 
Analytical 
Numerical 

10 
0 
0.1 
0.1 
0.35423 
0.35422747 
1.00337 
1.00336781 
0.39913 
0.39912839 
1.07059 
1.07058521 



0.3 
0.34768 
0.34767813 
0.97818 
0.97818444 
0.39186 
0.39186369 
1.04546 
1.045461295 



0.5 
0.34059 
0.34058452 
0.95061 
0.95060987 
0.38405 
0.38404555 
1.01815 
1.01814560 
10 
0 
0.1 
0.1 
0.35423 
0.35422747 
1.00337 
1.00336781 
0.39913 
0.39912839 
1.07059 
1.07058521 


0.3 

0.30478 
0.30478323 
1.05170 
1.05170420 
0.34438 
0.34437949 
1.12009 
1.12008705 


0.5 

0.26100 
0.26100299 
1.06575 
1.06575282 
0.29570 
0.29569842 
1.13521 
1.13520899 
10 
0 
0.1 
0.1 
0.35423 
0.35422747 
1.00337 
1.00336781 
0.39913 
0.39912839 
1.07059 
1.07058521 

1 


0.287352 
0.28735186 
0.86314 
0.86313586 
0.33916 
0.33915732 
0.95827 
0.95827136 

2 


0.24690 
0.24690692 
0.77458 
0.77458149 
0.30419 
0.30418486 
0.89014 
0.89013613 
10 
1 
0.1 
0.1 
0.287352 
0.28735186 
0.86314 
0.86313586 
0.33916 
0.33915732 
0.95827 
0.95827136 
12 



0.28715 
0.28714701 
0.93599 
0.93598525 
0.33888 
0.33888312 
1.03576 
1.03576257 
14 



0.28697 
0.28697711 
1.00044 
1.00044277 
0.33865 
0.33865419 
1.10433 
1.10433227 
Fig.17 Effects of the Brownian motion parameters, Lewis number, and the magnetic parameter on the local Sherwood number for Pr = 7 and Nr = 0.1. 
7. Conclusion
The hydromagnetic flow of an electrically conducting nanofluid past a vertical plate was studied both analytically and numerically. The governing equations for this investigation were solved analytically using the HAM, and the results were verified using numerical solutions obtained by the fourthorder Runge–Kutta method along with the shooting technique. The following specific conclusions were derived from this study:
 It was found that the dimensionless velocity increases with the decrease of the magnetic parameter. The temperature and the concentration profiles increase with the increasing values of the magnetic parameter. The increasing values of the Prandtl number increase the velocity profile and decrease the temperature and solid volume fraction profiles in the flow region.
 It was noted that the Brownian motion and thermophoresis parameters enhance the velocity of the temperature profiles but suppress the concentration profile.
 The increasing values of the buoyancy ratio decrease the velocity profile and increase the temperature and concentration profiles in the flow region.
 The Nusselt number decreases with the magnetic parameter, Brownian motion parameter, thermophoresis parameter, and Lewis number
 The local Sherwood number decreases with the magnetic parameter and increases with the Brownian motion parameter, thermophoresis parameter, and Lewis number.
Nomenclature
D_{B} Brownian diffusion coefficient
D_{T} Thermophoretic diffusion coefficient
f rescaled nanoparticle volume fraction
g gravitational acceleration vector
k thermal conductivity
Le Lewis number
M magnetic parameter
Nr buoyancyratio parameter
Nb Brownian motion parameter
Nt thermophoresis parameter
Nur reduced Nusselt number
Pr Prandtl number
Ra_{x}_{ }local Rayleigh number
T temperature
T_{w} temperature at the vertical plate
T_{¥} ambient temperature
(x,y) Cartesian coordinates
Greek symbols
a Thermal diffusivity
b volumetric expansion coefficient h similarity variable
dimensionless temperature
dynamic viscosity of the fluid
kinematic viscosity
fluid density
nanoparticle mass density
heat capacity of the fluid
effective heat capacity of the nanoparticle material
nanoparticle volume fraction
nanoparticle volume fraction at the vertical plate
ambient nanoparticle volume fraction, attained as y, tends to infinity
stream function
Acknowledgments
The authors wish to express their sincere thanks to the honorable referees for their valuable comments and suggestions to improve the quality of the paper.
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چکیده های فارسی 
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