Medical, Pharma, Engineering, Science, Technology and Business

^{1}Science and Technology University, Department of Mechanical Engineering, P.O. Box 484, Babol, Iran

^{2} Babol University of Technology, Department of Mechanical Engineering, P.O. Box 484, Babol, Iran

- *Corresponding Author:
- Ganji DD

Babol University of Technology

Department of Mechanical Engineering

P.O. Box 484, Babol, Iran

**Tel:**+98 111 32 34 501ddg_davood@yahoo.com

E-mail:

**Received date:** August 28, 2016; **Accepted date:** September 23, 2016; **Published date:** September 27, 2016

**Citation: **Gholinia M, Ganji DD, Poorfallah M, Gholinia S (2016) Analytical and
Numerical Method in the Free Convection Flow of Pure Water Non-Newtonian
Nano fluid between Two Parallel Perpendicular Flat Plates. Innov Ener Res 5:142.

**Copyright:** © 2016 Gholinia M, et al. This is an open-access article distributed
under the terms of the Creative Commons Attribution License, which permits
unrestricted use, distribution, and reproduction in any medium, provided the
original author and source are credited.

**Visit for more related articles at** Innovative Energy & Research

In this research, free convection of a non-Newtonian silver-water nanofluid between two infinite parallel perpendicular flat plates is investigated. The Maxwell Garnetts (MG) nanofluid model is used in this work. The basic partial differential equations are reduced to the ordinary differential equations which are solved analytically using Homotopy Perturbation Method (HPM). The impact of various physical parameters such as nanoparticle volume fraction (φ ), dimensionless non-Newtonian viscosity ( ) and Eckert number (Ec) on the velocity and dimensionless temperature profiles is studied. The comparison of the results of HPM, Akbari-Ganji’s Method (AGM), Collocation Method (CM), the fourth-order Runge-Kutta numerical Method (NUM) and FlexPDE software results shows excellent complying in solving this problem. Also, this research shows that AGM and HPM are powerful methods to solve non-linear differential equations, such as the problem raised in this research.

Free convection; Non-Newtonian; Nanofluid; Homotopy Perturbation Method; Akbari-Ganji’s method; Collocation method; FlexPDE software

In **fluid mechanics**, we study the particles’ behavior at any point
within the range of different physical conditions. Mathematical models
are used to describe physical phenomena in fluid mechanics for a
variety of fluids such as Newtonian and non-Newtonian fluids. More
engineering problems, especially some of the heat transfer equations
are non-linear. For this reason, resolving these difficult problems has
been a controversial issue for mathematicians, physicists and engineers.
Some equations are solved by numerical solutions; some are solved
using different analytical methods. Methods that can be introduced to
examine the non-linear problems such as the Differential Transform
Method (DTM) [1-2], Least Square Method (LSM) [3-4], Akbari-
Ganji’s Method (AGM) [5-8], Hamiltonian Approach [9], Variational
Iteration Method (VIM) [10], and Adomian’s Decomposition Method
(ADM) [11], many methods are not considered in this study because
of brevity. One of the semi-analytical methods which does not need
small parameters is the Homotopy Perturbation Method (HPM).
The homotopy perturbation method, proposed first by He in 1998
and was further developed and improved by He [12]. This method in
most cases provides fast convergence to solve series. Usually, due to
the small number of trial and error, leads to the achievement of highprecision
solutions. This new method is applied in a lot of researches
in engineering sciences [13-17]. Heat transfer by free convection
frequently occurs in many physical problems and engineering
applications such as **geothermal** systems, heat exchangers, chemical
catalytic reactors, fiber and granular insulation, packed beds, petroleum
reservoirs and nuclear waste repositories [18-21]. In view of its
importance, the flow of Newtonian and **non-Newtonian** fluids through
two infinite parallel perpendicular flat plates has been examined by a
large number of researchers. The natural convection problem between
vertical flat plates for a certain class of non-Newtonian fluids has been
carried out by Bruce and Na [22]. Other laminar natural convection
problems involving heat transfer have been also studied by Ziabakhsh
and Domairry [23]. Rajagopal and Na [24] presented an analysis for
the natural convection in non-Newtonian fluid flow between two
parallel plates. Yoshino et al. [25] presented a new numerical method for
incompressible non-Newtonian fluid flows based on the lattice Boltzmann
method (LBM). Pawar et al. [26] carried out an experimental study on
isothermal steady state and non-**isothermal** unsteady state conditions in
helical coils for Newtonian and non-Newtonian fluids. Also, numerous
models and methods have been proposed by different authors to study
convective flows of nanofluids and we mention here the papers written
by Khan and Pop [27] Vajravelu et al. [28] and Yacob et al. [29]. The
main aim of this work is to present the effects of nanoparticle volume
fraction, dimensionless non-Newtonian viscosity and Eckert number
on velocity profiles and temperature profiles in the flow of nanofluids
between two infinite parallel perpendicular flat plates. The Maxwell–
Garnetts (MG) nanofluid model [30] is used in this work. The reduced
ordinary differential equations are solved analytically using HPM. The
comparison of the results of HPM, Akbari-Ganji’s Method (AGM),
Collocation Method (CM), the fourth-order Runge-Kutta numerical
Method (NUM) and FlexPDE software results shows excellent
complying in solving this non-linear problem.

**Description of the problem**

The schematic theme of the problem is shown in **Figure 1**. This
figure includes two parallel plates, perpendicular to each other, in
which there is a non-Newtonian fluid flowing due to the free convection
between two parallel plates. The distance between the two plates is 2b.
The walls at x= +b and x= -b are held at constant temperatures T_{2} and
T_{1} , respectively, where T_{2} > T_{1} . The difference between the two walls temperature makes the fluid near the wall, to rise at x= -b and fall at x=
+b. The fluid is a water-based **nanofluid **containing silver. It is assumed
that the source of fluid and nanoparticles are in **thermal equilibrium**,
and no slip occurs between them. The thermo-physical properties of
the nanofluid are listed in **Table 1** [31].

Material | Density (kg/m_{3}) |
C_{p} (J/kg.k) |
K (w/m.k) |
β× 10^{-5}(k^{-1}) |
---|---|---|---|---|

Pure water | 997.1 | 4179 | 0.613 | 21 |

Silver | 10500 | 235 | 429 | 1.89 |

**Table 1: **Thermo-physical properties of water and nanoparticles [31].

The effective density ( ρ_{n f} ), the effective dynamic viscosity ( μ _{n f} ), the
heat capacitance ( ρC_{p})_{n f} and the thermal conductivity ( n f K ) of the
nanofluid can be expressed as where (φ ) is the solid volume fraction.

Rajagopal [24] has demonstrated that by using the similarity variables:

Under these assumptions and following the nanofluid model proposed by Maxwell–Garnetts (MG) model [30], the Navier-Stokes and energy equation can be reduced to the following pair of ordinary differential equations:

Where Prandtl number ( pr ), Eckert number ( Ec ),
dimensionless non-Newtonian viscosity (δ ) and A_{1} have the
following forms:

The appropriate boundary conditions are:

In this section three methods have been examined:

**Homotopy perturbation method (hpm)**

To explain the basic ideas of this method, we consider the following non-linear differential equation:

A(u) − f (r) = 0, r∈Ω (11)

With the boundary condition of:

Where A is a general differential operator, B a boundary operator, f (r) a known analytical function, (Γ ) is the boundary of the domain ( Ω ) and denotes differentiation along the normal drawn outwards from ( Ω ).

A can be divided into two parts which are L and N, where L is linear part and N is non-linear part. Eq. (11) can therefore be rewritten as follows:

L(u) + N(u) − f (r) = 0 (13)

**Homotopy** perturbation structure is shown as follows:

H(ν , p) = L(ν ) + L(u_{0} ) + pL(u_{0} ) + p(N(ν ) − f (r)) = 0 (14)

Where,

ν (r, p) :Ω×[0,1]→R (15)

In Eq. (14), p∈ [0, 1] is an embedding parameter and u_{0} is the first
approximation that satisfies the boundary condition. We can assume that
the solution of Eq. (14) can be written as a power series in p , as following:

ν =ν + pν_{1} + p_{2}ν_{2} .............. (16)

and the best approximation for solution is:

**Collocation method (cm)**

Weighted residual method was first introduced by Ozisk [32]
to solve the differential equation in **heat transfer**, Collocation and Galerkin method are analytical methods that are based on the weighted
residual method.

Suppose a differential operator D, is applied on a function u to produce a function p

D(u(x)) = p(x) (18)

u is approximated by a function , which is a linear combination of basic functions chosen from a linearly independent set. That is,

Now, when substituted into the differential operator, D, the result of the operations is not, in general, p(x) . Hence an error or residual will exist as

The main idea of the CM is to force the residual to zero in some average sense over the domain. That is:

Where the number of weight functions Wi is exactly equal to the
number of unknown constants c_{i} in function. The result is a set
of n **algebraic equations **for the unknown constants ci. For collocation
method, the weighting functions are taken from the family of Dirac
functions in the domain. That is, W_{i}x) =δ(x − x_{i}) . The Dirac
function has the property of:

Also, the residual function in Eq. (20) must be forced to be zero at specific points.

**Akbari-Ganji’s Method (AGM)**

Boundary conditions and initial conditions are required differential equation according to the physic of the problem. Therefore, we can solve every differential equation with any degrees. In order to comprehend the given method in this paper, two differential equations governing on engineering processes will be solved in this new manner. The nonlinear differential equation of p which is a function of u, the parameter u which is a function of x and their derivatives are considered as follows:

The non-linear differential equation p (which is a function of u ), the parameter u (which is a function of x ), and their derivatives are considered as follow:

Boundary conditions:

To solver the first differential equation, with respect to the boundary conditions in x = L in Eq. (23), the series of letters in the n th order with constant coefficients, which is the answer of the first differential equation, is considered as follows:

The boundary conditions are applied to the function as follows:

a) The application of the boundary conditions for the answer of differential Eq. (24) is in the form of

If x =0

And when x = L

b) After substituting Eq. (26) into Eq. (22), the application of the boundary conditions on differential Eq. (22) is done according to the following procedure:

With regard to the choice of n; (n < m) sentences from Eq. (24) and in order to make a set of equations which is consisted of (n + 1) equations and (n + 1) unknowns, we confront with a number of additional unknowns which are indeed the same coefficients of Eq. (24). Therefore, to remove this problem, we should derive m times from Eq. (22) according to the additional unknowns in the afore-mentioned set differential equations and then this is the time to apply the boundary conditions of Eq. (23) on them.

Application of the boundary conditions on the derivatives of the
differential equation p_{k} in Eq. (28) is done in the form of

The ( n +1) equations can be made from Eq. (25) to Eq. (30)
so that ( n +1) unknown coefficients of Eq. (24) for example,
a_{0},a_{1},a_{2},a_{3} ........... a_{n} can be computed. The answer of the nonlinear
differential Eq. (22) will be gained by determining coefficients
of Eq. (24).

According to different works with AGM method and similar methods have been shown the AGM method is powerful method to solve non-linear problems. Because this method solves different variable such as power series, Sine and exp function at the same time and shows acceptable outcomes.

**Application of described methods in the problem**

In this section, we will apply the HPM to non-linear ordinary differential Eqs. (6) and (7). According to the HPM, we construct a homotopy suppose the solution of Eqs. (6) and (7) has the form:

We consider V(x) and θ(x) as follows:

Substituting Eq. (32), into Eq. (31), and some simplification and rearranging on powers of P-terms, we have:

Solving Eqs. (33) and (35) with boundary conditions:

In the same manner, the rest of components were obtained by using the Maple package, that we obtain (32) parameters of it. According to HPM, we can conclude:

Since trial function must satisfy the boundary conditions in Eq. (10), so they will be considered as

We select the collocation locations x = 1/5 to 4/5 which are
evenly spaced throughout the domain. Introducing these values into
the residual Eq. (40). Thus we have eight algebraic equations for
the determination of the eight unknown coefficients c_{1} to c_{8} . For
example, Using collocation method with (Pr=0.5, Ec=0.5, φ =0.05, δ
=0.5) V (x) and θ (x) are as follows:

**Akbari-Ganji’s Method (AGM)**

In order to solver the differential Eqs. (6) and (7), an answer function is considered as a finite series in the form of

The given answer function has the constant coefficients c_{0} a to c_{3} a
and 0 c to 3 c , which can easily be computed by applying the initial
conditions from Eq. (10). It is notable that the more numbers of series
sentences of Eq. (42), the more precise the answer, and the answer is
tended to the exact solution [8]. For example, Using Akbari-Ganji’s
Method with (Pr=1, Ec =0.1, φ =0.01, δ =0.2) a_{0} to a_{3} and c_{0} to
c_{3} are as follows:

By substituting the constant coefficients from Eq. (43) into Eq. (42), the solution of the non-linear differential Eqs. (6) and (7) can be gained as

**Solution whit flexpde software**

In this study, we first introduce the **FlexPDE software**. FlexPDE
software is simple modeling software based on finite element method
for coding. This means that FlexPDE is a finite element method for
coding. This means that FlexPDE converts written codes and partial
differential equations to a finite element model. FlexPDE performs all
necessary functions for solving partial differential equations as follows.
Editing and preparation of texts, creating finite element network (grid),
finite element solver organization for answers, adjusting the graphical
output to delivered results. FlexPDE is a software that does not have a
predetermined range of issues. The user is free for selection of equation.
In addition, unlike commercial software, there is no doubt about what
is the deal because the text fully describes the system of equations and
the problem ranges. Another advantage of FlexPDE software is that
equations or phrases can be added based on problem requirements.
The problems that FlexPDE is capable of solving. Problems such as first
or second order partial differential equations in (one, two or three)
dimensional Descartes and **geometry**, one-dimensional spherical or
cylindrical geometry, sustainable or transition system [33], linear or
non-linear equations [34-35], eigen values problems, and several other
issues are the problems that this simple software is able to solve. In this
software, boundary conditions are applied when defining the problem
boundaries. The primary types of boundary conditions are boundary
are NATURAL, VALUE. NATURAL boundary conditions determine
the value of a variable on the boundary of the domain NATURAL
boundary conditions determine the amount of charge on the boundary
of the domain Problem definition is divided into different parts in
FlexPDE software. Determining variables, determining the geometry
of the problem, determining material properties, determining the
graphical output, applying boundary conditions include the parts
of this simple software. The coding rules in this software are that
they are not sensitive to lowercase or uppercase. A differentiation
such as is shown like . All coordinate systems names are
known valid such as second derivations like dxx (H) and differential
functions such as Curl, Grad and Div. In this study, we compare
the results of FlexPDE software with obtained results from HPM
by writing FlexPDE software codes for Eqs. (6) and (7). For brevity,
we evaluate only one case. For **Figure 7**, when (φ =0.03, δ=2, Ec
=2, Pr=2), we have evaluated the velocity and temperature profiles.
FlexPDE software codes for Eqs. (6) and (7) at (φ =0.03, δ=2, E =2,
Pr=2) are given in Appendix A.

The comparison of the results of HPM with the results of the AGM, CM, Num and FlexPDE software results was conducted. Also, this research shows that AGM and HPM are powerful methods to solve non-linear differential equations.

**Results of (HPM)**

**Comparison of numerical results with the results of (HPM)**

**Table 2**

V (x) | θ (x) | ||||
---|---|---|---|---|---|

x | Nu | HPM | Error | NuHPM | Error |

-1 | 0.000000 | -1.00e-12 | 1.00e-12 | 0.500000 0.500000 | 0.000000 |

-0.8 | 2.49e-02 | 0.024978 | -8.90e-05 | 0.404852 0.404687 | 0.000165 |

-0.6 | 0.034361 | 0.034499 | -0.000138 | 0.307571 0.307541 | 2.94e-05 |

-0.4 | 0.316000 | 0.031771 | -0.000171 | 0.210070 0.210225 | -0.000150 |

-0.2 | 0.020505 | 0.020717 | -0.000212 | 0.112157 0.112397 | -0.000240 |

0.0 | 0.005037 | 0.005283 | -0.000246 | 0.013034 0.013246 | -0.000210 |

0.2 | -1.09e-02 | -1.07e-02 | -0.000254 | -8.77e-02 -8.76e-02 | -0.000140 |

0.4 | -2.35e-02 | -2.32e-02 | -0.000243 | -0.189907 -0.189774 | -0.000130 |

0.6 | -2.85e-02 | -2.83e-02 | -0.000234 | -0.292665 -0.292458 | -0.000210 |

0.8 | -2.19e-02 | -2.17e-02 | -0.000206 | -0.395570 -0.395312 | -0.00026 |

1 | 0.000000 | 1.00e-12 | -1.00e-12 | -0.500000 -0.500000 | 0.000000 |

**Table 2:** Comparison between Numerical results and HPM for V(x) and θ(x) when δ=1, Pr=6.2, ϕ=0.01, Ec=1.

**Table 3**

V (x) | θ (x) | |||||
---|---|---|---|---|---|---|

x | Nu | HPM | Error | NuHPM | Error | |

-1 | 0.000000 | 2.00e-11 | -2.00e-11 | 5.00e-01 0.500000 | 0.000000 | |

-0.8 | 2.64e-02 | 0.026707 | -3.18e-04 | 0.412251 0.410966 | 1.28e-03 | |

-0.6 | 0.037801 | 0.038229 | -4.28e-04 | 0.318437 0.317645 | 7.92e-04 | |

-0.4 | 0.036652 | 0.037135 | -4.83e-04 | 0.223868 0.223923 | -5.43e-05 | |

-0.2 | 0.026641 | 0.027227 | -5.87e-04 | 0.128579 0.129006 | -4.28e-04 | |

0.0 | 0.011684 | 0.012360 | -6.77e-04 | 0.030749 0.030992 | -2.43e-04 | |

0.2 | -4.42e-03 | -3.73e-03 | -6.90e-04 | -7.09e-02 -7.10e-02 | 9.29e-05 | |

0.4 | -1.78e-02 | -1.72e-02 | -6.59e-04 | -0.176022 -0.176076 | 5.45e-05 | |

0.6 | -2.44e-02 | -2.37e-02 | -6.60e-04 | -0.282871 -0.282354 | -5.16e-04 | |

0.8 | -1.96e-02 | -1.90e-02 | -6.13e-04 | -0.390062 -0.389033 | -1.03e-03 | |

1 | 0.000000 | 0.000000 | 0.000000 | 0.500000 -0.500000 | 0.000000 |

**Table 3:** Comparison between Numerical results and HPM forV(x) andθ(x) when δ=2, Pr=6.2, ϕ=0.07, Ec=2.

**Comparison of (CM) result with the result of (HPM)**

**Comparison of (CM) results and (AGM) results with the
results of (HPM)**

Comparison of FlexPDE software results with the results of (HPM)

In the present study, free convection of a non-Newtonian nanofluid
between two infinite parallel perpendicular flat plates has been
investigated. These equations were solved analytically using the HPM,
CM and AGM. In order to verify the accuracy of the present results, we
have compared HPM results with numerical methods (the fourth-order
Runge–Kutta method) and FlexPDE software. Comparison between
HPM and numerical method are presented in Tables A_{1}nd 2. For brevity,
only two models, when (Pr=6.2, Ec=1, φ =0.01, δ=1) and (Pr=6.2, Ec=2,
φ =0.07, δ=2), are discussed for velocity and temperature profiles.
As seen in these Tables A_{1}nd 2, for different values of x in the range of
[-1, 1] error rate is very low, which indicates the good complying between
the two methods. Also in this study the effect of various parameters such
as nanoparticle volume fraction (φ ), dimensionless non- Newtonian
viscosity (δ) and Eckert number (Ec) on velocity and dimensionless
temperature profiles examined. **Figures 2a and 2b** show the effect of
nanoparticle volume fraction (φ ) on the velocity and temperature
profiles when (Ec=1, =1, Pr=6.2). According to the **Figures 2a and
2b**, by increasing nanoparticle volume fraction (φ ), the velocity and
temperature profiles decline. **Figures 3a and 3b**, are shown the effect
of dimensionless non-Newtonian viscosity (δ) on the velocity and
temperature profiles, when (Ec=1, φ =0.05, Pr=6.2). The dimensionless
non-Newtonian viscosity indicates the relative significance of the
inertia effect compared to the viscous effect. According to the **Figures 3a
and 3b**, by increasing dimensionless non-Newtonian viscosity (δ) velocity and temperature profiles decline. The effects of the Eckert number
(Ec) on velocity profiles and temperature profiles when (δ=1, φ =0.05,
Pr=6.2) are shown in **Figures 4a and 4b**, respectively. It is observed that
due to the increasing velocity and temperature Eckert number (Ec)
increases, when we ignore viscosity dissipation, the minimum values
for the velocity and temperature are achieved. **Figures 5a and 5b **show
the comparison between the results of HPM and Collocation method.
In these Figures, we investigated the velocity and temperature profiles,
when (Ec=0.5, δ=0.5, Pr=0.5, φ =0.05). The slight error between the
Figures indicate the good complying between the two methods. **Figures
6a and 6b** show the comparison between the results of HPM, AGM
and Collocation method. In these figures we examine one mode of
(Ec=0.1, δ=0.2, Pr=1, φ =0.01) for velocity and temperature profiles.
In accordance with the observed figures error is very low between
the Figures, this means there is complying between the two methods.

**Figures 7a and 7b **show the comparison between the results of HPM
and FlexPDE. In these figures we examine one mode of (Ec=2, δ=2,
Pr=2, φ =0.03) for velocity and temperature profiles. The slight error in
Figures indicate that HPM is a high accuracy method to solve these issues.

This paper has analyzed the phenomenon of free convection flow of a non-Newtonian nanofluid between two infinite parallel perpendicular flat plates using the HPM, AGM, CM and Num. The effects of the nanoparticle volume fraction (φ ), dimensionless non- Newtonian viscosity (δ) and Eckert number (Ec) on the velocity and dimensionless temperature profiles have been determined for a silver and water nanofluid with a Prandtl number of Pr=6.2. The results of this study indicate that increasing in nanoparticle volume fraction (φ ) leads to the decrease in the thickness of the boundary layer of the velocity and temperature. While increasing Eckert number (Ec) increases the velocity and temperature profiles. The results show that the HPM method hase excellent agreement with the fourth-order Runge-Kutta numerical Method (NUM) and FlexPDE software. As the most important result of this study it was observed that HPM and AGM are powerful methods to solve this kind of non-linear problems.

**FlexPDE software codes:**

Title ‘ Nanofluid between two parallel perpendicular flat plates’

Coordinates cartesian1 Variables V theta

Definitions Ks=42 Kf=0.613 phi=0.03 Pr=2 Ec=2 delta=2

A= (Ks+2*Kf-2*phi*(Kf-Ks))/ (Ks+2*Kf+2*phi*(Kf-Ks))

Equations

V: dxx (V) +6*delta* (1-phi) ^2.5 * (dx (V)) ^2 * dxx (V) +theta=0

Theta: dxx (theta) +Ec*Pr* (((1-phi) ^ (-2.5))/A) *(dx (V)) ^2 + (2) *delta*Ec*Pr*(1/A) *(dx (V)) ^4=0

Boundaries region 1 start (-1) point value (V) =0 point value (theta) =0.5

Line to (1) point value (theta) = (-0.5) point value (V) =0

Plots elevation (V) from (-1) to (1) elevation (theta) from (-1) to (1) tecplot (V) Tecplot (theta)

End

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