# APPLICATION OF HOMOTOPY PERTURBATION METHOD TO HEAT TRANSFER IN NANOFLUIDS.

1. INTRODUCTION

Nanofluids are formed when nanoparticles such as oxide ceramics, nitrides, graphites etc are mixed with base fluids like water, polymer solutions and lubricants. Nanofluids possess heat transfer properties that can help address the energy demand and emission issues of the present world. They can be used for industrial cooling purposes and this could result in great energy savings and significantly reduce emission.

These properties and their potential benefits have made nanofluids an important area of research. Wang et al. (1999) in their research into the thermal conductivity of nanoparticle-fluid mixture provided suggestions to improve the conductivity of nanofluids. Do and Jang (2010) analyzed the effects of thermophysical properties Aluminum Oxide on the heat transfer of a flat micro heat pipe. Uddin et al. (2012) discovered that an increase in Newtonian heating enhanced the heat and mass transfer rate of a nanofluid. The dimensionless governing equations were solved using the Runge-Kutta-Fehlberg method coupled with shooting technique.

Hamad (2011) studied free convective flow of a nanofluid over a linearly stretching sheet in the presence of magnetic field. Oahimire et al. (2016) extended the work of Hamad by incorporating a thermal radiation parameter into the flow equations and solved them using the Runge-kutta Fehlberg method together with shooting technique. To the best of our knowledge, HPM has not been applied to solve the flow equations of Oahimire et al. (2016).

In this present study, HPM is applied to study the effects of volume fraction, magnetic field and buoyancy force on the rate of heat transfer of natural convection flow of a nanofluid over linearly stretching sheet in the presence of magnetic field. The Homotopy Perturbation method (HPM) is a technique based on the concept of the Homotopy from topology that was introduced by Dr. Ji-Huan He in 1998. It is a simple but effective method for solving non-linear partial differential equations. The basic idea is illustrated below. Consider a non-linear differential equation

[A(u) - f(r)] = 0 (1)

Where f(r) is a known analytic function and A(u) is a nonlinear differential operator which can be separated into 2 parts, one linear part, L and a non-linear part, N. i.e.

A(u) = L(u) + N(u) (2)

We construct a homotopy as follows

[mathematical expression not reproducible] (3)

where p is an embedding parameter that lies in the unit interval [0, 1] and vo is an initial guess of the solution to the equation. Setting the value of our small parameter to 0, we have the initial guess while setting its value to 1 gives us the original equation. This process of changing p from 1 to 0 is called a deformation.

[mathematical expression not reproducible] (4)

[mathematical expression not reproducible] (5)

According to the HPM, we assume our solution is in form of a series

[mathematical expression not reproducible]

We solve for un iteratively and setting p = 1 we have

[mathematical expression not reproducible]

This is the approximate solution to Eq. (1). We have the freedom of choice for the operator L. However great care must be taken to choose an operator which simplifies the solution process as the solution depends entirely on the choice of the L and the initial guess vo. Ayati and Biazar (2015) showed that in most cases, the HPM solution is convergent.
```NOMENCLATURE
a = Constant
g = Acceleration due to gravity
k = Thermal Conductivity
Pr = Prandtl Number
T = Fluid Temperature
Tw = Surface Temperature
T8 = Free Stream Temperature
u,v = Velocity Components
x,y = Cartesian Coordinates
f(x) = Dimensionless Stream Function
Gr = Grashof Number
Bo = Magnetic Field of Constant Strength
Ks = Rosseland Mean Absorption Coefficient
K = Thermal Conductivity Coefficient
GREEK SYMBOLS
[beta] = Thermal Expansion Coefficient
[micro] = Dynamic Coefficient of Viscosity
[thera] ([bern]) = Dimensionless Temperature
[eta] = Similarity Variable
[rho] = Fluid Density
[PSI] = Stream Function
[sigma]' = Stefan-Bottzman Constant
```

2. MATHEMATICAL FORMULATION

Consider a steady, two-dimensional flow of an incompressible viscous nanofluid past a linearly semiinfinite stretching sheet. Magnetic field of strength B0 is applied perpendicularly to the sheet. The nanofluid under consideration is water-based and contains Copper, Silver, Aluminum oxide and Titanium Dioxide. The nanofluid is assumed to be in thermal equilibrium. Following Oahimire et al. (2016), the governing equations are:

[mathematical expression not reproducible] (6)

[mathematical expression not reproducible] (7)

[mathematical expression not reproducible] (8)

[mathematical expression not reproducible] (9)

Where qr is the radiative heat flux, T' is the temperature of the fluid, x'andy' are the coordinates along and perpendicular to the sheet while u' and v' are the velocity components in the x' and y' directions respectively and a is a constant. The effective density ([rho]nf), effective dynamic viscosity ([mu]nf), heat capacitance ([rho]Cp)nf and the effective thermal conductivity (knf) of the nanofluid, in that order, are given as

[mathematical expression not reproducible] (10)

Where A is the solid volume fraction (A [not equal] 1), [mu]f is the dynamic viscosity of the base fluid, while [rho]f and [mu]s are the densities of the pure fluid and the nanoparticle respectively. The constants kf and ks are the thermal conductivities of the base fluid and the nanoparticle respectively. Using Rosseland approximation given by [mathematical expression not reproducible]' with Taylor's series expansion and differentiation, Eq. (8) becomes

[mathematical expression not reproducible] (11)

The following variables are used for transformation

[mathematical expression not reproducible] (12)

Eq. (12) transforms Eq. (6), (7) and (11) into the following

[mathematical expression not reproducible] (13)

[mathematical expression not reproducible] (14)

[mathematical expression not reproducible] (15)

Where M = [[sigma][beta]0]/a is the magnetic field parameter, Pr = [vf]/[[mirco]]/[mu]f is the Prandtl number, [mathematical expression not reproducible] is the radiation parameter, [mathematical expression not reproducible] is the Grashof number and the corresponding boundary conditions are

u = x, v = 0,[theta] at = at y = 0 (16)

[mathematical expression not reproducible]

To satisfy Eq. (7) we apply the stream function [mathematical expression not reproducible] our equations reduce to

[mathematical expression not reproducible] (17)

[mathematical expression not reproducible] (18)

[mathematical expression not reproducible] (19)

3. METHOD OF SOLUTION

The transformed non-linear equations can be written as

[mathematical expression not reproducible]

Where [mathematical expression not reproducible]

We construct the homotopy of the transformed equations as follows

[mathematical expression not reproducible]

And

[mathematical expression not reproducible]

We assume f and [theta] in the following form

[mathematical expression not reproducible]

and group the terms according to the order: For order zero, we have

[mathematical expression not reproducible]

With boundary conditions

[mathematical expression not reproducible]

For order one, we have

[mathematical expression not reproducible]

[mathematical expression not reproducible]

With boundary conditions

[mathematical expression not reproducible]

For order two, we have

[mathematical expression not reproducible]

With boundary conditions

[mathematical expression not reproducible]

Solving the equations with their respective boundary conditions, we have the following solutions

[mathematical expression not reproducible]

Where

[mathematical expression not reproducible]

We can calculate the value of the constant coefficients using the boundary conditions. Following standard practice, we replace the boundary condition [eta] = [infin] with [eta] = 6.

4. DISCUSSION AND RESULTS

Numerical evaluation of the solutions was performed with mathematical software "Matlab" and the results are presented in tabular form. This was done to illustrate effect of some governing parameters involved. The rate of heat transfer for different value of volume fraction (A), magnetic parameter(M) and Grashof number(Gr) are obtained as shown in table 2. We notice that an increase in the values of A, M and Gr led to an increase in the values of the heat transfer coefficient -[theta](0).

The therrmophysical properties of nanoparticles used in the evaluation as given by Hamad (2011) are shown below.

These values where used together with the solutions to obtain the following table showing the effects of varying different flow parameters on the heat transfer of the nanofluid.

5. CONCLUSION

In this work, the dimensionless equations of the governing equations were solved with HPM and the effects of M, Gr and A on heat transfer are presented in Table 2. And we notice that increasing the values of M, Gr and A leads to a corresponding increase in the rate of heat transfer in all of the nanoparticles considered. This is in agreement with the solutions gotten using the Runge-Kutta-Fehlberg method by Oahimire et al. (2016). This shows that He's Homotopy Pertubation method is an effective method for solving similar flow problems.

REFERENCES

Ayati, Z. and Biazar, J. (2015). "On the Convergence of the Homotopy Perturbation Method." Journal of the Egyptian Mathematical Society, Issue 23. pp. 424-428

Do, K. H. and Jang S. (2010). "Effect of nanofluids on the thermal performance of a flat micro heat pipe with a rectangular grooved wick." International Journal of Heat and Mass Transfer, 53, pp 2183-2192

Hamad M.A.A. (2011). "Analytical solution of natural convection flow of a nanofluid over a linearly stretching sheet in the presence of magnetic field." International Communications in Heat and Mass Transfer, Vol. 38, Issue 4, pp 487-492.

Oahimire J.I, Bazuaye F.E and Harry T. H (2016). "Numerical method for the analysis of thermal radiation on heat transfer in nanofluid." Journal of Nanoscience and Technology, Vol.3, Issue 1, pp 1-4

Uddin M.J., Kahn W. A., and Ismail A.I. (2012). "MHD Free Convective Boundary Layer flow of a Nanofluid past a Flat Vertical Plate with Newtonian Heating Boundary condition." PLoS ONE 7(11): e49499 doi:10.1371/journal.pone.49499

Wang X., Xu X. and Choi S.U.S. (1999). "Thermal conductivity of nanoparticle - fluid mixture." Journal of Thermophysics and Heat Transfer, Vol. 13, No. 4, pp 474-80.

Jonathan Oahimire (*1) and Olusegun Adeokun (2)

(1) Department of Mathematics, Michael Okpara University of Agriculture, Umudike, Nigeria ORCID ID 0000-0003-3685-484 imumolen@yahoo.co.uk

(2) Department of Mathematics and Statistics, University of Port Harcourt, Port Harcourt, Nigeria ORCID ID 0000-0002-5498-1735 shegs637@gmail.com

(*) Corresponding Author

```Table 1. Thermo physical properties of water and nanoparticles. Hamad
(2011)

Compound               [rho]    Cp       k
(kg/m3)  (J/kgK)  (W/mK)

Pure water             997.1    4179     0.613
Copper (Cu)            8933     385      401
Alumina (Al2O3)        3970     765      40
Silver (Ag)            10500    235      429
Titanium Oxide (TiO2)  4250     686.2    8.9538

Table 2. Effects of variation of A, Gr and M on the rate of heat
transfer

-[theta](0)           -[theta](0)
A    B    Gr    Cu                    [Al.sub.2][O.sub.3]

0.2  0.5  0.2   -6.3599 x [10.sup.3]  -2.8533 x [10.sup.3]
0.3  0.5  0.2   -4.5049 x [10.sup.3]  -1.9933 x [10.sup.3]
0.4  0.5  0.2   -3.0301 x [10.sup.3]  -1.3219 x [10.sup.3]
0.5  0.5  0.2   -1.8996 x [10.sup.3]  -8.1678 x [10.sup.3]
0.6  0.5  0.2   -1.0754 x [10.sup.3]  -4.5554 x [10.sup.3]
0.1  0.6  0.2   -8.5324 x [10.sup.3]  -3.8252 x [10.sup.3]
0.1  0.7  0.2   -8.4332 x [10.sup.3]  -3.7259 x [10.sup.3]
0.1  0.8  0.2   -8.3339 x [10.sup.3]  -3.6266 x [10.sup.3]
0.1  0.9  0.2   -8.2346 x [10.sup.3]  -3.5274 x [10.sup.3]
0.1  1.0  0.2   -8.1353 x [10.sup.3]  -3.4281 x [10.sup.3]
0.1  0.5  0.3   -8.5332 x [10.sup.3]  -3.8260 x [10.sup.3]
0.1  0.5  0.4   -8.4348 x [10.sup.3]  -3.7275 x [10.sup.3]
0.1  0.5  0.5   -8.3363 x [10.sup.3]  -3.6290 x [10.sup.3]
0.1  0.5  0.6   -8.2378 x [10.sup.3]  -3.5306 x [10.sup.3]
0.1  0.5  0.7   -8.1393 x [10.sup.3]  -3.4321 x [10.sup.3]

-[theta](0)           -[theta](0)
A    B    Gr    Ag                    [T.sub.i][O.sub.2]

0.2  0.5  0.2   -7.4671 x [10.sup.3]  -3.0511 x [10.sup.3]
0.3  0.5  0.2   -5.2976 x [10.sup.3]  -2.1350 x [10.sup.3]
0.4  0.5  0.2   -3.5695 x [10.sup.3]  -1.4183 x [10.sup.3]
0.5  0.5  0.2   -2.2415 x [10.sup.3]  -8.7778 x [10.sup.3]
0.6  0.5  0.2   -1.2711 x [10.sup.3]  -4.9051 x [10.sup.3]
0.1  0.6  0.2   -1.0019 x [10.sup.4]  -4.0908 x [10.sup.3]
0.1  0.7  0.2   -9.9194 x [10.sup.3]  -3.9915 x [10.sup.3]
0.1  0.8  0.2   -9.8201 x [10.sup.3]  -3.8922 x [10.sup.3]
0.1  0.9  0.2   -9.7209 x [10.sup.3]  -3.7929 x [10.sup.3]
0.1  1.0  0.2   -9.6216 x [10.sup.3]  -3.6937 x [10.sup.3]
0.1  0.5  0.3   -1.0019 x [10.sup.4]  -4.0916 x [10.sup.3]
0.1  0.5  0.4   -9.9210 x [10.sup.3]  -3.9931 x [10.sup.3]
0.1  0.5  0.5   -9.8225 x [10.sup.3]  -3.8946 x [10.sup.3]
0.1  0.5  0.6   -9.7241 x [10.sup.3]  -3.7961 x [10.sup.3]
0.1  0.5  0.7   -9.6256 x [10.sup.3]  -3.6977 x [10.sup.3]
```
Author: Printer friendly Cite/link Email Feedback Oahimire, Jonathan; Adeokun, Olusegun Turkish Journal of Engineering (TUJE) Report May 1, 2018 2282 MODULAR APPROACH TO THE DESIGN OF PATH GENERATING PLANAR MECHANISMS. APPROACHES TO THE DESIGN OF A PLANAR PARALLEL MANIPULATOR. Differential equations Heat transfer Homotopy theory Mathematical models Perturbation (Mathematics) Perturbation theory