MHD Partial Slip Flow and Heat Transfer of Nanofluids through a Porous Medium Over a Stretching Sheet with Convective Boundary Condition

This paper investigates the boundary layer analysis for magnetohydrodynamic partial slip flow and heat transfer of nanofluids through porous media over a stretching sheet with convective boundary condition. Four types of nanoparticles, namely copper, alumina, copper oxide and titanium oxide in the ethylene glycol (50%, i.e., Pr = 29.86) and water (i.e., Pr = 6.58) based fluids are studied. The governing highly nonlinear and coupled partial differential equations are solved numerically using fourth order Runge-Kutta method with shooting techniques. The velocity and temperature profiles are obtained and utilized to compute the skin friction coefficient and local Nusselt number for different values of the governing parameters viz. nanoparticle volume fraction parameter, magnetic field parameter, porosity parameter, velocity slip parameter and convective parameter. It is found that the velocity distribution of the nanofluids is a decreasing function of the magnetic parameter, porosity parameter, and velocity slip parameter. However, temperature of the nanofluids is an increasing function of magnetic field parameter, nanoparticle volume fraction parameter, porosity parameter, velocity slip parameter and convective parameter. The flow and heat transfer characteristics of the four nanofluids are compared. Moreover, comparison of the numerical results is made with previously published works for special cases and an excellent agreement is found.


INTRODUCTION
The study of the boundary layer flow of an electrically conducting fluid has many applications in manufacturing and natural process which include cooling of electronic devices by fans, cooling of nuclear reactors during emergency shutdown, cooling of an infinite metallic plate in a cooling bath, textile and paper industries, glass-fiber production, manufacture of plastic and rubber sheets, the utilization of geothermal energy, the boundary layer control in the field of aerodynamics, food processing, plasma studies and in the flow of biological fluids . would be desired to combine the two substances to produce a heat transfer medium that behaves like a fluid, but has the thermal conductivity of a metal. A lot of experimental and theoretical researches have been done to improve the thermal conductivity of the natural fluids.
In 1993, during an investigation of new coolants and cooling technologies at Argonne national laboratory, Choi invented a new type of fluid called Nanofluid (Sarit et al., 2007).
Nanofluids commonly contain up to a 5% volume fraction of nanoparticles to see effective heat transfer enhancements. Nanofluids are studied because of their heat transfer properties: they enhance the thermal conductivity and convective properties over the properties of the base fluid.
Moreover, the presence of the nanoparticles enhance the electrical conductivity property of the nanofluids, hence are more susceptible to the influence of magnetic field than the conventional base fluids. The suspended metallic or nonmetallic nanoparticles change the transport properties and heat transfer characteristics of the base fluid. Nanofluids have enhanced thermo-physical properties such as thermal conductivity, thermal diffusivity, viscosity and convective heat transfer coefficients compared to those of base fluids. Typical thermal conductivity enhancements are in the range of 15-40% over the base fluid and heat transfer coefficient enhancements have been found up to 40% (Yu et al., 2008). Thermo-physical properties of nanofluids have been enormously studied by various workers, such as Kang et al. (2006); Velagapudi et al. (2008); Rudyak et al. (2010) and others.
After the pioneer investigation of Choi (Yu et al., 2008) various thriving experimental and theoretical researches were undertaken to discover and understand the mechanisms of heat transfer in nanofluids. The knowledge of the physical mechanisms of heat transfer in nanofluids is of vital importance as it will enable the exploitation of their full heat transfer potential. Musuda et al.
(1993) observed that the characteristic feature of nanofluid is thermal conductivity enhancement.
This observation suggests the possibility of using nanofluids in advanced nuclear systems (Buongiorno and Hu, 2005). A comprehensive survey of convective transport in nanofluids made by Buongiorno (2006) indicated that a satisfactory explanation for the abnormal increase of the © CNCS, Mekelle University 42 ISSN: 2220-184X thermal conductivity and viscosity is yet to be found. Buongiorno (2006) further focused on heat transfer enhancement observed in convective situations. Khan and Pop (2011) suggested a similar solution for the free convection boundary layer flow past a horizontal flat plate embedded in a porous medium filled with a nanofluid. Makinde and Aziz (2010) studied MHD mixed convection from a vertical plate embedded in a porous medium with a convective boundary condition. Wubshet et al. (2013) have studied MHD stagnation point flow and heat transfer due to nanofluid towards a stretching sheet, while Turkyilmazoglua and Pop (2013)  reported so far in the literature. Hence the study would help to supplement this unrevealed gap in the area.
The governing highly nonlinear partial differential equation of momentum and energy fields have been simplified by using a suitable similarity transformations and then solved numerically using fourth order Runge-Kutta method with shooting techniques. The effects of the governing parameters on the velocity and temperature have been discussed and presented in tables and graphs.

MATHEMATICAL FORMULATION
In this paper a steady two-dimensional laminar boundary layer flow of nanofluids is considered over a stretching sheet with a linear velocity ( ) = , where, a is a constant and x is the coordinate measured along the stretching surface and the flow takes place at ≥ 0, where y is the coordinate measured normal to the stretching surface. Two equal and opposite forces are applied along the x-axis so that the sheet is stretched keeping the origin fixed. The fluid is electrically conducting under the influence of an applied magnetic field 0 ( ) normal to the stretching surface. Since the magnetic Reynolds number is very small for most fluids used in industrial applications it is assumed that the induced magnetic field is negligible in comparison to the applied magnetic field. The fluid is water and ethylene glycol based nanofluid containing four different types of nanoparticles; namely copper, alumina, copper oxide and titanium oxide. It is assumed that the base fluids and the nanoparticles are in thermal equilibrium and no slip occurs between them. The thermo-physical properties of the base fluids and nanoparticles are given in table 1 (Oztop and Abu-Nada, 2008). Table 1. Thermo-physical properties of water and base fluids and nanoparticles. the continuity, momentum and energy equations are given by:

Physical properties Water Ethylene glycol Cu CuO
(2) Subjected to the boundary condition Where, u and v are the velocity components in the x and y directions, respectively.
T is the temperature of the nanofluid ∞ is the temperature of the nanofluid far away from the sheet 0 is the uniform magnetic field strength is the electrical conductivity 0 is the permeability of the porous medium A is the velocity slip factor h is a heat transfer coefficient k is the thermal conductivity is the convective temperature over the top surface of the plate. is the dynamic viscosity is density, and is the thermal diffusivity of the nanofluid (as given in Tiwari and Das (2007); and Ahmad et al., (2011) In which , , , and are the kinematic viscosity, dynamic viscosity, density, and thermal conductivity of the base fluid respectively; , , and ( ) are the density, thermal conductivity, and heat capacitance of the nanoparticle respectively; is the solid volume fraction of nanoparticles; and is thermal conductivity of the nanofluid.
To simplify the mathematical analysis of our study we introduce the following similarity Making use of equation (6), the continuity equation (1) is automatically satisfied and equations.
(2), (3) and the boundary condition (4) reduce to With boundary conditions Where, ( ) and ( ) are the dimensionless velocity and temperature, respectively. Primes denote differentiation with respect to the similarity variable . ℎ = ℎ √ is convective parameter, = 0 2 , 1 = 0 , = ( ) , and = √ represent the magnetic parameter, porosity parameter, Prandtl number and the velocity slip parameter, respectively and 1 , 2 , and 3 that depend on the nanoparticle volume fraction are given by The physical quantities of interest in this problem are the local skin friction coefficient and the Nusselt number which represents the rate of heat transfer at the surface of the plate, which are defined as: Where, is the skin friction and is the heat flux through the plate, which are given by Making use of equations (5) and (6) in (11), the dimensionless skin friction coefficient and wall heat transfer rates are obtained as Where, = is the local Reynolds number.

NUMERICAL METHOD
The nonlinear boundary value problem represented by equations (7) to (9) is solved numerically using the fourth order Runge-Kutta method with shooting technique. In solving the system of nonlinear ordinary differential equations (7) and (8)

RESULTS AND DISCUSSION
Heat transfer in MHD flow of nanofluids through a porous medium due to a stretching sheet with partial slip and convective boundary condition is studied considering four different types of nanoparticles namely, copper, copper oxide, alumina and titanium oxide, with water and ethylene glycol (50%) as the base fluids (i.e. with a constant Prandtl numbers Pr = 6.58 and Pr=29.86, respectively). The thermo-physical properties of the nanofluids were assumed to be functions of the nanoparticle volume fraction. The transformed nonlinear equations (7) and (8) (7) subjected to the boundary condition (equation 9) given in equation (14) indicate an excellent agreement.  Grubka and Bobba (1985), and the analytic solution given by Abramowitz and Stegun (1965) in terms of Kummer's functions, as given below:  As the values of magnetic parameter M increases, the retarding force increases and consequently the velocity decreases. It is also noted that the boundary layer thickness reduces as magnetic parameter M increases, and the Cu-water has higher value of velocity distribution than TiO 2 .   Variation of temperature distribution among the nanofluids with Cu, Al 2 O 3 , CuO and TiO 2 nanoparticles in water base fluid as shown in figure 4 indicates that the temperature profile is influenced by the type of the nanoparticles. It is also observed that the thermal boundary layer is higher for Cu compared to CuO, A1 2 O 3 and TiO 2 nanoparticles in water base fluid. This is because of the high thermal diffusivity attributed to its high thermal conductivity of Cu compared to the others.  Figure 5 show effects of ϕ on θ(η) for the Cuwater and CuO-water nanofluids. It illustrates that, the temperature distribution increases with .
Also, the temperature is lower for a regular fluid (ϕ = 0, water) compared to the nanofluids (ϕ ≠ 0), Cu-water and CuO-water. Moreover, the Cu-water has a thicker thermal boundary layer than CuO-water.   and velocity slip parameter on the temperature profile for different nanoparticles and base fluids.
The graphs show that, the temperature at a fixed value of is observed to decrease with an increase in the Prandtl number of the base fluid. This is due to the fact that a higher Prandtl number fluid has relatively low thermal diffusivity, which reduces conduction and thereby the thermal boundary-layer thickness and temperature of the nanofluid decreases.   In figures 5 to 9, temperature of the nanofluids is found to be an increasing function of , M, 1 , and ℎ . The temperature boundary layer thickness also increases with , M, 1 , and ℎ .  Values of Pr for water and ethylene glycol are 6.58 and 29.86 respectively. As the Prandtl number increases, the rate of heat transfer increase. It is seen that the higher local Nusselt number occurs in the pure base fluid (i.e, = 0) for water and ethylene glycol. In our case, the fluid with higher Prandtl number, ethylene glycol has the highest rate of heat transfer in the surface compared to water and the nanofluids. Moreover, the Nusselt number is a decreasing function of M.
Furthermore, the TiO 2 −water has the highest rate of heat transfer on the surface of the sheet compared to Cu − water, Al 2 O 3 −water and CuO −water nanofluids.  observed that the local Nusselt number is a decreasing function of and 1 . Hence, to achieve a high rate of heat transfer, less slip on the fluid-solid interface is desired.

CONCLUSION
In this paper, the MHD partial slip flow and heat transfer of nanofluids through a porous medium over a stretching sheet with convective boundary condition is investigated numerically for the Cu, Al 2 O 3 , CuO, and TiO 2 nanoparticles with base fluids water and ethylene glycol. The governing nonlinear partial differential equations were transformed into ordinary differential equations using the similarity approach and solved numerically using the fourth order Runge-Kutta method with shooting techniques. Our numerical results revealed, among others, the following.  An increment in the solid volume fraction , magnetic parameter M, porosity parameter 1 , velocity partial slip parameter and convective parameter ℎ yields an increment in the nanofluids temperature, this leads to a decrease in the heat transfer rates. The Cu − water has the highest thermal boundary layer thickness compared to the other nanofluids, this is due to the high thermal conductivity of Cu.  The heat transfer rate at the plate surface decreases with increasing , , 1 and .
Highest rate of heat transfer occurs in the ethylene glycol and the nanofluid with TiO 2 nanoparticle has the highest cooling performance than the other nanoparticles (Cu, Al 2 O 3 and CuO).
A comparison to validate the accuracy of the present results with previously published papers has shown an excellent agreement.

ACKNOWLEDGEMNETS
I would like to thank Prof. Bandari Shankar, Osmania University, Hyderabad, India; and Dr.
Gebregiorgis Abraha, Mekelle University, Mekelle, Tigray, Ethiopia for their continuous professional support and advice during the preparation of this article.