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We analyse the influence of Brownian motion and thermophoresis on a nonlinearly permeable stretching sheet in a nanofluid. The governing partial differential equations are reduced into a system of ordinary differential equations using similarity transformation and then solved numerically using the Runge-Kutta with shooting technique. Effects of Brownian motion and thermophoresis on the flow, concentration, temperature, and mass transfer and heat transfer characteristics are investigated. The local Nusselt number and the local Sherwood numbers are presented and compared with existing results and are found to be in good agreement.

Nanofluids attract a great deal of interests with their enormous potential to provide enhanced performance properties, particularly with respect to heat transfer. Nanofluids are used for cooling of microchips in computers and other electronics which use microfluidic applications. Using nanofluids as coolants would allow for the radiators with smaller sizes and better positioning. Das et al. [_{2}O_{3} or CuO nanoparticles over a small temperature range of 21˚C - 51˚C. A comprehensive survey of convective transport in nanofluids has been made by Buongiorno [

However, to the best of authors’ knowledge, no attempt has been made to analyse the simultaneous effects of thermal radiation, Brownian motion and thermophoresis on the rate of heat and mass transfer flow of nanofluids over a non-linear stretching sheet. Hence, it is the reason why this problem is investigated.

Here, consideration is given to a steady, laminar, and incompressible and two dimensional boundary layer flow and heat transfer of a nanofluid past a permeable stretching/shrinking sheet. The pressure gradient and other external forces are neglected. Applying the boundary layer approximation, the governing equations for the conservation of mass, momentum, thermal energy and nanoparticle concentration are expressed as follows:

The boundary conditions for Equations (1)-(4) are:

where l is the stretching/shrinking parameter, with l > 0 for a stretching surface and l < 0 for a shrinking surface.

By using Roseland approximation, the radiation heat flux q_{r} is given by:

where

In view of Equation (3) reduces to

Further, we seek for a similarity solution of Equations (1) to (4) subject to the boundary conditions (5). The governing partial differential forms can be solved by converting them to ordinary differential equations; this is done by using similarity functions:

, (9)

where prime denotes differentiation with respect to eta (η). To have similarity solution of Equations (1) to (5),

we assume:

these similarity variables on the governing partial differential equations, transformed conservation equations and boundary conditions are then obtained as follows:

However, the quantities of physical and engineering interest are the reduced Nusselt number (

The local heat transfer rate (Local Nusselt) number is given by

And then the local Sherwood number is;

where

The set of Equations (10) to (12) under the boundary conditions (16) have been solved numerically using shooting technique. We consider:

The results obtained shows the influences of the non-dimensional governing parameters, namely Radiation parameter R, Suction parameter S, Lewis number Le, thermophoresis parameter N_{t} and Brownian motion parameter N_{b} on temperature profile, nanoparticle concentration profile, a the local Nusselt number and the Sherwood number. For numerical results we used Le = 2, Ec = 0.5, n = 2, S = 2 and l = 2 for different values of N_{b}, N_{t} and R in entire study. These values are kept constant except the varied values shown in the figures. The numerical results obtained, i.e., the present results for Nusselt number and Sherwood number were compared with those obtained by Khairy, Anuar Ishak and Ioan Pop [_{b} = 0.5, N_{t} = 0.5 and Pr = 6.2.

In order to get a clear insight of the physical problem, numerical computation have been carried out as described above for various values of different parameters (

To assess the accuracy of the method, the results are compared with those reported in literature by Khairy, Anuar and Ioan Pop [

With increase thermophoresis parameter, both figures show that the boundary layer thickness increases, leading to increase in temperature. The rate of heat transfer increases rapidly initially (up to the point h = 0.4) and later decreases to a non-zero value. In

Khairyzaimi, Anuar and Ioan Pop [ | Present results | ||||
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S | l | ||||

2.5 | −0.5 | 7.887191 | −6.070289 | 7.88719 | −6.07029 |

3.0 | 2.0 | 7.984141 | −4.499616 | 7.9841 | −4.4996 |

3.5 | −0.5 | 10.790697 | −8.043106 | 10.7907 | −8.0431 |

4.0 | 2.0 | 10.750182 | −6.361137 | 10.7502 | −6.3611 |

5.0 | −0.5 | 15.266129 | −11.207552 | 15.2661 | −11.2076 |

2.5 | 2.0 | 7.151258 | −4.446270 | 7.1513 | −4.4463 |

3.0 | −0.5 | 9.681430 | −6.311584 | 9.6814 | −6.3116 |

3.5 | 2.0 | 9.366247 | −5.434047 | 9.3663 | −5.4305 |

4.0 | −0.5 | 12.274461 | −9.083765 | 12.2745 | −9.0838 |

5.0 | 2.0 | 14.788043 | −10.216636 | 14.7795 | −10.2166 |

We now concentrate on the effects of Brownian motion on the temperature, concentration, rate of heat transfer and the rate of mass transfer. In

Next, we look at the effects of the stretching parameter.

This study analysed the influence of Brownian motion and thermophoresis in nonlinearly permeable stretching

sheet in a nanofluid. The non-linear partial differential equations and their associated boundary conditions have been transformed to non-dimensional ordinary differential equations using the similarity transformations and the resultant initial value problem is solved by an iterative Runge-Kutta method along with shooting technique. The present results are compared with the existing results in literature and were found to agree well. The influences of the governing parameters on the temperature, concentration, heat and mass transfer rates have been systematically examined. From the present numerical investigation, the following conclusion can be made:

1) There is a rise in the temperature with an increase in the thermophoresis parameter or Brownian motion parameter or stretching parameter.

2) Species concentration decreases with an increase in Brownian motion while the concentration increases for an increase in the values of the thermophoresis parameter.

3) A rising value in N_{b} and the decreasing in N_{t} produce a decrease in the nanoparticle concentration, and as a result increase in the Sherwood number.

A.Falana,O. A.Ojewale,T. B.Adeboje, (2016) Effect of Brownian Motion and Thermophoresis on a Nonlinearly Stretching Permeable Sheet in a Nanofluid. Advances in Nanoparticles,05,123-134. doi: 10.4236/anp.2016.51014

a: a positive constant

u, v: velocity components in x and y directions respectively

l: the stretching/shrinking parameter (l > 0 for stretching surface and l < 0) for shrinking surface

Le: Lewis number

Pr: Prandtl number

m: wall mass flux

w: wall heat flux

Re_{x}: local Reynolds number

x: coordinate along the sheet

y: coordinate normal to the sheet

C: nanoparticle volume fraction

Greek symbols