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Two-dimensional, steady, laminar and incompressible natural convective flow of a nanofluid over a connectively heated permeable upward facing radiating horizontal plate in porous medium is studied numerically. The present model incorporates Brownian motion and thermophoresis effects. The similarity transformations for the governing equations are developed by Lie group analysis. The transformed equations are solved numerically by Runge-Kutta-Fehlberg fourth-fifth order method with shooting technique. Effects of the governing parameters on the dimensionless velocity, temperature and nanoparticle volume fraction as well as on the dimensionless rate of heat and mass transfer are presented graphically and the results are compared with the published data for special cases. Good agreement is found between numerical results of the present paper and published results. It is found that Lewis number, Brownian motion and convective heat transfer parameters increase the heat and mass transfer rates whilst thermophoresis decreases both heat and mass transfer rates.

Nanoparticles are made from various materials, such as oxide ceramics (

A good number of research papers have been published on nanofluids to understand their performance so that they can be used to enhance the heat transfer in various industrial applications. A review of convective transport in nanofluids was conducted by Buongiorno [

According to previous researchers, for example, Aboeldahab and Azzam [

Fluid flow and heat transfer in porous media have many engineering applications such as postaccidental heat removal in nuclear reactors, solar collectors, drying processes, storage of radioactive nuclear waste, heat exchangers, geothermal energy recovery and crude oil extraction, ground water pollution, thermal energy storage, building construction and flow through filtering media, separation processes in chemical industries [

Above investigators found similarity solutions via dimensional analysis which can find only one particular type of similarity independent variable of the form

The aim of the present study is to investigate the effect of thermophoresis, the Brownian motion, radiation and the thermal convective boundary condition on the boundary layer flow of a nanofluid over an upward facing radiating permeable horizontal plate numerically. A possible application of this problem is in the design of furnace where the transfer of heat from surfaces occurs simultaneously by radiation and convection. Also, the interaction of solar radiation with the earth’s surface fabricates complex free convection patterns and hence complicates the studies associated with the weather forecasting and marine environment for predicting free convection patterns in oceans and lakes. Using similarity transformations developed by Lie group analysis, the governing partial differential equations are reduced to a set of coupled nonlinear ordinary differential equations with the corresponding boundary conditions. The effect of emerging flow controlling parameters on the dimensionless axial velocity, the temperature, the nanoparticle volume fraction, the rate of heat transfer, and the rate of nanoparticle volume fraction is investigated and shown graphically and discussed.

We consider a two-dimensional

Coordinate system and flow model.

Here

Consider a steady state flow. In keeping with the Oberbeck-Boussinesq approximation and an assumption that the nanoparticle concentration is dilute, and with a suitable choice for the reference pressure, we can linearize the momentum equation and write (

The boundary conditions are taken to be [

The boundary conditions in (

By applying Lie group method to (

Thus the infinitesimals become

From (

Equations

Hence the similarity transformations are

Substituting the transformations in (

It is worth citing that in case of impermeable nonradiating plate (

The parameters of engineering interest are the local skin friction factor

The set of coupled nonlinear similarity Equations (

Comparison of present results with Gorla and Chamkha [

Present results | Gorla and Chamkha [ | |||

0.1 | 0.32578 | 1.48242 | 1.484164 | |

0.2 | 0.32385 | 1.46704 | 1.468161 | |

0.3 | 0.32188 | 1.45125 | 1.452664 | |

0.4 | 0.31985 | 1.43503 | 1.436392 | |

0.5 | 0.31777 | 1.41833 | 1.419499 | |

0.1 | 0.31777 | 1.41833 | 1.419499 | |

0.2 | 0.30486 | 1.41491 | 1.416536 | |

0.3 | 0.2927 | 1.41561 | 1.416866 | |

0.4 | 0.28125 | 1.41991 | 1.421582 | |

0.5 | 0.27046 | 1.42737 | 1.429226 | |

0.1 | 0.3672 | 1.32611 | 1.327454 | |

0.2 | 0.34271 | 1.39216 | 1.393615 | |

0.3 | 0.31777 | 1.41833 | 1.419499 | |

0.4 | 0.29399 | 1.43428 | 1.435464 | |

0.5 | 0.27161 | 1.44598 | 1.44772 |

Figures

Effect of (a) radiation parameter, (b) buoyancy ratio parameter on the dimensionless velocity for different values of suction/injection parameter.

Effect of (a) Biot number, (b) Lewis number on the dimensionless velocity for different values of suction/injection parameter.

Variation of the dimensionless temperature and corresponding thermal boundary layer thickness with radiation parameter, suction/injection parameter, the Biot number, thermophoresis, and Brownian motion parameters is shown in Figures

Effect of (a) radiation parameter, (b) Biot number on the dimensionless temperature for different values of suction/injection parameter.

Effect of (a) Brownian motion parameter, (b) thermophoresis parameter on the dimensionless temperature for different values of suction/injection parameter.

Note that the temperature increases with the increasing of the Brownian motion and thermophoresis parameters when the plate is permeable or not (Figure

Figure

Effect of (a) radiation parameter, (b) Lewis number on the dimensionless nanoparticle volume fraction for different values of suction/injection parameter.

The effect of various controlling parameters on the dimensionless heat transfer rate from a permeable horizontal upward facing plate with the thermal convective boundary condition in porous media is shown in Figure

Variation of local heat transfer rate with (a) Brownian motion, suction/injection, and radiation parameters, (b) thermophoresis, buoyancy ratio, and Biot number.

Figure

Variation of local mass transfer rate with (a) Brownian motion, radiation, and suction/injection parameters, (b) Lewis number, buoyancy ratio, and thermophoresis parameters.

We studied numerically a 2-D steady laminar viscous incompressible boundary layer flow of a nanofluid over an upward facing horizontal radiating permeable plate placed in the porous media considering the thermal convective boundary condition. The governing boundary layer equations are transformed into highly nonlinear coupled ordinary differential equations using similarity transformations developed by Lie group analysis, before being solved numerically. Following conclusions are drawn:

the dimensionless velocity, the temperature, and the concentration decrease in case of the suction and increase in case of the injection; the phenomenon is reversed,

the Brownian motion, radiation, thermophoresis, and buoyancy ratio parameters decrease the heat transfer rate whilst the suction parameter and the Biot number enhance the heat transfer rate,

the radiation, Lewis number, Brownian motion, and the suction parameters cause to enhance nanoparticle volume fraction rate whilst thermophoresis and buoyancy ratio parameters lead to decreasing nanoparticle volume fraction rate.