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A numerical study on boundary layer flow behaviour, heat and mass transfer characteristics of a nanofluid over an exponentially stretching sheet in a porous medium is presented in this paper. The sheet is assumed to be permeable. The governing partial differential equations are transformed into coupled nonlinear ordinary differential equations by using suitable similarity transformations. The transformed equations are then solved numerically using the well known explicit finite difference scheme known as the Keller Box method. A detailed parametric study is performed to access the influence of the physical parameters on longitudinal velocity, temperature and nanoparticle volume fraction profiles as well as the local skin-friction coefficient, local Nusselt number and the local Sherwood number and then, the results are presented in both graphical and tabular forms.

The industrial processes like hot rolling, wire drawing, spinning of filaments, metal extrusion, crystal growing, glass fibre production, paper production, cooling of a large metallic plate in a bath, which may be an electrolyte, etc. to require the study of flow and heat transfer over a stretching surface. In all these cases, the quality of final product depends on the surface heat transfer rate and the skin friction coefficient. So this study has gained considerable attention in the recent years. Choi [_{2}O_{3}, TiO_{2} and CuO), carbides (SiC), nitrides (AlN, SiN), or nonmetals (graphite, carbon nanotubes) and conductive fluids, such as water or ethylene glycol, or oil, other lubricants, bio-fluids, polymer solutions as base fluids are used in the manufacturing of nanofluids. 5% volume fraction of nanoparticles in these fluids ensure effective heat transfer enhancements which help them to exhibit enhanced thermal conductivity and the convective heat transfer coefficient compared with the base fluid. Routbort [

It is often assumed in the problems of boundary layer flow over a stretching surface that the velocity of the stretching surface is linearly proportional to distance from the fixed origin. However, Gupta [

Hitesh Kumar [

This paper provides the solution to the problem of flow and heat transfer of a nanofluid over an exponentially stretching porous sheet by considering the effect of chemical reaction, joule heating and thermal radiation parameters along with the suction parameter by adopting the Keller Box method.

The present problem is based on a steady two-dimensional incompressible viscous laminar flow of an electrically conducting nanofluid over a permeable exponentially stretching sheet in a porous medium as shown in

where, u & v are the velocity components in X and Y-directions respectively, ρ-density of the nanofluid, B_{0}-in- duced magnetic field, σ-electrical conductivity, υ-kinematic viscosity, K_{1}-thermal conductivity, α_{m}-thermal diffusivity parameter, (ρc)_{f}-the heat capacitance of the base fluid, τ-ratio between the effective heat capacity of the nanoparticle material and heat capacity of the nanofluid, D_{B}-the Brownian diffusion coefficient, D_{T}-the thermophoresis coefficient, q_{γ}-radiative heat flux, k_{1}-chemical reaction parameter, c_{p}-specific heat capacity of the nanoparticle, γ-chemical reaction parameter, α-the chemical reaction coefficient, T-temperature of the nanofluid, C-concentration of the nanofluid, T_{w} and C_{w}-the temperature and concentration along the stretching sheet,

The boundary conditions are:

By using the Rosseland approximation for radiative heat flux is defined as:

where

To examine the flow, the following transformations are used:

To determine the velocity, temperature distribution and rate of heat and mass transfer in the above boundary layer (5), we solve the equations related to the stretching sheet problem to obtain the following similarity equations using (8). In deriving these equations, the external electric field is assumed to be zero and the electric field due to polarization of charges in negligible.

The transformed boundary conditions take the following forms:

where,

For the type of boundary layer flow, the skin-friction coefficient, heat transfer coefficient and mass transfer coefficients are important physical parameters.

They defined as:

The dimensionless forms of these parameters are:

where the surface shear stress

and the thermal conductivity, respectively. The numerical values of

Equations (9)-(11) subjected to the boundary conditions (12) are solved numerically using implicit finite difference method that is known as Keller Box in combination with the Newton’s techniques as described by Cebeci and Bradshaw [

Pr | N.G. Rudraswamy [ | Present work |
---|---|---|

0.72 | 0.6180 | 0.6152 |

1.0 | 0.7097 | 0.7093 |

1.5 | 0.7862 | 0.7862 |

The principal steps in using the Keller Box method are:

1) Reducing higher order ODEs (systems of ODES) in to systems of first order ODEs;

2) Writing the systems of first order ODEs into difference equations using central differencing scheme;

3) Linearizing the difference equations using Newton’s method and wring it in vector form;

4) Solving the system of equations using block elimination method.

In order to solve the above differential equations numerically, we adopt Matlab software which is very efficient in using the well known Keller Box method. In accordance with the boundary layer analysis, the boundary condition (12) at

tion with five decimal place accuracy as the criterion of convergence. Obtained coupled ordinary non-linear Equations (9)-(11) are solved by Keller Box method for boundary condition (12). Accuracy of this numericalmethod shown in

Referring from

Figures 3-5 depict the effects of suction parameter S on velocity, temperature and concentration profiles, respectively at the boundary for exponentially stretching sheet. It is observed that velocity decreases significantly with increasing suction parameter where as fluid velocity is found to increase with blowing. It is observed that, when the wall suction (S > 0) is considered, this causes a decrease in the boundary layer thickness and the velocity field is reduced. Opposite behaviour is noted for blowing S < 0. In

Figures 7-10 show that an abnormal increase in the concentration

As shown in

As depicted in

Pr | Le | S | γ | R | Nt | Nb | M | G | H | θ'(0) | f''(0) | ϕ'(0) |
---|---|---|---|---|---|---|---|---|---|---|---|---|

0.72 | 10 | 1 | 0.01 | 0.01 | 0.45 | 0.45 | 0.3 | 0.5 | 0.06 | −0.5992 | −2.1252 | −10.3194 |

1 | −0.6919 | −2.1252 | −10.3194 | |||||||||

1.5 | −0.7677 | −2.1252 | −10.2214 | |||||||||

10 | −0.5992 | −2.1252 | −10.3194 | |||||||||

15 | −0.5934 | −2.1252 | −15.3972 | |||||||||

20 | −0.5903 | −2.1252 | −20.4433 | |||||||||

−0.2 | −0.1744 | −1.4741 | −1.0406 | |||||||||

0.2 | −0.3017 | −1.6648 | −3.5434 | |||||||||

0.4 | −0.3494 | −1.7701 | −5.1324 | |||||||||

0.1 | −0.5991 | −2.1252 | −10.3317 | |||||||||

1 | −0.5986 | −2.1252 | −10.4507 | |||||||||

0.1 | −0.5648 | −2.1252 | −10.3431 | |||||||||

0.2 | −0.5302 | −2.1252 | −10.3674 | |||||||||

0.3 | −0.4992 | −2.1252 | −10.3895 | |||||||||

0.4 | −0.6102 | −2.1252 | −10.2551 | |||||||||

0.8 | −0.531 | −2.1252 | −10.5394 | |||||||||

1.2 | −0.4669 | −2.1252 | −10.6288 | |||||||||

0.4 | −0.6182 | −2.1252 | −10.3511 | |||||||||

0.8 | −0.4838 | −2.1252 | −10.1787 | |||||||||

1.2 | −0.3829 | −2.1252 | −10.1445 | |||||||||

0.5 | −0.5965 | −2.1865 | −10.3149 | |||||||||

1.5 | −0.5856 | −2.4636 | −10.2956 | |||||||||

5 | −0.5626 | −3.2155 | −10.2502 | |||||||||

0.5 | −0.5992 | −2.1252 | −10.3194 | |||||||||

1.5 | −0.5875 | −2.4115 | −10.2991 | |||||||||

5 | −0.5636 | −3.1783 | −10.2522 | |||||||||

0.06 | −0.5992 | −2.1252 | −10.3194 | |||||||||

0.9 | −0.4896 | −2.1252 | −10.4189 | |||||||||

1 | −0.4765 | −2.1252 | −10.4308 |

which leads to decrease in the heat transfer rate. Thus, the radiation should be at its minimum in order to facilitate the cooling process. All these physical behaviour are due to the combined effects of the strength of the Brownian motion and thermophoresis particle deposition.

As shown in

profiles for temperature and concentration are decreasing.

A numerical study corresponding to the flow and heat transfer in a steady flow region of nanofluid over an exponential stretching surface and effects of chemical reaction, thermal radiation, magnetic, suction parameter, porosity parameter and joule heating parameters is examined and discussed in detail. The main observations of the present study are as follows.

An increase in suction parameter leads the velocity, temperature and concentration profiles to decrease. For larger values of Le suppress the concentration profile i.e. inhibit nanoparticle species diffusion, as observed. There will be a much greater reduction in the concentration boundary layer thickness. As Nt increases, temperature profile increases but the concentration profile decreases. With increasing values of Nb, both temperature and concentrations profiles increase. As increase in chemical reaction paramter γ leads the concentration profiles to decrease. The impact of porosity parameter shows that velocity profile is decreasing and temperature profile is increasing. This is because the porous medium inhibits the fluid not to move freely through the boundary layer. This leads the flow to increase thermal boundary layer thickness. Joule heating parameter reduces the temperature and concentration of nanofluid.

Jakkula AnandRao,GandamallaVasumathi,JakkulaMounica, (2015) Joule Heating and Thermal Radiation Effects on MHD Boundary Layer Flow of a Nanofluid over an Exponentially Stretching Sheet in a Porous Medium. World Journal of Mechanics,05,151-164. doi: 10.4236/wjm.2015.59016