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Recent studies carried out in terms of viscous flow and heat transfer of nano-fluids on the non-linear sheets. In this paper, detailed studies to understand the characteristics such as viscous flow and heat transfer of nano-fluids under the influence of thermal radiation and magnetic fields are studied using Keller-Box method. Various governing parameters affecting the viscous flow and heat transfers are drawn based on quantitative results. The raise in temperature affected the velocity to a negative value; however, the same observation was made even for the increasing magnetic field. The impact of radiation parameter is proportional seems to be proportional to temperature and it is observed to be inversely proportional with concentration.

A nanofluid is a fluid which contains nanometer sized solid particles. Nanometer-size solid particles have unique chemical and physical properties. Since suspending nanometer-size particles to the conventional heat transfer fluids lead a better heat transfer, nanofluids are proposed to be employed in several applications such as transportation, nuclear reactors and electronics [

Many engineering processing applications use the characteristics obtained by the flow studies over stretched surfaces includes extrusion, hot rolling, etc. For example, polymer sheets and filaments, in general use continuous extrusion of polymer form a die to the windup roller that is almost placed at a finite distance. The velocity in stretched surfaces almost differs from the plain surfaces and it will be proportional to the distance from orifice [

Apart from these, due to advancement of technology and introduction of nanotechnology the size of materials reduced to nanometres possesses special physical and chemical characteristics [

To study these aspects, Keller-Box method is used and is known to be one of the best numerical methods included with mixed finite volume that considers the average of conservation laws and related constitutive laws [

In this paper, the properties of viscous flow with respect to nano-fluid on a stretching sheets is considered. Generally the flow is possible when the condition

The following assumptions are made before getting into more mathematical formulations. Assume the temperature at stretching surface as a function of x (see Equation (5)), ambient temperature T as constant and at the sheet nano particle fraction C as a constant with a value

Now consider a nanofluid is flowing at

The boundary conditions for parameters velocity, temperature and nano-particle fraction are given below:

where,

u and v are the velocity components in x and y directions,

a is acceleration of components.

Thermal Diffusivity

However,

The

Here,

c is volumetric volume coefficient,

C is rescaled volume fraction of nanoparticles,

R is radiation parameter.

Now assume variable magnetic field

Using Rosseland approximation for radiation, we can write

where

Hence, Equation (7a) becomes

Hence

where

Equation (1) is satisfied identical. The governing Equations (2)-(4) or reduced by Equations (7).

The transformed boundary conditions are

Here, primes in the above equation represent the differentiation with respect to η, which is known to be an involved physical parameter and it is defined as follows:

Here,

This boundary value problem is reduced to the classical problem of flow and heat and mass transfer due to a stretching surface in a viscous fluid when

In this study the quantities of practical interest are listed below:

The heat and mass fluxes

Substituting Equation (7) into Equations (13) and (14), we obtain

where

Exact analysis for the reduced Equations (8)-(11) are not possible as they are nonlinear and coupled. However using Keller-Box method these problems can be evaluated for different values and parameters such as Pr, Nb, T and Le. Various effects are investi-

gated in this paper such as dimensionless velocity, temperature, skin function, mass transfer and rate of heat.

Some of the principal steps of Keller Box methods are listed below:

1) Reducing higher order ODEs to the first order ODEs

Let

Let

Let

In the following steps only first ODE statement will be considered for further steps to follow using Keller Box method.

2) First order ODEs are written into difference equations by using central differencing schemes

Let us apply the central differential equation for

3) Using the Newtons method to linearizing the difference equation for extending it into vector form

4) And finally using block elimination method the system equations are solved to draw the corresponding matrices.

MATLAB software is used to solve the differential equations numerically. This software is very efficient for using with Keller Box method.

Various comparisons are presented in

The magnetic parameter (M) effect is shown in

Nb | Hamad | Present | Hamad | Present |
---|---|---|---|---|

0.1 | 3.771628 | 3.7716 | 3.658830 | 3.5688 |

0.2 | 3.251502 | 3.2515 | 3.149618 | 3.1496 |

0.3 | 2.827929 | 2.8279 | 2.735630 | 2.7356 |

0.1 | 5.621979 | 5.6220 | 5.345776 | 5.3458 |

0.2 | 0.998139 | 0.9981 | 0.875024 | 0.8750 |

0.3 | 0.451703 | 0.4517 | 0.525092 | 0.5251 |

M | ||||
---|---|---|---|---|

0.0 | 0 | 1.2352 | 2.8279 | 0.4517 |

0.1 | 0 | 1.2756 | 2.8310 | 0.4314 |

0.3 | 0 | 1.3527 | 2.8370 | 0.3925 |

0.5 | 0 | 1.4254 | 2.8426 | 0.3559 |

1.0 | 0 | 1.5921 | 2.8555 | 0.2719 |

0 | 0 | 1.2352 | 2.8279 | 0.4517 |

0 | 0.1 | 1.2352 | 2.7356 | 0.5251 |

0 | 0.3 | 1.2352 | 2.5492 | 0.6734 |

0 | 0.5 | 1.2352 | 2.3604 | 0.8239 |

0 | 1.0 | 1.2352 | 1.8772 | 1.2096 |

both magnetic and electrical fields. The Lorentz force acts against the flow when magnetic field is applied normal to the direction. In such scenarios the resistive forces will slow down.

The positive changes in M allow increasing the temperature within the boundary layer as shown in

The influence of M, nonlinear stretching parameter n on dimensionless, and nanoparticle concentrations are shown in

parameter is slightly negligible for the variation of nanoparticle concentration. Note that these parameters are negligible for both positive and negative values of n.

The effect of R for fixed values of other parameters the dimensionless concentrations

are shown in

Behaviour of temperature with respect to different values of Pr numbers are shown in

diffused away from the heated surface as compared with the high values of Pr. Therefore the smaller Pr number due to thermal boundary layer leads to a thicker and hence the transferred heat will be reduced.

The temperature profiles for different values of Ec are plotted in

The effect of Ec on dimensionless concentration for various fixed values of other parameters is shown in

For the plot between behaviour of temperature and different value of Le are shown in

Effect of Le on dimensionless concentration for various fixed parameters is shown in

Behaviour of temperature with respect to different values of Nb are showin in

values of Nb will allow to an increase in thermal conductivity of fluids so that heat will be diffused from heated surface rapidly for higher values of Nb. Hence for smaller Nb the thermal boundary layer is thicker and rate of heat transfer will be reduced.

Effect of Nb for dimensionless concentration for various fixed values of other parameters is observed in

layer.

The Nt on temperature and concentration profiles are shown in

the thickness of thermal boundary layer.

Influence of Nt and n on dimensionless and nanoparticle concentrations are shown in

increase of Nt. However Lorentz force opposes the fluid motion hence the heat is produced. Due to such conditions thermal boundary and nanoparticle volume fraction boundary layer thickness will become thicker for strong Nt.

In this paper, various similarity solutions for viscous flow and heat transfer of nanofluids on nonlinear streached sheets under the influence of thermal radiation are shown in the results. However, similar solutions are depending on Pr, Le, M, n, Nb, T and Nt. The graphical representations of various governing parameters are obtained in different conditions to draw the importance of thermal radiation and some of the featured observations of the findings are listed below:

・ The velocity is decreasing with the increase in temperature.

・ Ec will increase proportionally with Eckert number.

・ The increase in M will lead to a decrease in velocity parameter.

・ The temperature will increase due to an increase in R.

・ The concentration will decrease due to an increase in R.

I thank MHRD, Govt. of India for supporting my work through UGC-BSR Fellowship.

Rayapole, S.P. and Jakkula, A.R. (2016) Characteristics of Na- nofluids over a Non-Linearly Stretched Sheet under the Influence of Thermal Radiation and Magnetic Field. World Journal of Mechanics, 6, 456-471. http://dx.doi.org/10.4236/wjm.2016.611032