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Study to analyze the MHD stagnation point flow of a Casson fluid over a nonlinearly stretching sheet with viscous dissipation was carried out. The partial differential equations governing this phenomenon were transformed into coupled nonlinear ordinary differential equations with suitable similarity transformations. These equations were then solved by finite difference technique known as Keller Box method. The various parameters such as Prandtl number ( Pr ), Eckert number ( Ec ), Magnetic parameter ( M ), Casson parameter ( β ) and non linear stretching parameter ( n ) determining the velocity and temperature distributions, the local Skin friction coefficient and the local Nusselt number governing such a flow were also analyzed. On analysis it was found that the Casson fluid parameter ( β ) decreased both the fluid velocity and temperature whereas an increase in ( β ) increased both the heat transfer rate and wall skin-friction coefficient.

The study of boundary layer flow over a stretching sheet has been a great challenge for researchers and it has immense applications in various industrial processes such as extraction of polymer sheet, paper production, wire drawing, glass-ﬁber production etc. The flow of incompressible fluid over a linearly stretching sheet was first investigated by Crane [

The numerical analysis of magnetic field on Eyring-Powell fluid flow towards a stretching sheet has been discussed by N S Akbar [

All the above investigations are restricted for the flow over a linearly stretching sheet but it is not necessary that the stretching sheet has to be linear. Vajravelu [

The study of stagnation point flow towards a stationary semi infinite wall was first introduced by Hiemenz [

In all these attempts, stagnation point flow due to the stretching sheet was analyzed. The boundary layer flow over a shrinking sheet was ﬁrst investigated by Wang [

Many fluids in industries resemble non-Newtonian behaviour. Non-Newtonian fluids are more appropriate than Newtonian fluids because of their varied industrial applications like petroleum drilling, polymer engineering, certain separation processes, food manufacturing etc.

For non-Newtonian fluids, the relationship between stress and the rate of strain is not linear and it is difficult to express all these properties in a single constitutive equation. Consequently, these fluid models [

Casson fluid is one such type of such non-Newtonian fluid, which behaves like an elastic solid, with a yield shear stress existing in the constitutive equation. This ﬂuid model has its origin in modelling of ﬂow of many biological ﬂuids especially blood. Examples of such fluids include foams, yoghurt, molten chocolate, cosmetics, nail polish, tomato puree etc. Casson [

Motivated by the above investigations on non-Newtonian fluids and its wide applications, the objective of the present study is to analyze MHD stagnation point flow of a Casson fluid over a non linear stretching sheet with viscous dissipation. We have extended the works of Vajravelu [

In Section 2 the problem is formulated and similarity transformation has been employed to transform the partial differential equation into a non linear ordinary differential equations. In Section 3 a numerical solution using Keller box method has been discussed. In Section 4 the numerical results have been discussed graphically. Finally, the main concluding observations are mentioned in Section 5.

Consider the steady two-dimensional stagnation point ﬂow of an incompressible Casson ﬂuid located at y = 0. The flow being confined in the region y > 0. We take the non-linear stretching sheet in the XOY plane (see

where

The continuity, momentum and energy equations governing the fluid flow are given by

where u, v are the velocity components in x, y direction respectively,

The suitable boundary conditions are given by

Here, c (

With the help of following similarity transformations

The Equations (1), (2) and (3) are transformed into coupled non linear ordinary differential equations as follows.

and the boundary conditions are transformed into

where

where prime denotes the differentiation with respect to

The quantities of practical interest are the Skin friction coefficient

where

Hence the dimensionless form of Skin friction

where

The numerical solution for the above coupled ordinary differential equations for different values of governing parameters is obtained using finite difference scheme called Keller-box method. This method involves the four main steps which are as follows:

1) Reduce the equation or system of equations to a first order system.

2) Write the difference equations using central differences.

3) Linearize the resulting algebraic equations (if they are non-linear) by Newton’s method.

4) Write them in matrix-vector form and use the block-tridiagonal-elimination technique to solve the linear system.

This method has been widely used in laminar and turbulent boundary layer flows. It seems to be much faster, easier to program, more efficient and flexible to use than other methods.

The numerical computations are carried out with the help of Keller box method. The effects of different parameters like Stagnation point, MHD, viscous dissipation on velocity and temperature profiles has been clearly analyzed including with its physical quantities of significance. The effect of Casson fluid parameter on skin friction and local Nusselt number has been examined. Apart from those the various quantities of non linear stretching parameter were also analyzed.

To verify the accuracy of the present results, comparison has been made with the previous results of Vajravelu and Cortell (

The effects of magnetic parameter M and velocity ratio parameter λ on the flow field velocity is displayed in

n | Vajravelu | Cortell | Present Study | ||||
---|---|---|---|---|---|---|---|

Pr | 0.71 | 7.0 | Pr | 0.71 | 7.0 | ||

1 | −1.0000 | −0.4590 | −1.8953 | −1.000000 | −1.0000 | −0.4590 | −1.8954 |

5 | −0.1945 | −0.4394 | −1.8610 | − | −0.1945 | −0.4395 | −1.8616 |

10 | −1.2348 | −0.4357 | −1.8541 | −1.234875 | −1.2348 | −0.4356 | −1.8547 |

stretching parameter n.

when n is large. This is because of the term

ity, as mentioned in the subfigures. Therefore the observations for the large values of n is not a study of interest.

decreases with the increase in Prandtl number. Since the Prandtl number is the ratio of momentum diffusivity to thermal diffusivity; it reduces the thermal boundary layer thickness. In general the Prandtl number is used in heat transfer problems to reduce the relative thickening of the momentum and the thermal boundary layers.

The influence of Eckert number Ec for linearly/non linearly stretching parameters i.e., n = 1, n = 10 is depicted in

In the present study, MHD stagnation point flow of a Casson fluid over a non linearly stretching sheet are investigated with viscous dissipation. The numerical solution is obtained by Keller box technique. The effects of various governing parameters on heat flow characteristics were analyzed. Briefly the above discussion can be summarized as follows.

・ The velocity boundary layer thickness reduces for magnetic parameter M.

・ An increase in Casson parameter β decreases the velocity of the fluid as well as the thermal boundary layer thickness.

・ The velocity of the fluid decreases and temperature increases with an increase in nonlinear stretching parameter..

・ The Eckert number increases the thermal boundary layer thickness where as the Prandtl number decreases it.

・ Both the skin friction coefficient and heat transfer coefficient increases with Casson parameter β.

Ojinga Gideon Omiunu,1 1,Monica Medikare,Sucharitha Joga,Kishore Kumar Chidem, (2016) MHD Stagnation Point Flow of a Casson Fluid over a Nonlinearly Stretching Sheet with Viscous Dissipation. American Journal of Computational Mathematics,06,37-48. doi: 10.4236/ajcm.2016.61005