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The steady two-dimensional laminar boundary layer flow and heat transfer of a viscous incompressible electrically conducting fluid over an exponentially stretching surface in the presence of a uniform magnetic field with thermal radiation are investigated. The governing boundary layer equations are transformed to ordinary differential equations by taking suitable similarity transformation and solved numerically by shooting method. The effects of various parameters such as magnetic parameter, radiation parameter, Prandtl number and Eckert number on local skin-friction coefficient, local Nusselt number, velocity and temperature distributions are computed and represented graphically.

The study of boundary layer flow and its applications are vital for advancement in the field of technology and engineering. The computation and computer coordinated applications of flow over a stretching surface are playing a pivotal role in different realm of industrial products of aerodynamics, polymers and metallurgy, such as liquid films in condensation process, artificial fibers, glass fiber, metal spinning, the cooling process of metallic plate in a cooling bath and glass, wire drawing, paper production, aerodynamic extrusion of plastic sheets, crystal growing, cable coating and many others, to get end product of desired quality and parameters. Sakiadis [

Boundary layer flow and heat transfer over an exponentially stretching surface have wider applications in technology such as in case of annealing and thinning of copper wires. Magyari and Keller [

At higher operating temperature, the effects of thermal radiation and heat transfer play a pivotal role on the fluid flow problem of boundary layer. The application of controlled heat transfer in polymer industries is very important to get final product of desired parameters. The modern system of electric power generation, plasma, space vehicles, astrophysical flows and cooling of nuclear reactors are governed by applications of thermal radiation and heat transfer of fluid flow. Elbashbeshy [

With reference to above significant studies and in view of importance of MHD applications in various field of technologies, the objective of present paper is to investigate the effect of thermal radiation on an electrically conducting two-dimensional boundary layer incompressible viscous fluid flow over an exponentially stretching surface in the presence of uniform magnetic field by using Rosseland approximation. Numerical results of the momentum and energy equations are computed by using shooting method. The promising results of velocity and temperature distributions, local skin-friction coefficient and surface heat transfer are discussed for various physical parameters and simplified their effects for different conditions.

Consider the steady two-dimensional laminar boundary layer flow

where T_{0} is the reference temperature, L is the reference length, U_{0} is the reference velocity,

ficient of kinematic viscosity, _{p} is the specific heat at constant pressure, T is the temperature, _{r} is the radiative heat flux. The other symbols have their usual meanings.

The boundary conditions are:

By using Rosseland approximation of the radiation for an optically thick boundary layer, the radiative heat flux q_{r} is expressed (Bidin and Nazar [

where _{r} is effective at a point away from boundary layer surface in an intensive absorption flow. Considering that the temperature variation within the flow is very small, the T^{4} may be expressed as a linear function of temperature T. Expanding T^{4} by Taylor’s series about temperature

Using Equation (5) and (6), equation (3) is reduced to:

The continuity Equation (1) is identically satisfied if we defined stream function

For the solution of momentum and energy Equations (2) and (7), introducing the following dimensionless variables:

Using Equations (8) to (11), Equations (2) and (7) are reduced to:

The boundary conditions are:

where prime (') denote differentiation with respect to η,

For numerical solution of the Equations (12) and (13), we use the following power series in terms of small magnetic parameter M as:

Substituting the values of

The corresponding boundary conditions are:

The Equation (17) is same as that obtained by Bidin and Nazar [

The important physical quantities are the local skin-friction coefficient C_{f} and the local Nusselt number Nu, which are defined as:

and

In the present case which can be expressed in dimensionless form as:

and

where

values of _{f} and local Nusselt number Nu at the surface respectively and these are presented by

Figures 3-6 show the temperature distributions

The values of the local skin-friction coefficient C_{f} and the local Nusselt number Nu in terms of _{f} and the local Nusselt number Nu decreases with increasing value of the magnetic parameter M. Moreover, the local Nusselt number Nu decreases with increasing value of the radiation parameter K, whereas the reverse phenomena occurs for the Prandtl number Pr. Further,

f'' (0) | ||||||
---|---|---|---|---|---|---|

M = 0.00 | M = 0.04 | M = 0.25 | ||||

−1.2821 | −1.3135 | −1.4642 | ||||

θ' (0) | ||||||

K | Pr | Ec = 0.0 | ||||

M = 0.00 | M = 0.04 | M = 0.25 | ||||

0.0 | 1 | −0.9559 | −0.9475 | −0.9080 | ||

2 | −1.4712 | −1.4627 | −1.4217 | |||

3 | −1.8689 | −1.8605 | −1.8202 | |||

0.5 | 1 | −0.6860 | −0.6786 | −0.6455 | ||

2 | −1.0737 | −1.0652 | −1.0246 | |||

3 | −1.3805 | −1.3720 | −1.3309 | |||

1.0 | 1 | −0.5528 | −0.5466 | −0.5192 | ||

2 | −0.8653 | −0.8571 | −0.8190 | |||

3 | −1.1215 | −1.1129 | −1.0721 | |||

The characteristic relationships among various parameters influencing viscous incompressible electrically conducting fluid over an exponentially stretching surface in the presence of a uniform magnetic field with thermal radiation have been analyzed and illustrated graphically. The similarity equations are determined and solved numerically by shooting method. It is observed that thickness of the velocity boundary layer, the local skin-friction coefficient and the local Nusselt number decreases with increasing value of the magnetic parameter. However, thickness of the thermal boundary layer increases with increasing value of the magnetic parameter. Further, it is observed that thickness of thermal boundary layer increases with increasing value of the radiation parameter or the Eckert number, whereas, reverse phenomenon observed for the Prandtl number. Moreover, the local Nusselt number decreases with increasing value of the radiation parameter, while reverse behaviour observed for the Prandtl number.