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The present study concentrates on the analysis of MHD free convection flow past an inclined stretching sheet. The viscous dissipation and radiation effects are assumed in the heat equation. Approximation solutions have been derived for velocity, temperature, concentration, Nusselt number, skin friction and Sherwood number using Nachtsheim-Swigert shooting iteration technique along with the six-order Runge-Kutta iteration scheme. Graphs are plotted to find out the characteristics of different physical parameters. The variations of physical parameters on skin friction coefficient, Nusselt number and Sherwood number are displayed via table.

In recent years, considerable interest has been shown in investigating radiation interaction with natural convection flow commonly known as free convection for heat transfer in fluid. This is due to the significant role of thermal radiation in the surface heat transfer when convection heat transfer is small particularly in free convection problems. Again the boundary layer flow on continuous surfaces is an important type of flow which occurs in a number of technical processes. Examples are paper production, crystal growing and glass blowing, aerodynamics extrusion of plastic sheets and fibers. Thus, the study of heat transfer has become important industrially for determining the quality of the final product. Laminar natural convection flow and heat transfer of fluid with and without heat source in channels with constant wall temperature have been extensively studied by Ostrach [

The viscous dissipation effect plays an important role in natural convection. Natural convection flow is often encountered in the cooling of nuclear reactors. Viscous dissipation effects on non-linear MHD flow in a porous medium over a stretching porous surface have been studied by S.P. Anjali Devi and B. Ganga [

The analysis and discussion of natural convection flow, the viscous dissipation effect is generally ignored but here considered the combined effect of viscous dissipation and radiation on free convection flow an inclined stretching sheet.

A steady-state two-dimensional heat and mass transfer flow of an electrically conducting viscous incompressible fluid along an isothermal stretching permeable inclined sheet with an angle _{0} plays a role which gives rise

to magnetic forces

ductivity assumed to be directly proportional to the x- translational velocity (u) of the fluid found by Helmy [

The fluid is considered to be gray, absorbing emitting radiation but non-scat- tering medium and the Rosseland approximation is used to describe the radia-

tion heat flux in the energy equation. The radiative heat flux in the -direction is negligible to the flux in the y-direction. The plate temperature and concentration are initially raised to

Under the usual boundary layer and Boussinesq approximations and using the Darcy-Forchhemier model, the flow and heat transfer in the presence of radiation are governed by the following equations.

Continuity equation

Momentum equation

Energy equation

Concentration equation

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

The corresponding boundary conditions are

where

By using Rosseland approximation,

where

Thus

Using the Equations (6) and (7) in Equation (3), we get

In order to obtain similarity solution for the problem under consideration, we may take the following suitable similarity variables

where

Since

tion (6) given by

where prime denotes the derivative with respect to

Now introducing the similarity variables from Equation (9) and using Equation (10), Equations (2), (8) and (4) are reduced to the dimensionless equations given by

where

The transformed boundary conditions are

where

The nonlinear ordinary differential Equations (11), (12) and (13) under the boundary conditions (14) are solved numerically for various values of the parameters entering into the problems.

The parameters of engineering interest for the present problem are the skin friction coefficient

where

where

to the skin-friction coefficient, Nusselt number and Sherwood number are

The numerical solutions of the non-linear differential Equations (11)-(13) under the boundary conditions (14) have been performed by applying a shooting method namely Nachtsheim and Swigert [

In order to verify the effects of the step size

For the purpose of discussing the results of the flow field represented in the

ters are chosen arbitrarily where

Due to free convection problem positive large values of

The effect of the angle of inclination

Figures 8-10 are drawn to discuss the influence of Eckert number Ec on velocity, temperature and concentration profiles.

cally. These figures indicate that since the boundary layer thickness getting smaller but the temperature and concentration gradient at the stretching sheet getting steeper, the increasing suction enhance the heat and mass transfer coefficient.

The influence of magnetic field parameter M on velocity, temperature and concentration profiles are plotted in Figures 14-16. Here from

Figures 17-19 describe the dimensionless velocity, temperature and concentration profiles for different values of radiation parameter N. A strong decline in the velocity and temperature profiles are caused by increasing N have found in

The behavior of Prandtl number Pr on velocity, temperature and concentration distributions are displayed in Figures 20-22. From the

For different values of heat source parameter Q, the velocity, temperature and concentration profiles are illustrated in Figures 23-25. Here we have plotted non-dimensional velocity, temperature and concentration profiles against

The main goal of this study was the mathematical and numerical study of the viscous dissipation and radiation effect on MHD free convection flow past an inclined stretching sheet. The numerical solutions of the governing differential equations were obtained by using the shooting method. We observed the behavior of the physical parameters

Finally, the effects of various parameters on the skin friction

Pr | Q | Ec | N | |||
---|---|---|---|---|---|---|

0.71 | 0.5 | 0.1 | 0.5 | 3.40388 | 0.26724 | 0.46977 |

0.71 | 0.75 | 0.1 | 0.5 | 3.5041 | 0.19956 | 0.47801 |

0.71 | 1.25 | 0.1 | 0.5 | 3.74353 | 0.03763 | 0.49644 |

7.0 | 0.75 | 0 | 0.5 | 2.25432 | 0.65081 | 0.33366 |

7.0 | 0.75 | 0.1 | 0.5 | 2.40807 | 0.45194 | 0.34297 |

7.0 | 0.75 | 0.3 | 0.5 | 2.81869 | -0.1642 | 0.36898 |

7.0 | 0.75 | 0.1 | 0.1 | 3.14799 | 0.27511 | 0.43487 |

7.0 | 0.75 | 0.1 | 0.2 | 2.83916 | 034083 | 0.39314 |

7.0 | 0.75 | 0.1 | 0.3 | 2.64517 | 0.38711 | 0.36661 |

We can make the following conclusions from the present study:

1. The effect of heat generation is remarkable. An increase in heat generation results in increasing velocity and temperature within the boundary layer.

2. Eckert number has effects on velocity as well as temperature profiles.

Radiation has significant effects on temperature profiles. So we can control the temperature field by using this parameter.

Hasan, M., Karim, E. and Samad, A. (2017) MHD Free Convection Flow past an Inclined Stretching Sheet with Considering Viscous Dissipation and Radiation. Open Journal of Fluid Dynamics, 7, 152-168. https://doi.org/10.4236/ojfd.2017.72010