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The main aim of this article is to introduce the approximate solution for MHD flow of an electrically conducting Newtonian fluid over an impermeable stretching sheet with a power law surface velocity and variable thickness in the presence of thermal-radiation and internal heat generation/absorption. The flow is caused by the non-linear stretching of a sheet. Thermal conductivity of the fluid is assumed to vary linearly with temperature. The obtaining PDEs are transformed into non-linear system of ODEs using suitable boundary conditions for various physical parameters. We use the Chebyshev spectral method to solve numerically the resulting system of ODEs. We present the effects of more parameters in the proposed model, such as the magnetic parameter, the wall thickness parameter, the radiation parameter, the thermal conductivity parameter and the Prandtl number on the flow and temperature profiles are presented, moreover, the local skin-friction and Nusselt numbers. A comparison of obtained numerical results is made with previously published results in some special cases, and excellent agreement is noted. The obtained numerical results confirm that the introduced technique is powerful mathematical tool and it can be implemented to a wide class of non-linear systems appearing in more branches in science and engineering.

The previous investigations ([

In this section, we will consider a steady, 2Dim boundary layer flow of an incompressible Newtonian fluid over a continuously impermeable stretching sheet. The origin is located at a slit, through which the sheet is drawn through the fluid medium (see

where

where A is a very small constant so that the sheet is sufficiently thin and m is the velocity power index. We must note that the proposed model is satisfied only for

In case of approximations for the usual boundary layer of the Newtonian fluid, we can write, the steady 2D im boundary-layer equations taking into account the thermal radiation effect in the energy equation in the following form:

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

The radiative heat flux

where

The physical and mathematical advantage of the Rosseland formula (4) consists of the fact that it can be combined with Fourier’s second law of conduction to an effective conduction-radiation flux

where

ing the net contribution of the radiation emitted from the hot wall and absorbed in the colder fluid, takes the form

To obtain similarity solutions, it is assumed that the magnetic field

where

The mathematical analysis of the problem is simplified by introducing the following dimensionless coor- dinates

where

In this study, the equation for the dimensionless thermal conductivity

where

Upon using these variables, the boundary layer governing Equations (1)-(3) can be written in a non-dimensional as follows

where

Also, the boundary conditions transformed to the following form

where

cates the plate surface. We can write the Equations (13)-(14) in a simple form and to facilitate the computation, if we take into the account the transformation

The physical quantities of primary interest are the local skin-friction coefficient

where

In this section, we implement Chebyshev spectral method to solve the resulting system of non-linear ODEs of the form (17)-(18) with boundary conditions (19)-(20). To achieve this aim, and since the Gauss-Lobatto nodes

inside

form

also the boundary conditions will transform to the following form

where

Also using the boundary conditions (24), the constants of integration

Now, the approximations to the Equations (25) and (26) will take the following forms

where

And with implementation the transformation (27), we can write the system (22)-(23) to a form of system of non- linear equations in the highest derivative as follows:

These equations are non-linear system of 2n + 2 algebraic equations in 2n + 2 unknowns

In

In

m | 10.00 | 9.00 | 7.00 | 5.00 | 3.00 | 1.00 | 0.50 | 0.00 | −1/3 | −0.50 |
---|---|---|---|---|---|---|---|---|---|---|

1.0603 | 1.0589 | 1.0550 | 1.0486 | 1.0359 | 1.0000 | 0.9799 | 0.9576 | 1.0000 | 1.1667 | |

Present work | 1.0603 | 1.0588 | 1.0551 | 1.0486 | 1.0358 | 1.0000 | 0.9798 | 0.9577 | 1.0000 | 1.1666 |

m | 10.00 | 9.00 | 7.00 | 5.00 | 3.00 | 1.00 | 0.50 | 0.00 | −1/3 | −0.50 |
---|---|---|---|---|---|---|---|---|---|---|

1.1433 | 1.1404 | 1.1323 | 1.1186 | 1.0905 | 1.0000 | 0.9338 | 0.78439 | 0.5000 | 0.0833 | |

Present work | 1.1433 | 1.1404 | 1.1322 | 1.1186 | 1.0904 | 1.0000 | 0.9337 | 0.7843 | 0.5000 | 0.0832 |

From

The influence of the velocity power index parameter m on the temperature profiles is displayed in

From

In

increase when the internal heat generation parameter

M | m | R | Pr | |||||
---|---|---|---|---|---|---|---|---|

0.0 | 0.2 | 0.5 | 0.1 | 0.5 | 1.0 | 0.1 | 0.924821 | 0.347439 |

0.5 | 0.2 | 0.5 | 0.1 | 0.5 | 1.0 | 0.1 | 1.168580 | 0.282221 |

1.0 | 0.2 | 0.5 | 0.1 | 0.5 | 1.0 | 0.1 | 1.369821 | 0.210946 |

0.5 | 0.0 | 0.5 | 0.1 | 0.5 | 1.0 | 0.1 | 1.13387 | 0.240928 |

0.5 | 0.25 | 0.5 | 0.1 | 0.5 | 1.0 | 0.1 | 1.17740 | 0.286494 |

0.5 | 0.5 | 0.5 | 0.1 | 0.5 | 1.0 | 0.1 | 1.22242 | 0.330877 |

0.5 | 1.0 | 0.5 | 0.1 | 0.5 | 1.0 | 0.1 | 1.31682 | 0.418037 |

0.5 | 0.2 | 0.0 | 0.1 | 0.5 | 1.0 | 0.1 | 1.04487 | 0.372139 |

0.5 | 0.2 | 0.5 | 0.1 | 0.5 | 1.0 | 0.1 | 1.16858 | 0.277498 |

0.5 | 0.2 | 5.0 | 0.1 | 0.5 | 1.0 | 0.1 | 1.32780 | 0.090113 |

0.5 | 0.2 | 0.5 | 0.0 | 0.5 | 1.0 | 0.1 | 1.16858 | 0.305911 |

0.5 | 0.2 | 0.5 | 0.2 | 0.5 | 1.0 | 0.1 | 1.16858 | 0.253150 |

0.5 | 0.2 | 0.5 | 0.5 | 0.5 | 1.0 | 0.1 | 1.16858 | 0.196673 |

0.5 | 0.2 | 0.5 | 0.1 | 0.0 | 1.0 | 0.1 | 1.16858 | 0.433209 |

0.5 | 0.2 | 0.5 | 0.1 | 0.5 | 1.0 | 0.1 | 1.16858 | 0.277498 |

0.5 | 0.2 | 0.5 | 0.1 | 1.0 | 1.0 | 0.1 | 1.16858 | 0.185307 |

0.5 | 0.2 | 0.5 | 0.1 | 0.5 | 0.7 | 0.1 | 1.16858 | 0.166469 |

0.5 | 0.2 | 0.5 | 0.1 | 0.5 | 1.0 | 0.1 | 1.16858 | 0.277498 |

0.5 | 0.2 | 0.5 | 0.1 | 0.5 | 3.0 | 0.1 | 1.16858 | 0.778337 |

0.5 | 0.2 | 0.5 | 0.1 | 0.5 | 1.0 | −1.0 | 1.16858 | 0.904339 |

0.5 | 0.2 | 0.5 | 0.1 | 0.5 | 1.0 | −0.5 | 1.16858 | 0.708445 |

0.5 | 0.2 | 0.5 | 0.1 | 0.5 | 1.0 | 0.0 | 1.16858 | 0.402232 |

0.5 | 0.2 | 0.5 | 0.1 | 0.5 | 1.0 | 0.05 | 1.16858 | 0.352254 |

0.5 | 0.2 | 0.5 | 0.1 | 0.5 | 1.0 | 0.1 | 1.16858 | 0.288756 |

In this work, we implemented the Chebyshev spectral method to solve the non-linear system of ODEs of the proposed model. The fluid thermal conductivity is assumed to vary as a linear function of temperature. Asystematic study on the effects of the various parameters on flow and heat transfer characteristics is carried out. It has found that the effect of increasing values of the magnetic parameter, the velocity power index parameter, thermal conductivity parameter and the radiation parameter reduce the local Nusselt number. On the other hand, it is observed that the local Nusselt number increases as the Prandtl number and wall thickness parameter increases. Moreover, it is interesting to find that as the magnetic parameter, wall thickness parameter and the velocity power index parameter increases in magnitude, causes the fluid to slow down past the stretching sheet, the skin-friction coefficient increases in magnitude. A comparison with previously published work is given to ensure that the obtained numerical results are in excellent agreement, and this confirms that the validity of the proposed method to solve the presented model. Finally, we can make the error is smaller by increasing the terms in the series (27). All computations in this paper are done using Mathematica 8.

The first three authors thank Deanship of Academic Research, Al Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh, KSA, for the financial support of the project number (351227).

Mohamed M.Khader,Mohammed M.Babatin,AliEid,Ahmed M.Megahed, (2015) Numerical Study for Simulation the MHD Flow and Heat-Transfer Due to a Stretching Sheet on Variable Thickness and Thermal Conductivity with Thermal Radiation. Applied Mathematics,06,2045-2056. doi: 10.4236/am.2015.612180