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This paper investigates the effects of Hall currents and radiation on free-convective steady laminar boundary-layer flow past a semi-infinite vertical plate for large temperature differences. A uniform magnetic field is applied perpendicular to the plate. The fluid density is assumed to vary exponentially and the thermal conducting linearly with temperature. The fluid viscosity is assumed to vary as a reciprocal of a linear function of temperature. The usual Boussinesq approximation is neglected. The nonlinear boundary layer equations governing the problem under consideration are solved numerically by applying an efficient numerical technique based on the shooting method. The effects of the magnetic parameter , the Hall parameter m, the density/temperature parameter * n*, the radiation parameter

*, the thermal conductivity parameter*

**N***, the viscosity temperature , and the temperature ratio parameter are examined on the velocity and temperature distribution as well as the coefficient of heat flux and shearing stress at the plate.*

**S**The study of natural convection boundary layer flow of an electrically conducting fluid over a continuously stretching heated semi-infinite plate is considered very essential to understand the behavior of the performance of fluid motion in several applications. This is because it serves understanding of some phenomenon occurring in several environmental and engineering fields. Prominent applications are the aerodynamic extrusion of plastic sheets, cooling of an infinite metallic plate in a cooling path, Fibers spinning and continuous casting, glass blowing, packed bed reactors or absorbent and others. The analysis of such flow forms the bases of a series of further investigations for laminar boundary layers. The first who presented boundary layer flow over a continuous solid surface with constant speed is Sakiadis [

Most of the effort in understanding fluid radiation is devoted to the derivation of reasonable simplifications. One of these simplifications was made by Cogley et al. [

Accordingly, Cogley et al. showed that for an optically thin nongray gas near equilibrium, the following relation holds:

In addition, they simplified (1) by assuming small temperature differences

where

For an optically thick gas, the gas self-absorption rises and the situation becomes difficult. However, the problem can be simplified by using the Rosseland approximation [

Previous studies of convective flow along vertical plates in the presence of radiation were restricted, in general, to the case where the temperature difference between the plate and the fluid was small. In this case, the fluid’s physical properties such as its viscosity and thermal conductivity may be taken as constant. Also, for small temperature differences, the Boussinesq approximation [

In situations where there is large temperature differences between the plate and the fluid, the fluid’s physical properties are affected by the high temperature and they can no longer be regarded as constant. Also, in this case, the Boussinesq approximation can no longer be used.

Some recent studies for radiating fluids have taken into account variations of the physical properties with temperature. For example, Aboeldahab [

Previous studies of convective flow along vertical plates in the presence of radiation were restricted, in general, to the case where the temperature difference between the plate and the fluid was small. In this case, the fluid’s physical properties such as its viscosity and thermal conductivity may be taken as constant. Also, for small temperature differences, the Boussinesq approximation [

It is worth mentioning that using the Cogley-Vincenti-Gilles model (2) depends on the assumption that the temperature differences

The above relation is more suitable for expressing the radiation term in the energy equation for the variable physical property problems.

Hence, in the present work, we study Hall currents effects on the MHD free-convective flow of an optically thin gray gas past a semi-infinite vertical plate with variable density, viscosity and thermal conductivity for high temperature differences neglecting the Boussinesq approximation. The nonlinear boundary layer equations, governing the problem, are solved numerically by applying an efficient numerical technique based on the shooting method. The velocity and temperature distributions as well as the coefficient of heat flux and the shearing stress at the plate are determined for different values of the Hall parameter m, the temperature ratio parameter

A steady laminar free-convective flow of a viscous gray gas in the optically thin limit past an isothermal semi- infinite vertical plate is considered. The

A uniform magnetic field is applied transversely to the direction of the flow. The magnetic Reynolds number is taken to be small enough so that the induced magnetic field can be neglected.

The viscous dissipation; the radiative heat flux in the

The density is assumed to vary exponentially with temperature as follows: [

where

The fluid thermal conductivity is assumed to vary as a linear function of temperature in the form

where

The fluid viscosity is assumed to vary as a reciprocal of a linear function of temperature in the form (see Lai and Kulacki, ref. [

or

where

Then the steady laminar two-dimensional free-convective flow is governed by the following boundary-layer equations:

The physical problem suggests the following initial and boundary condition

By using Equations (1), (3) and (5), Equations (7), (8) and (9) become

Introducing the following dimensionless variables

The continuity equation is satisfied by

From (14) and (15) we find that

Also, let

Using the above transformation the governing equations are reduced to:

The boundary conditions are transformed into

where

And primes denote differentiation with respect to

The most important characteristics of the flow are shearing stress at the plate

And the rate of heat transfer at the plate (Nusselt number)

Equations (22), (23) and (24) with the boundary conditions (25), are approximated by a system of nonlinear ordinary differential equations replacing the derivatives with respect to

The value of ^{−9} between any two successive iteration is employed as the criterion of convergence. We use the symbolic computational software Mathematica to solve this system. Solutions are obtained for the Prandtl number

In view of Equation (18) Equation (5) can be written in the form

Since

From this expression it is obvious that, since free-convection flow is studied,

It is worth mentioning that when the temperature difference

where the higher order terms are omitted, in addition, it is assumed that

Then according to the above relation the density can be treated as a constant in the continuity equation, energy equation and convective terms in the momentum equation and treated as a variable only in the buoyancy term of the momentum equation (Boussinesq approximation). Therefore, when

It is to be noted that, as

increase the rate of energy transport to the fluid and accordingly increases the fluid temperature. This increase in the fluid temperature increases the velocity of the fluid particles

Figures 9-11 show as expected, that the dimensionless velocities and temperature increase as the thermal conductivity parameter

In this paper, we have studied the effects of Hall currents and radiation on the MHD free convective steady lamina boundary layer flow past an isothermal semi-infinite vertical plate, for high temperature differences, the fluid is considered to be electrically conducting in the sence that it is ionized due to radiation.

The fluid density is assumed to vary exponentially and the thermal conductivity linearly with temperature the fluid viscocity is assumed to vary as a reciptocal of a linear function of temperature. Because of the high temperature differences between the fluid and the plate, the Boussinesq approximation is neglected the formula

This paper demonstrates the fact that the Boussinesq approximation gives substantial errors in the velocity and temperature distribution for high temperature differences. Therefore, to conclude more accurate results the density variation has to be taken into consideration in the continuity equation, energy equation and all terms of the momentum equation.

Besides, it is observed that:

1) The increasing in the radiation parameter

n | M | m | N | S | θ_{w} | θ_{r} | g | -θ' | |
---|---|---|---|---|---|---|---|---|---|

0.1 | 0.2 | 0.1 | 0.1 | 0.2 | 1.1 | 1.2 | 0.148867 | 0.000046254 | 0.512446 |

0.5 | 0.2 | 0.1 | 0.1 | 0.2 | 1.1 | 1.2 | 0.268777 | 0.000113357 | 0.664167 |

1 | 0.2 | 0.1 | 0.1 | 0.2 | 1.1 | 1.2 | 0.533357 | 0.000297222 | 0.95982 |

1 | 1 | 0.1 | 0.1 | 0.2 | 1.1 | 1.2 | 0.525044 | 0.00141242 | 0.952498 |

1 | 2 | 0.1 | 0.1 | 0.2 | 1.1 | 1.2 | 0.515072 | 0.0026532 | 0.943453 |

1 | 0.2 | 1 | 0.1 | 0.2 | 1.1 | 1.2 | 0.534418 | 0.00151062 | 0.960776 |

1 | 0.2 | 2 | 0.1 | 0.2 | 1.1 | 1.2 | 0.535081 | 0.00121293 | 0.961244 |

1 | 0.2 | 0.1 | 0.2 | 0.2 | 1.1 | 1.2 | 0.543009 | 0.000305804 | 0.921002 |

1 | 0.2 | 0.1 | 0.3 | 0.2 | 1.1 | 1.2 | 0.552471 | 0.000314226 | 0.883659 |

1 | 0.2 | 0.1 | 0.1 | 0.5 | 1.1 | 1.2 | 0.556452 | 0.00031661 | 0.854017 |

1 | 0.2 | 0.1 | 0.1 | 1 | 1.1 | 1.2 | 0.586593 | 0.000343934 | 0.743981 |

1 | 0.2 | 0.1 | 0.1 | 0.2 | 1.4 | 1.2 | 0.540372 | 0.000303249 | 0.929716 |

1 | 0.2 | 0.1 | 0.1 | 0.2 | 1.8 | 1.2 | 0.554563 | 0.000315406 | 0.870098 |

1 | 0.2 | 0.1 | 0.1 | 0.2 | 1.1 | 1.5 | 0.955568 | 0.000621407 | 1.012 |

1 | 0.2 | 0.1 | 0.1 | 0.2 | 1.1 | 2.2 | 1.40311 | 0.00101715 | 1.05857 |

2) The increasing in the Hall parameter m yields to a significant increasing in the secondary flow velocity, a slight increasing in the fluid velocities ^{ }and the fluid temperature the dimensionless wall-velocity gradients and the rate of heat transfer from the plate to the fluid.

3) The increasing in the magnetic parameter

4) The increasing in the thermal conductivity parameter s yields to an increasing in the fluid velocities, the fluid temperature the dimensionless wall-velocity gradient and the plate to the fluid.

5) The increasing in the viscosity-temperature parameter

6) The increasing in the density-temperature parameter

7) The increasing in the temperature ratio parameter

This research is funded by the Deanship of Research and Graduate Studies in Zarqa University/Jordan.