_{1}

^{*}

The effect of radiation on the flow over a stretching plate of an optically thin gray, viscous and incompressible fluid is studied. The fluid viscosity is assumed to vary as an inverse linear function of the temperature. The partial differential equations (PDEs) and their boundary conditions, describing the problem under consideration, are dimensionalized and the numerical solution is obtained by using the finite volume discretization methodology which is suitable for fluid mechanics applications. The numerical results for the velocity and temperature profiles are shown for different dimensionless parameters entering the problem under consideration, such as the temperature parameter, *θr*, the radiation parameter,* S*, and the Prandtl number, *Pr*. The numerical results indicate a strong influence of these parameters on the non-dimensional velocity and temperature profiles in the boundary layer.

At high temperature, radiation has significant effects on the flow field. These effects have substantial applications in many industrial areas, such as electrical power generation, solar power technology, and aerospace engineering.

There has been extensive research on the effects of radiation on fluid flow. The free convection flow in the presence of radiation has been previously studied by Ali et al. [

Bestman and Adiepong [

In the present study, we determine the effect of radiation on the flow field over a stretching plate of an optically thin gray fluid. We consider the fluid as viscous and incompressible, with temperature dependent viscosity.

The presented results are obtained after dimensionalization of the PDEs using a numerical approach. This approach is based on the finite volume (FV) discretization scheme. The discretization was performed with the use of a specialized symbolic package created in Mathematics.

We consider the flow of a viscous and incompressible fluid due to an isothermal stretching flat surface. The fluid properties are assumed to be isotropic and constant, except for the fluid dynamic viscosity. The x-axis is taken along the plate and the y-axis normal to it, as depicted in

The equations governing the problem are given by:

Continuity equation

Momentum equation

Energy equation

where are the components of the velocity in the x and y directions respectively, is the fluid density, is the dynamic viscosity, T is the fluid temperature, k is the thermal conductivity, is the specific heat of the fluid under constant pressure and is the radiative heat flux.

The dynamic viscosity is assumed to be an inverse linear function of temperature [

or

where

is a constant, is the dynamic viscosity at infinity, is the kinematic viscosity at infinity, is a reference temperature, is the temperature at infinity, is a constant which in general is positive for liquids and

negative for gases.

The boundary conditions are defined as follows:

where is a constant and is the temperature of the stretching flat surface. In the case of an optically thin gray fluid the local radiant absorption is expressed as [8, 25,28],

where is the absorption coefficient and is the Stefan-Boltzman constant. We assume that the temperature differences within the flow are sufficiently small such that may be expressed as a linear function of the temperature. This is accomplished by expanding in a Taylor series about and neglecting higher-order terms, thus

Equation (8) through (9) takes the form:

Introducing the following transformations

where a prime denotes differentiation with respect to. In view of (10) and (11), Equation (1) is satisfied identically and Equations (2) and (3) reduce to

The boundary conditions (7) are transformed to

The non-linear system of coupled differential Equations (12) and (13) subject to the boundary conditions (14) has been solved following a symbolic approach. For this purpose we have used the Computer Algebra System (CAS) Mathematica [

The analysis begins by obtaining the discretized form of the system of equations by using a symbolic package developed for that purpose [

In the present study we numerically investigate the effect of radiation on the flow field over a stretching plate of an optically thin gray fluid,

In

The effect of the radiation parameter S (S: 0.1, 1, 7) on the non-dimensional velocity is shown in

The effect of the radiation parameter S (S: 0.1, 1, 7) on the non-dimensional temperature is shown in

The effect of the Prandtl number, Pr, on the non-dimensional temperature is presented in

Finally, Figures 6 and 7 show the effect of the temperature parameter on the non-dimensional velocity and temperature when Pr = 0.7, S = 1 and for

three different values of the temperature parameter. Temperature increases with the increase of the temperature parameter, (: –2, –0.05, –0.01). However, the effect of the temperature parameter is more pronounced on the non-dimensional velocity and it decreases as the parameter increases,

The numerical results of this study could bring new insight on the effect of thermal radiation on the flow past a stretching plate with temperature dependent viscosity. These results could be utilized in many industrial and practical areas, including glass and semiconductor processing, atmospheric flows with application to global climate change, electrical power generation, solar power technology, and aerospace engineering.

The effects of thermal radiation on the flow and temperature fields over a stretching plate of an optically thin gray fluid were numerical investigated. The fluid was considered incompressible with temperature dependent viscosity. The main findings of this study could be summarized in the following: (1) Increase of the radiation parameter increases the velocity profile but decreases the temperature profiles. (2) On the other hand, increase of the temperature parameter decreases the velocity profile and increases the temperature profile in the boundary layer. (3) Increase of Prandtl number decreases the temperature profile in the boundary layer.