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We study the effects of thermal radiation of a viscous incompressible fluid occupying a semi-infinite region of space bounded by an infinite horizontal moving hot flat plate in the presence of indirect natural convection by way of an induced pressure gradient. The fluid is a gray, absorbing emitting radiation but a non scattering medium. An exact solution is obtained by employing Laplace transform technique. Since temperature field depends on Reynold number the flow is considered to be non-isothermal case (the temperature of the plate Tw ≠ constant) and for an isothermal case (Tw = constant) the flow is determined by the Reynold number which is equal to 1.

Thermal radiation of an optically thick gray gas is of great importance to the study of high temperature physics and space technology. Mentioning the study of this type of problem with a view to analyse the transient approach of a radiative heat-transfer aspects of an optically thick fluid it seems to be appeared in the literature as studied by many authors. England and Emery [

Although the radiation boundary layer thickness depends on Reynolds number the aim of the present investigation of the problem is to a study of thermal radiation of an optically thick gray gas in taking into account of an unsteady flow of an incompressible viscous fluid occupying a semi-infinite region of space bounded by an infinite horizontal moving hot flat plate in the presence of indirect natural convection by way of induced pressure gradient. Since the temperature field depends on Reynolds number the wall temperature does not constant (constant) as the temperature varies along the plate and the recovery factor is determined by the Reynolds number. An uniform wall temperature (constant) for an isothermal flat plate is fully understood if the value of Reynolds number is equal to 1. Thus it comes to a conclusion that since the temperature field depends on radiation layer thickness it is a decisive importance to an isothermal flat plate (constant) with regard to a finite thickness () [see Ghosh and Pop [

Consider the unsteady flow of a viscous incompressible fluid occupying a semi-infinite region of space bounded by an infinite horizontal plates moving with constant velocity with reference to indirect natural convection by way of induced pressure gradient. The flow is considered optically thick gray gas with indirect natural convection and radiation. We choose the cartesian coordinate system is such a way that x-axis is taken along the plate in the direction of the flow and y-axis is normal to it [see

The momentum equations in component form can take the form

where is the fluids density, p the pressure, the co-

efficient of viscosity and g the acceleration due to gravity.

The equation of energy is

where is the specific heat and k the thermal conductivity.

It is assumed that there is a temperature variation along the x-direction of the horizontal plate. The temperature of the flow can be written as

where T is the temperature of the fluid, the temperature of the fluid far away the plate and the dimensionless temperature and.

The equation of state becomes

where is the density of the fluid, the coefficient of thermal expansion and the other symbols have their usual meanings.

From (2) and (5) we have

Sine the temperature is uniform at infinity, it is reasonably assumed to be as. Thus

is zero everywhere in the flow. Hence (6) becomes

On the use of (7), the Equation (1) becomes

Using infinity conditions in (8), one find

Hence the Equation (8) reduced to

The initial and boundary conditions are

From (4), it is stated that the temperature of the flow is dependent on Reynolds number.

The dimensionless temperature with the help of (4), we get

where and the Reynolds number.

In comparison to the study of Ghosh and Pop [

, where δ the radiation layer thickness and the other symbols have their usual meanings with

(L is the characteristic length), it is rigorously stated that the radiation layer thickness depends on Reynolds number and the plate temperatures does not constant (constant). For an isothermal plate (constant), the thickness of the radiation layer should be taken finite value i.e.. In this situation, Ghosh and Pop [

Introduce the dimensionless quantities

where, , , k, g and are, respectively, the coefficient of viscosity, kinemetic coefficient of viscosity, specific heat at constant pressure, thermal conductivity, gravitational acceleration and the coefficient of thermal expansion and the other symbols have their usual meanings.

On the use of (13), the Equation (9) becomes

The radiation flux vector can be found from Isachenko et al. [

where and are, respectively, the Stefan-Boltzman constant and the spectral mean absorption coefficient of the medium.

It is assumed that the temperature differences within the flow are sufficiently small such that may be regarded as a linear function of the temperature. It can be established by expanding i.e. a Taylor series about and neglecting higher order term. Therefore, can be expressed in the following form

Using Equations (15) and (16), the energy Equation (3) can be written in a dimensionless form subject to (13) such as

where is the radiation paeameter.

The corresponding boundary conditions are

The solutions for the velocity and temperature distributions can be obtained by applying Laplace transform technique subject to the boundary conditions (18) and (19) together with the Equations (14) and (17) become

and

where

We shall now discuss some particular cases of interest Case I: In the absence of radiation parameter and the prandtl number, the solutions (20) and (21) reduce to

and

Case II: In the absence of radiation temperature

() and the pressure gradient the Equations (1)-(3) transform into a flat plate at zero incidence so that the velocity and the temperature fields are identical when the prandtl number.

The graphical representations of numerical results with different parameters, , , and for the velocity and temperature distributions are plotted against in Figures 2-8. There is steep decline from the wall for all profiles in Figures 2-8 and no velocity and temperature overshoot. The profiles of spatial dimensionless velocity with distance from the wall, at various time are shown in