_{1}

An analysis of the hydromagnetic free convective flow past a vertical infinite porous plate in a rotating fluid is carried out. The temperatures involved are assumed to be very large so that the radiative heat transfer is significant, which renders the problem very non-linear even on the assumption of a differential approximation for the radiative flux. The temperature and velocity fields are computed using a generic software tool based on the Nakamura finite difference scheme. The genericity of the software tool is in the sense that it is a common solution to the category of time dependent laminar fluid flows expressed in one spatial coordinate. The input equations, together with other relevant parameters, are transformed into postfix code which will be farther interpreted in the computation process. The influence of the various parameters entering into the problem is shown graphically followed by a discussion of results.

Extensive research efforts have been directed to the study of the theory of rotating fluids due to its application in Cosmical and Geophysical fluid dynamics, meteorology and engineering [

This paper therefore incorporates radiative transfer into the study of natural convection in a rotating fluid the- reby widening the applicability of the results. For an optically thin gas in a transparent medium with the absorption coefficient α (

where

Consider the unsteady convective flow of an electrically conducting incompressible viscous fluid past an infinite vertical porous flat plate at z' = 0. Let the fluid and the plate be in a state of rigid rotation with uniform angular velocity Ω about the z' axis which is normal to the plate. A uniform magnetic field B_{0} is imposed along z'-axis and the plate is assumed to be electrically non-conducting. The temperature of the plate is maintained at

where _{p} is the specific heat of the fluid; g is the acceleration due to gravity; β is the coefficient of volume expansion; σ_{c} is the electrical conductivity of the fluid and w_{o} is the constant suction velocity.

The boundary conditions are given by:

Introducing the following non-dimensional quantities

Equations (2)-(5) reduce to:

where

The above system of Equations (7)-(9) with boundary conditions (10) has been solved numerically by a generic software based on the Nakamura [

Equations (7)-(9) are coupled non-linear parabolic partial differential equations in u, v and q. First,

and

(11)-(13):

where

For the sake of simplicity, we write:

Using the above formulation, Equations (11)-(13) take the form:

Using the central difference scheme which is unconditionally stable, Equations (14)-(16) reduce to:

At time step j + 1, Equations (17)-(19) reduce to:

If the problem is well defined, Equations (20)-(22) admit a solution but cannot be solved individually for each grid point i. The equations for all the grid points must be solved simultaneously. The set of equations for

where

where

where

For each time step, the system of Equations (23)-(25) requires an iterative procedure due to the presence of non-linear coefficients. Successive substitution and iteration are continuously executed for each time step until convergence is reached.

In order to investigate the behavior of the velocity and temperature profiles, curves are drawn for various values of the parameters that describe the flow and are displayed in Figures 2-11.

From

1) the temperature profile (θ) increases due to an increase in time (t);

2) there is a fall in temperature profile (θ) due to an increase in Prandtl number (Pr);

3) there is a rise in temperature profile (θ) due to an increase in rotation parameter (Er);

4) there is an insignificant change in the temperature profile (θ) due to an increase in radiation parameter (R).

From

1) the transient primary velocity profile (u) is backward;

2) the transient primary velocity profile (u) decreases due to an increase in time (t);

3) there is a rise in the transient primary velocity profiles (u) due to an increase in rotation parameter (Er) whereas the transient velocity field (u) decreases due to an increase in Prandtl number (Pr);

4) there is an insignificant change in the transient primary velocity profile (u) due to an increase in radiation parameter (R).

From

1) the transient primary velocity profile (u) increases due to an increase in time (t);

2) there is a fall in the primary velocity profile (u) due to an increase in rotation parameter (Er) whereas u rises with an increase in Prandtl number (Pr);

3) there is an insignificant change in the transient primary velocity profile (u) due to an increase in radiation parameter (R).

From

1) the transient secondary velocity profile (v) increases due to an increase in time (t) and Prandtl number (Pr);

2) there is a fall in the transient secondary velocity profile (v) due to an increase in rotation parameter (Er)

and radiation parameter (R).

From

1) the secondary velocity profile (v) is backward;

2) there is a fall in the secondary velocity profile (v) due to an increase in time (t) and Prandtl number (Pr);

3) there is a rise in the transient secondary velocity profile (v) due to an increase in rotation parameter (Er) and radiation parameter (R).

In this work, a simulation was carried out on the hydromagnetic free convective flow past a vertical infinite porous plate in a rotating fluid. Very large temperatures were assumed in the analysis in order to make the radia- tive heat transfer significant. The velocity and the temperature fields were computed using a generic software tool based on the Nakamura finite difference scheme. From the results obtained, we observed that:

1) the development of generic tools can drastically simplify the solution of fluid flow problems;

2) only the input equations and the relevant parameters need to be defined in a generic file which will be used as input to the simulation system.

3) the postfix code is very efficient in the computation of arithmetic expressions.

Lev G. Biazrov,Harouna Naroua, (2016) A Computational Solution of Natural Convection Flow in a Rotating Fluid with Radiative Heat Transfer. American Journal of Computational Mathematics,06,108-119. doi: 10.4236/ajcm.2016.62012