FREE CONVECTIVE MHD FLOW PAST A VERTICAL PLATE THROUGH A POROUS MEDIUM WITH RAMPED WALL TEMPERATURE AND CONCENTRATION

This study attempts to examine the ramped wall temperature and concentration of an unsteady free MHD, viscous, incompressible, electrically conducting fluid past a vertical plate through a medium of porous nature under thermal radiation. Sets of dimensional governing equations are considered by taking suitable assumptions, which are then transformed into non-dimensional forms. Non-dimensional equations are being solved analytically with the help of Laplace Transform method. Resultant effects of some parameters concerning the problem on temperature, velocity, concentration, coefficient of skin-friction, Nusselt number and Sherwood number are discussed through different graphs and are physically interpreted. From the graphs, results are found out and related interpretations are made.


INTRODUCTION
Magnetohydrodynamics (MHD) is the science dealing with analysis of interaction between magnetic fields and electrically conducting fluids in motion. MHD principles are applied in engineering, plasma-physics, biotechnology, biomedical science, astrophysics, geophysics, 1207 FREE CONVECTIVE MHD FLOW electronics etc. So, many engineers and scientists are interested in its applications in their respective fields. MHD flows are important in many fields and due to its importance several researchers have put their attention in doing their research works on MHD field. Some notable among them are Babu et al. [1], Panezai et al. [2], Rajesh et al. [3], Basha et al. [4], Raju et al. [5] etc.
Convection and radiation are the two important ways of heat transfer. In the process of convection heat transfer takes through actual motion of matter, while in radiation transfer of heat occurs through electromagnetic waves. The processes of radiation and convections associated with fluid flows characterise the radiative convection flows which find suitable applications in various energy plants. Effects of radiation on MHD free fluid have also become more important in varied industrial activities and research. Due to its ever-growing importance, many researchers have carried out their research works on free convective incompressible viscous fluid flow taking thermal radiation into account. In this context, significant works are done by Lavanya [6], Ali Shah et al. [7],Chiranjibi et al. [8], Sambath et al. [9], Cogley et al. [10], Makinde et al. [11], Vasu et al. [12], etc.
Flow can easily be transmitted through the pores of a medium. Properties and behaviours of flow passing through a porous medium have become an important theme used in different fields of applied engineering and sciences. Considering the practical utility of research on fluid mechanics involving porous medium, volume of works done in this line has been increasing. The notable research works relating to this field include those conducted by Mishra et al. [13], Siddabasappa[14], Mehta et al. [15], Prasad and Reddy [16], Krishna et al. [17]. It needs to be mentioned here that Sinha et al. [18] studied MHD free convective flow through a porous medium past a vertical plate with ramped wall temperature. Importantly, the model of Cogley ' s et al. [10] has been applied by them to analyse radiative heat flux.
This work is an extension of the investigation carried out by Sinha et al. [18], where the effects of concentration have been examined in addition to temperature parameter. Because, it is essential to know the effects of concentration level of fluids on the rate of mass transfer.

FORMULATION OF THE PROBLEM
The present study is to investigate an unsteady MHD free convective radiative fluid flow of an optically thin viscous incompressible fluid past an infinite vertical plate through a porous medium in presence of temperature and concentration. A coordinate system is introduced, where 1208 MAUSHUMI MAHANTA, SUJAN SINHA X-axis is considered along vertical direction of the wall and Y-axis is considered along the normal to the wall as shown in Figure 1.
Here, fluid is considered as optically thin gray gas with free convection and radiation.
Following assumptions are taken into consideration in the investigation: i. Only fluid density varies, but other fluid properties are kept constant.
ii. As compared to Y direction, radiative heat flux in X direction is minimal. iii. Viscous dissipation of energy is also very minimum.
iv. In the case of plate being infinite in X-direction, all the physical variables become functions of y and t.
Considering above mentioned assumptions, dimensional governing equations are as follows: 1.

Energy Equation:
2 3. Concentration Equation: Initial and boundary conditions as regards to velocity, temperature and concentration fields are: y 0: u 0, T T , C C for t 0 Following Cogley ' s model [10], the radiative heat flux rate for a non gray gas in an optically thin fluid is as follows: Here, 0 K  denotes absorption co-efficient,  means wavelength, h e  represents Planck ' s function. Again subscript 0 means that all physical quantities are found out at temperature T  .

METHOD OF SOLUTION
Applying Laplace Transform technique in the equations (8) to (10), solutions can be written in following way: The boundary condition equations (11.1) to (11.3) are also transformed to equation (15) Applying Laplace Transform technique in the equations (12) to (14) and considering the boundary conditions defined in equation (15), the solutions of the problem are obtained.
Solutions are written using error function (erf) and complementary error function (erfc) as: Where, Co-efficient of heat transfer rate considering Nusselt number (Nu) is as follows: P r P r t , t t P r R a e r f tR a e 2 R a Now, co-efficient of mass transfer rate considering Sherwood number (Sh) is as follows:

Skin-friction co-efficient
Co-efficient of Skin-friction () is: Where, R a t P r , R a , t P r e P r R a e r R a t , t

COMPARISON OF RESULTS
Work of Sinha et.al. [18] is considered for comparing the results of the present paper.
Comparing figure 13 with figure 2 of the work done by Sinha et.al. [18], we observe the same kind of behaviour due to the implementation of magnetic intensity in velocity profile. i.e. there is a significant effect of magnetic parameter on this profile. Thus, there is an excellent agreement between the results obtained by Sinha et.al. [18] and those arrived at by the present authors.

CONCLUSIONS
1. Fluid motion gets reduced due to action of magnetic parameter M and Schmidt number.  Greek Symbol: : Kinematic viscosity,   : Co-efficient of thermal expansion,  : Electrical conductivity, : Density of fluid,  : Thermal conductivity, : Non-dimensional temperature, : Non-dimensional concentration.