CHEMICALLY REACTING MAGNETO-HYDRODYNAMICS (MHD) FLUID FLOW THROUGH A POROUS MEDIUM

This paper considered and investigated the effects of chemically reacting Fluid flow through a porous medium in a Magnetic field as a magneto-hydrodynamics fluid flow through a porous medium. We developed a mathematical model for the flow and using perturbation method, the modeled partial differential equations were reduced to a system of ordinary differential equations and solved. The geometry of the problem was considered where the x-axis is taken in the vertical direction along the plate while y-axis is normal to the plate so that all the characteristics of the fluid are independent of the variable x. The impact of Chemical reaction, Magnetic intensity, Prandt number, schmidt number, porosity and thus permeability of the physical parameters were investigated on velocity, temperature, and concentration profiles. More so, the impacts of the same parameters were investigated on the skin-friction, Nuselt number, and Sherwood number. The results showed that growing magnetic intensity leads to decrease in velocity and skin-friction of the flow field at every point, increase in chemical reaction and Schmidt number leads to decrease in temperature and concentration at every point of the flow field. An increase in Prandtl number, Magnetic intensity, and Permeability shows no effect on concentration profile at every point of the flow field. 5297 CHEMICALLY REACTING MAGNETO-HYDRODYNAMICS (MHD) FLUID


INTRODUCTION
It has been observed that the chemically reacting fluid flow in a magnetic field through a porous medium considered as a Magneto-hydrodynamic (MHD) fluid flow through a porous medium has gotten remarkable importance in many technical fields, so that many scientists and technologist are taking keen interest in the study of fluid flow so that their results so obtained may be applied with great accuracy in the respective fields. In this context, we have gone through the contributions made by many great researchers and their achievements in this field in a chronological order so that some new mathematical models governing such motion of a fluid may be innovated and improved results in this study based on the adopted theories and Mathematical methods can be found. We observed that, Chambre &Yonug [1] discussed the diffusion of a chemically reactive species. Also, the effect of magnetic field, heat, and mass transfer on unsteady two-dimensional laminar flow of a viscous incompressible fluid past a semi-infinite moving vertical porous plate with variable suction in the presence of homogeneous first order chemical reaction and temperature dependent heat generation was studied by [2]. Mbeledogu and Ogulu [3] presented the solution of unsteady hydromagnetic natural convection heat and mass transfer flow of a rotating fluid past a vertical porous plate in the presence of first order chemical reaction and radiative heat transfer. Pal & Mondal [4] in their pioneer work studied the effects of variable thermal conductivity, thermal diffusion, and diffusion thermo on mass transfer and MHD non -Darcy mixed convection heat over a non -linear stretching sheet with chemical reaction. Khan et al. (2014) proposed numerically MHD laminar boundary layer flow over a wedge because of heat generation, chemical reaction, and thermal radiation. Chamkha and Ahmed [6] studied on similarity solution for unsteady MHD flow near a stagnation point of a three -dimensional porous body with mass 5298 DIDIGWU, ONAH, MBAH transfer and heat, chemical reaction, and heat generation / absorption. Rao et al. [7] analyzed an unsteady MHD free convection heat and mass transfer flow past a semi-infinite vertical permeable moving plate with chemical reaction and soret effects, heat absorption, and radiation. Srinivasachanya and Reddy [7] made a detailed report on free convection in a non -Newtonian power law fluid over a saturated porous medium with chemical reactions and radiation effects. In this work, we investigated the chemically reacting flow through a porous medium in a magnetic field as a MHD fluid flow through a porous medium. The momentum, energy and diffusion equations that govern the flow field are solved using perturbation method.

Fig. 2.1, Geometry of the flow problem
In fig.1.1 above, we considered unsteady, two-dimensional, MHD natural convection with chemically reacting fluid flow over a porous vertical plate. The * -axis is taken along the vertical plate while the * -axis is perpendicular to the plate. A steady magnetic field is applied in the direction normal to the plate. By the application of magnitude analysis on the Navier-Stokes's equation, the MHD term is derived. The properties of fluid are assumed to be constant but the 5299 CHEMICALLY REACTING MAGNETO-HYDRODYNAMICS (MHD) FLUID influence of temperature on density variation has been considered in the body force. For the fact that the plate is vertical and continuous in  x -direction, all physical parameters depend only on y and t. Based on the above claim, the governing equations and boundary conditions for the flow field are given below as given by [9] are.
Continuity equation Momentum equation Diffusion equation Subject to the boundary conditions.
The parameters of the model are described as: where The negative sign shows that the flow is towards the plate.
We introduce the following dimensionless quantities and parameters. , Applying these dimensionless quantities and parameters in equations (2-4), we obtained the following equations With the following applicable conditions at the boundary

METHODOLOGY
To bring down the system of partial differential equations 8 -10 to a system of ordinary differential equations in the non-dimensionless form using perturbation method, we assume the following for velocity, temperature, and concentration distribution of the flow field as the amplitude ( ≪ 1) of the permeability variation is very small.  Substituting equation 12 in equations 8 -10, we get , Pr Pr Pr Pr Subject to the conditions at the boundary 0 = , 1 = 0, 0 = 1, 1 = 0, 0 = 1, 1 = 0, = 0,

SOLUTION OF THE PROBLEM
Solving the differential equations from 14 -19, using boundary conditions 20 and finally with the help of equations 12 and 13, we obtain the velocity, temperature, and concentration as follows ( , The Skin -friction, Nusselt number, and Sherwood number are important physical parameters for this type of boundary layer flow. These parameters can be defined and determined as follows

Skin -friction
Based on the velocity field, the skin -friction at the plate can be obtained, which in nondimensional form and is given by

Nusselt number
Based on the temperature field, the rate of heat transfer coefficient can be obtained, which in the non -dimensional form, in terms of the Nusselt number and is given by

Sherwood number
Based on the concentration field, the rate of mass can be obtained, which in the nondimensional form, in terms of the Sherwood number and is given by

RESULTS AND DISCUSSION
The chemically reacting flow of MHD fluid through a porous medium has been studied.  that as chemical reaction parameter rises, the temperature profile decreases. This is because heat is not being released but rather utilized to facilitate the reaction (endothermic reaction). shows that, there is no effect of chemical reaction on concentration profiles at k1=3, meaning that any increase in chemical reaction after k1=1 shows no effect on concentration profiles. It shows that rate of chemical reactioncannot be increased unabated.  This is because, at the free stream region, the skin-friction is mild (as a result of no flow competition) but at the boundary region, the skin-friction is high (as a resulty of porosity along the plate) thereby causing the skin-friction to reduce more at the stream region than at the boundary region.  Fig. 5.7 illustrates the influence of permeability on velocity profile. It can be seen that, velocity profile slowly increases as permeability increases. We also observed that at the free stream region, all velocity profiles are smoothly decayed to zero. This is because, at distant points from the plate (free stream region), nothing instigate velocity but at the boundary region, velocity is instigated by porosity and thereby causing velocity to rise close to the plate. in permeability from 2, 4, 6, and to the highest number of 8, results to increase on skin-friction coefficient. We also noticed that at a time in the increment of permeability, there will be no effect on skin-friction coefficient at both boundary and stream region. We as well observed that, the effect is more at the stream region than at the boundary region. This is because, at the boundary region, there is turbulence flow as a result of porosity along the plate, but far away from the plate (stream region), the flow is mild.    Prandtl number shows no effect on velocity and that velocity decayed to zero level at the distant points from the plate (free stream region). from the heated plate more rapidly. Hence, in the case of smaller Prandtl numbers as the thermal boundary layer is thicker and therefore the rate of heat transfer depressed. We also observed that at distant points from the plate (stream region), temperature is at the zero level. Fig. 5.17 shows the effect of prandtl number on concentration profile. It is observed that increase in prandtl number has no effect on concentration profile at both the boundary layer region and at the free stream region irrespective of the values of Prandti number.

Conclusion
The results of the physical interest on velocity, temperature, concentration, skin-friction, heat transfer, and mass transfer for the flow field are summarized herein as follow: 1. A growing magnetic intensity results to decrease in velocity, and skin-friction of the flow field at every point.
2. An increase in chemical reaction and Schmidt number leads to decrease on velocity at all point of the flow field. 5. An increase in Prandtl number, Magnetic intensity, and Permeability shows no effect on concentration profile at every point of the flow field. 7. An increase in permeability increases skin-friction of the flow field at certain points.

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.