THE COMBINED EFFECTS OF UNSTEADY ELECTRO-OSMOTIC AND MAGNETO HYDRODYNAMIC WITH VISCOSITY AND THERMAL CONDUCTIVITY IN REACTIVE FLUID FLOW COMBINED EFFECTS OF UNSTEADY ELECTRO-OSMOTIC AND MAGNETO HYDRODYNAMIC WITH VISCOSITY AND THERMAL CONDUCTIVITY IN REACTIVE FLUID FLOW.”

: This work examined the combined effects of unsteady electro-osmotic and magneto hydrodynamic when viscosity and thermal conductivity of the reactive fluid flow is assumed to vary exponentially with temperature. The dimensionless variables was use to dimensionalized the governing equations of the flow using suitable variables. The Galerkin weighted residue method was used to solve both the momentum and energy equations in the unsteady state for a constant viscosity and thermal conductivity. The graphical results were used to study the Thermo physical behavior of the unsteady flow of the model .


Introduction
Newtonian fluids are fluids that obey Newton's law of viscosity and for which  has a constant value. More precisely, a fluid is Newtonian only if the tensors that describe the viscous stress and strain rate are related by a constant viscosity tensor that does not depend on the stress state and velocity of the flow. Most common liquids and gases such as water and air can be assumed to be Newtonian for practical calculations under ordinary conditions. Zeta potential is the electrical potential at the shear plane. Application of an electric field along the length of the micro-plates causes an electrical body force to be exerted on the mobile ions in the diffuse layer. Then the ions move under the influence of electrical field and move the liquid by viscous forces. This type of flow is called electro-osmotic flow (EOF) [2, 5, and 10].
Magneto hydrodynamic (MHD) studies the magnetic properties of electrically conducting fluids such as Plasmas, liquid metals and saltwater or electrolytes. The fundamental concept behind MHD is that magnetic fields can induce currents in a moving conductive fluid which in turn polarizes the fluid and reciprocally changes the magnetic fluid itself. More so, reactive fluid flow have received increasing attention for studies of contaminant transport ground water quality, waste disposal, acid mine drainage, remediation, mineral deposits, sedimentary just to mention a few. However, a liquid is said to be viscous, if its viscosity is substantially great. Joule heating is the process by which the passage of an electric current through a conductor releases heat. The amount of heat release is proportional to the square of the current such that 2 l Q  . This is caused by interactions between the moving particles that form the current (but not always electrons) and the atomic ions that make up the body of the conductor [13][14][15].
This fluid has wide applications in many branches of science and engineering of most focus is the thermal behaviors of fluids whose viscosity changes with temperature and the flow is accompanied by a simultaneous transfer of mass, energy and momentum in the system due to reaction occurring between the fluids. The ability to adequately describe such system is necessary for the prediction of its thermal stability among others. Hence, Efforts have been devoted to the study of heat transfer and thermal stability of reacting Newtonian fluids that is of extreme importance not to compromise on safety of life and materials during handling and processing of such fluids and for quality control purposes in many manufacturing and processing industries. An improvement in thermal recovery and utilization during the convention flow in any fluids is one of the fundamental thermal integration of such systems that provide a better materials processing, energy conservation and more environmentally being process.
The possibility of the existence of a considerable resistance to heat transfer between the reacting fluids and system as a result of low conducting fluids or highly conductive vessel wall, resulting in significant temperature gradient, as was reported by Frank -Kamanetskii [1]. In recent time, the mathematical formation of thermally critical system mainly focuses on the determination of the critical regions separating the regions of explosivity and non explosivity of various works on stability of flows was examined by Billingham [2]. Yihao Zhery et al. [3] investigated the kinetic behaviour and hydrodynamics of pressure driven Poiseuille flow. Makinde [4] studied the thermal stability of a reactive thirdgrade liquid flowing steadily between two parallel plates with symmetrical convective cooling at the walls. Hence the study of electro-osmotic, magneto hydrodynamics with variable viscosity and thermal conductivity in reactive flow is significant for practical reasons.
The present study investigate combined electro-osmotic and magneto hydrodynamic with viscosity and thermal conductively in reactive fluid flow and determine the effect of fluid parameters on velocity profile and temperature profile for a steady, constant viscosity and thermal conductivity in a reactive fluid flow using Galerkin weighted residual method.

Mathematical Formulations
An incompressible viscous fluid flow of the combined effects of unsteady electro-osmotic and magnetohydrodynamics with viscosity and thermal conductivity in a reactive fluid flow between two parallel plates was investigated. The equations governing the motion of the fluid are momentum, energy and electrical potential written as: The boundary and the initial conditions of the flow are as follows: where  is the density,  the viscosity, p C heat capacity with constant pressure, U and V o velocity components along x and y axis respectively, T the temperature, P is pressure, K the thermal conductivity, x the co-ordinate in the direction of flow, E the activation energy, R the universal gas constant, Q heat released per unit mass during reactions.  the electric field conductivity,  the electrical potential, e where o  is bulk ionic concentration and z is valence of type -i ions Substituting (5) into (1) to (3) with the initial and boundary conditions gives; ), equations (6) and (7) . al., Vol.5 (Iss.1)

Method of Solution
The Galerkin Weighted Residual Method (GWRM) requires inner product, basis of a vector space of which is the same as the weight functions. So for GWRM, a weighted residual method uses a finite number of functions where 'L' is a differential operator and 'f' is a given function. A trial function of U was introduced to solve the problem: The Residual were defined as: Thereafter, an arbitrary weight functions ) (x w was choose from the basis functions j  , From the concept of inner product and orthogonally. These are the set of n-order linear equations which must be solve to obtain all the j C coefficients. Using GWRM-on the non-homogeneous equations (9) and (10) problems resulted into the Velocity and Temperature profile functions as given below respectively.
[Ajilore et. al., Vol.5 (Iss.1)   Temperature profile function is derived as follows: [131] so since the viscosity of this present work is constant, low velocity was observed for higher viscosity parameters base on Newton law of viscosity from Fig. 3. Little difference was seen in the electro-kinetic. Fig. 5 depict that electro-kinetic parameter increases from 2, 4 to 6, gives a very small decrease in the velocity.  Furthermore, from Fig. 1b, the temperature profile decrease as thermal conductivity (d) parameter increases. More so the temperature profile increases as different parameters of viscosity (g), reactive (f) and magnetic term (g) increases.

Conclusion
The combined effect of unsteady electro-osmotic, magneto hydrodynamic with viscosity and thermal conductivity shows a direct relationship with velocity profile and temperature profile of a reactive fluid flow. The influence of electro-osmotic and magnetic field on the flow fluid is significant as the parameters retarded the flow while thermal conductivity and viscosity enhances the temperature field due to the thickness in thermal boundary layer as the parameter increases.