Dissipative Heat Transfer of Micropolar Hydromagnetic Variable Electric Conductivity Fluid Past Inclined Plate with Joule Heating and Non-uniform Heat Generation

This work was carried out in collaboration between both authors. Author SOS formulated the problem and the derivation of equation and reviewed the literatures. Author EOF solved the problem, carried out the mathematical analysis, did the tables and graphs and discussions of findings. Both authors the final manuscript. ABSTRACT Computational simulations of hydromagnetic dissipative heat transfer of variable electric conductivity of micropolar fluid flow and non-uniform heat absorption or generation with joule heating have been studied in this work. The flow past an inclined surface with an unvarying heat flux. The transformed dimensionless equations of the governing model are solved by Runge-Kutta algorithm coupled with shooting scheme to depict the dimensionless temperature, microrotation and velocity distributions at the border layer. The substantial bodily quantities of the flow are conferred. The results depict that the significance of the coefficient of the skin friction and the Nusselt number increases for uneven electric conductivity and non- homogeneous heat sink or source at the plate.


INTRODUCTION
A micropolar fluid consists of gyratory microcomponents which make fluid to display non-Newtonian actions. The formulations of micropolar fluid flow was found helpful in examining body fluids, colloidal suspensions, lubricants, liquid crystals, fluid polymeric, flows in capillaries and microchannels, preservative suspensions and flow of shear turbulent. The presumption of micropolar fluids originated by Eringen [1] have been a lively research area for many years.
Ariman et al. [2] reviewed excellently the model and applications of micropolar fluid. Hoyt and Fabulan [3] carried out experimentally analysis to show that the present of minute Stabilizers polymeric in fluids can decrease the flow effect at the wall up to 25 to 30 percent. The decrease that was described by the hypothesis of micropolar fluids as reported by Power [4], the brain fluid that is an example of body fluids fluid can be sufficiently formulated as micropolar fluids.
Flow of convective fluids containing microstructure have several applications such as dilute polymer fluids solutions, liquid crystals and different kinds of suspensions. The fluid driven by buoyancy forces occur in wide-ranging of uniform fluid flow. Free convective flow of micropolar fluids past a curved or flat surfaces has fascinated the mind of scholars from the time when the flow model was devised. Several studies have accounted and analyzed outcomes on micropolar fluids. [5][6][7] examined the effect of radiation on magnetohydrodynamics convective heat and mass transfer flow. Ahmadi [8] considered micropolar boundary layer fluid flow along semi-infinite surface using similarity solution to transform the models to ordinary different equations. [9][10][11] verified the viscous dissipation influences on MHD micropolar flow with ohmic heating, heat generation and chemical reaction.
Many convective flow are caused by heat absorption or generation which may be as a result of the fluid chemical reaction. The occurrence of heat source or sink can affect the fluid heat distribution that alters the rate of deposition of particle in the structures such as semiconductor wafers, electronic chips, nuclear reactors etc. Heat absorption or generation has been assumed to be temperature dependent heat generation and surface dependent heat generation. Rahman et al. [12] examined the influence of non-homogenous heat absorption or generation and variable electric conductivity on micropolar fluid. It was noticed that the surface dependent heat generation is lower compared to temperature dependent heat generation. [13,14] examined the effect of thermal-diffusion and nonhomogenous heat sink/source on radiative micropolar hydromagnetic fluid past a permeable medium.
The influence of dissipation on magnetohydrodynamic fluid and energy transfer processes has become significant industrially. Several engineering practices happen at high temperature with viscous dissipation heat transfer. Such flows have been investigated by Siva and Shanshuddin [15] reported on viscous dissipation heat and mass transfer of hydromagnetic micropolar fluid with chemical reaction.
Rawat et al. [16] studied magnetodyrodynamic micropolar fluid of heat and mass transport in a permeable medium broadening plate, chemical reaction, heat flux and variable micro inertia. [17,18] verified steady MHD Micropolar flow fluid with joule heating, viscous dissipation and constant mass and heat fluxes. it was observed that the flow field rises initially within 1 0   as the microrotation parameter value increases. Later, the flow field gradually reduces for 1 >  as the microrotation parameter rises. Also, microrotation changes from negative to positive signs in the border layer. Moreover, Ajaz and Elangovan [19] considered the effect of inclined magnetic field on the oscillatory flow of micropolar fluid in a porous micro-channel under the action of alternating electric field.
Following the above cited literature, the aim is to study the viscous dissipation hydromagnetic micropolar fluid behavior heat transfer over inclined surface in permeable media with heat fluxes and joule heating for high speed fluid in non-even heat absorption/generation. The flow equations are made up of partial differential equations that can be transformed by similarity solution to nonlinear ordinary coupled differential equations. The obtained equations are simulated by Runge-Kutta technique coupled with shooting scheme. Therefore, it is obligatory to study the flow distributions, temperature and microrotation crosswise the border layer in toting up to the surface wall effect and nusselt number.

THE FLOW MATHEMATICAL FORMU-LATION
Deem convective flow of two-dimensional magnetohydrodynamic viscous, laminar, micropolar fluid through a semi-finite flat surface inclined at an angle  to the vertical. The magnetic field differs in potency as function of x which is taken in y-direction as defined The Reynolds number is small while the exterior electric field is taken as zero. Therefore, the applied external magnetic field is high contrasted to the stimulated magnetic field. The density with the buoyancy forces stimulated the convective motion. The fluid viscosity  is taken to be unvarying while the body forces and the pressure gradient are neglected.
Follow from the assumption above, the steady convective micropolar fluid flow follow the Boussinesq approximation may be explained by the subsequent equations and geometry. implies varnishing of the antisymmetric module of the stress tensor that stand for feeble concentration. This confirm that for a fine particle suspension at the wall, the particle swivel is the same as the fluid velocity but when 1 = a shows the turbulent boundary layer flows.
The non-uniform heat absorption or generation is represented as: Introducing the following dimensionless variables where 0 U is the reference velocity,  is the stream function and Using equations (7) and (8) The boundary conditions becomes The substantial quantities of engineering concern for this flow are the local skin friction The computational values for f C and

RESULTS AND DISCUSSION
The computational results for the nonlinear coupled differential equation are obtainable for the dimensionless microrotation, temperature and velocity distributions. In this study, the following default parameter values are chosen:    on the dimensionless velocity and microrotation distributions. It is evidenced that the flow and microrotation field decreases as the porosity parameter rises, this is as a result of the wall of the plate that gives an additional opposition to the flow mechanism by influencing the fluid to move at a decelerated rate.

CONCLUSION
The numerical simulations was carried out for dimensionless boundary layer equations of convective heat transfer in hydromagnetic joule heating, micropolar fluid past heated inclined surface with non-uniform heat sink/source and variable electric conductivity. It was observed from the study that, the magnetic field decreases the flow rate and angular velocity but increases the heat transfer phenomenon while the temperature dependent and surface temperature heat absorption or generation terms as well as viscous dissipation parameter increases the flow, angular velocity and temperature distributions.

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The vortex viscosity parameter decreases the flow and microrotation profiles near the wall but later increases the profiles distance away from the wall while porosity resist the flow and microrotation of the fluid by decreasing the profiles.