EXPLORING THE DYNAMICS OF AN EXOTHERMIC CHEMICAL REACTION CONTROLLED BY ARRHENIUS KINETICS WITH HALL EFFECT IN A MICROCHANNEL

The consequences of chemically reactive fluids have provoked many researchers due to their potential to enhance the behavior of heat. Hence, this article discussed a theoretical investigation to explore the actions of Arrhenius chemical reaction and Hall current on hydro-magnetic free convection of a viscous fluid flowing along an upstanding micro-channel. Subject to the required boundary conditions, the governing coupled equations representing the flow pattern in non-dimensional form were solved using the homotopy perturbation method (HPM). Line graphs are also used to generate expressions for energy, momentum, volume flow rate, drag force, and Nusselt number in both primary and secondary flow directions as a function of regulating parameters like chemical reaction, Hall current, rarefaction, and wall ambient temperature difference ratio. It is worthy of note that the fluid temperature and the fluid flow are substantially propelled by the viscous heating term in the Arrhenius chemical reaction of the system for growing values of the wall ambient temperature difference ratio parameter. Additionally, it is noticeable that volume flow rate performance is seen as a growing influence of viscous heating and rarefaction parameters. The findings of this study can be applied to a wide range of electrically controlled devices, thermal and petro chemical engineering, and the serviceability of industrial products.


INTRODUCTION
The phenomena of Hall effects appear when the introduced magnetic factor is extremely strong or when partly ionized gases have a high cyclotron frequency. Most of the electricity generated in a fluid is propelled by electrons, which repeatedly collide with other energized particles and are often more unsteady than ions. Because of the rapid movement and randomness of the charged particles, the conductivity parallel to the electric field decreases, and current is induced in the direction perpendicular to both the electric and magnetic fields. This occurrence is defined as the "Hall Effects." Magneto-hydrodynamics (MHD) flows having Hall currents have engineering applications such as Hall accelerations, power generators, turbine construction, and centrifugal machines (Sutton andSherman, 1965, Quader andAlam, 2021). In light of this, Urgun et al. (2022) discussed the analytical simulation of Von Karman flow and MHD heat transmission of an electrically conducting fluid influenced by Hall Effect through an infinitely radially shrinking rotating disk. Jha and Malgwi (2021) have published a paper on the role of conducting and non-conducting surfaces on the hydro magnetic free convection of viscous fluid along an upstanding micro-channel in the coexistence of Hall current and induced magnetic number.  evaluated the actions of Hall and ion slip currents on the time-dependent rotating flow of a conducting fluid having porosity effect. It was concluded that the presence of a magnetic number impedes flow reversal and that the inclined angle has a declining influence on fluid velocity.  then described the hydro-magnetic rotating flow of microscopic creature propulsion across a permeability medium in the coexistence of Hall and ion slip condition. It was discovered that the magnetic number, Darcy porous parameter, Hall parameter, and ion slip parameter all play vital roles in fluid flow characteristics. The combined effects of Dufour effect, radiation-absorption, and Hall and ion slip flow on the hydro magnetic free convective circulatory flow of nano-fluids (Ag and TiO2) across a semi-infinite porous non stationary plate with uniform heat emission were explored by Krishna and Chamkha (2019). Their findings reveal that increasing the rotation parameter reduces velocity while increasing the Hall and ion slip conditions and the Dufour effect increases velocity. In another article, Jha and Malgwi (2019) reported the actions of Hall conditions and induced magnetic number on steady hydro magnetic convection in a micro-channel subjected to an inclined magnetic field. Their computational analysis revealed that magnetic field inclination plays a substantial role on the flow characteristics in the microchannel. Rajput and Gupta (2021) explored the effects of Hall current and thermal radiation on transient free hydro magnetic flow with an inclined magnetic field. The key findings of their computational research revealed that the Hall Effect significantly increase both the primary and secondary flow distributions. Magnetic fields and thermal radiation, on the other hand, tend to boost secondary velocity while decreasing primary velocity. Jha et al. (2017) stressed the impact of Hall current on MHD convection in a micro-channel limited to two immeasurable upstanding parallel plates restricted to unequal heating. The study's key findings demonstrated that higher levels of Hall current and rarefaction characteristics increase the volume flow rate. In view of the numerous cooling functions in processing activities and fabrication, including micro-heat pipes, aerospace and high-power density chips in supercomputers and other devices, there has been an upsurge in interest in microfluidics and heat transfer in recent years. Comprehending the flow configuration is becoming extremely significant for exact modelling predictions since the bulk of these structures involve internal micro-channel flows (Al-Nimr andKhadrawi 2004, Jha et al. 2014a). To this end, Hamza et al. (2023) recently highlighted an analytical investigation on the effect of Arrhenius chemical reaction due to an induced magnetic field in a microchannel. Jha and Aina (2016) discussed the hydro magnetic convection in a microchannel confined to two electrically non-conducting immeasurable upstanding plates. Ltaifa et al. (2021) carefully investigated heat transfer in a rectangular inclined microchannel immersed with a water/Al2O3 nanofluid. Other references include (Chen and Weng 2005, Buonomo and Manca, 2012, Weng and Chen 2009, Jha et al. 2014b). The in-depth investigation of magneto-hydrodynamics (MHD) flow phenomenon is crucial and has attracted great intellectual curiosity attributable to its possible advantages in a number of industrial and engineering fields, like in MHD generators, fusion reactors, electric transformers, optical grafting, metal casting, crystal growth, MHD accelerators and metallurgical processes. Additionally, chemical energy technology already uses MHD pumps for propelling electrically conductive fluids in a number of atomic energy structures. Besides contributing to these uses, an applied magnetic field may profoundly improve free convection when the fluid is electrically charged (Zainal et al. 2021. Numerous investigations of MHD convection flows in several physical conditions have been performed. Ojemeri et al. (2023) recently inspected the steady MHD flow of an electrically conductive Casson fluid controlled by radiation effect in an upstanding porous channel. Applying the finite difference method (FDM), Omokhuale & Dange (2023) numerically analyzed the action of heat sink on time-dependent MHD Jeffery flow of a chemically reactive fluid. Hamza and Shuaibu (2022) deliberated on MHD convection of an exothermic chemical reaction affected by Newtonian heating in an upstanding channel. Ojemeri and Hamza (2022) proposed an analytical investigation of an Arrhenius kinetically affected by heat source/sink fluid with MHD effect in a microchannel. Gul et al. (2021) highlighted the hybrid nanofluid function on hydro magnetic boundary layer flow for a chemically reacting fluid in light of this. Saeed and Gul (2021) researched the hydro magnetic Casson nanofluid flow with heat and mass transfer along a non-stationary plate. Ziz and Shams (2020) carried out the entropy generation in the hydro magnetic Maxwell nanofluid flow affected by slip condition, thermal generation, and variable thermal conductivity. Gurivireddy et al. (2016), employed a semi-infinite non stationary upstanding porous plate with Soret factor to analyze the impacts of heat generation and chemical reactions on hydro magnetic flow. El-Aziz and Yahaya (2017) discussed the heat and mass transfer coefficient of a time-dependent hydro magnetic flow saturated with porosity effect across an upstanding plate with uniform layer heat condition. The equations governing the current flow problem are nonlinear and coupled, making it challenging to derive closed-form solutions. As a result, numerical schemes or approximate solution procedures can be used to tackle such problems. One of the most efficient approaches is the perturbation method. The perturbation method's solutions, on the other hand, are confined to small perturbation parameters.
Another method, known as the homotopy perturbation method, was developed to overcome this limitation. He (1999) utilized the method to solve partial or ordinary forms that are linear, nonlinear, or coupled equations. He investigated a nonlinear coupling method using a homotopy and a perturbation procedure (He 2000 andHe 2006). Later, He (2003) presented a new nonlinear analytical methodology based on homotopy perturbation. A closer look into the above-mentioned literature, however reveals a knowledge gap; the influence of chemically reactive fluid has not been considered, probably for the reason of having a simplified mathematical formulation. However, this effect is particularly significant due its relevance in emerging MEMS, engineering sciences and lubrication industries, since most of the lubricants used in engineering and industrial activities, like synthetic esters, hydrocarbon oils, polyglycols, are reactive. Therefore, the main target of this research is to build on the model of Jha et al. (2017) by incorporating the action of Arrhenius chemical reaction on steady free convection in an upstanding micro-channel affected by Hall effect confined by two immeasurable upstanding parallel plates subjected to unequal heating of the plates using the homotopy perturbation technique. The uniqueness of this work is that it utilizes HPM to establish some approximate solutions for a chemically reacting fluid regulated by Arrhenius kinetics between two vertical microchannel walls. This research is also motivated by the increased quest for various industrial working fluids and earlier researches on the relevance of Arrhenius energy and exothermic chemical reactions in chemical engineering, biotechnology, food processing, thermal sciences, and other fields.

Problem formulation
Under the consequences of the Hall Effect, the steady free convection of an Arrhenius-driven fluid in a vertical micro-channel restricted to two electrically non-conducting immeasurable upstanding plates is investigated with an applied magnetic field. The x-axis is normal to the gravitational force g but in the reverse direction, while the yaxis is orthogonal to the upward plate. The flow configuration is portrayed in Figure 1. A unaltered magnetic strength of 0 is assumed to be applied in the path normal to the flow movement. Also, it is assumed that the magnetic Reynolds number is negligible, meaning that the produced magnetic field is little in contrast to the one introduced externally. The surfaces are heated unequally, with one plate retained at 1 and the other at 2 , with 1 > 2 . Free convection in the microchannel is due to the temperature variance between the plates and the reactive nature of the fluid. The following assumptions are used to formulate this problem: i. Unlike buoyancy, the horizontal magnetic field has a considerable impact. ii. A uniform magnetic field B0 is provided perpendicular to the flow direction while ignoring the induced magnetic fields by assuming a very tiny magnetic Reynolds number. iii. The influence of the Frank-Kamenetskii parameter is taken into account. iv. The impact of Joule heating is neglected in favor of viscous dissipation.

EXPLORING THE DYNAMICS OF AN EXO…
Where Np is the Hall current, M is the magnetic number, is the Frank-Kamenetskii parameter and Ac is the activation energy. It is useful to clarify that, in the expansion of heat source term (1 + ) 1+ , the Arrhenius heating case (i.e. when m = 0) was investigated in this research. The dimensionless parameters employed in the flow configuration are described below:

Solution of the problem Homotopy Perturbation Method (HPM)
The closed form expressions for the nonlinear ordinary differential equations have been obtained.

}
Substituting equation (9) into equations (6), (7) and (8), and comparing the coefficient of like powers p, we obtain the solutions to the temperature and velocity as follows: Setting p to be 1, the solutions of the differential equations in the approximate form becomes The series in equations (17) typically converges even for few terms. On the other hand, the nonlinear operator determines the rate of convergence. Ayati and Biazar (2015) assert that just a few terms of the homotopy perturbation procedure can be utilized to get an approximative solutions that converges to the series solution.
Defining the complex velocity as = + , eqs (1) and (2) now becomes   (18) and (19) and assuming the solution of the complex velocity to be ϕ= ϕ0+p ϕ1+p 2 ϕ2+… We obtain the appoximate solution of ϕ as:

DISCUSSION OF RESULTS
This section is devoted to showcasing the deviations of the fluid parameters such as viscous heating term ( ), activation energy (Ac), wall-ambient temperature difference ratio ( ), rarefaction parameter ( ) and Hall current (Np) on free convection and heat transfer flow employing the homotopy perturbation method (HPM) in a vertical microchannel. The deviation of temperature, primary velocity (U), secondary velocity ( ), volume flow rate, sheer stress and heat transfer amount are described in Figures 2-12 to illustrate the impacts of these regulating factors. The present research was done in the continuum and slip flow region (Kn≤0.1) over meaningful interval of 0  ≤0.1, 1≤Np≤2 and 0≤In≤10. The product of show a level of departure from the continuum regime, and (In) implies a fluid-wall interaction factor. For this analysis, the selected default values for ( ) and (In) are 0.05 and 1.667, respectively, as employed by Chen and Weng (2005), and M=2 as used by Jha et al. (2014a) in addition, the reference values taken for this computations are =0.1, Np=1. M=2, =0.05, In=1.667 and Ac=0.01 to elucidate the influence of various flow characteristics.  Figure 2 show the impacts of chemical reaction parameter ( ), also known as viscous heating parameter, rarefaction parameter ( ) and wall-ambient temperature difference ratio ( ) on the temperature gradient under three cases of the wall-ambient temperature difference ratio ( = -1: one heating and one cooling; = 0: one heating and one not heating, =1: both walls are heated). It is abundantly obvious that when the values of and grow, the temperature jump increases significantly, notably in the region of the left proximity of the micro-channel. This is naturally true because uplifting the viscous heating term yields a minimization in the drag force, consequently weakening the flow resistance. The size of the viscous heating sources in energy equations is encouraged as fluid motion rises, leading to a stronger fluid temperature. Figure 3 depict the pattern of and ( ) on the velocity profile. A close examination at this figure reveals that both the primary and secondary flows are moving upward, notably towards the center of the micro-channel, as and are increased. In the case of the primary velocity, these impacts are more pronounced. We observed that with greater levels of reaction rate parameter drastically instigate the exothermic chemical reaction and substantially boost the fluid movement due to the temperature spike. Naturally speaking, the exothermic chemical reactions causes an upsurge in the heat generation due to growing levels of . Moreover, the speeds of Arrhenius provoked reactions encourages internal heat enhancement. Figure 4a and b describe the influence of and ( ) on the primary and secondary velocity. The fluid velocity is increased in both the primary and secondary directions when the levels of rarefaction and the wall ambient temperature differential ratio are increased. This effect can be described from the fact that when the rarefaction parameter grows, the temperature rises as well, reducing the amount of heat transferred from the walls to the fluid. The strengthening in the fluid motion caused by the petering out in the diminishing drag forces close to the surfaces make up for the reduction in fluid flow initiated by reducing the impact on the amount of heat transport. It's worth noting that in the case of the primary velocity, these affects are more evident. The functions of Np and ( ) on both the primary and secondary flows are demonstrated in Fig. 5a and b. It is clear that mounting values of Np and ( ) respectively causes a substantial growth in both the primary and secondary velocity, with the effects being more obvious in the case of the secondary velocity. Further, raising the Hall current parameter tends to reduce the magnetic field effect on the complex velocity, resulting in the upsurge in both the primary and secondary velocities. The flow pattern for the volume flow rate as vary is depicted in Fig. 6. It is plain that uplifting the values of against , the primary and secondary flows rises at the micro-channel walls. It is shown that, a rise in the level of rarefaction parameter encourages a strengthening in the volume flow amount. In addition, it is clearly noticeable that the chemical reaction parameter is seen to boost up the rate of volume flow. The plots for frictional forces at both plates affected by is illustrated in Figs. 7 and 8. As seen from Figs. 7a and b, at different growing levels of ( ), the skin friction is noticed to reduce for an increasing function of at Y=0, whereas a counter attribute is evident at Y=1 as portrayed in Figs 8a and b. The actions of ( ) versus ( ) on the heat transmission rate is plotted in Fig. 9. It is worthy of mentioning that an observable growth in the heat transfer rate is stronger as and rise respectively in the plate (Y=0), as noticed in Figure 9a, while at Y=1, an opposite behavior is established as depicted in Fig. 9b. This is so because strengthening (reducing) fluid temperature encourages (slows down) the exothermic chemical reactions, which in turn boosts (lowers) the heat transfer gradient.   Tables 1 and 2, a good agreement was recorded for the complex velocity and temperature gradients.