Theoretical study of ARRHENIUS-controlled heat transfer flow on natural convection affected by an induced magnetic field in a micro-channel

The current study analyzes the implications of an Arrhenius-controlled heat transfer fluid on free convection in a micro-channel confined by two immea-surable vertical parallel plates that are electrically non-conductive due to an induced magnetic field (IMF) effect. The governing coupled nonlinear equations are ordinary differential equations, and the dimensionless steady-state solutions were determined using the homotopy perturbation method (HPM). The derived results were discussed and represented graphically with the help of illustrative line graphs for momentum, IMF, temperature, and volume flow rate for the major controlling parameters, namely arrhenius kinetics, rarefaction, wall ambient temperature difference ratios, and Prandtl magnetic number. Thermo-physical properties that are of engineering interest, like sheer stress and Nusselt number, are also computed and displayed. It is pertinent to report that the velocity of the fluid increases as a result of chemical reactions and rarefaction factors, whereas strengthening the Prandtl magnetic number decreases the volume flow rate. Also, numerical data was obtained and presented in tabular form to compare this research outcome to those of Jha and Aina, and great consistency was found. Microelectronics and microfluidics are some areas where this study can find

are ordinary differential equations, and the dimensionless steady-state solutions were determined using the homotopy perturbation method (HPM). The derived results were discussed and represented graphically with the help of illustrative line graphs for momentum, IMF, temperature, and volume flow rate for the major controlling parameters, namely arrhenius kinetics, rarefaction, wall ambient temperature difference ratios, and Prandtl magnetic number.
Thermo-physical properties that are of engineering interest, like sheer stress and Nusselt number, are also computed and displayed. It is pertinent to report that the velocity of the fluid increases as a result of chemical reactions and rarefaction factors, whereas strengthening the Prandtl magnetic number decreases the volume flow rate. Also, numerical data was obtained and presented in tabular form to compare this research outcome to those of Jha and Aina, and great consistency was found. Microelectronics and microfluidics are some areas where this study can find relevance. micro-channel flows. 1,2 A number of comparable studies have recently been published that investigate the effect of fluid movement on microstructure in different physical conditions. Recently, using a semi-analytical method, Hamza et al. 3 presented the actions of free hydro-magnetic thermal transfer in a micro-channel saturated with porous material and the availability of variable thermal conductivity. It was reported from their findings that variable thermal conductivity improves fluid flow under both asymmetric and symmetric heating of the micro-channel boundary wall. Jha and Aina 4 carefully analyzed natural hydro-magnetic convection in a vertical micro-channel influenced by an induced magnetic field (IMF), made of two immeasurable vertical parallel plates that are electrically non-conductive. Jha and Gwandu 5 discussed the natural convection flow of a viscous reactive fluid past a vertical micro-channel of rectangular geometry. It was revealed from their results that the heat emission parameter increased the thermal gradient when the superhydrophobic surface was heated and retarded it when no slip surface was heated in the micro-channel. Under laminar flow, Ltaifa et al. 6 unambiguously examined heat transfer in a rectangular, inclined micro-channel saturated with a water/Al 2 O 3 nanofluid. They discovered that as the volume fraction of Al2O3 and the inclined angle increase, so does the heat transfer gradient. References [7][8][9][10][11][12] provide additional information on the literature discussed in this approach.
In recent times, the concept of magneto-hydrodynamics (MHD) study has attracted research interests in the branch of engineering and science because of its significance in several MHD applications, namely MHD generators and accelerators, electric transformers, and the cooling of metallic plates in cooling baths. Pumping electrically conductive fluids with MHD pumps is already in use in various atomic energy facilities as part of chemical energy technology. In addition to these applications, an applied magnetic field has a profound impact on free convection movement once the fluid is electrically conducting. 13 Various investigations of hydro-magnetic convective flow in a variety of physical conditions have been conducted. Gul et al. 14 outlined the hybrid Nano-fluid consequence on hydro-magnetic boundary layer flow for viscous fluids. The MHD Casson nanofluid flow was emphasized by Saeed and Gul 15 for heat and mass transmission over a shifting plate. Ziz and Shams 16 scrutinized the entropy production in the MHD Maxwell nano-fluid flow under variable thermal conductivity, slip condition, and heat radiation and generation. Using the HAM method, Jawad et al. 17 simulated the transient Maxwell MHD flow of nanofluid across a stretched layer having Soret, Dufour, and porosity parameters. Reddy et al. 18 used the Fehlberg method and nonuniform heat generation and absorption to investigate the actions of cross diffusion on the flow of non-convectively heated fluids towards an extended layer. Gurivireddy et al. 19 described the consequences of heat formation and chemical processes on hydro-magnetic flow by utilizing a vertical porous plate that moves indefinitely in one direction with a soret effect. El-Aziz and Yahaya 20 searched the heat and mass flow properties of a transient MHD convection flow filled with porous material over a vertical plate having constant layer heat properties. Jamil et al. 21 investigated a time-dependent heat transfer flow of incompressible viscoelastic Maxwell fluid instigated by a stretching surface. Anwar and Rasheed 22 carried out a numerical analysis of heat transfer through a Forchheimer medium in MHD to develop differential-type fluid flow. Rasheed and Anwar 23 discussed the fractional MHD viscoelastic two-dimensional unsteady fluid flow using the recently established finite difference technique along with the "L1 algorithm." Hayat et al. 24 explored the effects of viscous dissipation and joule heating on the MHD stagnation point flow of a second-grade fluid across a permeable stretching cylinder.
Incorporating an IMF into the analysis of MHD flows, either theoretically or experimentally, has created numerous possible applications due to its relevance in many scientific and technological industries, such as geophysics, crude oil purification, MHD electrical power generation, and glass manufacturing. IMFs have become increasingly important in many MHD processes, particularly when the Reynolds number is high. 25 27 deliberated on the actions of Eckert and Prandlt numbers on the MHD free convection of an electrically conducting, viscous fluid bounded by two nonconducting walls in the availability of a magnetic field induction through a micro-channel. According to their findings, raising the magnetic field intensity and Prandtl magnetic number yields a dramatic reduction in volume flow rate. Khan et al. 28 inspected the analysis of the stagnation point transmission of the second-grade fluid with linear stretching having the combined influence of variable thermal conductivity and an IMF. They reported that the IMF is stronger when raised. Raza et al. 29 examined the flow of salt water as a base fluid having nanoparticles of various shapes in an asymmetrically permeable channel using illustrative models, namely those of Hamilton and Crosser. Rostami et al. 30 presented a comprehensive analysis of the effects of a magnetic field and sinusoidal boundary conditions on the natural convective heat transmission of a non-Newtonian power-law fluid enclosed in a square with two constant-temperature impediments employing the lattice Boltzmann method. Their results show that by raising the levels of Rayleigh number, the heat transfer amount is improved, but with the growing power-law index and Hartmann number, the heat transfer rate diminishes for shear thinning, Newtonian, and shear thickening fluids. Using an efficient explicit finite difference method, Poddar et al. 31 investigated the analysis of MHD boundary layer convective flow of heat radiating and dissipative fluid past an immeasurable plate of vertical shape having an IMF and Soret effect. They revealed that a rising magnetic parameter suppresses the IMF. Additionally, the suction parameter lowers the quantitative flow of the velocity, temperature, and concentration. More authors who have scrutinized the significance of IMF effects in MHD flows are. [32][33][34][35][36][37] The goal of this study is therefore to modify the work of Jha and Aina 4 by using the homotopy perturbation method (HPM) to provide an analytical solution to the nonlinear coupled lead equations of an arrhenius-controlled heat transfer fluid in hydro-magnetic free convection due to an IMF restricted to two electrically nonconducting infinitely parallel vertical plates in a micro-channel. The HPM's convergence is so fast that only a few terms of the series solution are needed to attain great accuracy in the solutions. The novelty of this present investigation is revealed in the fact that an arrhenius-controlled heat transfer coefficient was incorporated in the extended model and the homotopy perturbation procedure was used to obtain the steady-state solutions of the formulated controlling equations. No such work has been reported in the previous literature to the best of the authors' knowledge, hence the motivation for this research. The interaction between chemical reaction and natural convection occurs widely in chemical engineering applications, for instance, in tubular laboratory reactors, chemical vapor deposition systems, the oxidation of solid materials in large containers, the synthesis of ceramic materials by a self-propagating reaction, and so on. On the other hand, a wide range of potential applications, including micro-devices made with microfabrication processes, micro-electro-mechanical systems (MEMS), biomedical sciences, and different medical treatment modalities, and particularly the heat therapeutic procedure, can all benefit from the outcomes of this study. Additionally, the findings in this work can be instrumental in providing a framework for validating the accuracy of numerical or empirical procedures.

STRUCTURE OF THE FLOW PROBLEM
• We investigate the steady, fully developed free convection of an electrically conducting, reactive fluid in a vertical microslit bounded by two electrically nonconducting infinite vertical parallel plates. Figure 1 describes the geometry of the problem under consideration. The flow is assumed to have a transverse velocity of zero. In order to build the mathematical model for this current analysis, we further make the following assumptions: • In contrast to the buoyancy force, the transverse magnetic field has a significant influence.
• In the instance that the magnetic field is significant, the IMF was taken into account.
• The action of the Frank-Kamenetskii parameter is considered.
• In favor of viscous dissipation, the influence of Joule heating is ignored.
The x-axis is perpendicular to the y-axis and runs vertically along the walls. The gap between the plates is denoted by b and. The plates are heated asymmetrically, with one plate retained at T 1 and the other at T 2 , with T 1 greater than T 2 . A free convection stream forms in the micro-channel as a result of the temperature ramp. Heat motion and fluid movement are more variable at the micro-scale than at the macro-scale. In this scenario, a slip condition for velocity and a jump condition for temperature should be used because the fluid no longer approaches the surface's velocity or temperature. In this investigation, the traditional continuum technique is used, with continuum equations applied to the two fundamental patterns of micro-scale events, velocity slip and temperature jump. The definition of velocity slip is as follows 38 : where T s represents the fluid temperature at the wall, T w the wall temperature, and ( t ) denotes the heat accommodation estimation, which varies depending on the gas and surface substances. However, in the case of air, it takes on normal values near unity. 38 For the remainder of the discussion, ( v ) and ( t ) will be treated as 1.
Thus, for an electrically conducting and chemically reactive fluid caused by the Arrhenius heating effect and IMF impact under relevant boundary conditions, the basic form of continuity, velocity, and thermal equations with the usual Boussinesq approximation are expressed below, following Jha and Aina 4 and Hamza 39 ; The non-dimensional quantities used are The governing equations in dimensionless form becomes Subject to the relevant boundary conditions in non-dimensional form as where M is the magnetic field, Pt is the Prandlt magnetic, B is the IMF, dU dZ and dB dZ are the convective terms of the velocity and IMF respectively, K c is used to represent essentially the viscous heating parameter known as Frank-Kamenetskii parameter. It is important to mention that, in the expansion of heat source term λ (1 + ) m e ( ∕(1+ )) , the Arrhenius heating case (i.e. when m = 0) was considered in this study.

Analytical solution
Equations (9)-(11) are a system of ordinary differential equations with constant coefficients that are coupled together. To solve our present problem, HPM has been used. Convex homotopy on the nonlinear coupled governing equations (9)- (11) has been constructed. Therefore, the velocity-IMF and temperature equations are as follows: We assume the solution of U, B and to be in the form: Substituting Equation (17) into Equations (14)- (16) and comparing same powers of p, the sets of ordinary differential equations with their transformed boundary conditions now becomes and The transformed boundary conditions now become The transformed boundary conditions now become The solutions of the temperature, momentum and IMF are obtained as follows: If we assume that p equals 1, then the solutions to the differential equations in their approximate form are as follows: Ayati and Biazar 40 investigated the series solution's convergence using HPM. It is obvious from the analysis that only a few terms from the HPM's series can be used to approximate the solution.
Two parameters that are vital for buoyancy-induced micro-flow are the volume flow rate (Fr) and the frictional drag force ( ). The volume flow rate is computed as follows: Steady-state skin frictions ( ) and heat transfer rate at Z = 0 and Z = 1 are obtained as follows:  Figure 2 shows the impacts of (Kc) and v Kn along with ( ) on the temperature gradient under three conditions in the wall-ambient temperature difference ratio ( = − 1: a heating and a cooling system; =0: one is heated while the other is one not, = 1: heated plates on both sides). It is quite evident that, as the values of Kc and v Kn increases, for fixed values of In = 1.667, M = 5, and Pt = 0.5, there is a substantial rise in the temperature surge, particularly at the region of the left vicinity as reflected in Figures 2A, B. This impact can be argued from the premise that, as rarefaction parameter rises, the temperature jump grows as well and this weakens the quantity of heat propagation from the micro-channel walls to the fluid. The decaying in fluid movement as a result of the diminishing effect in the amount of heat transmission is compensated by the improvement in the fluid motion occasioned by the minimization in the frictional drag forces near the micro-channel walls. Raising the levels of Kc in the temperature equation increases the chemical reaction and viscous heating term, resulting in a substantial increase in temperature, as ( ) grows.

RESULTS AND DISCUSSION
The impacts of rarefaction parameter are showcased in Figure 3A, B for the velocity and IMF for higher levels of ( ). A close look at these figures revealed that, both the velocity and IMF experience upward movement for increasing values of v Kn. It is noteworthy to report that, the flow on the IMF is more significant on the cold plate, while a dramatic rise in the fluid velocity is quite glaring in the middle of the micro-channel. Figure 4 depicts the influence of Kc on both the velocity and the IMF. Increasing the value of (Kc) has been found to improve the flow motion significantly as shown in Figure 4 A as well as an enlargement in the IMF as revealed in Figure 4 B. As expected, there is an upsurge in the temperature as a result of increased Kc leading to a diminishing effect in the measure of the fluid density, consequently yielding a rise in the fluid motion.
The flow characteristics for the volume flow rate with the variation of (Kc) and (Pt) is displayed in Figure 5. It is determined that, growing levels of , the volume flow significantly rise in the vicinity of the heated plate as depicted in Figure 5 A, as (Pt) is seen to have the same effect as displayed in Figure 5 B. It is expectedly so because the fluid velocity increases as the viscous heating parameter (Kc) grows, which leads to the escalation in the volume flow rate.
The deviations of skin frictions at both cold and hot micro-channel plates under the influence (Kc) and (Pt) are illustrated in Figures 6 and 7. As seen from Figure 6 A, at different ascending values of ( ), the shear stress is observed to be boosted more profusely at the right wall for an increasing function of Kc at Z = 0, whereas as shown in Figure 6 B, a reverse phenomenon occurs at the hot plate. Similarly, the impact of Prandtl magnetic on the frictional force at both micro-channel walls is shown in Figure 7. It is paramount to report that, at Z = 0, as revealed from Figure 7 A, a more pronounced increase is recorded in the vicinity of the right wall for higher values of (Pt) at different growing levels of ( ), whereas a reverse case is established at Z = 1 as displayed in Figure 7 B.
The effects of ( v Kn) and (Kc) on the Nusselt number is shown in Figure 8. It is pertinent to report that at Z = 0, a significant rise in the heat transfer amount is obvious as Kc and BvKn increases respectively, as observed in Figure 8 A, whereas at Z = 1, a contrast phenomenon is recorded as revealed in Figure 8 B.

Validation of results
In order to ensure that the current work is accurate, the numerical comparison between Jha and Aina 4 and the current analysis are computed. As shown in Table 1, an excellent agreement was established for the momentum and energy distributions.

CONCLUSIONS
A computational treatment of steady, fully developed natural convection of arrhenius-controlled heat transfer fluid in a micro-channel was performed, with IMF effects taken into consideration. Closed form expressions were used to estimate temperature, velocity, volume flow rate, sheer stress and heat transfer amount. The actions of major pertinent parameters such as (Kc), (Pt), ( v Kn), and ( ) parameters have been determined, and the findings are represented graphically in the form of line graphs. The summary of the noteworthy findings is outlined below: 1. The functions of Kc and v Kn effects is to drastically speed up the fluid motion and the amount of volume flow for various ascending levels of . 2. Similarly, for higher levels of Kc and , the temperature profile improves significantly. 3. When Pt and Kc are increased for growing values of v Kn, the skin-friction effect tends to rise at Z = 0, while the converse occurs at Z = 1. 4. When the chemical reaction term is neglected, this current study reduces to the work of Jha and Aina. 4 5. As the fluid is heated at Z = 0, the heat transfer amount is greatly improved with rising levels of Kc and , whereas a counter attribute is noticed at Z = 1.

CONFLICT OF INTEREST STATEMENT
The author declares that there is no conflict of interest.

DATA AVAILABILITY STATEMENT
The data that support the findings of this study are available from the corresponding author upon reasonable request. , ) ,  19 1 + 2BnKnIn , 10 .