# Time-Dependent Magnetohydrodynamic (MHD) Flow of an Exothermic Arrhenius Fluid in a Vertical Channel with Convective Boundary Condition

Authors: M. M. Hamza; S. Abdulsalam; S. K. Ahmad
Journal: Advances in Mathematical Physics
 (2023)

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## Abstract

The current study examined the effects of magnetohydrodynamics (MHD) on time-dependent mixed convection flow of an exothermic fluid in a vertical channel. Convective heating and Navier’s slip conditions are considered. The dimensional nonlinear flow equations are transformed into dimensionless form with suitable transformation. For steady-state flow formations, we apply homotopy perturbation approach. However, for the unsteady-state governing equation, we use numerical technique known as the implicit finite difference approach. Flow is influenced by several factors, including the Hartmann number, Newtonian heating, Navier slip parameter, Frank-Kamenetskii parameter, and mixed convection parameter. Shear stress and heat transfer rates were also investigated and reported. The steady-state and unsteady-state solutions are visually expressed in terms of velocity and temperature profiles. Due to the presence of opposing force factors such as the Lorentz force, the research found that the Hartmann number reduces the momentum profile. Fluid temperature and velocity increase as the thermal Biot number and Frank-Kamenetskii parameter increase. There are several scientific and infrastructure capabilities that use this type of flow, such flow including solar communication systems exposed to airflow, electronic devices cooled at room temperature by airflow, nuclear units maintained during unscheduled shutoffs, and cooling systems occurring in low circumstances. The current findings and the literature are very consistent, which recommend the application of the current study.


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## Body

## 1. Introduction

According to recent theoretical developments, mixed convection is a flow condition that includes both free and forced convection movement. Mixed convection flows develop when buoyant forces alter the circulation, temperature, and species composition regimes. When the attractive force on a force flow is significant, the impact intensities of forced and free convection are comparable, resulting in mixed convection. Mixed convection phenomena have been observed frequently in nature, including astronomical detectors exposed to weather systems, telecommunications devices cooled by fans, nuclear facilities cooled during forced outages, and heat exchangers deployed in low-velocity locations. Madhu and Kishan [1] employed a finite element method and a nonlinear MHD model to analyze the coupled heat and mass transfer flow of a non-Newtonian power-law nanofluid. Abdul and Amer [2] simulated the free and forced magnetohydrodynamic flow of a nanofluid using a computer. In their investigation of mixed convection over a vertically extended sheet, Halim and Noor [3] demonstrated that the aided flow had a higher rate of convective heat transfer than the opposing flow over a moving cylinder with a yawed axis controlled by buoyancy. Dinarvand [4] found that hybrid nanofluids are better than base fluids and fluids containing single nanoparticles. Dinarvand et al. [5] examine the mixed convection of incompressible viscous and electrically conducting hybrid nanofluid flow approaching the stagnation point of the planar flow. Gul et al. [6] reviewed the MHD combined flow movement in a vertical channel. Using the spectrum relaxation technique, Ahamed et al. [7] studied the effects of transient free and forced convection flow on the partial slip effect in MHD nanofluid. In [8, 9], one can create excellent mixed convection flow courses.The use of magnetohydrodynamics to solve different energy-related problems has lately received a lot of attention. Examples include heat energy, refrigeration of current electronic systems, geothermal electricity, hydroelectric power systems, and renewable energy detectors. Because of the field’s substantial success in a range of difficulties, these techniques have gained a lot of traction in recent years. Jha and Aina [10] examined the initial mass function of magnetohydrodynamic flow in a microchannel with infinite vertical parallel electrically isolated walls. In a preliminary study, Hamza et al. [11] investigated steady-state MHD natural convection slip movement in a vertical geometry and discovered that increasing the contaminant reactivity parameter had no effect on fluid momentum. Y. Daniel and S. Daniel [12] investigated how radiation and buoyancy affect MHD flow. Das et.al [13] studied hydromagnetic oscillatory reactive flow through a porous channel in a rotating frame subject to convective heat exchange under Arrhenius kinetics concluded that the results reveal that the combined effects of magnetic field, rotation, suction/injection, and convective heating substantially affect the flow characteristics in the channel. Ali et. al [14] scrutinized the analysis of bio-convective MHD Blasius and Sakiadis flow with Cattaneo-Christov heat flux model and chemical reaction. Jha et al. [15] predicted the influence of the radiation effect using a numerical approach. Khan and Mustafa [16] investigated and quantified the effects of nonlinear radiation on MHD flow. Hosseinzadeh et al. [17] investigated the influence of radiant radiation on MHD flow using the Runge-Kutta method. Yasin et al. [18] investigated the impact of radiant energy in magnetohydrodynamic flow with a convective boundary. Previously, Jha and Samaila [19] showed the influence of radiative heat flow on convective surface boundary conditions. Uddin et al. [20] employed a numerical method to calculate the combined effect of convective heating and radiation. Das et al. [21] solve their problem using empirical methods that account for Newtonian heating and radiation effects. Ojemeri et al. [22] methodically prepared the effects of the slip condition and Soret effect of a natural convection flow via a vertical porous cylinder with a radial magnetic field over time. Das et. al [23] examined the Hall effects on unsteady MHD reactive flow through a porous channel with convective heating at the Arrhenius reaction rate. Ullah et al. [24] use computational tools to investigate the impact of momentum slip in MHD flow on convective heating of walls. According to Kumar and Singh [25], who examined the impact of convective heating and cooling on induced magnetic fields analytically, the field may have been extensively embraced due to its importance in research and engineering. The researches by Chaudhary et al. [26], Das et al. [27], Gangadhar et al., and Afify and Elgazery [28] have conducted more extensive and useful research on Newtonian heating. Hamza [29] examined the free convection slip flow of an exothermic fluid in a convectively heated vertical channel and reported in his finding that a higher number of the Biot number represents a higher degree of convective heating at the lower channels. Ali et. al [30] analyzed a comparative study of unsteady MHD Falkner-Skan wedge flow for non-Newtonian nanofluids. Turkyilmazoglu [31] studied precise solutions for an incompressible, viscous, magnetohydrodynamic fluid of a porous rotating disk flow with Hall’s current and discovered some interesting results. Mahdi et al. [32] created a novel theoretical and experimental model for the Casson hybrid nanofluid flow caused by a stretching/shrinking sheet. A normal-to-the-sheet magnetic field increases the similarity velocity profiles in the hydrodynamic boundary layer by a factor of two, according to the study’s findings. Jabbaripour et al. [33] investigated aqueous aluminium-copper hybrid nanofluid flow over a sinusoidal cylinder in a short time while accounting for a three-dimensional magnetic field and slip boundary conditions. According to Dinarvand and Mahdavi Nejad [34] who demonstrated the flow of an aqueous Fe2O3-CuO hybrid nanofluid via a permeable stretching/shrinking wedge, concluding that the magnetic parameter and mass suction at the wedge’s surface both contribute to an increase in the local Nusselt number. Izady et. al [35] examined the flow of a CuO-Ag/water hybrid nanofluid at the nodal/saddle stagnation points boundary layer using a novel hybridity model. Ali et. al [36] inspect melting effect on Cattaneo-Christov and thermal radiation features for aligned MHD nanofluid flow comprising microorganisms to leading edge. Recently, Hamza and Abdulsalam [37] reported that the Lorentz force opposing the fluid flow causes fluid velocity to drop as the Hartmann number grows. This results in a decrease in boundary layer thickness, which causes a decline in momentum region.Only a few assessments, including the results of transitory MHD mixed movement of flow with a thermal breakdown fluid under convective heating of the channel, have been published as a result of prior discoveries and researchers’ improved knowledge. Hamza [29] investigated the effects of momentum, Navier slip, and convective heating on the natural convection flow of an exothermic fluid between vertical channels in a manner similar to our work. In this work, we investigated the influence of MHD flow on an exothermic fluid in a vertical channel. An implicit finite difference scheme technique is used in the case of the transitory state. At first, the analytical solution for the steady state was made using HPM, the basic equations of momentum, energy, and boundary conditions, and it had no units. It is not uncommon for the previously stated phenomena to be significant in areas such as geothermal, oil reservoir engineering, MHD closed cycle power generation, and MHD open cycle power production. In addition to the experimental investigations mentioned above, theoretical work is needed to foresee the effects of mixed convective flow conditions such as atmospheric boundary layer flow, heat recovery, solar panels, nuclear fuel, and telecommunication equipment. This sort of flow is used in many industrial processes, such as the manufacture and excavation of polypropylene and plastic layers, paper manufacturing, deep drawing and acrylic production, molten metal spinning, centrifugal casting, and so on.


## 2. Mathematical Structure

Figure1 depicts the steady, laminar, incompressible, viscous hydromagnetic flow with mixed convection across a vertical channel separated by a distance H under the effect of Newtonian heating and velocity slip. The fluid flows due to fluid’s reactivity and convective heating of the channel wall.Figure 1
An instance of the considered model in geometric form.The following presumption has been made:(1)
The buoyancy force pales in comparison with the influence of the transverse magnetic field(2)
The infinite vertical parallel walls are nonconductive(3)
Newtonian heating of the plate from bottom channel shall be considered(4)
Joule heating and viscous dissipation have been overlooked in this studyUnder Boussinesq’s assumption, Hamza [29] states that the nondimensional model equation is thus
(1)∂u′∂t′=ν∂2u′∂z′2−∇B02ρ+gβθ′−θ′0−1∂ρ′ρ∂x′,(2)∂θ′∂t′=kpCp∂2θ′∂z′2+QC0∗ApCpe−E/RT′.The relevant boundary conditions to be satisfied in a dimensionless form are(3)t′≤0:u′0,θ′⟶θ′0,0≤z′≤H,t′0:u′=γ∗∂u′∂z′,−k∂θ′∂z′−hθ′0−θ′=0,atw′=0,u′=0,θ′=θ′0,asz′⟶H,where C0 denotes the starting concentration, β stands for heating value, and Q indicates reaction rates. R also stands for gas constant and denotes flow properties E. A fluid’s thermal efficiency is often expressed by K, whereas gravity’s force is represented by g, heat capacity at constant pressure by Cp, and the fluid’s strength by ρ.To solve equations (1) and (2), we apply dimensional parameters presented below.
(4)T=Eθ′−θ0Rθ02,λ=QC0∗AEH2RT02e−ER0,ε=Rθ0E,U=u′μ0EgβH2RT02,Gr=gβθ1−θ0v2,Pr=μ0pCpk,γ=γ∗H,Ta=Eθa−θ0Rθ02,Hp=hHk,Re=u0v,A=∂ρ∂x,Ha2=∇B02ρ,t′=t′μ0H2,z=z′H.Using (4), assume the following structure for equations (1)–(3):
(5)∂u∂t=∂2u∂z2−H2aU+GrReT−∂ρ∂x,(6)∂2T∂t=1Pr∂2T∂z2+λPreT/1+εT.The governing flow boundary is given as(7)U=0,T=0,0≤z≤1,t≤0,t>0:U=γ∂u∂z,∂T∂z=HpT−Ta,atz=0,U=0,T=0,asz=1.


## 3. Homotopy Perturbation Method for Steady State

To demonstrate the fundamental principles of the homotopy perturbation technique, often known as the HPM, we will use the preceding nonlinear differential equation as an example.(8)Au−fr=0,r∈Ω,with respect to the boundaries
(9)Bu,∂u∂n=0,r∈Γ,where A is a generic differential operator, B is a boundary operator, fr is a known analytical function, and correspondingly Γ is the boundary of the domain. In a general sense, the operator A may be broken down into two pieces, which are L and N, where L is the linear portion of the operator andN is the nonlinear part of the operator. Because of this, (8) may also be written as
(10)Lu+Nu−fr=0,r∈Ω,To build a homotopy, we use the technique of homotopy.vr,p:Ω×0,1⟶R which obeys
(11)Hv,p=1−pLv−Lu0+pAv−fr=0.There are limits on the initial approximation in equation (11), which is in conformity with the imbed parameter p∈0,1 in equation (8). We can see from equation (11) that we have
(12)Hv,0=Lv−Lu0=0,(13)Hv,1=Av−fr=0.We may suppose that the rebuttal to equation (11) can be expressed as a power series in
(14)v=v0+pv1+p2ν2+⋯,Settingp=1 gives the approximate solution of equation (8) as
(15)u=limp⟶1v=v0+v1+v2+⋯.Utilizing the homotopy perturbation approach, as described in [14], it is possible to determine the convergence of the series solution. The research shows that the solution may be approximated by using a few terms from the homotopy perturbation method series.
(16)d2Udz2−H2U+GrReT=A,(17)d2Tdz2+λeT/1+εT=0.At the fluid wall limit, the operating requirements that account for the velocity slip and temperature jump properties are as follows:(18)U−γdUdz=0,dTdz−HpT−Ta=0,atz=0,U=0,T=0,atz=1,to solve this governing problem in this context by constructing a uniform homotopy to equations (16) and (17).
(19)HU,p=1−pd2Udw2−pν0∂u∂w+Ha2U−GrReT=0,HT,p=1−pd2Tdw2+p1+T+12−εT2=0.Assume the solution to be in the form of(20)Tw=T0+pT1+p2T2+⋯,Uw=u0+pu1+p2u2+⋯.By substituting equation (13) into equations (8) and (9) and equating the ratio of equal powers p, the following differential equations and related boundary conditions are obtained:
(21)d2u0dz2+pd2u1dz2+p2d2u2dz2+⋯=pεν0∂u0∂z+pν0∂u1∂z+p2ν0∂u2∂z+Ha2u0+pu1+p2u2−GrReT0+pT1+p2T2,d2T0dz2+pd2T1dz2+p2d2T2dz2+⋯=+pε1+T0+pT1+p2T+12−εT0+pT1+p2T2=0,p0:d2T0dz2=0,d2u0dz2=0.The corresponding transformed boundary condition now become(22)dT0dz−BrT0−Ta=0u0−γdu0dz=0atz=0,T0=0u0=0atz=1,p1:d2T1dz2+λ1+T0+2−εT02=0,d2u1dz2−Ha2u0+GrReT0=A.The corresponding transformed boundary condition now become(23)dT1dz=BrT1u1=γdu1dzatz=0,T1=0u1=0atz=1,p2:d2T2dz2+λT1−2εT1T0+4T1T0=0,d2u2dz2−Ha2u1+GrReT1=0.The corresponding transformed boundary condition now become(24)dT2dz=BrT2u2=γdu2dzatz=0,T2=0u2=0atz=1.If we assume thatp equals 1, then the solutions to the differential equations in their approximation form are as follows:
(25)T=limTp⟶1=T0+T1+T2+⋯,U=limUp⟶1=u0+u1+u2+⋯,T0=k0+k1z,T1=−λz22+k0z36+k1z22+Qk02z412+k0k1z33+k12z22+k2z+k3,T2=−4λ−2ελλz424+k0z5120+k1z424+Qk02z6360+k0k1z560+k12z424+k2z36+k3z22+k4z+k5,u1=−GrRek0z36−k1z22+k8z+k9,u2=Ha2−GrRek0z5120−k1z424+Az424+k8z36+k9z22−GrRe−λz424+k0z5120+k1z424+Qk0z6360+k0k1z560+k12z424−k2z36−k3z22+k10z+k11.Calculating the stress distribution as well as the heat transfer rate across both plates requires differentiating the solution found for the temperature and momentum with respect toz.
(26)Nu0⇔dTdzz=0=k0+k2+k4,Nu1⇔dTdzz=1=k0−λ1+k02+k1+QA023+k0k1+k1+k2−4λ−2ελk06+k026+k0k124+Q+k0260+k0k112+k126+k22+k3+k4,Sk0⇔dudzz=0=k8+k10,Sk1⇔dudzz=1=−GrRek02+k1+A+k8+Ha2−GrRek024+k16+A6+k82+k9+−GrRe−λ124+k024+k16+Qk0260+k0k112+k126+k22+k3+k10.The constants can be found in the appendix.


## 4. Numerical Solution

There are two tentative differential equations with physical laws that can be solved using a time-fractional technique. We approximated the first and second derivatives using second-order central differences and the forward difference formula. To account for the model parameters, the last nodes are recalculated in the equations. Here are the recipes for equations (5) and (6)’s implicit finite difference equation.
(27)−rUi−1j+1+1+2rUij+1−rUi+1j+1=1−M2ΔtUij+GreΔtθij+ΔtA,−rTi−1j+1+Pr+2rTij+1−rTi+1j+1=PrTij+λΔtexpTij1+εTij.


## 5. Discussion of Findings

As part of a previous study, we solved the governing equations (8) and (9) at the same time with the same boundary condition (10); however, the IFDS approach is used to solve equations (5) and (6) with relation to the boundary condition (7). Table 1 displays the numerical validation of reference work [29] and present work. Graphs depict several parameters influencing momentum and energy flow formation, friction coefficient, and heat transfer characteristics. Except as otherwise specified, the fallout of the reactive viscous parameter (λ), Hartmann number Ha2, Navier slip parameter (γ), free and forced convection parameter Gre, and local Biot number (Hp) on fluids profile is verified in Figures 2–9. The parameters’ numerical values are assumed to be Ha2=2, λ=0.1, γ=0.1, Hp=0.1, Gre=5, Pr=0.71, and e=0.1. The time intervals utilized are 0.2≤t≤0.6.Table 1
Numerical computations comparing Hamza’s [29] work with the current work. γ=0.1, λ=0.1, Gre=1, Hp=0.1, Ha=0, and A=0.
zHamza [28]UzPresent workUz0.10.00840.00840.20.01110.01110.30.01260.01260.40.01300.01300.50..01330.0134Figure 2
Velocity and temperature in a steady position forλ.
(a)(b)Figure 3
Velocity in a steady position forGre and Ha2.
(a)(b)Figure 4
Unsteady-state velocity forGre and Ha2.
(a)(b)Figure 5
Temperature and velocity in an unsteady state forHp.
(a)(b)Figure 6
Unsteady-stateλ for velocity and temperature.
(a)(b)Figure 7
Skin friction forHa2 in proportion to t.
(a)(b)Figure 8
Skin friction forGre in proportion to t.
(a)(b)Figure 9
The Nusselt number forHp in proportion to t.
(a)(b)Figures2(a) and 2(b), respectively, elucidate the impact of the reactive viscous fluid parameter on both energy and momentum flow at the steady-state region; clearly, an increase in λ tends to viscous heating in the energy equation, which generates a surge in temperature. As the temperature goes up, the viscosity of the fluid lowers, causing the momentum profile to climb. That is what Hamza [29] contends.The impact of combined convection parameterGre on the flow at steady-state region is depicted in Figure 3(a), where a slight increase in Gre fuels the growth in the velocity component. This is because buoyancy forces are far more effective than viscous forces, resulting in an increase in convection at the lower plate z=0, as portrayed in Figure 3(a). However, Figure 3(b) emphasizes the importance of MHD in the governing equation. It has been shown that the momentum and boundary layer thickness are trending downwards ratios of Ha2, leading to a drop in boundary layer thickness, which leads to a decline in the momentum profile.The effects ofHa2 and Gre on velocity along through dimensionless time t are depicted in Figures 4(a) and 4(b). A rise in Ha2 leads to the presence of an opposing force termed as the Lorentz force, which resists the fluid flow, resulting in a drop significantly in momentum fluid flow, according to Figure 4(a). However, as time t passes, the fluid velocity increases due to a reduction in the number of drag forces opposing the fluid flow. The effect of the mixed convection parameter on fluid velocity is conveyed in Figure 4(b). It was noticed that a slight increase in Gre results in an increase in velocity distribution, because buoyancy forces are much more effective than viscous forces, and this led to an increase in convection at the bottom surface z=0; as time t increases, the fluid momentum increases due to higher number of combined forces.In Figures5(a) and 5(b), the effect of the local Biot number (Hp) is examined. As seen in Figures 5(a) and 5(b), as Hp increases, the fluid velocity at the bottom surface channel increases, as described by Hamza [29]. Despite the fact that as time t passes, the energy and momentum fluid flow rise; this is due to the fact that the larger Biot numbers (Hp) indicate a greater degree of convective heating, meaning that the flows improved at a lower plate energy and momentum profile.In Figures6(a) and 6(b), the effect of viscous reactive fluid parameters λ on velocity and temperature is shown. Both temperature and velocity fluid flow increase and notice an increase at the lower channel z=0. Although adding value of λ raises the reaction mixture’s strength and the sticky heating source substances in the thermostat’s composition, the momentum and energy fluids improve over time t, leading in a significant rise in temperature. The considerable temperature increased as a result of a decrease in apparent flow, resulting in a rise in fluid velocity. Furthermore, due to the plate’s lopsided heating, the temperature rises in the lower channel while falling in the upper channel.Figures7(a) and 7(b) show how important skin friction is in forecasting the flow characteristics within a wall. Figures 7(a) and 7(b) reveal that surge in Ha and dimensionless time t, respectively, leads to increase in the rate of shear stress. This is due to a higher number of the Hartmann numbers; this causes an increase in the momentum boundary layers along the wall’s edge. As time t increases, the skin friction rises at the upper plate z=1 in both Figures 7(a) and 7(b).The variability of skin friction overGre and dimensionless time t was depicted in Figures 8(a) and 8(b). Once the time t is increased at a higher value of the mixed convection parameter, a stronger force is created, resulting in more kinetic energy, which speeds up the fluid flow at the upper plate z=1.Figures9(a) and 9(b) demonstrate the variation of the Nusselt number for the local Biot number over dimensionless time. A higher number of local Biot number (Hp) corresponds to convective heating at the upper plate, according to Hamza [29].


## 6. Conclusion

The major purpose of this research is to determine the effects of MHD, velocity slip, and convective heating on both transient and persistent mixed convection flows in a vertical channel involving an exothermic fluid. Implicit finite difference was used to account for unsteady energy and momentum. In contrast, the approximation solutions employ the homotopy perturbation approach. The significance of the following criteria in influencing the formation’s influence has been demonstrated.(1)
The momentum profile surges with an increase in the Biot number (Hp), viscous reactive fluid parameter (λ), and mixed convection parameter (Gre), while it decreases with an increase in Ha(2)
The temperature profile enhances with an increase in the local Biot number (Hp) and viscous reactive fluid parameter (λ)(3)
The skin friction coefficient increases with an increase inGre at both plates(4)
It was also noticed that the skin friction increases as timet passes in both plates with an increase in Ha(5)
It was noticed from the outcome of the finding that the Nusselt number is higher on the platez=0 than that of z=1 as time t passes



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