J. Nig. Soc. Phys. Sci. 4 (2022) 130–137 Journal of the Nigerian Society of Physical Sciences Analysis of Hydromagnetic Double Exothermic Chemical Reactive Flow with Convective Cooling through a Porous Medium under Bimolecular Kinetics F. O. Akinpelu, R. A. Oderinu, A. D. Ohaegbue∗ Department of Mathematics, Ladoke Akintola University of Technology, Ogbomoso, Nigeria Abstract In this study, the analytical solution of steady hydromagnetic double exothermic combustible reaction fluid flow in a porous medium with con- vective cooling wall is presented. The viscous heating reactive liquid is totaling exothermic without consumption of material. The combustion reaction of the fluid takes place in a Poiseuille device, and it is been propelled by pressure gradient and pre-exponential bimolecular kinetics. The device is exposed to convective cooling to keep the reactive hydromagnetic fluid from distortion. The weighted residual method (WRM) is analytically used to get the numerical values for the dimensionless nonlinear governing equations. The solution to temperature and velocity distribution is carried out and the result is graphically depicted. The Nusselt number and skin friction coefficient is also showed for some sig- nificant parameters engrained in the flow and the solution obtained is compared with numerical method. As obtained in the study, the second exothermic reaction term increases the combustion process; hence the term will assist in reducing toxic discharge from the engines that pollute the environment. The Frank-Kamenetskii parameter contributes highly to system thermo-fluid destruction; as such it must be monitored. DOI:10.46481/jnsps.2022.525 Keywords: Exothermic reaction, Convective cooling, Bimolecular kinetics, Porous medium Article History : Received: 19 December 2021 Received in revised form: 06 February 2022 Accepted for publication: 19 February 2022 Published: 28 February 2022 c©2022 Journal of the Nigerian Society of Physical Sciences. All rights reserved. Communicated by: J. Ndam 1. Introduction The idea of laminar flow of fluid particles along a medium continues to receive tremendously attention because of its ap- plication to fluid mechanics particularly in industries. Its signif- icant has been influenced by many engineering analyst, biolog- ical science, atomic control, aerodynamics of plasma and mag- netohydrodynamic (MHD) system, Ellahiand Afzal [1]. The flow of exothermic reactive fluid is useful in industrial and engi- ∗Corresponding author tel. no: +2348061198971 Email address: ohaegbueanthony@gmail.com (A. D. Ohaegbue ) neering system and is frequently accompanied with heat trans- fer which has long been called combustion, Makinde [2]. Some convective fluids flows are caused by the generating or absorb- ing of heat which results to a fluid chemical reaction. Heat source encourages the distribution of heat which then changes the amount of particles deposition within the arrangements such as electric clip, nuclear reactor, semiconductor etc. Generation or the absorption of heat has been assumed to depend on tem- perature. The effect of heat production or absorption of non- homogenous state in a micro polar fluid with variable conduc- tivity was considered by Chen [3]. The results show the de- pendent generation of heat at the surface is less to temperature 130 Akinpelu et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 130–137 131 dependent generation. The authors [4, 5] studied the impact of non-uniform heat supply on hydromagnetic micro polar fluid with radiative in permeable media. Many studies have been carried out on exothermic chemical re- active fluid flows and its applicability in energy storage sys- tems, atomic reactor plan, geophysics, polymer expulsion and MHD reactive fluid flow. Consequent to its applications, Gbadeyan and Hassan [6] worked on the reaction of Couette variable vis- cosity fluid flow for Arrhenius case. Transfer of exothermic heat flow through a pipe was investigated by Marcello et al. [7] under different chemical kinetics, Adomian decomposition was used to solve the problem. The comprehensive survey on generation of heat by hydromagnetic fluid flow considering all the basic physical properties was reported in [8-9]. Among the numerous suggested models is a single step exothermic reac- tion while some are under convective cooling. Hazarika et al. [10] worked on the relevance of variable properties on MHD flow through a perpendicular plate. They declared that the ve- locity outline rises and the parameter of heat conductivity de- creases. Dulal & Hiranmoy [11] analyzed the effect of heat on variable heat conductivity and MHD of non-Darcy mixed con- vective flow of concentration over an extending sheet. The im- pact of MHD on heat distribution over an extented walls fixed in porous medium with heat source, variable viscousity and vis- cous dissipation was discussed by Hunegnaw & Naikoti [12]. Hassan and Maritz [13] investigated the chemical reaction on hydromagnetic flow in stationary parallel convective cooling walls. The result shows that a rise in magnetic intensity and porous medium values result to a fall in the velocity profiles and a rise in the values of porous medium, activation energy and magnetic force with convective cooling terms decreases the temperature of the fluid. Dulal [14] examined temperature and mass transfer in a fixed flow of viscous fluid on a distending per- pendicular sheet, taking buoyancy and heat radiation into con- sideration. Hazarika and Utpal [15] investigated the effect of energy conductivity and variable properties on hydromagnetic flow across a perpendicular plate. They observed that when the heat conductivity value is reduced, the velocity distribution rises. Adesanya and Folade [16] investigated the flow of hy- dromagnetic material through a porous media at the third grade level. Bala and Suneetha [17] investigated the effect of homoge- neous and heterogeneous chemical reactive stagnation flow on the MHD of a micropolar fluid through a permeability shrink- ing/stretching surface plate fixed in a porous wall. Salawu [18] reported on the heat transfer on third grade combustive echem- ical reactive flow and absorption of heat with convective cool- ing in a porous medium. Seth et al. [19] provided a computa- tional model for the MHD natural convection casson fluid flow with nth order of chemical species and Newtonian heating in porous channel. Their research concentrated on the steady-state one-step exothermic reactions of hydromagnetic flows. How- ever, analysis of steady hydromagnetic double exothermic com- bustible reaction fluid flow through a porous channel with con- vective cooling which takes place in a Poiseuille device is yet to be studied. Double exothermic reaction deals with an in- termediary reaction. The second exothermic reaction term in- creases the combustion process. Makinde et al. [20] reported on the heat stability of double steps combustive chemical reac- tion of a slab. They considered the diffusion reactant by assum- ing a chemical kinetics of variable factor of the unsteady and steady state. Salawu and Okedoye [21] provided a solution to the unsteady double step exothermic reaction along a conduit but they never considered the case when it’s under convective cooling. They studied second law of thermodynamics double step hydromagnetic exothermically combustible chemical reac- tive fluid flow considering viscous and gravity with heat absorb- ing along a channel. In the study, they observed that magnetic force decreases the flow rate and undesirable alterations in the fluid viscosity caused by an increase in temperature. Labelo et al. [22] works on two-step cylindrical stockpiles combustible reaction. Hence, the purpose of this research is to examine the steady hydromagnetic double reaction of pre-exponential combustible reactive fluid flow in porous channel with convective cooling. However, the important of double combustion reaction processes assist in improving the hydrodynamics lubricants that can leads to engine efficiency. Thus, this study is significant to pollution control, enhancing of industrial machine output and improving lubricant viscosity. An analytical weighted residual technique is employed to provide solutions to the dimensionless viscous dissipative exothermic reaction flow equations. Important of some properties of temperature and velocity field with heat gra- dient and skin friction are discussed. The present study will help in improving the performance of industrial engines and in reducing toxic discharge to the environment due to incomplete combustion. This study is motivated by previous report on the important and application on two step exothermic reaction to the chemical in- dustries. Introduction and background to the study is presented in the first section. The problem formulation equations are de- scribed in the second section. The collocation weighted residual method of solution is presented and utilized in the third section. The results and discussion is presented in fourth section. The final remark is stated in the last section. 2. The Flow’s Mathematical Formulation Consider viscous heating in a reactive fluid of double step combustible reaction between stationary parallel walls. The combustion reaction of the fluid takes place in a Poiseuille de- vice, and it is been propelled by pressure gradient and pre- exponential bimolecular kinetics. The flow is assumed to be in the direction of x with y − axis normal to it as illustrated in Figure 1. The device is exposed to convective cooling to keep the reactive hydromagnetic fluid from distortion. According to Chinyoka and Makinde [23], and ignoring time-dependent and the fluid reactive viscose consumption. The velocity and tem- perature balance equation governing the flow are as follows: − d p d x + µ d2u dy2 −σB2ou − µ k1 u = 0 (1) 131 Akinpelu et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 130–137 132 Figure 1: Flow coordinates schematic α d2T dy2 + µ ( du dy )2 + σB2ou 2 + µ k1 u2 + Q1C1 A1  KT vl m e− E1RT + Q2C2 A2  KTvl m e− E2RT = 0 (2) The imposing boundary conditions are: y = a, u = 0, −k dTdy = h ( T − Tw ) y = 0, u = 0, k dTdy = h ( T − Tw ) (3) Where B0, x , y, p,σ,u,µ,k1,α,Q1, C1, A1, E1, kb, T , l ,v,R, Q2, C2, A2, E2, Tw, m, are magnetic field, dimensional coor- dinate flow alongside the surface, dimensional coordinate flow along the normal surface, modify pressure, electrical conduc- tivity, axial fluid velocity, fluid viscosity, Darcy porosity, ther- mal conductivity, first step heat reaction, concentration species of step one reactant, constant rate of step one reactant, energy activation of step one, Boltzmann’s constant, dimensional fluid temperature, Plank’s number, vibration frequency, general gas constant, step two heat reaction, concentration species of step two reactant, constant rate of step two reactant, energy activa- tion of step two, temperature at the wall, computational index (m=0.5) respectively. Introducing the following dimensionless quantities: p = µupa , u = Uu , θ = E1 ( T−Tw ) RT 2w , x = ax, Ha2 = σB 2 o a 2 µ , G = −d pd x , y = y a , Br = µu2 E1 αRT 2w , λ = Q1C1 A1 a2 E1 αRT 2w ( kb Tw vl )m e 1 ε , Bi = ahk , Ha 2 = σB2o U 2 a2 µ , K = a 2 k1 , γ = Q2C2 A2 Q1C1 A1 e− 1 ε e 1 ε e rθ (1+εθ) (4) Substituting the dimensionless quantities of equation (4) into equation (1) to (2) gives: d2u dy2 − ( Ha2 + K ) u + G = 0 (5) d2θ dy2 + Br (K + Ha2) u2 + ( du dy )2 + λ [ (1 + εθ)m ( e θ (1+εθ) + γe rθ (1+εθ) )] = 0 (6) alongside with boundaries conditions u(0) = 0, dθ dy = −Biθ, u(1) = 0, dθ dy = Biθ (7) where γ, λ, G, Br, Ha2, r, ε and Bi are two-step exothermic reaction, frank-kamenetskii, pressure gradient, Brinkman num- bers, Hartmann numbers, activation energy ratio, activation en- ergy and Biot number parameter respectively. 3. Numerical setup The analytical method of solution used is Weighted Residual Method (WRM). The goal of weighted residual method [24, 25], is to look for an estimated result in the form of polynomial considering the given differential equation. Φ [u (v)] = f in the domain D,Bµ [u] = γµon∂D (8) whereΦ [u] is the non-linear differential operator relating of the dependent variables u and f the function of a known point, Bµ [u]is the estimated number of boundary conditions with do- main D and ∂D the boundary. The solution to the problem of the above estimated boundary isoften carried out by taking an estimate of the solutionu (v). The expression u (v) ≈ w (v, a1, a2, ...at) (9) relies on the number of parameters a1, a2, ....atand the arbi- trary value aljof the boundary conditions are satisfied. In prac- tice w (v, a1, a2, ...at) take the form w (v, a) = ϕo (v) + t∑ j=1 a jϕ j (10) where the functionϕ j(v) are given to satisfy the boundary conditions Qµ (ϕo) = γµ, Qµ ( ϕ j ) = 0, j = 1, 2. 3. ...t (11) For arbitrary values of al s, w (v, a1, a2, a3, ...at) fulfills the bound- ary condition (8). When equation (9 is substituted for equation (8), the differential equation’s residual becomes φ (v, a) = L (w (v, ai)) − f (v) (12) where a = (a1, a2, ....at).This yields the measure whereby the function w(v, a) satisfies the differential equation’s degree, as the number t of the function ’j is enhanced in subsequent esti- mations. The aim is to minimize the residual φ (v, a) to zero on an average basis throughout the domain. Which is∫ c φ (v, a) W jdv = 0, j = 1, 2, ...t (13) 132 Akinpelu et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 130–137 133 where the unknown number of constants a jin w is the same as the value of weight functions W j.The outcome consist an algebraic equations of set t for all the unidentifiedconstantsa j. So, functions of the weighted are taken as Dirac delta. That is,W j (v) = δ ( v − v j ) such that the chosen error at the nodev j is zero. Integrating equation (13) with W j (v) = δ ( v − v j ) result in φ ( v, a j ) = 0. Employing WRM to equations (5), (6) and (7), assuming that the parameters or polynomial with undefined coefficients will be found later, this function is referred to as the trial function. u (v) = 10∑ j=0 a jv j,θ (v) = 10∑ j=0 b jv j (14) Introducing the boundary condition (7) on (14) which is the trial functions and then substitute the trial function on the equation (5) and (6) to get the residual of the momentum and energy equation as: ur = 90y8a10 + 72y7a9 + 56y6a8 + 42y5a7 + 30y4a6 + 20y3a5 + 12y2a4+ 6ya3 + 2a2 − ( Ha2 + K )  y10a10 + y9a9 + y8a8 + y7a7 + y6a6 + y5a5+ y4a4 + y3a3 + y2a2 + ya1 + a0  + G (15) θr = 90y8b10 + 72y7b9 + 56y6b8 + 42y5b7 + 30y4b6 + 20y3b5 + 12y2b4 + 6yb3 + 2b2+ Br  ( 10y9a10 + 9y8a9 + 8y7a8 + 7y6a7 + 6y5a6 + 5y4a5 + 4y3a4 + 3y2a3+ 2ya2 + a1 )2 +( Ha2 + K )( y10a10 + y9a9 + y8a8 + y7a7 + y6a6 + y5a5 + y4a4 + y3a3+ y2a2 + ya1 + a0 )2  + λ  ( 1 + ε ( y10b10 + y9b9 + y8b8 + y7b7 + y6b6 + y5b5 + y4b4 + y3b3+ y2b2 + yb1 + bo ))m  e y10 b10 +y 9 b9 +y 8 b8 +y 7 b7 +y 6 b6 +y 5 b5 +y 4 b4 +y 3 b3 +y 2 b2 +yb1 +bo (1+ε(y10 b10 +y9 b9 +y8 b8 +y7 b7 +y6 b6 +y5 b5 +y4 b4 +y3 b3 +y2 b2 +yb1 +bo)) + γe r(y10 b10 +y9 b9 +y8 b8 +y7 b7 +y6 b6 +y5 b5 +y4 b4 +y3 b3 +y2 b2 +yb1 +bo) (1+ε(y10 b10 +y9 b9 +y8 b8 +y7 b7 +y6 b6 +y5 b5 +y4 b4 +y3 b3 +y2 b2 +yb1 +bo))   (16) The residual for the set of collocation points within the domain at regular interval is reduced to zero when m = 0.5, Br = 1,ε = 1,γ = 1,λ = 0.5, G = 1, Ha = 1, K = 1. That is Nn = (q−p)n M where n = 1, 2, 3, .., M − 1andp = 0, q = 1, M = 9. The unknown constant coefficients are gotten with the aid of MAPLE software. So, the dimensionless equations are as follow: u(y) = −0.000003521116925y10 + 0.00001760557764y9 − 0.0001970055654y8+ 0.0006823888193y7 − 0.005555067018y6 + 0.01435078358y5− 0.08333329281y4 + 0.1435095221y3 − 0.49999999994y2 + 0.4305285858y (17) θ(y) = 0.000317754724y10 − 0.001588772547y9 + 0.007075226439y8− 0.01876827406y7 + 0.05145794359y6 − 0.09535771165y5+ 0.1277972545y4 − 0.1153837668y3 + 0.2601529854y2− 0.2157026397y + 0.2157026397 (18) The process of WRM scheme is used for difference value of. K, λ,Ha2,G, γ, ε and Br. The physical quantities that are of interest to engineering are the heat gradient (Nu)and skin friction ( C f ) respectively as [23, 27, 28]. Nu = − dθ dy and C f = du dy (19) 133 Akinpelu et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 130–137 134 4. Results and Discussion The numerical result in Table 1 depicts the influence of many physical factors on skin friction and nusselt numbers. A rise in values of the term G and λ accelerate the temperature gradient and skin friction at the wall since temperature at the boundary layer increases. Also, an increase in values of Ha and K decreases the effect flow at the wall surface. Table 2 displays the relationship between the existing related work and the cur- rent results. Thus, the current results are consistent with the previous results. Table 3 demonstrates the comparison results of the exact and the existing work. The outcome shows that they are in good agreement.Figure 2 demonstrates the profile of velocity over pressure gradient(G). An increase in(G) term en- hances the fluid flow in the channel and heat within the system rises which cause the breaking of the force bonding in the fluid. Hence, pressure gradient aids changes in formation of fluid in the system. Figure 3 shows effect of fluid velocity on Hartmann number (Ha). The Lorentz force present in (Ha) increases the damping magnetic strength which then accelerate the fluid re- sistance and by that decelerate the motion of the fluid on con- vective cooling wall. Thus, the velocity profile is affected by a very small reading scale system which helps in fluid viscosity of many industries. Figure 4 demonstrates porosity parameter (K) which generates resistance force in the fluid as shown in the plot. Increase in porosity (K) values lead to an increase of the resistance in the fluid. This is due to the saturated of the medium with pores that decreases the free flow of the fluid. The term enhances viscosity fluid as a result of decrease within the heat source terms and so reduces the velocity flow. Figure 5 displayed the result of Frank-kamenetskii param- eter (λ). A rise in (λ) correspond to an increase in the rate of viscous heating in a reaction which enhanced the chemically reaction rate and consequently increase the distribution of the temperature within the system. This is due to enhancement in the momentum viscosity coupling that results in rational rises in the flow temperature. The parameter (λ) can cause a blow up if not well monitored. Figure 6 shows the strength of activa- tion energy (ε) on heat transfer in an exothermic chemical re- active system. The least energy which exists in a chemical sys- tem with potential reactants is known as activation energy. As the parameter term (ε) rises the level of heat transfer decreases. This is because of the presence of an appreciable number of molecules with translational energy equal to or greater than the activation energy. Thus, this help the industries to inspect the spark of activation energy required to break the bond in a chem- ical reaction. Figure 7 reveals that temperature falls while the porosity (K) values rises because the resistance to fluid motion offered by porous medium causes the fluid to heat up, so the temperature decreases within the boundary layer. The behavior is as a result of the wall of the plates that provides a supporting resistance to the fluid flow. Figure 8 presents the influence of second step term (γ) on the temperature field. It reveals that the double step exothermic combustive reaction increases heat transfer in the system. Thus, this leads to increase in the com- bustion process that in turn encourages complete burn of hydro- carbon in the engines. Hence, second step exothermic reaction Figure 2: Profile of velocity against (G) Figure 3: Profile of velocity against(Ha) Figure 4: Profile of heat against(K) reduces the release of carbon-monoxide that pollutes our envi- ronment as a result of unburned hydrocarbon. 134 Akinpelu et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 130–137 135 Table 1: Comparison of C f and Nu at various values of G, Ha2, K and λ Parameter Values Weighted Resid- ual method 4th order of R-K Absolute error C f Nu C f Nu C f Nu G 1.0 0.430529 0.290443 0.430529 0.290443 7.0 × 10−8 1.4 × 10−8 2.0 0.861057 0.344304 0.861057 0.344304 1.7 × 10−9 1.2 × 10−7 3.0 1.291586 0.434728 1.291586 0.434729 3.0 × 10−9 7.3 × 10−7 4.0 1.722114 0.564353 1.722114 0.564357 1.8 × 10−9 4.0 × 10−6 Ha2 1.0 0.430529 0.290443 0.430529 0.290443 8.0 × 10−10 1.4 × 10−8 3.0 0.290544 0.283715 0.290544 0.283717 5.1 × 10−8 1.1 × 10−6 5.0 0.193735 0.278940 0.193737 0.278951 2.4 × 10−6 1.0 × 10−5 7.0 0.141156 0.276443 0.141181 0.276472 2.5 × 10−5 2.9 × 10−5 K 1.0 0.430529 0.290443 0.430529 0.290443 2.8 × 10−8 1.4 × 10−8 1.5 0.416618 0.289792 0.416618 0.289792 1.4 × 10−9 2.1 × 10−8 2.0 0.403769 0.289188 0.403769 0.289187 1.9 × 10−9 3.3 × 10−8 2.5 0.391863 0.288624 0.391863 0.288624 2.6 × 10−9 5.0 × 10−8 λ 1.0 0.430529 0.388391 0.430529 0.388391 8.0 × 10−10 1.7 × 10−8 3.0 0.430529 0.526687 0.430529 0.526686 2.0 × 10−10 3.7 × 10−7 5.0 0.430529 0.599584 0.430529 0.599524 1.0 × 10−10 6.0 × 10−5 7.0 0.430529 0.703732 0.430529 0.707863 1.0 × 10−10 4.1 × 10−3 Table 2: Comparison of temperature profile for G = λ = 1, ε = Ha = Br = 1 y Makinde and Beg [28] Hassan and Maritz [13] Present study 0.00 0.3544502181 0.3581076494 0.3582159913 0.25 0.3323243479 0.3358417798 0.3358879051 0.50 0.2660663845 0.2691056991 0.2690827375 0.75 0.1556934861 0.1577379743 0.1578252719 1.00 0 0.0002178316 0.0000182633 Table 3: Comparison of the exact results with WRM for u(y) when G = K = Ha = 1 y Exact solution for u(y) Weighted residual method for u(y) 0.0 0 0 0.2 0.0671247049 0.0671247051 0.4 0.0993879030 0.0993879032 0.6 0.0993879029 0.0993879032 0.8 0.0671247053 0.0671247050 1.0 0.0000000001 0 135 Akinpelu et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 130–137 136 Figure 5: Profile of heat against(λ) Figure 6: Profile of heat against (ε) Figure 7: Profile of heat against (K) 5. Conclusion The analysis of steady hydromagnetic double exothermic chem- ically reactive flow with convective cooling through a porous medium under bimolecular kinetics was mathematically formu- lated. The weighted residual collocation integration method were used to solve the dimensionless viscous flow equations and the computational result obtained were used to compare with the fourth order of Runge-schemes coupled by shooting Figure 8: Profile of heat against (γ) techniques, and found to have very low absolute errors. As obtained in the study, the second exothermic reaction term in- creases the combustion of hydrocarbon due to rise in the heat distribution within the system. Hence the term will assist in reducing toxic discharge from the engines that pollute the en- vironment. 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