Magneto-Exothermic Catalytic Chemical Reaction along a Curved Surface

In the current study, the physical behavior of the boundary layer flows along a curved surface owing exothermic catalytic chemical reaction, and the magnetic field is investigated. The mathematical model comprised of a part of momentum, energy, and mass equations, which are solved using a finite difference method along with primitive variable formulation. Numerical solutions, using the method of quantitative differentiation, are made with the appropriate choice of dimensionless parameters. Analysis of the results obtained shows that the field temperature and flow of fluids are strongly influenced by the combined effects of catalytic chemical reactions and the magnetic field. The effects of skin friction, heat transfer, mass transfer, mass concentration, and temperature distribution along the curved surface are illustrated in the plots and in the form of tables. By setting the controlling parameters at the boundaries, the boundary conditions at the surface and away from the surface are determined in each graph. With a larger range of body shape parameter n , skin friction and heat transfer are improved, but mass transfer is reduced. Due to the increasing values of the exothermic parameter, the fluid velocity and mass concentration are decreased gradually and the temperature distribution is increased dramatically.


Introduction
A mechanism that occurs due to temperature uctuations because of the temperature di erence is known as natural convection.
e integrated approach to evolutionary and chemical reactions is embedded in both exploratory research and theoretical diversity and ow patterns due to its wide range of engineering and industrial applications. Natural convection has many applications in terms of temperature, solar energy, and dispersion of chemical pollutants, chemical catalytic reactors, and pottery. However, the combined process of natural convection ow and exothermic catalytic chemical reaction and the incorporation of magnetohydrodynamic into the curved area have not yet been investigated by researchers. e contributions of the research community to these approaches are highlighted below.
Magnetohydrodynamic (MHD) -free convection boundary layer ow of an incompressible steady viscous and electrically conducting uid with uniform mass and heat ux in the presence of strong cross magnetic eld along a vertical at heated plate was studied by D'sa [1]. Kuiken [2] investigated the problem of magnetohydrodynamic-free convection of an electrically conducting uid in a strong cross eld. He highlightedtheimportantcasesofboth theliquidmetals andionized gases. Hossain et al. [3] considered the MHD natural convection ow along a vertical porous plate in the presence of variable magnetic field. e combined mechanism of magnetic field and chemical reaction for natural convection flow over a stretching sheet was investigated by Afify [4]. Later, Makinde and Aziz [5] numerically studied the mechanism of heat and mass transfer from a vertical plate embedded in a perforated area by taking into account the chemical reaction and the magnetic field. e study of MHD two immiscible fluids present between the isothermal and insulated moving plates wascarriedoutbyStamenković etal. [6].Ashrafetal. [7] studied the combined effects of synthetic and radiation in the natural convection flow past the permissible magnetic plate. Rongy [8] addressed that intense changes occur in the exothermic autocatalytic front that is generating in the presence of Marangoni flows. Rout et al. [9] considered the simultaneous heat transfer over a straight moving plate to quantify the combined effects of chemical reactions and thermal production on static boundary conditions. Numerical studies on the reaction of exothermic/ endothermic chemicals and the ability to regenerate Arrhenius in MHD-free convection flow by meeting the effects of radiation were performed by Maleque [10]. Later, he studied the engineering of the binary chemical reaction associated with the natural MHD flow of convectionover anaccessible platein [11].
e study on production of heat due to exothermic reaction in a fully developed electrically conducting mixed convection flow was carried out by Jayabalan et al. [12].
Ashraf et al. [13] proposed a model of mixed convective flow in a specific magnetic field to study the effects of thermal conductivity and viscosity that vary in temperature. Anwar et al. [14,15] introduced the heat transfer and Joule heating in a magnetized flow model. Daniel et al. [16] analyzed the flow of MHD radiations to discuss the effects of nanofluid formation on an expanded sheet with a flexible layer. e transcendent and stable natural flow of hot liquid in a vertical station considering the conditions of slipperiness and temperature of the Newtonian was investigated by Hamza [17]. e chemical reaction that causes heat dissipation was considered obtained by computer analysis of free-flow convection in a curved area by Ashraf et al. [18].

Governing Mathematical Model and Coordinate System
Consider a steady and two-dimensional natural convective flow in the presence of chemical reaction on a curved surface, as shown in Figure 1. A nondimensional Cartesian coordinate system is taken along and normal to the curved surface, respectively. Here, surface temperature is assumed to be greater than the ambient temperature. Moreover, the gravitational acceleration g x is acting parallel to the curved surface, r(z) is a fixed side, and dr/dz � 0 indicates that there is no fluid flow along r(z), thus representing an external flow. Considering the above all assumptions, the dimensionless transport boundary layer equations for two-dimensional steady electrically conducting and incompressible natural convection flow of along a curved surface in the presence of exothermic binary chemical reaction (by following Ashraf et al. [18]) are, dr/dz � 0. With all the above assumptions, the dimensionless transport boundary layer equations for natural convection flow driven along the curved surface are e dimensionless variables used in the above equations are x � x l , where U s � (g x βT s l)/2 is the velocity scale defined in Ashraf et al. [18]. In the above equations, Pr � v/α, Sc � v/D B , and M � σB 2 0 l/ρU s are Prandtl, Schmidt, and Hartmann numbers, respectively. Here, β is the exothermic parameter, λ 2 � K 2 r l/U s is the dimensionless chemical reaction rate constant, where k 2 r is the chemical reaction rate constant and l is the characteristic length. e symbol, c � T w − T ∞ /T ∞, is the temperature relative parameter, and E � E a /kT ∞ is the dimensionless activation energy, with E a as the activation energy and k � 1.380649 × 10 − 23 JK − 1 is the Boltzman constant. e notations P(x) denotes the wall temperature function and Q(x) represents the body shape function which are defined as (see details in Ashraf et al. [18]) where g x is the acceleration due to gravity along the surface, and it is defined as which subjected to the boundary conditions which are u � 0, θ ⟶ 0, ϕ ⟶ 0 as y ⟶ ∞.
By following Ashraf et al. [18], it is noted that equation (2) is in a simple form that can be solved numerically for any values of P(x) and Q(x). us, the conservation equations along with boundary conditions (by dropping bars) take the following form: under dimensionless boundary conditions:

Solution Methodology
Before applying the finite difference method (FDM) used by Ashraf et al. [18] and many other, for the proposed dimensionless model presented in equations (8)-(11) along with boundary conditions (12), it is first transformed into a convenient form by applying primitive variable formulation (PVF) to make the algorithm smooth. en, the dimensionless transformed model is discretized by using a finite difference scheme. e central difference scheme is used along y-direction and the backward difference scheme is used along x-direction. To transform the above system of equation into a primitive form, we use the following transformation: e transformed boundary conditions are (14)- (18) are solved with the help of a finite difference scheme. e derivative terms involved in these equations are transformed into convenient form by using difference formulation. e whole discretization process is given below:

Discretization. Equations
Here, the subscripts i and j are the unit vectors along and perpendicular to the surface, respectively. Using (11) into equations (14)- (17), we obtain a tridiagonal matrix which is then solved with the Gaussian elimination technique. e system of algebraic equations subjected to the boundary conditions (18) is given below: where A k , B k , and C k (k � 1, 2, 3) are the coefficient matrices and D 1 , D 2 , D 3 are the column vectors for the unknown variables U, θ, and Φ. e discretized boundary conditions are Mathematical Problems in Engineering Furthermore, the above system of equations given in (12) is solved by using Gaussian elimination technique. With the help of this technique, we determine the quantities U, V, θ , and Φ. Moreover, skin friction and rate of heat and mass transfer along the curved surface can be computed by using the following laws:

Results and Discussion
e current section is devoted to the physical behaviors of the material properties such as velocity distribution, U, temperature distribution, θ, and mass concentration, ϕ, under   Figures 3(a)-3(c) depict that the velocity, temperature distribution, and mass distributions decrease with increasing values of Pr. e above mechanism happens because, with an increase of the Prandtl number, the kinematic viscosity (or momentum diffusivity) increases, which slows down the motion. Also, the increasing Pr decreases the thermal conductivity which results in low-temperature distribution. e effect of varying M on velocity distribution, temperature distribution, and mass concentration are explained in Figures 4(a)-4(c). We observed that the fluid  Now, in the preceding paragraph, we will converge our efforts to the interpretation of the results, particularly the effect of dimensionless numbers on local skin friction, the rate of heat, and the mass transfer along the surface of body shape. Figures 8(a)-8(c) display the performance of β on the skin friction, heat, and mass flux. ese figures illustrate that, with an increase in β, the physical property, τ w , has decreased, whereas θ w and ϕ w have increased. e behavior of τ w , θ w , and ϕ w for dimensionless chemical reaction rate constant λ is analyzed in Figures 9(a)-9(c). From these sketches, it is noted that τ w is decreasing, but meanwhile, θ w and ϕ w are increasing for increasing values of λ. It is noteworthy in the case of the mass transfer rate that, after attaining its peak value at 0.25, the graph becomes constant for all values of λ. Figures 10(a)  varying n on τ w , θ w , and ϕ w . In these plots, it is observed that, with the increase of n, τ w has decreased whereas θ w and ϕ w have increased. e skin friction and the heat transfer rate have minimum value for n � 0.3, but the mass transfer rate represents its maximum magnitude for n � 0.3. Table 1 presents skin friction, heat transfer, and mass transfer versus the distance X along the curved surface for Pr by taking two different values 0.71 and 7.0, i.e., for air and water. It is clearly observed that, with the increase of Pr on the skin, friction is increasing at the surface for different values of Pr, but it is also worth mentioning that the skin friction is decreasing downstream, whereas both heat and mass transfer rates show the opposite trend. ey have increased along the surface and exhibit gradually decreasing behavior at the surface. In Table 2, it is noted that there is a mild decrease in the skin friction with enhancing values of M at the leading edge, and it also kept on decreasing along the surface. Since the Hartmann number is the ratio of magnetic force to the inertial force, so with the increase of M, magnetic force increases, due to which electric current is generating in the fluid causing more the skin friction at the surface. It is also observed that, at the surface, heat transfer rate is increased with the increasing value of M and is also increasing along the surface. A completely different behavior is examined in the case of mass flux. It is increasing at the surface with the increasing values of M and has also increased downstream, but for M � 0.4, it descends at the beginning but then ascends very slowly beyond X � 4.0. For various values c, the behavior of skin friction, heat transfer, and mass transfer is illustrated in Table 3. It is seen that there is a mild decrease in skin friction with increasing values of c and that the skin friction is decreasing downstream. Apart from it, both the heat transfer rate and mass transfer rate have increased along the surface, but they have decreased gradually with the increasing values of c.

Concluding Remarks
e problem of natural convection flow is carried out in a curved surface along with exothermic catalytic chemical reaction, and a magnetic field is investigated by different physiological parameters. It is noteworthy that, by increasing the Schmidt number Sc, velocity field, temperature profile, and mass concentration have increased, although, an increase in Pr has reduced the field fluid velocity, temperature profile, and maximum concentration. With the sheer size of the Prandtl number, skin bumps are perceived to be rejected and temperature and mass transfer are increased. e size of the speed field, the distribution of temperature, and the concentration of weight are greatly enhanced by the high values of the Hartmann number which we have seen the skin collision decrease due to the increasing number of Hartmann numbers and away from the surface the impact of the magnetic field becomes much smaller. With increasing value β, the speed and concentration of the mass are determined by the magnitude, and the reverse behavior is observed in the case of temperature distribution. With a wide range of body shape parameters n, skin friction and heat transfer are noted as improved but weight transfer decreases. Due to the increasing number of exothermic parameters, the fluid velocity and concentration are gradually reduced and the temperature distribution increases dramatically, which satisfies the boundary condition. Skin tension decreases while heat and bulk transfer increases above the rising values in the fluid velocity; the distribution of temperature and the concentration of the weight is greatly increased by a very high rate of noninvasive chemical reaction rate. e various impact of the temperature-related parameter reduces skin friction, heat transfer, and mass transfer freely over the surface. Moreover, from the systematic variation of the control parameters, the convective stability of the numerical solution is achieved using a finite difference method. e future work of the current study can be extended as Dimensionless velocity along y direction x: Dimensionless distance along the surface y: Dimensionless distance normal to the surface T: Dimensioned surface temperature T ∞ : Ambient temperature E: Dimensionless activation energy M: Hartmann number ϕ: Dimensionless temperature ∅: Dimensionless mass concentration β: Exothermic parameter c: Temperature relative parameter λ: Chemical reaction rate constant.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.  Table 3: e results of τ w � (zU/zY) y�0 , θ w � (zθ/zY) y�0 , and