Chemical Reaction and Generalized Heat Flux Model for Powell–Eyring Model with Radiation Effects

In the current research, the numerical solutions for heat transfer in an Eyring–Powell fluid that conducts electricity past an exponentially growing sheet with chemical reactions are examined. As the sheet is stretched in the x direction, the flow occupies the region y > 0. MHD, radiation, joule heating effects, and thermal relaxation time are all used to represent the flow scenario. The emergent problem is represented using PDEs, which are then converted to ODEs using appropriate similarity transformations. The converted problem is solved numerically using the SLM method. The main goal of this paper is to compare the results of solving the velocity and temperature equations in the presence of K changes through SLM, introducing it as a precise and appropriate method for solving nonlinear differential equations. Tables with the numerical results are created for comparison. This contrast is important because it shows how precisely the successive linearization method can resolve a set of nonlinear differential equations. Following that, the generated solution is studied and explained in relation to a variety of engineering parameters. Additionally, the thermal relaxation period is inversely proportional to the thickness of the thermal boundary layer and the temperature, but the Eckert number Ec is the opposite. As Ec grows, the temperature within the channel increases.


Introduction
Non-Newtonian uids are widely encountered and are used in a wide variety of engineering applications. Some of these applications are notable and are applied in the paper, food, personal care products, textile coating, and suspension solutions industries. ese uids have mostly been divided into three categories: di erential, rate, and integrals. Recent technological and engineering advancements have resulted in the development of a diverse range of non-Newtonian uids with a number of major di erences from viscous uids. Ziegenhagen [1] explored the slow ow of a Powell-Eyring type uid and used variation techniques to obtain results. He studied the behavior of Oldroyd and Powell-Eyring uids and discovered that both uids behave identically in situations involving extremely slow uid ow. Sirohi et al. [2] studied it by observing the ow of Powell-Eyring uid around the accelerating plate. ey compared three distinct techniques. Yoon and Ghajar [3] pioneered the concept of a stretched sheet by providing a precise solution to the resulting di erential system. Recent academics have investigated this topic from a variety of perspectives [4][5][6][7][8][9][10][11][12]. Mushtaq et al. [13] investigated the Powell-Eyring uid ow and heat transport past a stretched sheet exponentially. ey discovered that increasing the velocity ratio parameter results in a thinned boundary layer. Malik et al. [12] examined the Powell-Eyring uid ow and heat transport with varying viscosity over a stretching cylinder by examining the steady condition. ey concluded that as Prandtl and Reynolds numbers increase, the boundary layer shrinks. Sher Akbar et al. [14] studied the e ect of magnetic factors on Eyring-Powell uid ow past a stretched surface. ey investigated ow resistance as the magnetic and hydrodynamic properties of the uid under study increased. Kumar and Srinivas [15] investigated the Powell-Eyring nano uid passing via an inclined permeable sheet. ey demonstrated that temperature increases as thermophoresis parameter values increase. While the contrary is true for nanoparticle concentration due to higher chemical reactions and Brownian parameters, increasing thermophoresis parameter values results in an increase in concentration. Pal and Mondal [16] demonstrated magneto-bioconvection of the Powell-Eyring nanofluid via a vertically stretched sheet that is convectively heated and also contains motile, gyrotactic microorganisms. ey discovered that as the Schmidt number and chemical reaction parameters increase, the concentration of nanoparticles drops. ermal relaxation time is the time required for fluid to return to its original temperature after being heated. It is a frequently used parameter for determining the time required for heat to leave a fluid. Hayat and Nadeem [17] investigated the effects of mass flux models on Eyring-Powell fluid flow in three dimensions. ey discovered that temperature and thermal-relaxation time have an inverse relationship. Reddy et al. [18] studied the effect of chemical reactions on the activation energy of the Eyring-Powell nanofluid flow via a stretching cylinder. ey concluded that as the relaxation parameter increases, the temperature curves decrease in shape. It takes a long time for an increase in the relaxation parameter assessment to transfer heat to neighboring material particles. Additionally, the Nusselt number improves behavior when nondimensional thermal relaxation calculations are performed.
Mustafa [19] researched the Maxwell fluid with a generalized heat flux model for rotating flow and heat transfer. ey also discovered that the thermal relaxation period is inversely proportional to temperature and thermal boundary thickness. Ishaq et al. [20] demonstrated that the entropy production of the Eyring-Powell fluid flow with nanofluid thin film flow can be calculated by considering the heat radiation and MHD impact. ey discovered that when the Brinkmann, Hartmann, and Reynolds numbers grow, so does the entropy profile. For increasing values of the Eyring-Powell and radiation parameters, the entropy profile reduces.
e Eyring-Powell nanofluid flow with nonlinear mixed convection and entropy generation was explored by Alsaedi et al. [21]. ey arrived at the conclusion that entropy generation showed a falling tendency for some fluid parameter values while increasing for others. rough a permeable stretching surface, Bhatti et al. [22] studied the irreversibility of the MHD Eyring-Powell nanofluid. More interesting articles can be seen in [23][24][25][26][27][28][29][30] and cross references.
According to the existing literature, no attempt has been made to investigate the electrically conducting Eyring-Powell fluid with radiation, thermal relaxation time, and joule heating effects beyond an exponentially stretched sheet with chemical reaction. is work visually depicts and tabulates the impacts of various flow parameters encountered in the governing equations. e SLM technique is used to solve the issue numerically, which is more computationally efficient. e relevant results are graphed and quantitatively analyzed. is research fills a void in the literature and lays the groundwork for future researchers to contribute their perspectives to the open literature. is is structured as follows: Section 1 contains the literature survey; Section 2 contains the mathematical formulation; Section 3 contains the methodology; Section 4 has the results; and Section 5 contains the conclusion.

The Problem's Formulation
Consider an incompressible Powell-Eyring fluid flowing across an exponentially stretched surface subjected to magnetic, joule heating, thermal radiation, and thermal relaxation periods, as illustrated in Figure 1. e sheet is put on the x-and y-axes, respectively, and the flow is restricted to y ≥ 0. Let U w (x) � ae (x/l) , represent the sheet velocity, U ∞ � be (x/l) represent the external fluid velocity, and T w (x) � T ∞ + ce (x/2l) represent the surface temperature, with T ∞ being the ambient temperature. e governing equations so obtained are given as (see for example, [13], [21], [22], [31]).
where ], ρ, u(x, y), v(x, y), β, C 0, , T, k, q rad , C p , B 2 International Journal of Mathematics and Mathematical Sciences parameters, temperature, thermal conductivity, thermal radiation, specific heat at constant-pressure, strength of the magnetic field, heat flux, the consternation field, diffusion coefficient, and chemical reaction rate, respectively, which satisfy the relation e appropriate boundary conditions are Using the similarity transformations as follows: e continuity equation is satisfied in the same way using (6), and (2)- (5). is transformed into the following form: Here, λ and Pr denote velocity ratio and Prandtl number, respectively. Where, S c Schmidt number, C R chemical reaction parameter, and Γ are the dimensionless fiuid parameters. Since Γ is a function of x, therefore, we use a local similarity solution of (8)-(10) that allows us to analyze parameter behavior. For K � 0, we have the case of a Newtonian fluid. e C f and the local Nu are mathematically described as follows: Here, τ w and q w are mathematically described as follows: e mathematical form of the local Nusselt number and skin friction coefficient are given as follows: where the local Reynolds numbers are

Solution Methodology
Bhatti et al. [22] solved a non-Newtonian model known as the Powell-Eyring fluid model using the collocation approach. Rahimi et al. [23] addressed this model numerically by using a sequential linearization approach and the Chebyshev spectral collocation method. Agrawal and Kaswan [24] solved the Eyring-Powell fluid model using a fourthorder precision methodology (BVP4C) and the homotopy analysis method (H.A.M). Jafarimoghaddam [25] studied the Eyring-Powell model and described fluid flow and heat transfer over a stretching sheet. He then solved the governing PDEs by using homotopy perturbation and homotopy analysis methods to convert them to ODEs. e thirdorder nonlinear ordinary differential equations (7) and the second-order nonlinear ordinary differential equations (8) are expressed as differential equations and solved using the successive linearization technique (SLM) [26,31] in this article.

Procedure of Computational.
SLM is used to find the numerical solutions for the nonlinear systems (8)-(10) that conform to the boundary condition (11). We choose the initial guess functions for the SLM solution, i.e., f(η), θ(η) and ϕ(η) are in the form Here, the two functions f i (η) and θ i (η) are representative unknown functions. F m (η), m ≥ 1, θ m (η), m ≥ 1 are successive approximations, which are obtained by recursively solving the linear part of the equation that results from substituting (15) in the governing equations. e mean idea of the SLM is that the assumption of unknown function f i (η), θ i (η), and ϕ i (η) are very small when i becomes larger; therefore, the nonlinear terms in f i (η), θ i (η), and ϕ i (η), and their derivatives are considered to be smaller and thus neglected. e intimal guess functions F o (η), θ i (η), and ϕ i (η), which are selected to satisfy the boundary conditions Which are taken to be in the form Table 1 illustrates the convergence for the numerical values of the skin friction coefficient, the local Nusselt number, and the local Sherwood number for various values of the parameters involved in using SLM,

Numerical Scheme Testing.
Here, we test the validity of our numerical results and contrast them with those of published works as limiting examples. As a result, we compare the current results to those obtained in reference [13], and we discover that they are in reasonable agreement, as shown in Table 2.

Result and Discussion
e velocity ratio parameter, the fluid parameter k, the magnetic parameter M, the nondimensional fluid parameter, and the velocity profile are all monitored for variation. Additionally, this section discusses the influence of the Prandtl number Pr, the velocity ratio parameter, the fluid parameter k, the Eckert number Ec, the radiation parameter Rd, the thermal relaxation time T, and the magnetic parameter M on the dimensionless temperature θ(η). Lastly, this section shows the effect of the velocity ratio parameter, the fluid parameter k, the magnetic parameter M, the Schmidt number S c , and the chemical reaction parameter C R on the dimensionless concentration ∅(η). Two types of boundary layers near the sheet have evolved in a flow with exponentially changing free stream velocity over an exponentially stretched sheet. is means that they are dependent on the velocity ratio parameter b/a, for values of b/a greater than or equal to one. Additionally, it's worth noting that when b/a � 1, no velocity boundary layer arises near the  sheet. e velocity pro les for various values are depicted in Figure 2. e in uence of the uid parameter K on the velocity is seen in Figure 3. A rise in K can be interpreted as either a fall in viscosity or a decline in the Powell-Eyring uid's rheological e ects. Here, we see that velocity and the thickness of the boundary layer are rising functions of K when λ < 1. is observation leads to the conclusion that the increase in the elastic e ects of the Powell-Eyring uid leads to a thinner momentum boundary layer. However, an opposite trend is noticed when λ > 1. increasing K results in a drop in uid viscosity, which results in an increase in velocity. Additionally, as K increases, the viscosity of the uid becomes lower due to which the increase in the velocity of the uid accrues. e velocity pro le declines as Γ grows but changes toward the border, indicating that the boundary layer's thickness has decreased, which is depicted in Figure 4. As the magnetic eld intensity increases, the velocity pro le in Figure 5 drops. is is because an increase in the Lorentz force creates resistance to uid ow, resulting in a drop in the velocity pro le. e uctuation of the velocity ratio parameter on the temperature pro le is depicted in Figure 6. e temperature is discovered to be a decreasing function of λ. is data may imply that a greater sheet velocity results in a thicker thermal boundary layer. As K increases, there is a slight reduction in temperature, as illustrated in Figure 7. Due to the lack of viscous dissipation e ects, the uid parameter K is not explicitly included in the energy calculation, and hence has a reduced e ect on the thermal boundary layer. Figure 8 illustrates the e ect of Pr on temperature θ(η). e temperature pro le falls as Pr μC p /k increases. Additionally, rising values of Pr decreases the thickness of the thermal boundary layer. As a result, heat travels rapidly, leading to a decrease in uid temperature.  International Journal of Mathematics and Mathematical Sciences e influence of radiation on temperature distributions can be seen in Figure 9. Increases in Rd result in an increase in heat fluxes from the sheet, which results in a rise in temperature. Ec ′ s effect on the temperature profile θ(η) is depicted in Figure 10. As the Ec value grows, the sheet's wall temperature increases. Due to the fact that when Ec is high, the rate of heat transfer at the surface is low, and the thickness of the thermal boundary layer increases. Frictional heating happens at the surface, raising the fluid's temperature. e effect of thermal relaxation time c on the temperature profile is illustrated in Figure 11. Temperature and thermal relaxation time have been found to have an inverse connection. Physically, when we increase the pressure, the fluid elements have to work harder to transfer heat to their neighboring components, resulting in a temperature drop. When c � 0, heat rapidly spreads throughout the fluid. Figure 12 illustrates the effects of the magnetic parameter M on the temperature profile. When K increases, there is a slight reduction in concentration, as seen in Figure 13. e fluid parameter K is not explicitly included in the energy calculation since there are no effects of viscous dissipation, which reduces its impact on the concentration boundary layer. Figure 14 depicts the effect of the magnetic field M on dimensionless concentration. e increase in M is thought to raise the concentration profile. Figure 15 shows how the velocity ratio parameter varies in relation to the concentration profile. It is shown that the concentration decreases as it increases. According to these findings, a thicker concentration boundary layer is produced by a higher sheet velocity. e effect of the Schmidt number S c on dimensionless concentration is shown in Figure 16. It is seen that as the Schmidt number S c increases, the concentration falls. Figure 17 shows how the chemical reaction C R affected the   concentration profile. e concentration decreases as the C R of the chemical reaction rises. e local Nusselt number is listed in Table 2, and was estimated using the SLM. In Table 3, the skin friction coefficient increases as k increases. As a result, as Γ increases, the coefficient of friction on the skin lowers. According to Mushtaq et al. [13], on an exponentially stretched surface, the magnitude of the skin friction coefficient decreases significantly as the velocity ratio grows. It has already been noted that when k grows, the thermal boundary layer's thickness decreases. As a result, the heat transfer rate at the stretching sheet is increased. Additionally, as Γ grows, the size of the local Nusselt population decreases dramatically. Additionally, it increases as the values of k and λ increase.

Concluding Remarks
In this article, the numerical solution for thermal transport in the Powell-Eyring model via generalized heat flux over an exponentially stretching sheet with a chemical reaction is obtained. By resolving expressions for velocity, temperature, and concentration distributions, the SLM approach is utilized to numerically solve the flow equations. e impact of the Powell-Eyring fluid parameter K, magnetic parameter M, Eckert number Ec radiation parameter Rd, thermal relaxation time c, and chemical reaction was investigated and presented in tables. e validity of the current results was tested, and they were contrasted with those that had previously been published [13]. Table 2 shows a limited example where there is strong agreement. e study's most important features are listed as follows (i) e velocity increases as the fluid parameter K is increased, while reverse behaviour is noticed for the temperature profile. (ii) For increasing values of the magnetic parameter M, the velocity profile falls while the temperature rises. In addition, as the resistance to flow increases, the magnetic field intensity and K increase. (iii) e temperature and thickness of the thermal boundary layer are inversely related to the thermal relaxation time c, whereas the Eckert number Ec has the opposite trend. With an increase in Ec, the temperature within the channel rises. (iv) Increasing values of the Rd (radiation parameter) increase the heat fluxes from the surface, which will cause an increase in the fluid's temperature and velocity. (v) Simulations of local Nusselt number are verified with published work.