Modelling Entropy in Magnetized Flow of Eyring–Powell Nanofluid through Nonlinear Stretching Surface with Chemical Reaction: A Finite Element Method Approach

The present paper explores the two-dimensional (2D) incompressible mixed-convection flow of magneto-hydrodynamic Eyring–Powell nanofluid through a nonlinear stretching surface in the occurrence of a chemical reaction, entropy generation, and Bejan number effects. The main focus is on the quantity of energy that is lost during any irreversible process of entropy generation. The system of entropy generation was examined with energy efficiency. The set of higher-order non-linear partial differential equations are transformed by utilizing non-dimensional parameters into a set of dimensionless ordinary differential equations. The set of ordinary differential equations are solved numerically with the help of the finite element method (FEM). The illustrative set of computational results of Eyring–Powell (E–P) flow on entropy generation, Bejan number, velocity, temperature, and concentration distributions, as well as physical quantities are influenced by several dimensionless physical parameters that are also presented graphically and in table-form and discussed in detail. It is shown that the Schemit number increases alongside an increase in temperature, but the opposite trend occurs in the Prandtl number. Bejan number and entropy generation decline with the effect of the concentration diffusion parameter, and the results are shown in graphs.


Introduction
The non-Newtonian fluids have wide application in engineering, technology, various sciences, the processing and devolatilization of polymers, and industrial processes. They also find essential use in processes within the chemical industry. Examples of non-Newtonian fluids include: paint, blood, liquid crystals, the thixotropic-like ink in a Fisher Space Pen, grease or lubricants, the gravity acting to provide lubrication under the shear force fluids, etc. The different types of non-Newtonian fluids can be categorized by consenting nonlinear stress-strain constitutive models like the Casson model [1], Carreau-Yasuda model [2], power-law model [3], cross model [4] and Maxwell model [5], but these types of models do not describe the individual features of rheological fluids. Eyring-Powell fluid is one type of non-Newtonian fluid with contained plasticity in its accumulation viscosity. Several researchers have deliberated the importance of non-Newtonian fluids [6][7][8][9][10].
Eyring and Powell first considered Eyring-Powell fluid [11] in the year 1944. The amount of energy is defined as entropy (an irretrievable loss of thermal energy) rather than characterizing the total system as entropy generation. It is concluded that the thermal   ∂u ∂x with consistent boundary conditions are After substituting the above transformation and identically satisfying Equation (1), Equations (2)-(6) are reduced as 1 Pr Here, flow variables are defined as magnetic parameter is M = σB 2 0 /ρa ; ratio between concentration and thermal buoyancy force is The physical quantities are heat surface, mass transfer and drag force and can be represented as Here, τ w , q w and q m are given as The non-dimensional parameters are the skin friction coefficient (C f ), Nusselt number (Nu x ) and Sherwood number (Sh x ) taken as: Here Re x = ax 2 υ is a Reynolds number.

Entropy Generation
The entropy generation contains four factors: Joule dissipation, heat transfer, mass transfer and viscous dissipation. The entropy generation volumetric rate of viscous fluid for the magnetic and electric fields are obtained as [43][44][45][46]. The entropy equation for the Eyring-Powell fluid follows as where the dimensionless volumetric entropy rate of generation owing the fluid friction and heat transfer of the form Here, the dimensionless parameters represented as the temperature difference parameter is α 1 (= ∆T/T ∞ ) = (T w − T ∞ )/T ∞ ; the concentration difference parameter is α 2 (= ∆C/C ∞ ) = (C w − C ∞ )/C ∞ ; the diffusion parameter is L 1 (= R * D(C w − C ∞ )/k); local entropy generation is N G (= T ∞ S G υ/k∆T); and Brinkman number is Br = µa 2 x 2 /k∆T . The Bejan number (Be) is formed as

FEM Solution
The third order dimensionless differential equation is transformed into a second order dimensionless differential equation by using F = h. the dimensionless differential Equations (7)-(9) are written in residual form. These residuals are multiplied with weighted functions and integrated with a typical two node element (η e , η e+1 ) given by where w t , (t = 1, 2, 3, 4) are weight functions and the variational functions are taken as F, h, θ, Φ, respectively. The unknown functions are approached by Galerkin approximations. The finite element model Equations (17)- (20) are achieved by substituting of finite element approximation form.
The unknown nodal values are F j , h j , θ j , Φ j and the linear shape function for ψ j , a typical line element (η e , η e+1 ) are presented as, The model of the finite element equation is expressed as The matrices of [K mn ] and [b m ](m, n = 1, 2, 3, 4) are defined as Φψ i ψ j dξ , where The whole domain is divided into 500 linear elements which are of equivalent sizes. It is solved iteratively. The three functions F , θ, Φ are examined at each node. The assumed known functions are F, h, θ and Φ, and are used for the linearized system. The velocity, temperature and concentration are set as equal to one for the first iteration. This process is repeated until the accuracy value is 10 −5 . The convergence results are calculated as the number of elements for n = 10, 20, 40, 80, 160, 320, 400 and 500. The convergent results are shown in Table 1.

Results and Discussion
Here, the aspects of the non-dimensional governing equations and the corresponding boundary conditions are solved numerically using the finite element method (FEM). The numerical solution and results that are carried out for the influence of non-dimensional flow .08 on Bejan number (Be); entropy generation (N G ); velocity (F (ξ)); temperature (θ(ξ)); and concentration (Φ(ξ)) distributions. Numerical results of skin friction (C f ), Nusselt number (Nu x ), and Sherwood number (Sh x ) are presented in Table 2. Figure 2a-c express the influence of a magnetic parameter (M = 1.1, 3.2, 6.3, 9.4) on velocity (F (ξ)), temperature (θ(ξ)), concentration (Φ(ξ)) distributions. In Figure 2a it is observed that the velocity boundary layer thickness decreases with increased values of the magnetic parameter. This occurs when the expanding values of the magnetic parameter magnify the Lorentz force which resists the fluid motion and intensity in the velocity profile. Figure 2b,c shows that the behavior of a magnetic parameter on temperature and concentration distributions increases with the increased values of the magnetic parameter.   Impact of N * on velocity (F (ξ)) and concentration (Φ(ξ)) distributions are captured in Figure 3a,b. For the higher approximation of N * (0.2, 2, 4, 6) the velocity distribution increases and the concentration distribution declines, meaning that both types work in opposing ways. The mixed convection parameter λ(3, 5, 10, 15) influence on temperature distribution (θ(ξ)) is revealed in Figure 4a. It is observed that an enlarged mixed convection parameter causes an identical slight decline in the temperature distribution. The dimensionless fluid parameter's impact on velocity distribution (F (ξ)) leads to a higher estimation of the non-dimensional fluid parameter ε(0.1, 1.0, 1.5, 2.0). How to enlarge the velocity distribution is shown in Figure 4b.
Impact of * N on velocity ( ( )) F ξ ′ and concentration ( ( )) ξ Φ distributions are tured in Figure 3a,b. For the higher approximation of * (0.2,2,4,6) N the velocity di bution increases and the concentration distribution declines, meaning that both ty work in opposing ways. The mixed convection parameter (3,5,10,15) λ influence temperature distribution ( ( )) θ ξ is revealed in Figure 4a. It is observed that an enlar mixed convection parameter causes an identical slight decline in the temperature di bution. The dimensionless fluid parameter's impact on velocity distribution ( ( F ′ leads to a higher estimation of the non-dimensional fluid parameter (0.1,1.0,1.5, 2 ε How to enlarge the velocity distribution is shown in Figure 4b.    Impact of * N on velocity ( ( )) F ξ ′ and concentration ( ( )) ξ Φ distributions are c tured in Figure 3a,b. For the higher approximation of * (0.2,2,4,6) N the velocity dis bution increases and the concentration distribution declines, meaning that both ty work in opposing ways. The mixed convection parameter (3,5,10,15) λ influence temperature distribution ( ( )) θ ξ is revealed in Figure 4a. It is observed that an enlar mixed convection parameter causes an identical slight decline in the temperature dis bution. The dimensionless fluid parameter's impact on velocity distribution ( ( F ′ leads to a higher estimation of the non-dimensional fluid parameter (0.1,1.0,1.5, 2. ε How to enlarge the velocity distribution is shown in Figure 4b.   The influence of (N b ) is analyzed in Figure 5a,b for several values of the Brownian motion parameter (N b = 10, 20, 30, 40) on temperature distribution (θ(ξ)), and declines in concentration distribution (Φ(ξ)) that need to be increased. The Brownian motion parameter plays a key role on the surrounding liquids during the heat transfer. Figure 6a,b depicts that the increased values of the thermophoresis parameter (N t = 3, 5, 7, 10) are produced in temperature (θ(ξ)) and concentration (Φ(ξ)) distributions. An enhancement of thermophoresis parameters causes an increment in the force of thermophoresis. This is a decline in temperature (θ(ξ)) and concentration (Φ(ξ)) distributions. clines in concentration distribution ( ( )) ξ Φ that need to be increased. The Brownian motion parameter plays a key role on the surrounding liquids during the heat transfer. Figure   6a,b depicts that the increased values of the thermophoresis parameter ( 3,5,7,10)       The formulation of the Schmidt number is the ratio of the viscous diffusivity rate to the molecular diffusivity rate. The increasing values of the Schmidt number lowers the rate of molecular diffusion and sources the diffusion of higher density species in air. Here, we achieve the results of an increase in temperature distribution (θ(ξ)) and a decrease in fluid concentration distribution (Φ(ξ)). Figure 8a shows that the increasing values of the Eckert number (Ec = 0.2, 0.4, 0.6, 0.8) with the concentration distribution (Φ(ξ)) have declined.
number lowers the rate of molecular diffusion and sources the diffusion of higher densit species in air. Here, we achieve the results of an increase in temperature distributio ( ( )) θ ξ and a decrease in fluid concentration distribution ( ( )) ξ Φ . The temperature distribution described in Figure 8b shows that the greatest value of the Prandtl number decrease the temperature distribution ( ( )) θ ξ due to the invers properties of the thermal and Prandtl numbers. Figure 9a,b signifies the influence of th chemical reaction parameter. The important note is that the enhanced homogeneou chemical reaction parameter enhances the velocity ( ( ))  The temperature distribution described in Figure 8b shows that the greatest values of the Prandtl number decrease the temperature distribution (θ(ξ)) due to the inverse properties of the thermal and Prandtl numbers. Figure 9a,b signifies the influence of the chemical reaction parameter. The important note is that the enhanced homogeneous chemical reaction parameter enhances the velocity (F (ξ)) and concentration (Φ(ξ)) distributions. The temperature distribution described in Figure 8b shows that the greatest values of the Prandtl number decrease the temperature distribution ( ( )) θ ξ due to the inverse properties of the thermal and Prandtl numbers. Figure 9a,b signifies the influence of the chemical reaction parameter. The important note is that the enhanced homogeneous chemical reaction parameter enhances the velocity ( ( ))      The increased values of L 1 thatt increase the entropy generation (N G ) and Bejan number (Be) decrease one surface of frequency after increasing another frequency. This is because a higher irreversible diffusion parameter increases the rate of nanomaterials. Entropy generation (N G ) decreases with the impact of an increased fluid parameter (δ). Table 2 describes the numerical values of skin friction (C f Re 1/2 x ), Nusselt number (Nu x Re −1/n+1 x ) and Sherwood number (Sh x Re −1/n+1 x ) for various physical parameter effects. The measured skin friction increases for the mixed convection parameter (λ), fluid parameter (δ), buoyancy force parameter (N * ) and chemical reaction parameter (C h ) and decreases for the fluid parameter (ε), thermophorasis (N t ) and Brownian motion parameter (N b ). The Nusselt number increases for λ, δ, N * , and the reverse process occurs with ε, N t , N b , C h . The Sherwood number decreases for δ, ε, N t and increases with the mixed convection parameter (λ), buoyancy force parameter (N * ), Brownian parameter (N b ) and chemical parameter (C h ).

Conclusions
The key points of this study are listed below. I The Bejan number (Be) shows a decreasing influence for larger values of the dimensionless parameter (L 1 ) and concentration diffusion parameter (α 2 ).