Physical Aspects of Homogeneous-Heterogeneous Reactions on MHD Williamson Fluid Flow across a Nonlinear Stretching Curved Surface Together with Convective Boundary Conditions

(is article is concerned with the fluid mechanics of MHD steady 2D flow of Williamson fluid over a nonlinear stretching curved surface in conjunction with homogeneous-heterogeneous reactions with convective boundary conditions. An effective similarity transformation is considered that switches the nonlinear partial differential equations riveted to ordinary differential equations. (e governing nonlinear coupled differential equations are solved by using MATLAB bvp4c code. (e physical features of nondimensional Williamson fluid parameter λ, power-law stretching index m, curvature parameter K, Schmidt number Sc, magnetic field parameter M, Prandtl number Pr, homogeneous reaction strength k1, heterogeneous reaction strength k2, and Biot number c are presented through the graphs. (e tabulated form of results is obtained for the skin friction coefficient. It is noted that both the homogeneous and heterogeneous reaction strengths reduced the concentration profile.


Introduction
e fluid is subdivided into two main categories: non-Newtonian fluid and Newtonian. One of its types of non-Newtonian fluid is a shear-thinning (pseudoplastic) fluid [1]. Pseudoplastic takes attention due to its large commercial applicability. Polymer solutions as well as molten polymers, complex fluids, and suspensions like nail polish, whipped cream, blood, ketchup, and paint are the industrial and everyday applications of pseudoplastic fluids. Gogarty [2] considered the porous media to study the rheological properties of pseudoplastic fluids. Researchers use different models to investigate the behaviour of non-Newtonian fluid like the Ellis model, Williamson model, cross model, Carreaus model, and the power-law model, but the Williamson model for fluid flow takes more attention for the study of pseudoplastic fluid. Williamson [3] provides an experimentally verified model for the analysis of pseudoplastic fluids. In the last decade, several researchers [4][5][6][7][8][9] investigated the behaviour of pseudoplastic fluid by using the Williamson fluid model. Hayat et al. [10,11] used the Homotopy analytical method to examine the impact of joule heating, thermal radiation, and Ohmic dissipation in the two-dimensional flow of Williamson fluid over a stretching surface.
From the last two decades, several investigators have focused on non-Newtonian fluid across nonlinear and linear stretching of a plate, flat surface, cylinder, or disk [12][13][14][15][16]. Flow across a curved surface is firstly introduced by Sajid et al. [17]. Later on, Abbas et al. [18] analyzed the heat transfer flow of MHD fluid across stretching curved surface. Ahmad et al. [19] examined the boundary layer flow across a curved surface embedded in a porous medium. Sanni et al. [20] investigated the flow of viscous fluid due to a nonlinear stretching curved surface. e effect of mass and heat transfer across a curveshaped surface is numerically examined by Ramana et al. [21]. Saleh et al. [22] investigated the flow of unsteady micropolar fluid flow over a permeable curved stretching/shrinking surface. e transfer of heat and mass of an electrically conducting micropolar fluid with MHD effect across a curved stretching sheet is presented by Yasmin et al. [23]. Recently Kamran et al. [24] numerically investigated the flow of Williamson fluid over an exponential curved stretching surface. Gowda et al. [25] used Li (KKL) correlation, Koo-Kleinstreuer, and modified Fourier heat flux model to investigate the nanofluid flow over a curved stretching sheet.
Chemical reactions involve h-h (homogeneous and heterogeneous) reactions in many chemically reacting processes. In homogeneous reactions, reactants and products are in the same phase, whereas the reactants involve two or more phases in heterogeneous reactions. Williamson fluid flow across a stretching cylinder with h-h reactions is studied numerically by Malik et al. [26]. e effect of h-h reactions on boundary layer flow across a nonlinear stretching curved surface with convective boundary conditions is studied by Saif et al. [27]. Khan et al. [28] utilized the concept of h-h reactions on Oldroyd-B fluid flow between stretching disks. Ahmed et al. [29] explored the stagnation flow of h-h reactions in single-walled carbon nanotubes nanofluid towards a plane surface. Ali et al. [30] analyzed the effect of mass and heat transport on cross fluid with h-h reactions to investigate the behaviour of a chemical process. Javed et al. [31] examined the melting heat transfer with radiative effects and h-h reaction in thermally stratified stagnation flow embedded in a porous medium. Sreedevi et al. [32] implemented the FEM to examine the impact of h-h reactions on Maxwell nanofluid with heat and mass transfer flow over a horizontal stretched cylinder.
In Williamson fluid flow problem, researchers [33][34][35][36][37][38][39] used different similarity transformations to change governing nonlinear partial differential equations into ordinary differential equations. e literature survey shows that the Williamson fluid model is more effective than other models for studying pseudoplastic fluid, but the researcher studied locally similar Williamson fluid parameter. e MHD Williamson fluid flow across a curved surface with convective boundary condition and h-h reactions is not studied until now. e objective of this study is to examine the effect of h-h reactions with convective boundary conditions on MHD Williamson fluid over a curved surface. is problem has many applications in engineering processes such as wire drawing, continuous casting, metal extrusion, paper production, glass fibre production, hot rolling, and crystal growing. Anappropriate similarity transformation [20,27] is used to convert governing PDEs to ODEs.By fixing the value of nonlinear index parameterm = 1/3,which provides an entirely similar Williamson fluid parameter. e impact of different parameters, i.e., curvature parameter K, nondimensional Williamson fluid parameter λ, Biot number c, magnetic field parameter M, Schmidt number Sc, Prandtl number Pr, homogeneous reaction strength k 1 , heterogeneous reaction strength k 2 , power-law stretching index m on velocity, pressure, temperature, and concentration profiles, is presented through the graphs, whereas the results of skin friction coefficient and Nusselt number are depicted in a tabulated form.

Formulation of Problem
We examined the 2-dimensional steady, incompressible, fully developed magnetohydrodynamic Williamson fluid flow across a nonlinear stretching curved surface having radius H. We consider (r, s) coordinate system. e s-axis represents the flow direction, whereas the radial direction is taken along r. e stretching of the curved surface is along the s-axis with velocity u � b 1 s m where b 1 is the initial stretching rate. e variable magnetic field B � B 0 s m− 1 is applied in the radial direction. For h-h reactions, we consider two chemical species A and C, respectively. Figure 1 shows the geometry of flow. In the case of Williamson fluid flow, Cauchy stress tensor is defined as S � − pI + τ in which τ represents extra stress tensor and defined as and μ ∞ are the first Rivlin-Erickson tensor; positive time constant; limiting viscosities at zero and infinite shear stress rates, respectively; and _ c is defined as _ c � ��� 1/2 √ π, whereas π � trace(A 1 ) 2 . Here we consider the case in which Γ _ c < 1 and μ ∞ � 0. e homogeneous reaction for cubic autocatalysis with two chemical species A and C is represented by the equation below: (1) For cubic autocatalysis, the heterogeneous reaction on the catalyst surface is mathematically represented as follows: where h c and h s are the rate constants and a and c represent the concentrations for chemical species A and C. Under these conditions, the governing boundary layer equations [9,[17][18][19][20][21][22][23][24][26][27][28][29][30][31][32] are e accompanying boundary conditions are [20,27] where u symbolizes the velocity component in the s direction and ] describes the velocity component in the r direction. Furthermore, ρ represents the density, ] and σ are the kinematic viscosity and the electrical conductivity, respectively. We introduce the following dimensionless variable transformations: where f shows the dimensionless velocity, η shows the similarity variable, and p is the pressure. Using equation (10) in equations (3)-(9), equation (3) satisfies identically, we get Mathematical Problems in Engineering represents the parameter for Williamson fluid; magnetic field parameter is expressed Abolishing P(η) from the equations (11) and (12) produces the following equation: When D A � D C , then δ � 1, we have us equations (14) and (15) take the form Along with boundary conditions e local skin friction coefficient C f and Nusselt number Nu s are defined as Shear stress τ w and heat flux q w near to the surface are mathematically represented as 4 Mathematical Problems in Engineering Using equation (10) in equation (21), equation (20) takes the form where Re s � u w s/].

Solution Procedure
e coupled nonlinear system of ODEs, (11), (13), (17), and (18), along with boundary conditions (16) and (19), are solved by using built-in MATLAB code bvp4c. In order to find out the solution of coupled ODEs, first of all, we rewrite equations (11) and (13), (17) and (18), with boundary conditions (16), (19) as an equivalent system of first-order differential equations by using the substitutions f � z 1 , f ′ � z 2 , f ″ � z 3 , f ″′ � z 4 , P � z 5 , θ � z 6 , θ ′ � z 7 , ϕ � z 8 , and ϕ ′ � z 9 . In the next step, we code these systems of first-order ODEs and the boundary conditions with function names "ex1ode" and "ex1bc" in MATLAB. Furthermore, we choose the interval of integration from 0 to 4 and divided it into 30 mesh points which formed the initial guess structure. Mesh selection and error control are built on the residual of the continuous solution. e relative error tolerance considered in this study is 10 − 6 . Finally, we call "bvp4c" function sol � bvp4c(@exlode, @ex1bc, solinit, options).
e "deval" built-in MATLAB function is used to evaluate the solution at a specific point. e Comparison of skin friction coefficient from the present study with the work of Kumar et al. [38] and Sajid et al. [17] by fixing λ � K � M � k 1 � k 2 � c � Pr � Sc � 0 and m � 1 shows the acceptable result in Table 1. We assumed m � 1/3, M � 1.5, Wb � 0.2, Pr � 7.0, Sc � 0.6,k 1 � 0.5, k 2 � 0.5, g � 0.3, and K � 5 in all discussions. ��� Re s C f and the decrease in the value of − θ (0). By raising the value of Pr and c, − θ (0) is rising for both parameters, but − C f remains unchanged because the velocity profile is independent of these parameters. e effect of K on f ′ (η) is observed in Figure 2(a). e graph exhibits the obvious results that, by an increase in the value of K, the radius of the surface enhances and hence boosts the velocity of the fluid. Figures 2(b)-2(d) display the changes of θ(η), P(η), and ϕ(η) profiles, for increasing radius of curvature K. It is seen that there is a decrease in θ(η), P(η), and ϕ(η) for larger values of K. is is because rising curvature makes the curved surface flat. e influence of M on f ′ (η), θ(η), P(η), and ϕ(η) is shown in Figures 3(a)-3(d). It is noted that fluid velocity, pressure, and concentration for higher values of M reduced. However, the temperature profile increases for greater values of M. Practically, this effect is shown when the magnetic field is applied perpendicular to the flow direction, which creates resistance for the fluid flow. Figures 4(a)-4(d) illustrate the effect of nonlinearity parameter m on f ′ (η), θ(η), P(η), and ϕ(η) profiles. It is easy to discern that the velocity, temperature, and pressure decrease, whereas concentration profile increases by increasing m which is as expected practically.   [17] and [38].

K
Ref. [17] Ref. [    Mathematical Problems in Engineering 9 observed for higher values of Pr. Hence for conducting fluids, Pr is responsible for enhancing the cooling rate. Figure 7(a) shows the effect of homogeneous parameter k 1 on concentration distribution ϕ(η). From this figure, we observe that the concentration profile decreases as there is a rise in k 1 . Hence it is concluded that reaction rate dominates diffusion coefficients. Figure  7(b) shows the strength of a heterogeneous reactionk 2 onϕ(η). e plot shows that the concentration of the fluid and associated boundary layer thickness is reduced for higher values of k 2 . Figure 8 illustrates the effect of Sc on ϕ(η). Sc is the ratio of momentum to mass  diffusivity. An increase in Sc means momentum diffusivity is dominated,resulting an increment in the concentration profile.

Conclusions
In this article, we have modelled the MHD flow of Williamson fluid across a nonlinear stretching curved surface with h-h reactions and convective boundary conditions. e governing partial differential equations are adapted into ordinary differential equations by using suitable similarity transformation.
e impact of involving parameters on pressure, velocity, concentration, and temperature profiles are examined. Some examples of this problem in biochemical science and engineering processes are blood flow, plasma flow, lubrication flow, wire drawing, continuous casting, metal extrusion, paper production, glass fibre production, hot rolling, crystal growing, etc. e main consequences are noted below: Data Availability e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.  Mathematical Problems in Engineering 11