Impacts of Chemical Reaction and Suction/Injection on the Mixed Convective Williamson Fluid past a Penetrable Porous Wedge

The impacts of chemical reaction and suction/injection in Williamson fluid flow along a porous stretching wedge are discussed in the present paper. Recently, a lot of numerical and theoretical studies are accessible for illustrating the chemical reaction impact on non-Newtonian fluids with different geometries and conditions. Considering this fact, we inspect the heat transport behavior of Williamson fluid due to a stretching porous wedge with suction or injection. The governing PDEs are converted into ODEs with reliable similarity transformation. These ODEs are solved numerically by BVP4C based in MATLAB system, and the assessment of outcomes for the validation tenacity is presented in Table. Combined plots are sketched to discern the impact of dominant sundry parameters on the flow fields. Along with them, the Sherwood number, skin friction factor, and the rate of heat transfer are also bestowed in graphs.


Introduction
Numerous authors chose to work with non-Newtonian uids rather than Newtonian uids until they understood the applications of non-Newtonian models [1][2][3] in the era of bio sciences, material processing, food industries, ceramic products, wire coating, lubricants, polymeric liquids, detergents, engineering and petroleum industries, etc. Several constitutive models with various non-Newtonian models are presented in [4]. e Williamson model is a non-Newtonian uid rstly introduced by Williamson [5]. e governing equations of this model describe the pseudo-plastic features of the uids which piqued the interest of many researchers to work on Williamson uid with di erent geometries and physical conditions. For instance, Malik et al. [6] searched the numerical results of Williamson uid over a stretched cylinder in the presence of heat generation absorption and variable thermal conductivity. Kumaran et al. [7] investigated the melting heat transfer phenomenon in MHD radiative ow of Williamson uid in the existence of non-uniform heat source/sink. Hayat et al. [8] utilized modi ed Darcy's law on the ow of Williamson uid in a channel. ey achieved the solution using built-in ND-solver command in Mathematica software. Heat and mass transfer characteristics in 3D Williamson-Casson uid past a stretching sheet had been highlighted by Raju et al. [9]. Noreen et al. [10] measured the performance of heat in electro-osmotic Williamson uid past a microchannel.
Meanwhile, Subbarayudu et al. [11] revealed the time dependent assessment of radiative blood ow of Williamson uid against a wedge. RK 4th order with shooting method was utilized by them to nd the solution of the governing equations. Hussain et al. [12] highlighted the homogeneousheterogenous reaction on the convective ow of Williamson fluid over sheet and cylinder. Inquiry of variable conductivity, viscosity, and diffusivity on the magneto cross Williamson fluid had been analyzed by Salahuddin et al. [13].
Involvement of blowing or suction past a porous wedge/ surface significantly changes the flow field behavior. Injection or withdrawal of fluid over a permeable bounding wedge/surface is of prodigious concern in everyday problems such as glazing of wires and film presentations, polymer fiber, coating, and so on. During the design of thrust bearing, thermal oil recovery, and radial diffusers, suction or injection plays a vital role. Numerous other researchers have successfully produced and discussed the results [14,15] in the area of Darcian porous wedge/surface.
In chemical reactions, suction is useful to remove reactants, whereas blowing is useful to eliminate reactants, whereas blowing is advantageous to avoid corrosion, add reactants, and cool the wedge or shrink. Zahmatkesh et al. [16] investigated the entropy generation in axisymmetric stagnation flow of nanofluid through a cylinder with constant wall temperature and uniform suction blowing at the surface. Singh et al. [17] have considered the flow of micropolar fluid past a permeable wedge in the presence of Hall, ion slip current, and chemical reaction effects. Analytic solutions were achieved by implementing the differential transform method (DTM). Saleem et al. [18] discussed the blowing suction effects on temperature and velocity distribution of flow past a flat plate. ey showed that factor of drag force enhanced with the increasing values of suction and reduced when blowing is applied. e problem of chemical and diffusion reaction in a isothermal laminar flow along semi-infinite plate has been discussed by Fairbanks and Wike [19]. Ahmed et al. [20] provided the numerical and analytical solution to 3D channel flow in the existence of chemical reaction and sinusoidal fluid injection. Sulochana et al. [21] scrutinized the frictional heating on chemically reacting mixed convective Casson nanofluid past an inclined porous plate with radiation. Zaib et al. [22] made a numerical treatment of second law analysis of magnetocross nanofluid past a wedge with binary chemical reaction and activation energy. Nandi and Kumbhakar [23] scrutinized the chemical reaction and viscous dissipation effects on tangent hyperbolic nanoliquid over a stretching wedge with various conditions. is discussion addressed the shortcoming flow of Williamson liquid bounded above the stretched porous wedge. e chemical reaction phenomenon is explained in this study. BVP4C [24][25][26] built-in MATLAB solver is used to solve the coupled non-linear equation. Impacts of sundry pertinent parameters are explained through graphs. e study of chemically reacting Williamson fluid over a stretchable porous wedge has not been discussed so far. e flow fields are characterized by expanding velocities, skin friction, and Sherwood and Nusselt number graphs. is work aims to provide basic ground for the researchers to explore the flow of the Williamson model over a porous wedge with chemical reaction.
In the current exploration, the contribution is highlighted by the following.
(1) e 2D Williamson fluid over porous stretching wedge is considered. (2) e flow is exposed due to suction or injection.

Model
We investigate 2-dimensional (x, y) mixed convective flow of the Williamson fluid over a wedge. It is presumed that velocity of the possible flow away from boundary layer is of the wedge are fixed and higher than the ambient concentration and temperature (C ∞ , T ∞ ), respectively (see Figure 1). e following assumptions are taken into account.
(i) Steady, laminar, incompressible, and mixed convective flow of the Williamson fluid is considered. (ii) Suction injection is considered at the boundary. (iii) e chemical reaction is also considered. (iv) Buoyancy is present in the leading equations.
Within the background of the above-mentioned deductions, the leading equations are as follows [27,28].
e boundary conditions are 2 Journal of Mathematics e velocity components u and v take the form where stream function ψ de nes e above expression also satis es the continuity equation (1). From (2), (3), and (4), we have the transformed equations 1 Pr e transformed velocity boundary conditions (5) and (6) can be written as Here, primes signify the di erentiation w.r.t η, G rx , λ, S c , and K 1 , these are de ned by

Numerical Method and Verification of Code
Dimensionless equations (10), (11), and (12) corresponding to boundary conditions (13) and (14) have been solved using the BVP4C scheme. e solution for the ow over the wedge is obtained using MATLAB software with Core i7 processor. e owchart is provided in Figure 2. To con rm the validity of the existing numerical system, we have matched the numerical outcomes through those provided by Su et al. [27], Yih [29], and Ishak et al. [30] on the distribution of C₁ for surface drag force of the stretched porous wedge and accomplished an identical decent agreement. Table 1 demonstrates that our consequences are well validated.

Discussion
is section highlights the impact of active parameters like ow parameter m, ratio of mixed convective parameter N, Prandtl and Eckert numbers Pr, Ec, suction injection parameter, velocity ratio parameter R, Schmidt number S c , and chemical reaction parameter K 1 on velocity f ′ , temperature h, concentration g, skin factor, and heat and mass transfer rate, via graphs. Solid and dashed lines represent when wedge stretches faster or slower than free stream velocity. e black, red, and blue lines represent the velocity, temperature, and concentration elds. Figure 3 is plotted to explore the e ect of We on f ′ , h , and g. It is vivid from this gure that the larger values of We increase the temperature and concentration elds and decrease velocity pro le. Figure 4 depicts the impact of m on f ′ , g, and h. From this gure, it is found that g and h are minimized while enriching the values of m, but f ′ lessens at R 1.2 and increases at R 0.8. e graph in Figure 5 shows that N results in decline in g and h, whereas an opposite behavior is noted in f ′ for the values of N. Figure 6 exhibits the in uence of Ec on the three ow elds. At R 0.8, a growing tendency is described for larger values of Ec similar to h and g elds. ere is a decay at R 1.2 in the pro les of velocity. Physically, the Eckert number represents the ow of the kinetic energy relative to the enthalpy di erence through the thermal boundary layer. e e ects of C 1 on f ′ , h, and g are demonstrated in Figure 7. From this gure, it is noted that there is a decay in all the velocity pro les for bigger values of suction injection parameter. e impact of S c for distinct values of R is drawn in Figure 8.
e growing values of S c decrease the concentration pro les. Figure 9 portrays the values of K 1 (chemical reaction parameter). For constructive or destructive values of chemical reaction parameter, g decreases. e variations of m and N of C f against λ 1 are represented in Figure 10. It is found that C f shows an increasing behavior of uplifted values of N, but a reverse behavior is noted for uplifted values of m. Figure 11 shows the in uences of S c and We on C f . It is noted that C f increases for augmented values of We and decreases for Sc values. e combined e ects of m and N on heat transfer coe cient are depicted in Figure 12. From this gure, N u x increases for escalating values of N and m. Figure 13 elucidates N u x for

Conclusions
is study was carried out to investigate the chemical reaction and suction/injection e ects on Williamson  Nomenclature W e : Weissenberg number R: Velocity ratio parameter f ′ : Dimensionless velocity C: Concentration velocity (x, y): Coordinates (m) g: Dimensionless concentration velocity h: Dimensionless temperature velocity (u, v): Velocity component (ms − 1 ) T w : Wall temperature C w : Wall concentration m: Flow parameter T: Temperature of the uid C ∞ : Concentration of free stream T ∞ : Temperature of free stream u w : Stretching velocity (ms − 1 ) υ: Fluid kinematic viscosity C p : Speci c heat (J K − 1 kg − 1 ) g c : Gravitational acceleration S c : Schmidt number k: ermal conductivity of the uid (W K − 1 m − 1 ) C 1 : Suction/injection parameter K 1 : Chemical reaction parameter E c : Eckert number U: Free stream velocity G r : Grashof number R e : Reynolds number µ: Fluid dynamic viscosity (pas) λ: Mixed convection parameter ρ : Density of uid (kg m − 3 ) β 0 : ermal expansion coe cient η : Similarity variable σ: Electrical conductivity of the uid Ω: Wedge angle parameter ODEs: Ordinary di erential equations