**5. Results and Discussion**

The MATLAB built-in function bvp4c was applied to integrate the numerical solution for the system of Equations (14), (15), and (17) with initial and boundary conditions, Equations (16) and (18), for numerous values of *K*1, *M*, *Rd*, λ1, and Sc graphically. For this technique, we first changed differential equations with the higher order to the equations of order one by utilizing new variables. The function bvp4c needs an initial guess for the solution and with the tolerance of 10−7. The guess we chose needed to satisfy the boundary conditions (Equations (16) and (18)) and the solution. The validation of our presented results is depicted in Table 2. An excellent agreement with Sanni et al. [23] was observed when *M* = 1, ϕ = 0.0, and in the absence of temperature and concentration profile.


**Table 2.** Comparison of presented results for skin friction coefficient <sup>1</sup> <sup>2</sup>*Cf*(Re*x*) 1 <sup>2</sup> when *M* = 1 and ϕ = 0.0.

Figures 2 and 3 show the impression of solid volume fraction ϕ on velocity and temperature profiles. Both velocity fields increased with increasing values of solid volume fraction ϕ. Further, the momentum and thickness of the thermal boundary layers were boosted with a larger value of ϕ. The values given to other parameters were *Pr* = 10, S*<sup>c</sup>* = 0.5, *K*<sup>1</sup> = 10, *Rd* = 0.5, θ*<sup>w</sup>* = 0.5, λ<sup>1</sup> = 0.5, and *M* = 0.1.

**Figure 2.** Upshot of ϕ on velocity distribution *f* **(**ζ**)**.

**Figure 3.** Upshot of ϕ on temperature field θ**(**ζ**)**.

The impact of the curvature parameter *K*<sup>1</sup> on velocity, concentration, and temperature profiles are depicted in Figures 4–6. Increasing values of *K*<sup>1</sup> resulted in an increase in fluid velocity and temperature field, while the concentration profile diminished. This was because of the radius of the surface augment when curvature parameter *K*<sup>1</sup> was increased. As a result, the flow increased but it offered more resistance, therefore the temperature rose. The values of other parameters were fixed as *Pr* = 10, S*<sup>c</sup>* = 0.5, ϕ = 0.1, *Rd* = 0.5, θ*<sup>w</sup>* = 0.5, λ<sup>1</sup> = 0.5 and *M* = 0.1.

**Figure 4.** Upshot of *K*<sup>1</sup> on velocity profile *f* **(**ζ**)**.

**Figure 5.** Upshot of *K*<sup>1</sup> on temperature profile θ**(**ζ**)**.

**Figure 6.** Upshot of *K*<sup>1</sup> on concentration field *h***(**ζ**)**.

Figure 7 demonstrates the variation in the velocity field for numerous estimates of magnetic parameter *M*. Here, increments in *M* led to a decline in the magnitude of fluid's velocity. This was because of the resistive force (called Lorentz force) triggered by the magnetic field, which lowered the velocity of the fluid's velocity flow. The values of the other parameters were fixed as *K*<sup>1</sup> = 10, Sc = 0.5, ϕ = 0.1, *Rd* = 0.5, θ*<sup>w</sup>* = 0.5, λ<sup>1</sup> = 0.5, and Bi = 0.1.

**Figure 7.** Upshot of *M* on velocity profile *f* **(**ζ**)**.

The characteristics of Biot number Bi and heat generation/absorption parameter λ<sup>1</sup> on temperature field are displayed in Figures 8 and 9, respectively. Figure 8 shows that the convective heat transfer coefficient intensified for higher estimates of Bi and the temperature subsequently rose. Figure 9 illustrates the behavior of λ1. To increase the estimation values of heat absorption/generation parameter, the temperature profile and thermal boundary layer thickness were increased. The values of other parameters were fixed as *K*<sup>1</sup> = 10, *M* = 0.3, ϕ = 0.5, *Rd* = 0.1, θ*<sup>w</sup>* = 0.5, and *Pr* = 10.

**Figure 8.** Upshot of Bi on temperature field θ**(**ζ**)**.

**Figure 9.** Upshot of λ<sup>1</sup> on temperature distribution θ**(**ζ**)**.

Figures 10 and 11 show the impacts of nonlinear radiation parameter *Rd* and Prandtl number Pr on temperature distribution, respectively. It can be seen that the temperature field fell with increasing Prandtl number Pr. As Prandtl number is linked in a reciprocal way to the thermal diffusivity, a quick augmentation in the Prandtl number Pr lessened the temperature and thickness of the thermal boundary layer. The temperature profile increased with increment in nonlinear radiation parameter *Rd*. Physically, the radiative heat flux increased with increasing values of *Rd* which ultimately boosted

the temperature of the fluid. The values assigned to other parameters were *K*<sup>1</sup> = 10, *M* = 0.3, ϕ = 0.5, Bi = 0.1, θ*<sup>w</sup>* = 0.5, and Sc = 5.

**Figure 10.** Upshot of *Rd* on temperature field θ**(**ζ**)**.

**Figure 11.** Upshot of Pr on temperature field θ**(**ζ**)**.

The impression of Schmidt number Sc on concentration distribution is portrayed in Figure 12. A decrease in concentration field was detected with increasing values of Sc. As the Schmidt number has a converse proportion with the Brownian diffusion coefficient, an increment in the Sc yielded a decay in Brownian diffusion coefficient that brought about a diminishment in concentration and its interrelated boundary layer thickness. The values allocated to other parameters were *K*<sup>1</sup> = 10, *M* = 0.3, ϕ = 0.1, Bi = 0.1, θ*<sup>w</sup>* = 0.5, and *Rd* = 0.1.

**Figure 12.** Upshot of Sc on concentration profile *h***(**ζ**)**.

The influence of curvature parameter *K*<sup>1</sup> and magnetic parameter *M* on skin friction coefficient −1 2*Cf*Re1/2 *<sup>x</sup>* is depicted in Figure 13. It can be noticed that the surface drag force diminished with increasing value of *K*1. A contradictory trend was demonstrated in case of *M*. In Figure 14, the consequence of magnetic parameter *M* and solid volume fraction ϕ on shear wall stress is demonstrated. The skin friction profile rose with increase in magnetic parameter *M* and solid volume fraction ϕ for fixed values of parameters *Pr* = 10, Sc = 0.5, *Rd* = 0.5, θ*<sup>w</sup>* = 0.5, λ<sup>1</sup> = 0.5 and Bi = 0.1.

**Figure 13.** Upshot of *<sup>K</sup>*<sup>1</sup> and *<sup>M</sup>* on wall shear stress <sup>−</sup><sup>1</sup> <sup>2</sup>*Cf* Re1/2 *<sup>x</sup>* .

**Figure 14.** Upshot of *<sup>M</sup>* and <sup>ϕ</sup> on wall shear stress <sup>−</sup><sup>1</sup> <sup>2</sup>*Cf* Re1/2 *<sup>x</sup>* .

Figure 15 shows the effect of Biot number Bi and solid volume fraction ϕ on Nusselt number Nu*x*(Re*x*) − 1 <sup>2</sup> . It was detected that for higher value of Bi and ϕ, the surface heat transfer rate upsurged when values of parameters were given as *K*<sup>1</sup> = 10, *M* = 0.3, Sc = 5.0, *Rd* = 0.1, θ*<sup>w</sup>* = 0.5, λ<sup>1</sup> = 0.5 and *Pr* = 10.

**Figure 15.** Upshot of Bi and on ϕ Nusselt number NuRe−1/2 *<sup>x</sup>* .

The outcome of curvature parameter *K*<sup>1</sup> and nonlinear radiation parameter *Rd* on Nusselt number Nu*x*(Re*x*) − 1 <sup>2</sup> is examined in Figure 16. Here, a reduction in Nusselt number was noted for increasing values of curvature parameter *K*<sup>1</sup> and the opposite trend was seen for nonlinear radiation parameter *Rd* for fixed values of ϕ = 0.1, *M* = 0.3, Sc = 5.0, θ*<sup>w</sup>* = 0.5, λ<sup>1</sup> = 0.5 and *Pr* = 10.

**Figure 16.** Upshot of *<sup>K</sup>*<sup>1</sup> and *Rd* on Nusselt number NuRe−1/2 *<sup>x</sup>* .

The impact of the Brinkman number (Br) on entropy generation is portrayed in Figure 17. From the illustration, it can be seen that entropy optimization was boosted with increasing values of (Br). The reason behind this was that more heat was generated between the layers of the fluid because of augmented values of (Br). Figure 18 displays the relationship between the magnetic parameter (*M*) and the entropy generation. Again, the same trait as depicted in case of (Br) was witnessed here. Higher values of (*M*) meant stronger Lorentz force and ultimate strengthening of the dissipation energy, and this was the main cause of irreversibility.

**Figure 18.** Upshot of *M* on *NG* (η).

Table 3 shows the behavior of Sherwood number Sh*x*(Re*x*) − 1 <sup>2</sup> for varied values of Sc (Schmidt number), *K*<sup>1</sup> (curvature parameter), and *M* (magnetic parameter). It can be seen that for snowballing values of Sc, the Sherwood number Sh*x*(Re*x*) − 1 <sup>2</sup> increased; however, it diminished for increasing value of *K*<sup>1</sup> and *M*.

**Table 3.** Numerical value of Sherwood number Sh*x*(Re*x*) − 1 <sup>2</sup> for various value of parameter with fixed value of *Pr* = 10, *Rd* = 0.1, ϕ = 0.1, Bi = 0.1, θ*<sup>w</sup>* = 0.5, λ<sup>1</sup> = 0.5 *f* **(**ζ**)**.

