**3. Solution Methodology**

By considering suitable boundary conditions on the system of equations, a numerical solution is developed using NDSolve in Mathematica. Shooting method is used via NDSolve. This method is very helpful in case of small step-size featuring negligible error. As a consequence, both *x* and *y* varied uniformly by a step-size of 0.01 [40].

#### **4. Graphical Results and Discussion**

Effects of magnetic parameter *M*, homogeneous-reaction strength *K*, Schmidt number *Sc*, Schmidt number (for heterogeneous–homogeneous reactions) *Scb*, thermophoresis parameter *Nt*, heterogeneous-reaction strength *Ks*, Prandtl number Pr and Brownian motion parameter *Nb* on concentration *φ* (*ζ*), concentration rate *r* (*ζ*) and temperature *θ* (*ζ*) for both second grade and elastico-viscous fluids are sketched in Figures 2–12.

Figure 2 depicts impact of magnetic parameter *M* on temperature *θ* (*ζ*). Here *M* = 0 is for hydromagnetic flow situation and *M* = 0 corresponds to hydrodynamic flow case. Temperature *θ* (*ζ*) is higher for hydromagnetic flow in comparison to hydrodynamic flow for both second grade and elastico-viscous fluids. Physically magnetic parameter depends upon Lorentz force. Lorentz force is an agent which resists the motion of fluid and therefore temperature *θ* (*ζ*) enhances.

Figure 3 displays variations in temperature *θ* (*ζ*) for increasing Prandtl number Pr. Temperature *θ* (*ζ*) decays for larger Pr for both second grade and elastico-viscous fluids. Physically Prandtl number involves thermal diffusivity. Larger Prandtl number corresponds to weaker thermal diffusivity which produces a decay in temperature *θ* (*ζ*).

Figure 4 depicts impact of Brownian motion parameter *Nb* on temperature *θ* (*ζ*). Larger *Nb* produces an increment in temperature *θ* (*ζ*) for both second grade and elastico-viscous fluids. Larger Brownian motion parameter *Nb* has stronger Brownian diffusivity and weaker viscous force which increased the temperature *θ* (*ζ*).

Figure 5 shows that larger thermophoresis parameter *Nt* leads to higher temperature *θ* (*ζ*) for both second grade and elastico-viscous fluids. Larger *Nt* causes strong thermophoresis force which tends to shift nanoparticles from hot to cold zone and therefore temperature *θ* (*ζ*) increases.

Impact of magnetic parameter *M* on concentration *φ* (*ζ*) is displayed in Figure 6 Concentration *φ* (*ζ*) is upgraded for increasing estimations of *M* for both second grade and elastico-viscous fluids. Furthermore, the concentration *φ* (*ζ*) shows similar trend for both second grade and elastico-viscous fluids.

Figure 7 depicts that concentration *φ* (*ζ*) is decreased for larger Schmidt number *Sc* for both second grade and elastico-viscous fluids. Schmidt number *Sc* has an inverse relation with Brownian diffusivity. Larger Schmidt number leads to weaker Brownian diffusivity which produces weaker concentration *φ* (*ζ*).

Impact of Brownian motion *Nb* on concentration *φ* (*ζ*) is shown in Figure 8 Bigger *Nb* produces a reduction in concentration *φ* (*ζ*) for both second grade and elastico-viscous fluids. Physically Brownian force tries to push particles in opposite direction of concentration gradient and make nanofluid more homogeneous. Therefore, higher the Brownian force, lower the concentration gradient and more uniform concentration *φ* (*ζ*).

Figure 9 displays that how thermophoresis *Nt* affects concentration *φ* (*ζ*). Here concentration *φ* (*ζ*) is upgraded for higher estimations of *Nt* for both second grade and elastico-viscous fluids. Furthermore, the concentration *φ* (*ζ*) shows similar trend for both second grade and elastico-viscous fluids.

Figure 10 displays that how Schmidt number *Scb* affects concentration rate *r*(*ζ*). Here concentration rate *r*(*ζ*) is upgraded for higher estimations of Schmidt number *Scb* for both second grade and elastico-viscous fluids. Furthermore, the concentration rate *r*(*ζ*) shows similar trend for both second grade and elastico-viscous fluids.

From Figure 11 it is noted that larger homogeneous-reaction *K* displays a decay in concentration rate *r*(*ζ*) for both second grade and elastico-viscous fluids. Larger homogeneous-reaction *K* corresponds to higher chemical reaction which consequently decreases the concentration rate *r*(*ζ*).

Figure 12 depicts that larger heterogeneous-reaction *Ks* produces higher concentration rate *r*(*ζ*) for both second grade and elastico-viscous fluids. Here heterogeneous-reaction parameter *Ks* has an inverse relation with mass diffusivity which produces an enhancement in concentration rate *r*(*ζ*).

Table <sup>1</sup> displays skin-friction <sup>−</sup>*Cf*Re1/2 *<sup>x</sup>* subject to varying *k*<sup>∗</sup> <sup>1</sup> and *M*. Here skin-friction has higher estimations for larger *M* for both second grade and elastico-viscous fluids. Table 2 depicts comparison for various estimations of *k*∗ <sup>1</sup> with homotopy analysis method (HAM). Table 2 presents a good agreement of numerical solution with existing homotopy analysis method (HAM) solution in a limiting sense. Table 3 depicts local Nusselt number *Nux*Re−1/2 *<sup>x</sup>* subject to varying *k*<sup>∗</sup> <sup>1</sup>, *Nb* and *Nt*. Here larger *Nb* and *Nt* correspond to lower local Nusselt number for both second grade and elastico-viscous fluids. Table 4 shows local Sherwood number *Shx*Re−1/2 *<sup>x</sup>* subject to varying *k*<sup>∗</sup> <sup>1</sup>, *Nb* and *Nt*. Here larger *Nt* produces lower local Sherwood number while opposite trend is noted via *Nb* for both second grade and elastico-viscous fluids.

**Table 1.** Skin-friction coefficient for various estimations of viscoelastic and magnetic parameters.


**Figure 2.** Variations of temperature for magnetic parameter when *Nb* = 0.2, *Nt* = 0.1 and *Sc* = Pr = 1.0.

**Figure 3.** Variations of temperature for Prandtl number when *Nb* = 0.2, *Nt* = 0.1, *Sc* = 1.0 and *M* = 0.2.

**Figure 4.** Variations of temperature for Brownian motion parameter when *Nt* = 0.1, *Sc* = Pr = 1.0 and *M* = 0.2.

**Figure 5.** Variations of temperature for thermophoresis parameter when *Nb* = 0.2, *Sc* = Pr = 1.0 and *M* = 0.2.

 **Figure 6.** Variations of concentration for magnetic parameter when *Nb* = 0.2, *Nt* = 0.1 and *Sc* = Pr = 1.0.

**Figure 7.** Variations of concentration for Schmidt number when *Nb* = 0.2, *Nt* = 0.1, Pr = 1.0 and *M* = 0.2.

**Figure 8.** Variations of concentration for Brownian motion parameter when *Nt* = 0.1, *Sc* = Pr = 1.0 and *M* = 0.2.

**Figure 9.** Variations of concentration for thermophoresis parameter when *Nb* = 0.2, *Sc* = Pr = 1.0 and *M* = 0.2.

**Figure 10.** Variations of concentration rate for Schmidt number (for heterogeneous–homogeneous reactions) when *K* = 0.2, *Ks* = 0.5 and *M* = 0.2.

**Figure 11.** Variations of concentration rate for homogeneous-reaction strength when *Scb* = 1.0, *Ks* = 0.5 and *M* = 0.2.

**Figure 12.** Variations of concentration rate for heterogeneous-reaction strength when *K* = 0.2, *Scb* = 1.0 and *M* = 0.2.

**Table 2.** Comparative data of skin-friction coefficient for various estimations of viscoelastic parameter when *M* = 0.



**Table 3.** Local Nusselt number for various estimations of viscoelastic, Brownian motion and thermophoresis parameters when *Sc* = Pr = 1.0 and *M* = 0.2.

**Table 4.** Local Sherwood number for various estimations of viscoelastic, Brownian motion and thermophoresis parameters when *Sc* = Pr = 1.0 and *M* = 0.2.

