**3. Results and Discussion**

The homotopy analysis method (HAM) is an analytical procedure which is employed for solving the nonlinear coupled DEs. From its introduction in 1992 [80], HAM has been heavily used by investigators for solving the nonlinear coupled ODEs. The wide range of uses and applications of HAM are because of its convergence properties and initial guess wide range [71,81,82]. The procedure that HAM follows is based on the transformation <sup>Ψ</sup>˜ : *<sup>X</sup>*<sup>ˆ</sup> <sup>×</sup> [0, 1] <sup>→</sup> *<sup>Y</sup>*ˆ, where *<sup>X</sup>*<sup>ˆ</sup> and *<sup>Y</sup>*<sup>ˆ</sup> are the topological spaces. The linear operators are defined as follow:

$$L\_{\hat{f}}(\hat{f}) = \hat{f}^{\prime\prime\prime},\ L\_{\hat{\mathfrak{F}}}(\hat{\mathfrak{g}}) = \hat{\mathfrak{g}}^{\prime\prime},\ L\_{\emptyset}(\hat{\theta}) = \hat{\theta}^{\prime\prime},\ \text{and}\ L\_{\hat{\mathfrak{G}}}(\hat{\mathfrak{g}}) = \hat{\mathfrak{G}}^{\prime\prime}.\tag{25}$$

We have employed HAM in this study for solving Equations (15)–(19). The achieved results are depicted through different graphs and the effects of related parameters over the Carreau fluid hydromagnetic behavior are investigated and explained in detail. Furthermore, the present study results are compared with the published work and the agreement ascertains the accuracy of HAM.

The dependence of *f* (*η*) (gradient in the velocity x-component) and *g*(*η*) (velocity ycomponent) on augmenting magnetic parameter *M* are respectively depicted in Figure 2a,b. The values of *M* used in this Figure are = 0.5, 1.0, 1.5, 2.0. It is clear from Figure 2a, that at fixed *M*, *f* (*η*) declines with the rising *η*. The decline in *f* (*η*) is much faster at smaller *η* values. Furthermore, the increasing magnetic parameter *M* values result in a downfall in the *f* (*η*) profile. It is obvious from the Figure 2a that reduction in the *f* (*η*) profile is more visible in the range *η* = 0.4 to *η* = 2.6. The downfall in the *f* (*η*) profile may be associated with the augmenting Lorentz forces due to the enhancing *M*, which causes to reduce the non-uniformity in the fluid velocity. Figure 2b shows that the velocity *g*(*η*) changes inversely with the rising *η* at fixed *M*. The velocity field augments with the enhancing *M*. The enhancing behavior of *g*(*η*) with uplifting *M* is more dominant upto *η* = 2.6. Thus, the augmenting Lorentz forces due to rising *M* accelerate the fluid flow.

The impact of the Hall parameter (*m*) on the velocity gradient *f* (*η*) and velocity *g*(*η*) is displayed respectively in Figure 3a,b. The different values of *m* used in this computation are 0.5, 1.0, 1.5, 2.0. It is evident from Figure 3a, that initially *f* (*η*) augments and then drops with the increasing *η* values at fixed Hall parameter value. It is further observed that the enhancing *m* results in the increase of the *f* (*η*) profile. The enhancing behavior of *f* (*η*) is more apparent from *η* = 0.4 to *η* = 2.4. The variation of the velocity *g*(*η*) with enhancing *m* is displayed in Figure 3b. It can be seen from this figure that at smaller *m*, the velocity drops with augmenting *η*. As the value of *m* is increased to *m* = 1.5 and *m* = 2.0, the trend in the *g*(*η*) profiles changes. Now initially the velocity increases, reaches to a maximum, and then declines with the increasing *η*. The variation in the velocity profiles is more dominant at smaller *η* as can be seen from the figure. The increasing trend with the

augmenting *m* is due to the higher Hall potentials produced in the fluid which augment the fluid velocity as well as the gradient in the velocity.

**Figure 2.** (**a**) *f* (*η*) variation with *M*, and (**b**) *g*(*η*) dependence on *M*.

**Figure 3.** (**a**) *f* (*η*) dependence on *m* and (**b**) *g*(*η*) variation with *m*.

Figure 4a,b show the variations of *f* (*η*) and *g*(*η*) with varying Weissenberg number (*We*). The values of *We* used in the present computation are 0.30, 0.50, 0.70, 0.90. From these two figures, it is observed that both *f* (*η*) and *g*(*η*) display almost similar decreasing trend with the increasing *We*. Thus, it is clear that the enhancing viscous nature of the Carreau fluid associated with the rising *We* constricts the fluid flow and hence reduces the fluid velocity.

**Figure 4.** (**a**) *f* (*η*) dependence on *We* and (**b**) *g*(*η*) dependence on *We*.

The variation of *f* (*η*) with shrinking parameter () and the index of power law (*n*) is depicted respectively in the Figure 5a,b. The values of used are 4.0, 5.0, 6.0, 7.0, while those of *n* are 1.6, 2.2, 2.5, 2.8. It is observed from Figure 5a that at fixed , *f* (*η*) first drops, reaches to minimum and then enhances with increasing *η*. Similarly, the *f* (*η*) profiles first drop and then augment with enhancing . Thus, due to suction ( > 0) during the Carreau fluid flow, the *f* (*η*) profiles rise beyond *η* = 1. Beyond *η* = 3.8, all the curves for different overlap with one another. Figure 5b shows the variation of *f* (*η*) with changing values of the power law index *n*. The Figure shows that, at fixed *n*, *f* (*η*) augments with higher *η* values. The rate of enhancement in *f* (*η*) is larger for lower *η* values in comparison with larger *η*. By enhancing the values of *n*, the *f* (*η*) profiles drop. The spacing between *f* (*η*) curves at different *n* increases with rising *η* as clear from the Figure.

**Figure 5.** (**a**) *f* (*η*) dependence on and (**b**) dependence of *f* (*η*) on *n*.

The variation in *f* (*η*) and *φ*(*η*) (fluid concentration) with the enhancing values of *χ*<sup>1</sup> are displayed respectively in Figure 6a,b. The different values of *χ*<sup>1</sup> used are *χ*<sup>1</sup> = 0.25, 0.50, 0.75, 0.95. From Figure 6a, it is observed that at a given *χ*1, *f* (*η*) changes inversely with *η*. The rate of decline of *f* (*η*) is faster at lower *η* in comparison with larger *η* values. Furthermore, by augmenting the values of *χ*1, the *f* (*η*) profiles drop to smaller values. The spacing between the curves for different *χ*<sup>1</sup> reduces with larger values of *χ*1. The different curves overlap beyond *η* = 4.0. The concentration field (*φ*(*η*)) variation with changing *χ*<sup>1</sup> is depicted in Figure 6b. It can be seen that at fixed *χ*1, the concentration field drops with enhancing *η*. The rate of decline of *φ*(*η*) is much larger at smaller *η* values. An enhancement in the *φ*(*η*) profiles is observed with the rising *χ*1. The rate of enhancement of *φ*(*η*) is larger for the larger *χ*1. The *φ*(*η*) curves for different *χ*<sup>1</sup> overlap with one another beyond *η* = 3.6.

**Figure 6.** (**a**) *f* (*η*) variation with *χ*<sup>1</sup> and (**b**) dependence of *φ*(*η*) on *χ*1.

The variation of the fluid temperature *θ*(*η*) with increasing *Pr* (Prandtl number) and *Rd* (radiation parameter) is displayed in Figure 7a,b. The values of *Pr* are taken as 7.0, 10.0, 13.0, 16.0, while those of *Rd* are taken as 1.0, 2.0, 2.5, 3.0. Figure 7a shows that the Carreau fluid temperature drops with the rising *η* at fixed *Pr*. The rate of decrease of *θ*(*η*) is much faster at smaller *η*. As the *Pr* values are increased, the temperature field profiles drop. The spacing between *θ*(*η*) curves is more prominent at the intermediate values of *η*. The drop in the fluid temperature with the enhancing Prandtl number is due to the smaller thermal diffusivity of the Carreua fluid, which causes a reduction in the temperature of the fluid. Figure 7b depicts the dependence of the temperature field on *Rd*. It can be observed that the fluid temperature augments with the rising *Rd* values. The rate of enhancement in *θ*(*η*) with increasing *Rd* is more drastic for smaller *η* values. The *θ*(*η*) curves overlap beyond *η* = 4.0. The augmenting fluid temperature with the higher *Rd* is due to the stronger heat source.

**Figure 7.** (**a**) *θ*(*η*) dependence on *Pr* and (**b**) *θ*(*η*) dependence on *Rd*.

The influence of augmenting values of Weissenberg number (*We*) and Dufour number (*Du*) on *θ*(*η*) is displayed respectively in Figure 8a,b. The *We* and *Du* values are taken as 0.3, 0.5, 0.7, 0.9 and 0.25, 0.45, 0.65, 0.95, respectively. It is clear that *θ*(*η*) declines with the augmenting *η* at constant *We*. The temperature field profiles of the fluid upsurge with the enhancing *We* values. This means that the augmenting viscous nature of the Carreau fluid associated with the rising *We* enhances the fluid temperature. The different *θ*(*η*) curves overlap beyond *η* = 4.0. Figure 8b displays the variation of the Carreau fluid temperature with the enhancing *Du* (Dufour number). It is observed that the fluid temperature profiles rise with the enhancing *Du* values. Thus, the accumulation of the fluid particles due to increasing *Du* raises the fluid temperature.

**Figure 8.** (**a**) *θ*(*η*) dependence on *We* and (**b**) *θ*(*η*) dependence on *Du*.

Figure 9a depicts the fluid concentration *φ*(*η*) with varying Schmidt number (*Sc*). From this Figure, we can observe that *φ*(*η*) changes inversely with rising *η* at constant *Sc*. As *Sc* changes from 0.10 to 0.40, 0.70, and 0.90, a decreasing behavior in the *θ*(*η*) profiles is seen. The different curves overlap beyond *η* = 3.6. The higher value of *Sc* is analogous to smaller value of mass diffusivity, that causes the concentration of the Carreau fluid to drop as can be seen from the Figure. The dependence of *φ*(*η*) on the increasing *Sr* (Soret number) is plotted in Figure 9b. The fluid concentration declines with augmenting *η* at fixed *Sr*. An increase is observed in the fluid concentration with the enhancing *Sr*. As *Sr* is related with the Carreau fluid temperature gradient, hence, higher *Sr* denotes greater temperature difference, which causes an enhancement in the concentration.

**Figure 9.** (**a**) *φ*(*η*) dependence on *Sc* and (**b**) dependence of *φ*(*η*) on *Sr*.
