**4. Results and Discussion**

Results and discussion provide the analysis of the problem through the impacts of all the relevant parameters. The non-dimensional Equations (20), (22), (28) and (30) with boundary conditions in Equations (25), (26) and (29) are analytically computed. The performances of different parameters on the velocity profiles with heat and concentration of homogeneous-heterogeneous chemical reactions are shown in the relevant graphs. The streamlines show the internal behaviors of flow. The physical representation of the problem is shown in Figure 1. Liao [61] introduced *h*¯-curves for the convergence of the series solution to get the precise and convergent solutions of the problems. *h*¯-curves are also called the convergence controlling parameters for solution in the homotopy analysis method (used for solution in the present case). These *h*¯-curves specify the range of numerical values. These numerical values (optimum values) are selected from the valid region in straight line. These optimum values of *h*¯-curves are selected from the straight lines parallel to the horizontal axis (please see carefully Figures 2–5) to control the convergence of problem solution. In the present case, the valid region of each profile *h*¯-curve is specified. Therefore, the admissible *h*¯-curves for *f*(*ζ*), *g*(*ζ*), *θ*(*ζ*) and *ϕ*(*ζ*) are drawn in the ranges −10.00 ≤ *h*¯ *<sup>f</sup>* ≤ −4.00, −10.00 ≤ *h*¯ *<sup>g</sup>* ≤ −5.00, −3.5 ≤ *h*¯ *<sup>θ</sup>* ≤ −2.50 and −1.50 ≤ *h*¯ *<sup>ϕ</sup>* ≤ −0.50 in Figures 2–5, respectively.

**Figure 2.** Illustration of the ¯*h<sup>f</sup>* -curve of *f*(*ζ*).

**Figure 3.** Illustration of the ¯*hg*-curve of *g*(*ζ*).

**Figure 4.** Illustration of the ¯*h<sup>θ</sup>* -curve of *θ*(*ζ*).

**Figure 5.** Illustration of the ¯*hϕ*-curve of *ϕ*(*ζ*).
