*Table Discussion*

The modelled equations with boundary conditions) are solved analytically as well as numerically. The comparison between analytical and numerical solutions is shown graphically as well as numerically in Tables 1–5 for velocities and temperature. From these tables, an excellent agreemen<sup>t</sup> between HAM and Numerical (ND-Solve Techniques) are obtained. The comparison for velocity and temperature profiles for different behavior of fluid between HAM and numerical method are shown in Tables 2–6, which shows an excellent agreemen<sup>t</sup> with HAM solution. The numerical values of *M*, *k*, β and *St* on skin friction *Cf* are given in Table 7. From this table it is clear that increasing values of *M*, *k* and β decrease *Cf* while increasing *We* increase skin friction. The numerical values of the surface temperature θ(β) for the dissimilar value of *M*,*Rd* and *St* are given in Table 8. It is observed that the increasing values of *M*, *Rd* and *k* increase the surface temperature <sup>θ</sup>(β), where opposite effect is found for *Pr*, that is the largest value of *Pr* reduces the surface temperature <sup>θ</sup>(β). The gradient of wall temperature θ(0) for dissimilar values of embedded parameters *Rd*, β, *Pr*, *S* has been shown in Table 8. It is perceived that larger values of Rd, β and *Pr* fall the wall temperature and *St* increase the wall temperature gradient <sup>Θ</sup>(0).


**Table 2.** The relationship between Homotopy Analysis Method (HAM) and Numerical techniques for *f*(η) in case of *n* = 0, when β = ε = 1, *St* = *M* = 0.1, ξ = 0.8.

**Table 3.** Relationship between Homotopy Analysis Method (HAM) and Numerical techniques for *f*(η) in case of *n* = 1, when β = ε = 1, *St* = *M* = 0.1, ξ = 0.8.


**Table 4.** Relationship between Homotopy Analysis Method (HAM) and ND-Solve for *f*(η) in case of *n* = 2, when β = ε = 1, *St* = *M* = 0.1, ξ = 0.8.



**Table 5.** Association between Homotopy Analysis Method (HAM) and Numerical techniques for <sup>θ</sup>(η) in case of *n* = 1, when β = 0.1, ε = 0.2, *St* = 0.5, Pr = 0.5, ξ = 0.3, *M* = 1.

**Table 6.** Association between Homotopy Analysis Method (HAM) and Numerical techniques for <sup>θ</sup>(η) in case of *n* = 1, when β = 0.1, ε = 0.2, *St* = 0.5, *Pr* = 0.5, ξ = 0.3, *M* = 1.


**Table 7.** Coefficient of Skin friction for dissimilar values of *M*, *k*, β and *S*.



**Table 8.** Values of θ(β) dissimilar values of *M*, *Pr*,*Rd* and *S*.
