## 3 Numerical Approach: SHAM
The contribution of the viscoelastic term (\(\beta\)) (i.e., Weissenberg numeric) is plotted in \[F_{L}=[f_{l}(\xi_{0},\ldots),f_{l}(\xi_{N}),\theta_{l}(\xi_{0},\ldots),\theta_ {l}(\xi_{N}),\theta_{l}(\xi_{0}),\ldots,\phi_{l}(\xi_{N})]^{T}\]3 The viscoelastic parameter \(\beta\) describes the influence of the coefficient of normal stress on the motion. Figure 10a portrays the contribution of the viscoelastic term on the velocity graph. It is detected from the graph that at any moment in the flow regime, an increase in \(\beta\) produces a drastic degeneration in the velocity of the fluid, whereas we observed in this study that a higher value of the viscoelastic term has no contribution to the temperature and concentration plot as depicted in Figure 10b,c. The result in Figure 10a signifies that a higher value of \(\beta\) has the tendency to decrease the hydrodynamics layer thickness. \[F_{L}=[f_{l}(\xi_{0},\ldots),f_{l}(\xi_{N}),\theta_{l}(\xi_{0},\ldots),\theta_ {l}(\xi_{N}),\theta_{l}(\xi_{0}),\ldots,\phi_{l}(\xi_{N})]^{T}\]4 Effect of Soret parameter Sr on (**a**) velocity, (**b**) temperature, and (**c**) concentration profiles. We plotted the contribution of the heat absorption/generation term (\(\alpha\)) on the velocity, concentration, and temperature distributions in \[F_{L}=[f_{l}(\xi_{0},\ldots),f_{l}(\xi_{N}),\theta_{l}(\xi_{0},\ldots),\theta_ {l}(\xi_{N}),\theta_{l}(\xi_{0}),\ldots,\phi_{l}(\xi_{N})]^{T}\]5 It is detected from Figure 11a,b that a higher \(\alpha\) produces an acceleration in the fluid velocity and temperature. This signifies that temperature together with velocity in the boundary layer increases when \(\alpha>0\) while the reverse is the case when \(\alpha<0\). Thus, when \(\alpha>0\), more heat is generated and the temperature within the layer is enhanced. The contribution of the thermal Grashof number (Gt) is elucidated in \[F_{L}=[f_{l}(\xi_{0},\ldots),f_{l}(\xi_{N}),\theta_{l}(\xi_{0},\ldots),\theta_ {l}(\xi_{N}),\theta_{l}(\xi_{0}),\ldots,\phi_{l}(\xi_{N})]^{T}\]6 Gt defined the ratio of the force of buoyancy to that of viscous material affecting the liquid. The thermal Grashof number can also be called the thermal buoyancy force term. As expected, in Figure 12a. the fluid velocity elevates as the thermal buoyancy force parameter is intensified. Furthermore, from Figure 12b, the temperature field decreases with a higher value of thermal Grashof. It is interesting to observe a slight degeneration in the concentration field when Gt is increased. \[F_{L}=[f_{l}(\xi_{0},\ldots),f_{l}(\xi_{N}),\theta_{l}(\xi_{0},\ldots),\theta_ {l}(\xi_{N}),\theta_{l}(\xi_{0}),\ldots,\phi_{l}(\xi_{N})]^{T}\]7 Effect of heat generation parameter \(\alpha\) on (**a**) velocity, (**b**) temperature, and (**c**) concentration profiles. Figure 13 represents the effect of the mass Grashof number (Gm) on the velocity, concentration, and temperature graphs. In Figure 13a, it is discovered that an increment in the Gm number allows a rise in the velocity of the fluid. In the same vein, as seen in Figure 13b,c a higher mass Grashof decreases both the temperature and concentration. \[F_{L}=[f_{l}(\xi_{0},\ldots),f_{l}(\xi_{N}),\theta_{l}(\xi_{0},\ldots),\theta_ {l}(\xi_{N}),\theta_{l}(\xi_{0}),\ldots,\phi_{l}(\xi_{N})]^{T}\]8 Effect of thermal Grashof number Gt on (**a**) velocity, (**b**) temperature, and (**c**) concentration profiles. Table 1 shows the numeric local skin friction, local Nusselt (Nu), and local Sherwood number (Sh). For distinct values of the viscoelastic parameter (\(\beta\)) and thermal Grashof number (Gt), they produce a notable elevation in the skin friction, local Nusselt, and local Sherwood numbers, respectively. In Table 2, we present the numeric local skin friction, local Nusselt, and Sherwood numbers for distinct values of the heat generation parameter (\(\alpha\)) and Soret number (Sr). The results in Table 2 revealed that an increase in both heat generation \(\alpha\) and the Soret number causes an increase in the skin friction, Nusselt, and Sherwood numbers.