To scrutinize the flow features of the second-grade fluid along with the effect of the magnetic field, radiative heat flux, viscous dissipation, and a double diffusion (Dufour and Soret) solution of Equations (9)–(11) is essential. Before getting to the solution of the coupled system, it is highly important to look at the nature and complexity of it. After sighting the system, it is clear that the momentum equation mentioned in Equation (9) is a third order equation, whereas Equations (10)–(11), i.e., the temperature and concentration equations, are second-order equations. So, there are various approaches (finite difference, finite element and finite volume) to get to the solution of these equations. Among them the best is the finite difference approach for such type of problems, so the Keller Box differencing scheme is implemented.

Figure 1,

Figure 2 and

Figure 3 are displayed to portray the behavior of the velocity distribution against the relative parameters. To see the impacts of viscoelastic parameter

$\beta $ on second grade velocity field SVF,

Figure 1 is plotted. Variation in velocity against (

$\beta )$ is measured for

$\beta =0.0,0.2,0.4,0.6$. An increasing trend in velocity is sketched against an increasing

$\left(\beta \right)$. This behavior is justified by the mathematical representation of

$\left(\beta \text{}=\frac{{\alpha}_{1{U}_{o}}}{\rho \nu l}\right)$ that by increasing the magnitude of

$\left(\beta \right)$, viscosity decreases as a result of the velocity of the fluid mounts and chaotic behavior uplifts. Here, it is productive to mention that for

$\beta =0.0$, the present problem reduces to the Newtonian case. From a boundary layer point of view, the thickness of the fluid increases with an increase in

$\left(\beta \right)$.

Figure 2 indicates the incrementing behavior of velocity against inciting the magnitude of the curvature parameter

$(\gamma =0.0,0.2,0.4,0.6,$). The reason is that by the increasing curvature parameter, bending of the surface as well as the radius decreases. Hence, less friction will be offered to fluid molecules by the surface and velocity uplifts. A declining trend in velocity is depicted against the Darcy parameter (Da) in

Figure 3. Since the Darcy parameter (Da) represents the presence of a porous medium, suggests porosity creates a high resistance to fluid molecules and as an outcome velocity declines. Variation in the thermal profile against influencing parameters are disclosed in

Figure 4,

Figure 5 and

Figure 6.

Figure 4 is manifested to excogitate the impression of radiation parameter (

R) on the thermal field. An increasing pattern in temperature is observed by choosing

R = 0.5, 1.0, 1.5, 2.0 and by fixing

$\gamma =0.1,\mathrm{Pr}=1,Du=0.5,Ec=0.1$. It is because of the fact that energy flux increases and consequently fluid temperature increases. In

Figure 5, the effect of the thermal Biot number (

${B}_{i1}$) on

θ(

η) is presented. By definition, the thermal Biot number is directly related to the heat transfer coefficient generated by the hot fluid. Thus, as the thermal Biot number increases, the convection due to the hot fluid raises and the temperature mounts. Impacts of the Dufour (Du) and Soret (Sr) aspects on the thermal field is disclosed in

Figure 6. In the present graph, the values of (Du) and (Sr) are selected in such a way that their product will give a constant magnitude. By growing the Dufour number (i.e., by declining the Soret number) the thermal change between the hot and ambient fluid increases, which enhances the temperature.

Figure 7,

Figure 8 and

Figure 9 are plotted to predict the changes in the concentration profile with respect to the involved parameters like

$\gamma ,{B}_{i2},DuandSr$.

Figure 7 presents the impression of the curvature parameter

γ on the concentration profile. Duality in concentration features are interpreted against the curvature parameter. For small values of

η, i.e., (

η < 2), the concentration decreases whereas it increases when

η > 3. The upshot of the concentration Biot number

$({B}_{i2})$ on

ϕ(

η) is captured in

Figure 8. From the drawn curves it is portrayed that by increasing the

$({B}_{i2})$ concentration field the associated boundary layer thickness increases. The joint conspiration of (Sr) and (Du) on the concentration field is divulged in

Figure 9. It is interpreted that by decreasing (Sr) and increasing (Du) the strength of the intermolecular forces weakens and as a consequence the concentration field declines.

Table 1,

Table 2 and

Table 3 are enumerated to record the variation in the Nusselt number (

$Nu)$, Sherwood number

$\left(Sh\right)$ and coefficient of skin-friction

$\left({C}_{f}\right)$ for numerous values of involved parameters. Excellent correlation is noticed between our results and findings made by Hayat et al. [

31]. This comparison between attained figures assures the credibility of the current work.