numerical results and discussions

Numerical solutions to the dimensionless Eqs. (9) – (13) are obtained using the Runge-Kutta-Gills method with the shooting technique and the embedded parameters are varied to study their effects on the dimensionless velocity \(\left(f^{{}^{\prime}}\right)\), temperature\(\left(\theta\right)\) and concentration \(\left(\phi\right)\)functions. Numerical results are obtained for the local skin-friction coefficient \(f^{{}^{\prime\prime}}\left(0\right)\), local Nusselt number\(-\theta^{{}^{\prime}}\left(0\right)\), local Sherwood number\(-\phi^{{}^{\prime}}\left(0\right)\) and results displayed in Tables (1) – (3).

Effects of various parameters on velocity profiles

The influence of the flow parameters on velocity profiles are illustrated in Figures (1) to (6). Figure 1 (a-b) shows that Dufour number \(\left(D_{f}\right)\) has similar effect on the velocity profile of both Blasius flow and Sakiadis flow. It is observed that the velocity profile decreases with increase in \(D_{f}\) for both the Blasius and the Sakiadis flows. As can be seen from Figure 2 (a-b), Eckert number \(\left(\text{Ec}\right)\) has opposite effects on the velocity profiles of Blasius and Sakiadis flows. For Blasius flow, the velocity profile decreases with increase in Ec whereas, the velocity profiles increase in Sakiadis flow with increase inEc. Effect of the local Grashof number\(\left(Gr_{x}\right)\) on the velocity profiles is shown in Figure 3 (a-b) and it is observed that the velocity profiles increase with increase in the Grashof number \(\left(\text{Gr}\right)\) for both the Blasius flow and the Sakiadis flow. In Figure 4 (a-b), we observed that increase in the local magnetic field parameter \(\left(M_{x}\right)\)causes a decrease in the velocity profile for both the Blasius flow and the Sakiadis flow due to the presence of Lorentz force which acts against the flow and Schmidt number due to low molecular diffusivity. Figure 5 (a-b) shows the influence of Prandtl number\(\left(\Pr\right)\) on the velocity profiles and it is observed that the velocity profile increases with increase in \(\Pr\) for both the Blasius flow and the Sakiadis flow. The influence of Soret number\(\left(\text{Sr}\right)\) is illustrated in Figure 6 (a-b). Soret number \((Sr)\) has opposite effects on the Blasius and Sakiadis flows such that velocity profile decreases with increase in Sr for the Blasius flow but increase for the Sakiadis flow.