## 4 Discussion  
In this work, numerical values are assigned to the physical parameters involved in the velocity, temperature, and concentration profiles. The numerical outcomes for velocity, temperature, and concentration profiles are presented in this section. An efficient numerical method called the ND-solve method has been used to solve the transformed Equations (8)-(10) subject to the boundary conditions in Equation (11). The paper examined the effects of governing parameters on the transient velocity profile, temperature profile, and concentration profile. For this purpose the SRM approach has been applied for various values of flow controlling parameters \(S=0.7\), \(P_{r}=10\), \(D_{u}=0.7\), \(S_{r}=0.7\), \(S_{c}=0.7\), \(\Lambda=0.7\), \(\lambda=0.7\) to obtain a clear insight into the physics of the problem. Therefore, all the graphs and tables correspond to the values above and the rest will be mentioned. The behavior of the non-dimensional unsteady parameter \(S\) for velocities, temperature, and concentration field during fluid motion is studied in Figures 7-9. The unsteady parameter \(S\) is inversely related to the stretching constant of the velocity field, whereas it is directly related to the stretching constants of the temperature and concentration fields. Therefore, by increasing the values of \(S\) the value of the velocity field is decreased while the values of the temperature and concentration fields increase. An increase in \(P_{r}\) leads to an increase in kinematic viscosity and a decrease in velocity. The reason is that the rise in viscosity tends to increase the resistance force and as a result the velocity profile descends (Figure 10). Figure 11 shows the effect of Prandtl number \(P_{r}\) in temperature fields; the same effect is observed for velocity fields. The thermal diffusion falls with the rise in Prandtl number \(P_{r}\) and as a result the thermal boundary layer becomes thinner and the temperature decreases. This variation in thermal diffusivity is due to the difference of temperature fields; the fluid is highly conductive. Therefore, a fluid with greater \(P_{r}\) and larger heat capacity increases the heat transfer, the same as in [21]. This variation in thermal diffusivity is due to the difference of temperature fields. The same effect for concentration field is exposed in Figure 12. The behavior of Dufour number \(D_{u}\) is discussed in Figures 13-15. The Dufour number is actually the ratio of temperature and concentration difference. The Soret effect is a mass flux due to a temperature gradient and the Dufour effect is enthalpy flux due to a concentration gradient and appears in the energy equation. It was also observed that the effect of \(D_{u}\) and \(S_{r}\) on the temperature and concentrationfields is opposite. In Figure 13 it is shown that increasing the value of Dufour number \(D_{u}\) decreases the velocity profile. Since the Dufour number \(D_{u}\) has an inverse relationship with thermal diffusion, we conclude that the falls in fluid velocity are due to the smaller thermal diffusion. However, it is clear in Figure 14 that the temperature field increases for greater values of \(D_{u}\). Physically, the Dufour effect has a direct relationship with the concentration gradient of energy flux and, as a result, temperature increases for larger values of \(D_{u}\). The concentration field decreases with increasing values of the Dufour number \(D_{u}\), as shown in Figure 15. The Soret number is the reciprocal ratio of the Dufour number; due to this property the reverse physical behavior of the Soret \(S_{r}\) and Dufour numbers \(D_{u}\) has been noticed in the concentration field and is shown in Figure 16. The effects of Schmidt number \(S_{c}\) on the temperature field and concentration field are discussed in Figures 17 and 18, respectively. Figure 17 exhibits the effect of Schmidt number on temperature fields: an increase in the value of \(S_{c}\) increases the temperature field.