**3. Results and Discussions**

In this special case of unsteady flow, the slot is moving with constant speed −*Uw* cot(*α*). For *α* = *π*/2, the surface velocity is zero as in the case of Sakiadis flow [22]. For 0 < *α* < *π*/2, the slot is moving with the constant speed *Uw* cot(*α*) in the opposite direction of stretching surface and the situation is termed as leading-edge accretion. For *α* ∈ (*α<sup>L</sup>* , 0) ∪ (*π*/2 , *αU*), the direction of slot motion is same as stretching sheet and the situation is termed as leading-edge ablation. As *α* → 0, the speed of slot approaches infinity in opposite direction to the stretching surface, which correspond to the Rayleigh starting-plate problem. The analytical solution for this case has been obtained using the perturbation method (see Appendix A). Since the exact analytical solution of the system (8)–(11) is not available for general α, we adopt the numerical method for the solution. In Table 1, the comparison of numerical results of skin friction with results of Fang [20] is tabulated. In Table 2, the comparison of the analytical result for *α* = 0 is given with the numerical solution. Tables 1 and 2 establish the reliability of our results.


**Table 1.** Comparison of Fang [20] and Present study for values of different moving slot parameters.

**Table 2.** Comparison of analytical and numerical solutions for Nusselt and Sherwood number for *α* = 0.


The numerical solution domain of *α*, (*α<sup>L</sup>* < *α* < *αU*), for the skin friction and Nusselt number mentioned by Fang [20] also hold for Sherwood number. In this study, we focus on the effects of nanoparticles on the heat transfer and behavior of nanoparticles concentration for the surface accretion and ablation.

Figure 2 demonstrates numerical solutions of velocity profile for various values of slot moving constant *α* ranging between −*π*/4 < *α* < *αU*. In Figure 3 the dual solution for the velocity profile is plotted for *α* = −48◦. The thickness of boundary layer is much greater for lower solution branch as compared to upper solution branch.

**Figure 2.** Velocity profiles for varying slot moving parameter α.

**Figure 3.** Profiles of velocity for different branches at α = −48◦.

Figure 4 represents the dual solution for a fixed value of moving slot parameter *α* = −48◦, with two distinct values of Prandtl number. For the above-mentioned values of parameters, both solutions show maximum temperature gradient which can be viewed in the region away from the wall. The change of heat transfer at the wall is less for lower solution as compared to the upper solution. The thermal layer thickness is greater for lower solution as compare to upper solution branch.

**Figure 4.** Temperature and its flux profiles for several branches at *α* = −48◦ for varying *Pr* with *Nb* = 0.01 and *Nt* = 0.001.

Figures 5 and 6 illustrate the numerical solution domain of reduced Nusselt number as a function of *α* for different values of Brownian and thermophoretic diffusion parameters, *Nb* and *Nt* respectively. For Nusselt number, the correlation expression in the form of *Nb* and *Nt* has also been written by applying the linear regression on the set of 2401 numerical values. The values of coefficients and constant of the correlation expression in the form

$$-\theta'(0) = \mathcal{C} + \mathcal{C}\_B \mathcal{N}\_b + \mathcal{C}\_T \mathcal{N}\_t$$

for *Nb* ∈ (0.01, 0.5) and *Nt* ∈ (0.0, 0.5) is given in Table 3 with maximum percentage error for different Prandtl number and moving slot parameter.

**Figure 5.** Effects of slot moving parameter α on reduced Nusselt number for varying *Nb* with *Pr* = *Le* = 1.0 and *Nt* = 0.1.

**Figure 6.** Effects of slot moving parameter α on reduced Nusselt number for varying *Nt* with *Pr* = *Le* = 1.0 and *Nb* = 0.1.

**Table 3.** Correlation expression for reduced Nusselt number and maximum percentage error defined for varying Prandtl number and moving slot parameter considering values of Brownian and thermophoresis diffusion parameters in the interval (0.01, 0.5).


It is observed that the Nusselt number decreases with an increase in parameters *Nb* and *Nt*, since higher temperatures correspond to higher Brownian and thermophoretic diffusion which resultantly reduces the surface heat flux. The same observation can be made from the correlation expressions since the coefficients of *Nb* and *Nt* are negative for all value of *Pr* and α. Furthermore, it is seen that dual solutions exist for a certain interval of slot moving parameter *α* and that interval can be viewed in Figures 5 and 6. The important observation is that the range of *α* reduces dramatically with an increase of *Nt* and the duality of solution vanishes for *Nt* = 0.05. For this reason, the correlation expression for *<sup>α</sup>* = −49<sup>o</sup> is derived for *Nt*∈ (0.0, 0.01). The variation of *Nb* has no effect on the duality of the solution.

For a fixed value of moving slot parameter *<sup>α</sup>* = −49o, Figures <sup>7</sup> and <sup>8</sup> show the dual solution for the variation of *Nb* and *Nt*. The thickness of concentration boundary layer is greater for the smaller solution branch. As the value of *Nb* increases, the concentration boundary layers become thinner for upper as well as for lower solution domains. The concentration thickness of boundary layer is less for the lower solution branch. As the value of *Nt* increases, the concentration boundary layers become thicker for upper and lower solution domains. In Figures 9–11, the effects of Lewis number, thermophoretic diffusion and Brownian diffusion on the nanoparticles concentration flux at the surface are plotted. The Sherwood number is plotted against the moving slot parameter α. Dual solution for Sherwood number is observed in the interval (−53◦, −49.5◦). Figure depicts that Sherwood number is growing function of α in the interval (−49.5◦, 30◦), and decreasing function in the interval (30◦, αU). As Le increases, i.e., the dominancy of viscous diffusion increases over the Brownian diffusion, the mass flux at the surface increases. Similar effects of Brownian diffusion and opposite effects of thermophoretic diffusion on Sherwood number are observed. In dual solution range, the effects of thermophoretic and Brownian diffusions on Sherwood number are found negligible.

**Figure 7.** Dual solutions of nanoparticles concentration profile for *α* = −49◦ and varying *Nb*, with *Pr* = *Le* = 1.0, *Nt* = 0.001.

**Figure 8.** Dual solutions of nanoparticles concentration profile for α = −49◦ and varying *Nt*, with *Pr* = *Le* = 1.0, *Nb* = 0.05.

**Figure 9.** Effects of slot moving parameter *α* on reduced Sherwood number for varying *Le* with *Pr* = 1.0, *Nb* = 0.05, *Nt* = 0.001.

**Figure 10.** Effects of slot moving parameter α on reduced Sherwood number for varying *Nb* with *Pr* = *Le* = 1.0, *Nt* = 0.001.

**Figure 11.** Effects of slot moving parameter α on reduced Sherwood number for varying *Nt* with *Pr* = *Le* = 1.0, *Nb* = 0.05.
