## 5 Results and Discussion
Figure 5b displays the opposite trend, where the heat transfer coefficient increases as \(\varepsilon\) increases for both solutions. This indicates that the heat transfer rate over shrinking surface is higher than that over a stretching surface for each \(s\) parameter. In addition, it is clearly seen in Figure 5 that the solution for the second solution was easier to find with a higher value of \(s\). Furthermore, as the effect of the suction parameter \(\left(s>0\right)\) becomes stronger, or the effect of injection parameter \(\left(s<0\right)\) becomes weaker, the skin friction coefficient increases for the first and second solutions \(\left(s<1.5\right)\), and the heat transfer coefficient increases for both solutions. Figure 4: (**a**) Skin friction coefficient and (**b**) Local Nusselt number for different \(A\) with \(\varepsilon\) in Copper-water nanofluid when \(s=2\), \(\sigma=0.1\) and \(\varphi=0.1\). Figures of local skin friction coefficient \(C_{f}(2\text{Re}_{x})^{1/2}\) and local Nusselt number \(Nu(2/\text{Re}_{x})^{1/2}\) from Equations (16) and (17) are illustrated in Figure 6 with different nanoparticle types, namely, Copper, Alumina, and Titania, for various nanoparticle volume fractions in the base fluid \((0\leq\varphi\leq 0.2)\) over both stretching \((\varepsilon>0.5)\) and shrinking \((\varepsilon<0.5)\) surfaces. The local skin friction for the shrinking surface is much higher than that over the stretching surface, which indicates that the shear stress between the surface and fluid is larger for the flow over a shrinking sheet than a stretching sheet. Meanwhile, the local Nusselt number for the stretching surface is obviously higher than that for the shrinking surface. This denotes that the heat transfer rate at the surface is greater when the sheet is stretched. The increasing of nanoparticle volume fraction from 0 to 0.2 in water affects both the local skin friction coefficient as well as the local Nusselt number, where it increases the local skin friction for each nanoparticle over both stretching and shrinking surfaces. However, only the Copper-water nanofluid showed an increase in transferring the heat at the surface; the two other types of studied nanoparticles, Alumina and Titania, caused a decrease in the value for the local Nusselt number. These figures also show that different types of nanoparticles have a significant affect on shear stress and the heat transfer rate at the surface. The Copper-water nanofluid was found to have the highest local skin friction coefficient and local Nusselt number for each value of the nanoparticle volume fraction over both surfaces. This means that the Copper-water nanofluid causes larger shear stress at the surface, and also has the ability to transfer heat at the surface more effectively than Titania and Alumina, since Copper has the highest thermal conductivity, as can be seen in Table 1.  
Figure 5: (**a**) Skin friction coefficient and (**b**) Local Nusselt number for different \(\varepsilon\) with \(s\) in Copper-water nanofluid when \(A=0.1\), \(\sigma=0.1\) and \(\varphi=0.1\). The profiles of velocity and temperature of the fluid for different values of \(\sigma\), \(\varepsilon\), and \(s\) are displayed in Figures 7-10 in order to support the results from Figures 2-6, where dual solutions are obtained. These profiles also are depicted to show that the far field boundary conditions from Equation (13) are asymptotically fulfilled. Each profile illustrates that the velocity and thermal boundary layer thicknesses for the first solution are smaller than for the second solution for every tested parameter, implying that the second solution is unstable, as mentioned by [PERSON] et al. [7]. Figure 7a shows that by increasing \(\sigma\), the fluid velocity increases for the first solution but decreases for the second. Meanwhile, Figure 7b presents the opposite behaviour, i.e., where the thermal boundary layer thickness is decreasing for the first solution and increasing for the second as the slip parameter becomes more significant.