**5. Discussion**

A three-dimensional nanofluid film flow with heat and mass transfer through a steady rotating inclined surface was analyzed. The impact of *M*, *S*, Ω, *Sc*, *Nt*, *Nb*, Pr and *K* were explored for radial velocity *k*(η), axial velocity *f*(η), induced flow *s*(η), and drainage flow *g*(η) respectively. Furthermore, the impact of these parameters was also described for the temperature profile θ(η) and concentration profile θ(η). The impact of the magnetic parameter *M* over the velocity profiles *f*(η), *k*(η), *g*(η), and *s*(η) are presented in Figures 2–5. The impact of the magnetic parameter on the axial and the radial profiles of the velocity look similar for up to η = 2.0 as shown in Figures 2 and 4. For greater values of the magnetic parameter these profiles decline sharply and the peak point is at η = 4.0. Physically, the larger values of the magnetic parameter reduce the rotational parameter, and as a result both the profiles decline. Figures 3 and 5 both show an increasing trend in the drainage and induced flow with the smaller and larger values of the magnetic parameter respectively. For η = 2.3 all the curves coincide, and after this point the induced velocity jumps in increasing order as shown in Figure 5, while on the other hand smaller values of the magnetic parameter enhance the drainage velocity as presented in Figure 3. The variation in Figures 3 and 5 as a consequence of layer thickness and internal velocity of nanofluid definitely reduce. The direction of this force is perpendicular to both the fields. Also *M* endorses the ratio of viscous and hydromagnetic body forces, the fluid flow is decreased due to greater values of *M* which require further hydromagnetic body forces. Lorentz force theory defines that *M* has a reverse influence on velocity function. Figures 6–9 show the effect of *K* on *f*(η), *k*(η), *g*(η), and *s*(η). Figure 6 displays the axial velocity which accelerates in response to an increases in the couple stress

parameter. This is mainly due to the decrease in friction, which rises from the particle (i.e., base-fluid particles) additives that create a size-dependent influence in couple stress liquids. The influence of *K* on *k*(η) is demonstrated in Figure 7. The disk rotation increases *K*, due to which the flow velocity also rises. For *K* = 0 the flow retains Newtonian fluid. Figure 8 displays the influence of *K* on drainage flow *g*(η). The flow is perceived to rise with rising values of *K*, since the rise in *K* causes a reduction in the dynamic viscosity of the liquid and, hence a rise in the molecular distance among the liquid particles. Figure 9 displays the influence of *K* on induced flow *s*(η). Here a rise in *K* leads to a rise in the stress among all the coupled fluid atoms, which further leads to a reducttion in the induced velocity. Figure 10 depicts the Pr impact on the temperature profile θ(η) since Pr has an inverse relation to the thermal diffusivity and direct relation to the momentum diffusivity. Greater values of Pr mean that there is robust momentum diffusivity which is associated with the thermal diffusivity and as a result the thermal diffusivity decreases the temperature profile θ(η). These variations almost look similar up to η = 1.8. Hence, the greater values of Pr drop the boundary layer of heat. The influence of *Nb* on θ(η) is demonstrated in Figure 11. It is clear that larger values of *Nb* enhance the thermal boundary layer after η = 2.7. Physically, when Brownian motion increases the interaction between the particles enhances and as a result the energy transfers rapidly from one point to another, which as a result increase the thermal boundary layer. The impact of the thermophoresis parameter *Nt* over θ(η) is given in Figure 12. It can be seen that a rise in *Nt* leads to augment the liquid temperature. Thermophoresis forces generate temperature gradient that further causes a degenerate flow away from the surface. The influence of Ω on θ(η) is demonstrated in Figure 13. The larger values of the rotation parameter Ω enhance the temperature profile θ(η). This is due to the larger values of Ω, the fluid temperature increases. Physically, greater values of Ω increase the kinetic energy, which in consequence increases θ(η). Influence of *Nb* on θ(η) is demonstrated in Figure 14. For greater values of *Nb* the concentration profile θ(η) increases. This variation is very effective for greater values of η. In practice, rising values of *Nb* cause an enhancement of the random motion amongst the nanoparticles, and consequently reduces θ(η) of the liquid. Higher values of *Nb* decrease the boundary layer thicknesses, which as a result reduces the concentration profile Φ(η). Figure 15 displays the performance of the *Nt* on Φ(η). Increasing values of *Nt* push the nanoparticles far away from the hot sheet, and consequently enhance the concentration profile Φ(η). Since *Nt* varies with the gradient of the temperature of the nanofluids, the kinect energy ncreases of the nanofluids due to the increase of *Nt*, which further increases Φ(η). The effect of *S* on Φ(η) is presented in Figure 16. It is seen that Φ(η) varies directly with *S*. Augmenting *S* increases the concentration, which as a result raises the kinect energy of the liquid, which further enhances the speed of fluid film. It is evident from the graph that the rising values of *S* reduce Φ(η). This impact of *S* over Φ(η) is due to the stretching sheet, steady flow, and the greater concentration, and hence rising *S* shows a converse influence. The effect of the Schmidt number *Sc* on the concentration field is presented in Figure 17. The concentration boundary layer reduces due to the rise in *Sc*. In practice *Sc* decreases the molecular diffusivity, which reduces the concentration boundary layer. It is observed that the reduction in heat transfer at the sheet leads to a rise in the values of *Sc*

#### **6. Conclusions**

The flow past an exponential stretching sheet of couple stress nanofluid with the impacts of MHD, viscous dissipation, and Joule heating was investigated analytically. The influences of zero mass flux and convective heat condition were also considered. HAM was applied for the solution of the non-linear differential equations. The effects of the numerous constraints on velocity field, temperature profile, and concentration are portrayed by the graphs. The important results of the current study are described below as follows:


• The higher values of the Brownian motion parameter increase the concentration distribution and decline with increasing values of the thermophoresis parameters.

**Author Contributions:** Conceptualization, methodology, software, validation, writing—original draft preparation, writing—review and editing, A.T., Z.S., I.A. and M.S.; conceptualization, methodology, software, visualization, writing—review and editing, F.A., A.U. and S.I.; writing—review and editing, visualization, project administration, funding acquisition, investigation, resources, A.T., M.S. and Z.S. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work is funded by the Deanship of Scientific Research at Majmaah University under Project Number (RGP-2019-28).

**Acknowledgments:** The authors extend their appreciation to the Deanship of Scientific Research at Majmaah University for funding this work under Project Number (RGP-2019-28).

**Conflicts of Interest:** The authors declare no conflict of interest.
