**3. Solution by Homotopy Analysis Method**

The optimal approach is used for the solution process. Equations (9) to (14) with boundary conditions (15) are solved by HAM. Mathematica software is used for this aim. The basic derivation of the model equation through HAM is given in detail below.

Linear operators are denoted as *L* <sup>ˆ</sup> *<sup>f</sup>* , *L*θ<sup>ˆ</sup> and *L*φ<sup>ˆ</sup> is represented as

$$\begin{array}{l} \mathcal{L}\_f(\hat{f}) = \hat{f}^{\prime \prime}, \mathcal{L}\_{\hat{k}}(\hat{k}) = \mathbf{k}^{\prime}, \mathcal{L}\_{\hat{\upwp}}(\hat{\mathfrak{g}}) = \mathbf{g}^{\prime \prime}, \\\mathcal{L}\_{\hat{\upwp}}(\hat{\mathfrak{s}}) = \mathbf{s}^{\prime}, \mathcal{L}\_{\hat{\upwp}}(\hat{\mathfrak{o}}) = \boldsymbol{\theta}^{\prime \prime}, \mathcal{L}\_{\hat{\upwp}}(\hat{\mathfrak{o}}) = \boldsymbol{\phi}^{\prime \prime} \end{array} \tag{22}$$

The modelled Equations (9) to (14) with boundary conditions (15) are solved analytically as well as numerically. The comparison between the analytical and numerical solution is shown graphically as well as numerically in Tables 1–6 for the velocities, temperature, and concentration profiles. From these tables, an excellent agreement between the HAM and numerical (ND-Solve Techniques) methods is obtained.


**Table 1.** Comparison of HAM and numerical solution for *f*(η).

**Table 2.** Comparison of HAM and numerical solution for *k*(η).


**Table 3.** Comparison of HAM and numerical solution for *g*(η).


**Table 4.** Comparison of HAM and numerical solution for *s*(η).



**Table 5.** Comparison of HAM and numerical solution for θ(η).

**Table 6.** Comparison of HAM and numerical solution for φ(η).


#### **4. Results and Discussion**

The three-dimensional flow of the liquid film through a steady rotating inclined surface with mass and heat transmission was examined. The influence of the embedded parameters, magnetic field, *M*, Casson parameter, γ, Schmidt number, *Sc*, Brownian motion parameter, *Nb*, and thermophoretic parameter, *Nt*, was investigated for the axial velocity, *f*(η), radial velocity, *k*(η), drainage flow, *g*(η), and induced flow, *s*(η), temperature field, θ(η), and concentration profile, φ(η), respectively. Figures 2–5 display the influence of the Casson fluid parameter, γ, on *f*(η), *k*(η), *g*(η), and *s*(η). Rising γ generates resistance in the flow path and decreases the flow motion of nanoparticles. It is observed that an increase of the Casson fluid parameter, γ, leads to a decrease of *f*(η), *k*(η), *g*(η), and *s*(η). The opposite trend is found in case of the z-direction, that is the enormous value of γ decreases the *f*(η), *k*(η), *g*(η), and *s*(η). The influence of *Pr* on θ(η) is displayed in Figure 6. It is interesting to note that θ(η) decreases with large values of *Pr* and increases with smaller values. In fact, the thermal diffusivity of nanofluids has greater values by reducing *Pr*, and this effect is inconsistent for larger *Pr*. Hence, the greater values of *Pr* drop the thermal boundary layer. The influence of the radiation parameter, *R*, on θ(η) is presented in Figure 7. It is observed that if *R* increases, then the boundary layer area θ(η) is augmented. The effect of *Nb* on θ(η) is displayed in Figure 8. The converse influence was created for φ(η) and θ(η), which means augmented *Nb* decreases the concentration profile, φ(η). The concentration boundary layer thickness decreased due to the rising values of *Nb* and as a result, the concentration field, φ(η), declined. The features of the thermophoretic parameter, *Nt*, on the concentration profile, φ(η), are presented in Figure 9. The enhancement of *Nt* increases φ(η). Thus, *Nt* depends on the temperature gradient of the nanofluids. The kinetic energy of the nanofluids rises with the increasing value of *Nt*, and as a result, φ(η) increases. Figure 10 identifies the influence of *Sc*. The dimensionless number, *Sc*, is stated as the ratio of momentum and mass diffusivity. It is obvious that the amassed *Sc* reduces the φ(η) and as a result, the boundary layer thickness is decreased.

**Figure 2.** The influence of γ on *f*(η) when Ω = 1, ρ = 1, σ = 0.5, *k* = 1, *M* = 1.

**Figure 3.** The influence of γ on *g*(η) when Ω = 1, ρ = 1, σ = 0.5, *k* = 1, *M* = 1.

**Figure 4.** The influence of γ on *k*(η) when Ω = 1, ρ = 1, σ = 0.5, *k* = 1, *M* = 1.

**Figure 5.** The influence of γ on *s*(η) when Ω = 1, ρ = 1, σ = 0.5, *k* = 1, *M* = 1.

**Figure 6.** The influence of *Pr* on θ(η) when Ω = 1, ρ = 1, σ = 0.5, *k* = 1, *M* = 1.

**Figure 7.** The influence of *R* on θ(η) when Ω = 1, ρ = 1, σ = 0.5, *k* = 1, *M* = 1.

**Figure 8.** The effect of *Nb* on φ(η) when *Nt* = 0.6, *Sc* = 0.6, *S* = 0.7.

**Figure 9.** The influence of *Nt* on φ(η) when *Nb* = 0.6, *Sc* = 0.7, *S* = 0.7.

**Figure 10.** The influence of the Schmidt number (*Sc*) on φ(η) when *Nb* = 0.6, *Nt* = 0.5.

Figures 11 and 12 demonstrate the effects of *Pr* and *R*. It can be seen that rising values of *Pr* and *R* increase *Nu*. In fact, the coaling phenomenon is enhanced with increased values of these parameters. Figure 13 identifies that *Nu* reduces for the amassed values of *k*.

**Figure 11.** The impact of the Prandtl number (*Pr*) on the Nusselt number.

**Figure 12.** The influence of radiation parameter (*R*) on the Nusselt number.

**Figure 13.** The influence of *k* on the Nusselt number.

#### **5. Conclusions**

In this article, the three-dimensional thin-film Casson fluid flow over an inclined steady rotating plane was examined. The thin film flow was thermally radiated and the suction/injection effect was also considered. By the similarity variables, the PDEs were converted into ODES. The obtained ODEs were solved by the HAM with association of the MATHEMATICA program. The main features of the study are highlighted as:


**Author Contributions:** A.S., Z.S. and S.I. modeled the problem and wrote the manuscript. P.K. and T.Z. thoroughly checked the mathematical modeling and English corrections. A.S., M.J. and A.U. solved the problem using Mathematica software, S.I., T.G. and P.K. contributed to the results and discussions. All authors finalized the manuscript after its internal evaluation.

**Funding:** This research was funded by the Center of Excellence in Theoretical and Computational Science (TaCS-CoE), KMUTT.

**Acknowledgments:** This project was supported by the Theoretical and Computational Science (TaCS) Center under Computational and Applied Science for Smart Innovation Research Cluster (CLASSIC), Faculty of Science, KMUTT.

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