**4. Results and Discussion**

In this section impact of physical parameters on velocity and temperature profiles are discuses. In Figure 2, h-curve for velocity and temperature profiles are displayed. In Figures 3–7, the physical influence of the embedded parameters on the thin layer flow of Darcy-Forchheimer nanofluid over a nonlinear radially extending porous disc is presented. Figure 3 depicts the impacts of positive integer *n* and magnetic parameter *M* on *f*- (η). It is determined here that both parameters show a declining behavior in the velocity profile.The nonlinearity stretching phenomena of the thin film flow reduced the thin layer with the escalation in *n*, because the bulky magnitude of *n* produced an opposing force to reduce the fluid motion. Therefore, the fluid velocity was reduced with the escalated *n*. Moreover, the large amount of *M* decreased the fluid velocity. Basically, the Lorentz force says that the resists the fluid motion on the liquid boundary which, in result, diminishes the velocity of the fluid. Figure 4 reveals the impacts of κ and *Fr* on *f*- (η). The porous medium performed a key role during fluid flow occurrences. Significantly, the porosity parameter disturbed the boundary layer flow of liquid which, as a result, produced opposition to the fluid flow and, hereafter, a decline the velocity of the fluid. Furthermore, *Fr* diminished the fluid flow at the surface of the radially extending disc. This behavior occurred because the porous medium was added to the flow phenomena which decreased the coefficient of inertia, and consequently, the fluid velocity was decreased. The influence of fluid layer thickness β on *f*- (η) and θ(η) is shown in Figure 5. Physically, the resistive force to fluid flow increased with the increase in fluid layer thickness β. The increased fluid layer thickness increased the velocity and a smaller amount of energy was needed for the motion of the fluid. Consequently, the velocity profile was reduced with an increase in fluid layer thickness. Similarly, the increase in fluid layer thickness was reduced θ(η). Figure 6 reveals the influences of γ and *R* on θ(η). Physically, γ acted like a heat producer which increased the boundary layer thickness and released heat to the fluid flow phenomena. Therefore, the increase in γ increased θ(η). The increase in *R* increased θ(η). The upsurge in *R* enhanced the thermal boundary layer temperature of the fluid flow; consequently, increased behaviour in θ(η) is observed. The impact of Pr and Ec is revealed in Figure 7. The increased Eckert number increased the temperature of the thin film flow. Actually, the Eckert number produced viscous resistance due to the presence of a dissipation term which increased the nanofluid thermal conductivity to increase the temperature field. The enhanced Prandtl number Pr reduced the temperature of the thin film flow. The higher Pr numbers (e.g.,Pr = 7.0) possess lower thermal conductivity which result in a decline in temperature of the boundary layer flow. Conversely, the lower Pr numbers possess higher thermal conductivity which consequently increases the temperature of the boundary layer flow.

Figures 8 and 9 display a comparison of the homotopy analysis method (HAM) and numerical (ND-Solve) techniques *f*- (η) and θ(η). The agreement of the HAM and numerical techniques is observed here.

The influence of entrenched parameters on*Cf* and*Nu* are displayed in Tables 1 and 2. The increasing fluid layer thickness increases the opposing force to fluid flow which, as a result, improves the *Cf* of the thin film flow. The escalating magnetic field increases the Cf. This influence is due to the increasing magnetic field which boosts the resistive force to the flow of fluid, called Lorentz force. The κ and *Fr* increase the Cf. The porosity parameter disturbs the boundary layer flow of the thin film flow which increases the resistive force to the fluid. The coefficient of inertia is directly proportional to the porosity parameter. The increase in the porosity parameter increases the coefficient of inertia which, in result, boosts the opposing force to fluid flow. The increasing positive integer boosts the nonlinearity which produces resistance to the fluid and increases the Cf. The increase in *R* increases the *Nu*. The thermal boundary layer temperature of the fluid flow increases with the increase in *R* which increases the heat transfer of the thin film flow. The increase in Pr increases the *Nu*. Usually, the large amount

of Pr reduces the nanofluid thermal conductivity. Therefore, the *Nu* increases with the increase in Pr. The larger amount of γ increases the *Nu*. This effect is due to the fact that the γ increases the boundary layer thickness of the nanofluid which, in result, increases the *Nu*. The increasing values of Ec reduces the *Nu*. The Eckert number is usually composed of the nanofluid thermal conductivity term to increase the temperature profile which, in turn, gives the opposite influence for cooling processes. The escalating positive integer increases the *Nu*.

Tables 3 and 4 display the assessment of the homotopy analysis method (HAM) and numerical (ND-Solve) techniques for *f*- (η) and θ(η). The agreement of the HAM and numerical techniques is observed here.

**Figure 2.** -curves for *f*- (η) and θ(η).

**Figure 3.** Impression of *n* and *M* on *f*- (η).

**Figure 4.** Impression of κ and *Fr* on *f*- (η).

**Figure 5.** Impression of β on *f*- (η) and θ(η).

**Figure 6.** Impression of γ and *R* on θ(η).

**Figure 7.** Impression of Pr and Ec on θ(η).

**Figure 8.** The assessment of HAM and ND-Solve for *f*- (η).

**Figure 9.** The assessment of HAM and ND-Solve for θ(η).

**Table 1.** The effect of embedded parameters on *Cf* at 15th order approximations of the homotopy analysis method (HAM).



**Table 2.** The effect of embedded parameters on *Nu* at 15th order approximations of the HAM.

**Table 3.** The assessment of the HAM and ND-Solve for *f*- (η).


**Table 4.** The assessment of the HAM and ND-Solve for θ(η).


#### **5. Conclusions**

The thin layer flow of Darcy-Forchheimer nanofluid over a nonlinear radially extending disc has been examined in this study. The nonlinear disc with a variable thickness of the nanofluid has been varied with the help of positive integer *n*. The magnetic field has been executed in a direction vertical to the nanofluid flow. The influences of magnetic field parameter, positive integer, porosity parameter, coefficient of inertia, fluid layer thickness, Prandtl number, heat source/sink, thermal radiation, and Eckert number on the fluid flow problem have been observed in this study. The key findings can be stated as follows:


**Author Contributions:** Conceptualization, A.D. and Z.S.; Methodology, A.D., Z.S. and W.K.; Software, A.D., Z.S. and S.I.; Validation, A.D. and W.K.; Resources, P.K.; Writing—Original Draft Preparation, A.D.; Writing—Review & Editing, A.D., Z.S., W.K. and S.I.; Visualization, Z.S., P.K. and S.I.

**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 author declares that they have no competing interests.
