**3. Entropy Generation**

The entropy generation under the aforementioned assumptions is given as below:

$$E\_{\rm gren}^{\prime\prime\prime} = \frac{k\_f}{T\_0} \left[ \frac{k\_{nf}}{k\_f} + \frac{16T\_{\rm ox} \overline{\alpha}^\* \sigma^\*}{3k^\* k\_f} \right] \left( \frac{\partial T}{\partial y} \right)^2 + \frac{\mu\_{nf}}{T\_0} \left( \frac{\partial u}{\partial y} \right)^2 + \frac{\sigma\_{nf}}{T\_0} B^2(t) u^2 \cos^2 \varepsilon \tag{18}$$

where all terms defined in Equation (15) portray the usual meaning. The entropy generation *NG* is defined as

$$N\_{\rm G} = \left(\frac{k\_{nf}}{k\_f} + \frac{4}{3}R\right) \text{Re}\,\varepsilon \boldsymbol{\theta}^{\prime} + \frac{1}{\left(1-\phi\right)^{2.5}} \frac{\text{Br}\,\text{Re}\_{\rm x}}{\alpha} \text{f}^{\prime\prime 2} + \frac{\text{BrM}}{\alpha} \frac{\sigma\_{nf}}{\sigma\_f} \cos^2\varepsilon f^{22} \tag{19}$$

where *S*<sup>0</sup> -- and *Sgen*-- are the characteristic entropy generation rate and the entropy generation rate. The parameters defined in the above equation are given as

$$\alpha = \frac{\Delta T}{T\_{\text{w0}}}, Br = \frac{\mu\_f \mu\_w}{k\_f \Delta T}, Re\_x = \frac{\mu\_w \chi}{\mathbf{v}\_f} \tag{20}$$

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

This section is devoted to witnessing the impression of numerous parameters on the involved profiles whilst keeping in view their physical significance. The MATLAB built-in function bvp4c is utilized to address the differential Equations (9), (10), and (16) with the associated boundary conditions of Equation (11). To solve these, first we have converted the 2nd and 3rd order differential equations to the 1st order by introducing new parameters. The tolerance for the existing problem is fixed as 10<sup>−</sup>6. The initial guess we yield must satisfy the boundary conditions asymptotically and the solution as well. The results show the influence of solid volume fraction (ϕ), dimensionless film thickness (λ), magnetic parameter (*M*), unsteadiness parameter (*S*), radiation parameter (*R*), thermal relaxation parameter (γ), and non-uniform heat source/sink parameter on the velocity, temperature and entropy generation profiles. Further, the numerical values for the Skin friction and Nusselt number are given in Tables 3 and 4 for different parameters. The numerical values of the parameters are fixed as ϕ = 0.1, *A*\* = *B*\* = λ = γ = 0.5 = *S*, *R* = 1.0 = *M*, and *Pr* = 6.2. Figures 2 and 3 display the impact of solid volume fraction (ϕ) on axial velocity and temperature distribution. For incremented values of the solid volume fraction (ϕ), the velocity and temperature profiles enhance in case of both SWCNTs and MWCNTs. Actually, the convective flow and the solid volume fraction are directly proportionate with each other and this is the main reason behind the enhancement of axial velocity and the temperature of the fluid.


**Table 3.** The comparison table of –θ'(0) with Sandeep [17] for varied estimates of *S* when *R* = *M* = γ = 0, *Pr* = 1.0.



**Figure 2.** The illustration of ϕ versus *f*'(η).

**Figure 3.** The illustration of ϕ versus θ(η).

Figures 4 and 5 depict the behavior of axial velocity and the temperature field for the growth estimates of the film thickness parameter λ. It is found that both velocity and temperature profiles diminish for increasing values of the film thickness parameter λ. In fact, the more the film thickness, the lesser the fluid motion. This is because of the fact that higher values of film thickness dominate the viscous forces, eventually diminishing the fluid velocity. Similar behavior is observed for the temperature field.

**Figure 4.** The illustration t of λ versus *f*'(η).

**Figure 5.** The illustration t of λ versus θ(η).

The effect of the magnetic parameter *M* on the velocity and temperature fields can be visualized in Figures 6 and 7. Figure 6 displays the impact of the magnetic parameter *M* on axial velocity. The is the axial velocity of the declining function of the magnetic parameter *M*. Physically, by enhancing the magnetic parameter *M*, the Lorentz force is strengthened in the flow, which has a tendency to resist the fluid's motion and slow it down. This force also creates heat energy in the flow. Consequently, the temperature distribution increases both the SWCNTs and MWCNTs, which is displayed in Figure 7.

**Figure 7.** The illustration of *M* versus θ(η).

Figures 8 and 9 show the effect of the unsteadiness parameter *S* on the velocity and temperature distributions. It is found that with the increase of the unsteadiness parameter *S*, the axial velocity diminishes. Physically, the bouncy effect acts on the flow and diminishes it due to the increase in the unsteadiness parameter *S*. Therefore, the thermal and momentum boundary layer thicknesses decrease.

**Figure 8.** The illustration of *S* versus *f*'(η).

**Figure 9.** The illustration of *S* versus θ(η).

Figure 10 determines the consequence of the thermal relaxation parameter γ on the temperature of the fluid. It is concluded that the temperature diminishes for increased values of the thermal relaxation parameter γ. The temperature tends to be sharper near the boundary as the value of γ is higher than the points on the growth in the wall slope of the temperature profile.

**Figure 10.** The illustration of γ versus θ(η).

Figure 11 demonstrates the impact of the radiation parameter *R* on the temperature profile. It is comprehended that the temperature field is an increasing function of the radiation parameter *R*. It is also concluded that the thermal boundary layer thickness for both carbon nanotubes is increased. In fact, larger estimates of the radiation parameter reduce the mean absorption coefficient and enhance the radiative heat flux's divergence. Due to this, the temperature of the fluid is upsurged.

**Figure 11.** The illustration of *R* versus θ(η).

The influence of non-uniform heat source/sink parameters *A*\* and *B*\* on the temperature distribution is shown in Figures 12 and 13. It can be understood that the temperature profile augments the boosted estimates of non-uniform heat source/sink parameters.

**Figure 12.** The illustration of *A*\* versus θ(η).

**Figure 13.** The illustration of *B*\* versus θ(η).

The effect of Brinkman number (*Br*), magnetic parameter (*M*) and Reynolds number (*Rex*) on the averaged entropy generation number is demonstrated in Figures 14–16. It is concluded that the entropy generation number increases for mounting estimations of Brinkman number (*Br*), magnetic parameter (*M*) and Reynolds number (*Rex*) for both SWCNT and MWCNT.

**Figure 14.** The illustration of *Br* versus *NG*(η).

**Figure 15.** The illustration of *M* versus *NG*(η).

**Figure 16.** The illustration of *Rex* versus *NG*(η).

Table 3 is erected to envision the precision of the presented model by comparing it with Sandeep [17] who discusses the flow of nanofluids past a thin film under the influence of the magnetic field. To make a comparison, we have neglected the impacts of the volume fraction, electrical conductivity, and thermal relaxation parameters. Excellent alignment is achieved between both results.

Table 4 shows the estimates of the Skin friction coefficient for different parameters. It is seen that the Skin friction coefficient increases for growing values of the magnetic parameter, solid volume fraction, unsteadiness parameter, and film thickness. Table 5 demonstrates the numerical values of Nusselt number for numerous parameters. It is determined that the Nusselt number increases with augmented values of the dimensionless film thickness, radiation parameter, solid volume fraction, and unsteadiness parameter, while it diminishes for growing values of non-uniform heat source/sink.


**Table 5.** The numerical value of the Nusselt number with γ = 0.1, *Pr* = 6.2.

#### **5. Conclusions**

The thin film flow of nanofluid comprising of CNTs of both types (SWCNTs/MWCNTs) is studied whilst keeping in view the important applications of CNTs in many engineering applications. The flow is supported by the additional effects like C-C heat flux and entropy generation. The model is solved numerically with the support of the MATLAB software function bvp4c. The highlights of the existing study are


**Author Contributions:** Data Curation, D.L.; Funding Acquisition, F.H.; Investigation, M.M.; Project Administration, M.R.; F.H.; Resources, D.L.; Software, M.M.; Supervision, M.M.; Validation, F.H.; Visualization, J.D.C.; Writing—Original Draft, J.D.C.

**Funding:** This research was funded by Zayed University research fund, Abu Dhabi, UAE.

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