## 7 Results and Discussion  
This segment was devoted to envisioning the physical insight for graphical illustration Figures 2-10. We took the fixed values of the parameters throughout the study as (\(\alpha=0.01\), \(\varepsilon=0.3\), \(\phi=0.04\), \(\phi_{d}=F_{r}=\lambda=Ec=0.1\), \(\alpha=m=\gamma=1\), \(\beta=\beta_{T}=0.5\), \(M=0.2\), \(N_{r}=6\), \(\theta_{r}=1.1\),) and \(Pr=6.8\). Figure 2a,b exemplify the impacts of needle's size \"\(a\)\" on the nanofluid velocity and velocity of the dust phase, respectively. It was comprehended that velocities were declining functions of the needle size in the case of CNTs of both types. Physically speaking, both velocities were highly dependent on the size of the needle. Increasing the needle's size lowered the velocities, which was obvious. An opposite trend was witnessed in the case of Figure 2c,d. It was witnessed that temperature was dominant in the case of SWCNTs as compared to MWCNTs. This was because MWCNTs have lower thermal conductivity than SWCNTs. The impact of radiation parameter \(Nr\) on the nanofluid temperature and temperature of the dust phase could be seen in Figure 3a,b. Owing to higher radiation, more heat was transmitted to both nanofluid and the dust phase. Eventually, the augmented temperature in both cases, i.e., nanofluid and dust fluid, was witnessed. Variation in temperature ratio parameter \(\theta_{r}\) on temperatures of the nanofluid and dust fluid is depicted in Figure 4a,b, respectively. \(\theta_{r}\) is the quotient of the wall temperature to ambient temperature. Larger values of \(\theta_{r}\) meant sturdier wall temperature than the ambient temperature. Higher estimates of \(\theta_{r}\) resulted in a rise in the temperature of both fluids in the case of both CNTs. Figure 5a,b revealed the nanofluid velocity and dust phase for the Darcy-Forchheimer parameter \(Fr\). It is learned that both velocity functions are diminishing for the growing values of \(Fr\)[29]. Actually, higher estimates of \(Fr\) produced resistance in nanofluid motion for both CNTs that ultimately dropped nanofluid and dust fluid velocities. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(F_{r}\) & \(\alpha\) & \(\beta\) & \(m\) & \(N_{r}\) & \(\theta_{r}\) & \ \begin{tabular}{c} **Nusselt Number** \\ **SWCNT** \\ \end{tabular}  & \begin{tabular}{c} **Number** \\ **MWCNT** \\ \end{tabular}  \\ \hline
0.10 & & & & & & 1.10739 & 1.05489 \\
0.25 & & & & & 1.12048 & 1.06695 \\
0.40 & & & & & 1.13570 & 1.08087 \\
1.0 & & & & & 1.10739 & 1.05489 \\
2.0 & & & & & 1.288  
Figure 3: Impact of \(N_{r}\) on (**a**) nanofluid temperature and (**b**) temperature of the dust phase. Figure 2: Impact of \(a\) on (**a**) nanofluid velocity, (**b**) the velocity of the dust phase, (**c**) nanofluid temperature, and (**d**) temperature of the dust phase. The upshot of magnetic parameter \(M\) on associated distributions is described in Figure 6a-d. Upon increasing the number of dust particles into the fluid, the drag force was strengthened, and more resistance to the fluid flow was experienced, and, eventually, a decrease in both velocities was witnessed. An opposing trend was identified for the temperature field and the dust fluid temperature, which was obviously owing to sturdier \(M\). Figure 7a,b are outlined to perceive the impact of Hall current parameter \(m\) on both velocities. It was detected that velocities were mounting functions of \(m\). Larger estimates of \(m\) enforced the damping force, and, eventually, velocities were strengthened. The impression of Hall current \(m\) on entropy generation and the Bejan number is shown in Figure 8a,b. Larger values of \(m\) lowered the temperature that resulted in a drop of entropy generation as well. An inverse behavior was seen for the Bejan number against \(m\). Figure 4: Impact of \(\theta_{r}\) on (**a**) nanofluid temperature and (**b**) temperature of the dust phase. Figure 5: Impact of \(F_{r}\) on (**a**) nanofluid velocity and (**b**) the velocity of the dust phase.