## 3 Results and Discussion  
The formulated problem (Equations (24)-(33)) and boundary conditions (Equation (34)) were treated numerically using the bvp4c function because of its efficiency and accuracy. The function was applied to the three-stage Lobatto IIIa formula, which gave fourth-order \(C^{1}\) continuous solutions depending on the collocation formula. The residual of the continuous solution controls the error and adjusts the mesh size. Figures 2-12 show the effects of various parameters on the temperature and velocity profiles. Here, the solid line shows the thermal and flow field profiles for the dusty phase, and the dashed line shows those for the nanofluid phase. The ranges of the parameters were taken as \(0.2\leq\beta_{v}\leq 1.3\), \(0.01\leq\phi\leq 0.03\), \(0.01\leq\delta\leq 0.03\), \(0.50\leq l\leq 0.52\), \(0.50\leq\alpha\leq 0.70\), \(1.2\leq Rd\leq 1.4\), \(0.5\leq\theta_{w}\leq 0.7\), and \(0.5\leq\delta_{1}\leq 1.5\). Figure 2 shows the impact of the momentum dust parameter \(\beta_{v}\) on the fluid with dusty particles, while keeping the mass concentration of the dust particles fixed as \(l=0.5\), \(\delta=0.5\), \(Rd=0.5\), and \(\theta_{w}=0.5\). Figure 2 shows the moment the dust phase is aggrandized for increases in \(\beta_{v}\). The velocity increased until it approached the nanofluid velocity. This increase occurred because the dust particles had a higher momentum. Figure 3 shows the impact of nanoparticle volume fraction on the velocity profile for the nanofluid and dust phases. Increasing the nanoparticle volume fraction causes an increase in particle-particle interactions, which lowers the velocity. Moreover, due to the presence of dust particles, the decrease in velocity was greater than in the nanofluid phase. Figure 4 illustrates the impact of film thickness on the velocity profile with constant spraying or condensation velocity; it shows that the velocity increased with increasing film thickness, but because of the presence of dusty particles, the increase in velocity of the dusty fluid was lower compared to that of the nanofluid. The velocity decreased with the increasing mass concentration of the dusty particles for the dusty and nanofluid phases, as shown in Figure 5. The additional dust particles created a resistive force that influenced the speed of the fluid. Figure 6 shows the draining velocity behavior for various spraying velocities in the nanofluid and dusty phases. The inclination of the disk causes gravitational flow. Gravity causes the downward flow of the fluid. The free surface flow on an inclined plane will be strongly affected by the spray rate or rotation of the disk. An influence of the momentum of the dust on the velocity of the nanofluid and dusty phases, \(g(\eta)\) and \(G(\eta)\), respectively, is observed. It can be seen in the figure that the velocity of dusty phase \(G(\eta)\) showed a smaller increase with increasing dust momentum in comparison to the nanofluid phase. Increasing the dust momentum increased the  
Figure 4: Impact of \(\delta\) on velocity profile \(f^{\prime}\) and \(F^{\prime}\) of fluid with dusty particles. Figure 5: Impact of \(l\) on velocity profile \(f^{\prime}\) and \(F^{\prime}\) of fluid with dusty particles. Figure 6: Impact of \(\alpha\) on draining velocity \(k\) and \(K\) of fluid with dusty particles. Figure 7: Impact of \(\beta_{v}\) on lateral velocity profile \(g\) and \(G\) of fluid with dusty particles. Figure 11: Impact of \(S_{1}\) on temperature profile \(\theta\) and \(\theta_{p}\) of fluid with dusty particles. Figure 8: Impact of \(Rd\) on temperature profile \(\theta\) and \(\theta_{p}\) of fluid with dusty particles. Figure 10: Impact of \(\delta_{1}\) on temperature profile \(\theta\) and \(\theta_{p}\) of fluid with dusty particles. Table 2 shows the numerical values for the rate of heat transfer as according to the Prandtl number \(Pr,\gamma,\theta_{w},Rd,\beta_{t}\), while the values of \(\delta_{1}=0.5,S_{t}=0.5,S_{1}=0.5\) remain fixed.