## 5 Results and Discussion
### Distributions of Non-Spherical Submicron Particle Velocity  
Figure 4b shows the large temperature difference in the micronozzle. Particles tend to move from the higher-temperature area to the lower-temperature area through thermophoresis when there is a relatively large gradient in the fluid temperature. The thermophoretic force is formulated as [25]:  
\[F_{T}=4.5\pi\frac{\mu^{2}}{\rho}d_{p}\frac{1}{1+3\frac{2\lambda}{d_{p}}}\frac {\frac{\zeta_{s}}{C_{p}}+2.48\frac{2\lambda}{d_{p}}}{1+2\zeta_{p}^{\zeta_{s}} +4.48\frac{\zeta_{s}}{C_{p}}}\frac{\
abla T}{T} \tag{15}\]  
where \(C_{s}\) represents the thermal conductivity of air, \(C_{p}\) represents the thermal conductivity of the particle and \(\lambda\) represents the mean free path. According to the temperature distribution in Figure 4b, there is a more than 200 K temperature difference in the nozzle which induces the thermophoretic force to affect the particles. At present, the influence of the thermophoretic force on submicron particle focusing in a micronozzle is vague. Therefore, we compared the velocity in the \(y\)-direction (\(u_{p\_y}\)) of the particle (\(d_{p}=0.6\)\(\upmu\)m) released from the same initial position under two working conditions in Figure 6. One condition is that the thermophoretic force acting on the particles is considered, where \(F=F_{T}\) in Equation (8), and another condition is that the thermophoretic force is ignored, where \(F=0\). The result shows that there is almost no effect of thermophoretic force on aerodynamic focusing, so the thermophoretic force can be ignored in this paper. The factor that determines particle focusing is inertial force, which will be discussed later. Figure 6: The change in the particle velocity in the nozzle considering the effect of the thermophoretic force. Most particles are non-spherical, which leads to different aerodynamic properties [26; 27]. According to the definition of the shape factor (Equation (13)), particles with smaller shape factors have lower sphericity. It is expected that those particles are more rod-like or disk-like in shape. Table 2 shows the sphericity of the regular-shaped particles. The submicron particle velocity is obtained by injecting Al particles (\(\rho_{Al}\) = 2688 kg/m\({}^{3}\)) with different shape factors and sizes into the nozzle inlet, as shown in Figure 7. The gas velocity in the flow field centerline and the particle velocity are defined as \(u_{c}\) and \(u_{p}\), respectively. With the rapid growth in the gas velocity, particle velocities have different degrees of increase with different shape factors. \(u_{p}\) with \(S_{f}\) = 0.4 coincides relatively well with \(u_{c}\) for \(d_{p}\) = 0.3 \(\upmu\)m, while this coincidence degree becomes lower as \(S_{f}\) increases. This implies that particles with small shape factors show a stronger momentum exchange between the gas and particles, which makes it easier for particles to be accelerated. As a result, the outlet velocity of particles with small shape factors is relatively greater. In addition, the difference between \(u_{p}\) and \(u_{c}\) with a different \(S_{f}\) becomes more obvious with the increase in \(d_{p}\), because the shape factor has little effect on the particle velocity of small particles (Figure 7a,d). We can also observe that as the particle size increases, the coincidence degree becomes lower between \(u_{p}\) and \(u_{c}\), which means that large particles show poor airflow followability due to their large inertia. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Particle Shape** & **Model** & **Shape Factor (\(S_{f}\))** \\ \hline Sphere & & 1 \\ Cube & & 0.806 \\  & (\(h=r\)) & 0.827 \\ Disk-like & (\(h=r/3\)) & 0.594 \\  & (\(h=r/10\)) & 0.323 \\  & (\(h=3r\)) & 0.86 \\ Rod-like & (\(h=10r\)) & 0.691 \\  & (\(h=20r\)) & 0.580 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The sphericity of several regular-shaped particles.