# Numerical Investigation of Air Entrapment Dynamics for High-Speed Thermal Spraying
## 4 Results and Discussion
Considering solidification, as shown in Figures 4 and 5, the SP bubbles' entrapment depends on both the spreading and the freezing processes. Therefore, the air entrapments under different impacting speeds are investigated, and the results are shown in Figure 6a. For a low impacting velocity (\(v\) = 50 m/s), the formed splat was thick and there were several small bubbles near the edge, together with a large center pore. For a medium impacting velocity (\(v\) = 75 m/s, 100 m/s and 125 m/s), larger entrapped SP bubbles could be detected due to the extended finger's fallback on edge. Lastly, for a high impacting velocity (\(v\) = 200 m/s), the thinner splat made the finger land quickly back onto the substrate without any large bubble entrapments on the edge. The entrapped bubble size of \(v\) = 100 m/s was the largest, with a diameter of about 3.5 \(\mu\)m.  
The entrapped-bubble ratio \(P_{\text{b}}=V_{\text{b}}/V_{\text{sp}}\)= (\(V_{1}+V_{\text{s}}\))/\(V_{\text{sp}}\) is introduced here to evaluate the entrapped bubbles. \(V_{\text{b}}\) is the total volume of the entrapped bubbles, including IM bubbles, \(V_{\text{i}}\), and SP bubbles, \(V_{\text{s}}\), \(V_{\text{sp}}\) is the volume of the final formed intact splat. The result is shown in Figure 6b. Both low (L) and high (H) zones could achieve a low entrapped-bubble ratio \(P_{\text{b}}\), compared to the medium zone (M). As shown in Figure 6a, this difference is mainly attributed to the bubbles entrapped on the edge, i.e., the SP bubbles, which resulted from the fallback of the pointed-out fingers. The behavior of the finger is decided by the competition of the solidification rate \(r_{sd}\) and spreading rate \(r_{\text{sp}}\). The non-dimensional solidification \(r_{\text{sd}}\) rate could be expressed by Equation (6) [31]:  
\[r_{sd}=\sqrt{\frac{2}{Re}\frac{Ja}{Pr}\frac{k_{s}}{k_{l}}} \tag{7}\]  
\(Ja=C_{\text{p}}(T_{\text{m}}-T_{\text{sub}})/L_{\text{m}}\) is the Jakob number, where \(T_{\text{sub}}\) is the temperature of the substrate, \(L_{\text{m}}\) is the latent heat of fusion; \(Pr\) is the Prandtl number, \(Pr=\mu/\rho a_{1}\), where \(a_{1}\) is the thermal diffusivity of the melt; \(k_{\text{l}}\) and \(k_{\text{s}}\) are the thermal conductivity of the melt and solid, respectively. The spreading rate could be expressed as the reciprocal of the time when the melt has reached its maximum spreading radius \(R_{\text{max}}\), \(r_{\text{sp}}=1/tR_{\text{max}}\). As shown  
Figure 6: (**a**) Final entrapped bubbles and formed splat for different initial impacting speeds from low (\(v\) = 50 m/s) to high (\(v\) = 200 m/s), (**b**) entrapped-bubble ratio \(P_{\text{b}}\), (**c**) solidification rate \(r_{sd}\) and spreading rate \(r_{sp}\), with different initial impacting speeds: Low zone (L) (\(v\) = 50 m/s), medium zone (M) (\(v\) = 75 m/s, 100 m/s, 125 m/s) and high zone (H) (\(v\) = 200 m/s). in Figure 6c, with the increase in the impacting velocity \(v\), \(r_{\rm sd}\) decreases, and \(r_{\rm sp}\) increases, i.e., the solidification proceeds more slowly and the spreading carries on more quickly. For Zone L, \(r_{\rm sd}\) is high; therefore, before the molten droplet spreads far away, the solidification has already completed, leading to a thick splat with high rims, cut off from the fingers. Since the splat's fingers have been cut off, no extra bubbles are entrapped, resulting in a smaller \(P_{\rm b}\). On the contrary, for Zone H, spreading carries on more quickly than the solidification, so the molten droplet has enough time to spread further away, and a thinner splat could be obtained. The still-molten droplet causes the finger land smoothly back to the substrate, and the newly entrapped SP bubble will contract due to the surface tension on the bubble/melt interface. Thus, a much lower \(P_{\rm b}\) could be achieved. For the M zone, the \(r_{\rm sd}\) and \(r_{\rm sp}\) are comparable, and the finger could still fall back to the substrate, even with a sharp rim on edge.