# Numerical Investigation of Air Entrapment Dynamics for High-Speed Thermal Spraying
## 4 Results and Discussion
These air bubbles that are newly entrapped during the spread of the molten  
Figure 4: Evolution of a molten nickel droplet’s solidification and air entrapment impacts a smooth stainless-steel substrate (room temperature \(T_{0}=293.15\) K) with an initial speed of \(50\) m/s. (**A1**) Detailed view of the impacting (IM) bubbles entrapping process. (**A2**) Detailed view of the spreading (SP) bubbles entrapping process: Entrapped by (**I**) falling back of fingers and (**II**) connecting with the splashed secondary droplet; (**III**) cluster of SP bubbles entrapment by falling back of a long finger. \(\phi_{\mathrm{d}}\) is the conductive heat flux. represents the mushy zone of the molten droplet. Figure 3: (**a**) Meshing of the model and (**b**) mesh independence check. droplet along the substrate surface are named SP bubbles. As shown in Figure 4A2, in Figure 4(A2-I), the pointed-out finger falls back to the substrate, entrapping a small bubble on edge. In Figure 4(A2-II), the finger just touches the previously splashed secondary droplet, leading to a new bubble entrapment. Finally, in Figure 4(A2-III), a cluster of bubbles is entrapped due to a longer finger falling-back. Unlike IM bubbles, SP bubbles are unique to molten droplets, which undergo phase transition during spreading. The formation of SP bubbles should be attributed to the inhomogeneous solidification of the droplet. Here, two kinds of heat transfer are involved, i.e., heat conduction via direct contact and heat convection via fluid movement. As shown in Figure 4, heat conduction occurs at the contact interfaces, including droplet/substrate interface, droplet/splat interface, and splat/substrate interface. While for heat convection, it refers to heat transfer during the movement of molecules within the fluids [29]. Accordingly, heat convection mainly occurs within the molten droplet's liquid phase. As shown in Figure 5a, at \(t=0.70\) us, the most severe heat convection occurs within the long pointed-out finger, while there is almost no heat convection within the mushy zone, and the corresponding velocity is zero (Figure 5b), indicating that the mushy zone mainly consists of solidified splat. Accordingly, the shear stress at the droplet/splat interface is large, restricting the further spread of the liquid part above the mushy zone. However, due to the non-zero dynamic energy, it continues to spread along the radial direction, and finally falls back to the substrate with new air entrapped. In contrast, at \(t=0.90\) us, the liquid part above is not continuous, and a half-solidified cutting point is formed. This contracts due to the surface tension, creating a wedge-shaped rim with a height of 4 um and bringing the cut-off of the finger. Consequently, the inhomogeneous solidification is the key to the behavior of the pointed-out finger: either falling back to the substrate with newly entrapped SP bubbles, or breaking up as secondary droplet. It is known that for a droplet without phase transition, the Ohnesorge number is used to describe its spreading dynamics after impact by comparing viscous forces with inertial and surface tension forces, \(Oh=\mu/(\rho d_{0}\sigma)^{0.5}=(We)^{0.5}/Re\), where \(We=\rho v^{2}d_{0}/\sigma\), \(Re=\rho vd_{0}/\mu\). We reflect the driving force for the spreading, and \(Oh\) reflects the resistance. For this case,  
Figure 5: (**a**) Convective heat flux \(\phi_{c}\) and (**b**) velocity \(v\) ✗ shear stress \(\tau\) distribution of fall-back and cut-off splash, respectively. (**b**) represents the mushy zone. \(d_{0}=54\)\(\mu\)m, \(v\)= 50 m/s, \(Oh\) = 0.071 and \(We\) = 634.58. With a high \(We\) (\(We>1\)) and low \(Oh\) (\(Oh<1\)), it is impact-driven [26] and inertia-resisted, i.e., the impact speed has a great influence on its behavior. For plasma spraying, the particles are dozens of microns and will be accelerated to an even higher speed of about several hundred meters per second [30]. Omitting solidification, the impacting speed greatly affects the spreading and, thus, the SP bubbles' entrapment.