Dynamics of a single free-settling spherical particle driven by a laser-induced bubble near a rigid boundary

In view of the mechanism of the enhanced cavitation erosion of hydraulic machinery operating in sand-laden water, we carry out a systematic investigation on the interaction between a laser-induced cavitation bubble and a free-settling spherical particle near a rigid boundary. We document different types of particle-bubble interactions and experimentally discover two important phenomena where the particle ends up impacting on the boundary at a relatively high velocity, driven by the explosive growth of the bubble and the microjet of the nonspherical bubble collapse respectively. By combining numerical and analytical analyses with experiments, we identify two limiting conditions of the particle-bubble bounce and the particle-boundary impact respectively in terms of the initial relative distance between the particle and the bubble, and the stand-off distance of the bubble against the boundary. These results provide guidance in controlling the particle-bubble dynamics in applications, such as alleviating cavitation erosions in sand-laden water, improving the efficiency of surface cleaning and drug delivery by cavitation and other means.

Figure 1

Configurations of experiments. (a) Schematic of experimental setup. (b) Experimental notation and particle-bubble interaction types in terms of initial relative positions of the particle and the bubble. Type I: the particle-bubble interaction is weakly influenced by boundary, where the particle is initially positioned near the northern hemisphere of the bubble. Type II: the particle-bubble interaction is strongly influenced by boundary, where the particle is initially positioned near the southern hemisphere of the bubble.

Figure 2

Computational domain (not to scale) and the mesh system.

Figure 3

Numerical flow velocities at different standoff distances on the $oz$ axis around a bubble near a rigid boundary ($u_f^{*}=u_f\sqrt{{\rho }_f/\left(p_{\infty }-p_v\right)}, p_{\mathrm{i}}=140\mathrm{MPa}$ and $h=2.000\mathrm{mm}$). They are compared with flow around a spherical bubble, on the Rayleigh solution with the same equivalent maximum radius $R_{\max }=2.339\mathrm{mm}$.

Figure 4

Type I interactions (weakly influenced by boundary) between the PMMA particle and the bubble: High-speed recordings are taken at 215 873 frames/s. Time $t=0$ is the beginning of the last image frame before a laser-generated bubble forms. The blue dashed line represents the position of the rigid boundary, and the red circle indicates the initial particle position. (a) The particle oscillates in sync with the bubble and moves along a straight line ($l_0/R_{max}=1.051, h/R_{max}=1.017$, and $a/R_{max}=0.104$). (b) The particle “bounces off” the bubble, with almost no backward motion during bubble collapse and a cusp-shaped deformation of the bubble occurs (indicated by “1”), when the bubble is near the maximum radius ($l_0/R_{max}=0.384, h/R_{max}=0.766$, and $a/R_{max}=0.102$). (c) The particle is ejected from the bubble with a high speed and a cusp-shaped deformation of bubble (indicated by “1”) occurs when bubble is growing ($l_0/R_{max}=0.282, h/R_{max}=1.513$, and $a/R_{max}=0.126$). Details of the particle-bubble dynamics are available in the Supplemental Material [30].

Figure 5

Comparison among the simulation, the experiment and the result of the force balance model. (a) The left part in a single frame is the simulation result ($p_{\mathrm{i}}=140\mathrm{MPa}, l_0=0.884\mathrm{mm}, h=1.764\mathrm{mm}$, and $a=235\mu \mathrm{m}$), while the right part is the experimental result ($l_0/R_{max}=0.384, h/R_{max}=0.766$, and $a/R_{max}=0.102$). (b) Bubble radius and PMMA particle position as a function of time during the bubble growth. (c) Comparison between the results of simplified model and the results including the boundary influence ($l_0/R_{max}=1.051, h/R_{max}=1.017$, and $a/R_{max}=0.104$). In (b) and (c), the errors bars are comparable to the size of the data points.

Figure 6

Comparison between the simulation and the experiment. (a) Images of particle-bubble dynamics. The left part in a single frame is the experimental result ($l_0/R_{max}=0.733, h/R_{max}=1.275$, and $a/R_{max}=0.108$), while the right part is the simulation result ($p_{\mathrm{i}}=120\mathrm{MPa}, l_0=1.589\mathrm{mm}, h=2.764\mathrm{mm}$, and $a=235\mu \mathrm{m}$). The bubble reaches its maximum radius at $T^{*}=0.961$ (i.e., $t=0.474T_b$). (b) Bubble radius and PMMA particle position as a function of time during the bubble growth. The errors bars are comparable to the size of the data points. (c) Simulated particle velocity.

Figure 7

Limiting condition of single particle-bubble bounce ($a/R_{\max }\sim 0.087$). (a) $h/R_{\max }–l_0/R_{\max }$ diagram. The black curve marks the limiting conditions based on the force balance model as described in Sec. 2. (b) Comparison between the experimental particle-bubble dynamics and the results based on the force balance model by plotting the bubble radius and PMMA particle displacement against time ($l_0/R_{max}=0.733, h/R_{max}=1.275$, and $a/R_{max}=0.108$). The initial azimuthal angle ${\theta }_0$ in the experimental cases are smaller than 10°. The errors bars are comparable to the size of the data points.

Figure 8

The initial relative position between the PMMA particle and the bubble during the bubble growth strongly influences the particle-bubble interaction. High-speed recordings are taken at 215 873 frames/s. Time $t=0$ is the beginning of the last image frame before a laser-generated bubble forms. The blue dashed line represents the position of the rigid boundary. (a) The particle is ejected by the bubble with a small cavity (indicated by “1”) and then impacts on the rigid boundary ($l_0/R_{max}=0.217, h/R_{max}=1.022$, and $a/R_{max}=0.094$). (b) The bubble pushes the particle away and hits the rigid boundary ($l_0/R_{max}=0.358, h/R_{max}=1.124$, and $a/R_{max}=0.070$). (c) The particle contacts the bubble during its growth and stays on the bubble surface until bubble collapses ($l_0/R_{max}=0.733, h/R_{max}=1.275$, and $a/R_{max}=0.108$). (d) The bubble presses the particle on the rigid boundary ($l_0/R_{max}=0.739, h/R_{max}=0.959$, and $a/R_{max}=0.068$). Details of the particle-bubble dynamics are available in the Supplemental Material [30].

Figure 9

Simulation results of limiting conditions when particle hits the boundary during the bubble growth and first collapse (a) $h/R_{\max }–l_0/R_{\max }$ diagram. (We added five more cases than Fig. 8 to present the limiting condition between Regime B and A/C.) (b) Particle impact velocity against $l_0/R_{\max }\left(h/R_{\max }\sim 1.3\right)$.

Figure 10

Examples where the PMMA particle motion is mainly driven by the collapse of the bubble. Time $t=0$ is the beginning of the last image frame before a laser-generated bubble forming. The blue dashed line represents the position of the rigid boundary. The red circle indicates the initial particle position, and the red dashed line shows the trajectory of the particle. (a) $l_0/R_{max}=1.047, h/R_{max}=1.251$, and $a/R_{max}=0.080$. The particle moves toward the boundary slightly and almost contacts with the boundary when the bubble expands and then moves backward when the bubble shrinks. After the bubble collapses, the microjet (indicated by “1”) accelerates the particle to hit the boundary with a high velocity $\sim 12.7\mathrm{m}/\mathrm{s}$. (b) $l_0/R_{max}=1.334, h/R_{max}=1.905$, and $a/R_{max}=0.136$. The particle comes in contact with the bubble surface when bubble rebounds and the collapse of the toroidal bubble drives the particle to move. (c) $l_0/R_{max}=2.212, h/R_{max}=3.005$, and $a/R_{max}=0.167$. The particle stays almost still until the remnant microbubbles from cavitation bubble collapse passes by (indicated by “2”). The particle then rotates counterclockwise and scrapes the boundary multiple times. Details of the particle-bubble dynamics are available in the Supplemental Material [30].

Figure 11

Simulation results of the interaction between the microjet and the particle ($p_{\mathrm{i}}=105\mathrm{MPa}, l_0=1.589\mathrm{mm}, h=2.764\mathrm{mm}$, and $a=235\mu \mathrm{m}$). (a) Images of the interaction between the microjet and the particle. In each frame the left part is the velocity field, while the right part is the pressure field. (b) Particle velocity as a function of time. (c) Pressure on the north pole and the south pole of the particle as a function of time.