Three different initial positions of the bubble inception are investigated near elastic and rigid boundaries, namely γ = 0.81, 1.20 and 1.68. We study the “mushroom” bubbles in the first group with γ = 0.81 and 1.20 for elastic boundary. As for the second group, the companion of two cases with γ = 1.68 is displayed near the elastic and rigid boundaries to illuminate the mechanism of its opposite bubble migration.

#### 4.2. The Bubble Migration

The bubble migration is one of most important characteristics of a bubble. The previous works focus on the bubble migration away from the boundaries aiming at preventing cavitation at the surface of the fluid machinery.

Figure 11 shows the comparison of bubble initial position

γ and collapse position

b_{RMIN}/L near elastic and rigid boundaries, where

b_{RMIN} is the height above the wall at which the north and south poles of the bubble meet each other in

Figure 1. It is noted that collapse position

b_{RMIN}/L > 1 indicates bubble moves away from the boundary,

b_{RMIN}/L < 1 represents bubble migrates closely to the boundary, and

b_{RMIN}/L = 1 is that the center of bubble keep motionless. As observed, when the initial position

γ > 1.25, the collapse position

b_{RMIN}/L is always larger than 1 for the elastic boundary, while

b_{RMIN}/L is always smaller than 1 for the rigid boundary. That indicates the bubble always moves away from the elastic boundary, and migrates towards the rigid one when

γ > 1.25.

To further demonstrate the detail bubble shapes,

Figure 12 shows the temporal evolution of typical bubble shapes near the elastic and rigid boundaries for

γ = 1.68 and

R_{m} = 20.0 mm. As observed, two bubbles both oscillate at a far distance from the elastic and rigid boundaries, the Bjerknes force between the bubbles and boundary tends to be weak, resulting in a nearly spherical bubble oscillation for the first period. As shown in

Figure 12a, once reaching at the minimum volume at frame 5, the bubble near the elastic boundary has a bulge toward the interior of a bubble at the bottom of the bubble surface, and it moves upwards evidently with several oscillation cycles. As for the bubble near the rigid boundary shown in

Figure 12b, the bubble has a sunken at the top of the bubble surface after frame 5, and then it is attracted by rigid boundary and fiercely impacts on the rigid wall with two period of bubble oscillation.

To further demonstrable the relationship between the bubble motion and the boundary vibration,

Figure 13a shows the comparison of the temporal evolution of two bubble margins and mid-span displacement of elastic boundaries for

γ = 1.68 and

R_{m} = 20.0 mm, and its temporal shapes are shown in

Figure 12a. When the bubble expands from

t = 0 ms to

t = 2.5 ms, the elastic boundary is compressed by the bubble, and its mid-span deflection keep decreasing trend to the minimum value of

δ = −2.0 mm at

t = 2.5 ms. After that, the bubble begins to shrink and the elastic boundary also pushes back towards the equilibrium state. At

t = 4.0 ms, the elastic boundary reaches to the equilibrium state, but it still has an upwards motion due to vibration force. When the bubble reaches to the minimum volume at

t = 4.36 ms, the min-span deflection presents a positive value of

δ = 0.5 mm. As for the bubble near the rigid boundary, the mid-span deflection of the boundary is neglected due to its large stiffness, as shown in

Figure 13b.

Figure 14 shows the PIV-measured velocity fields and calculated pressure contours around a shrinking bubble, where

Figure 14a,b are from the case of elastic boundary for time

t = 3.90 and 4.25 ms, and

Figure 14c,d are from the case of rigid boundary for time

t = 3.90 and 4.25 ms. Due to the similarity with

Figure 9, the expansion stages in this figure are neglected, and the shrink stages of bubbles become the focus of attention in this part. It can be seen that there exists a high-pressure region at the bottom of the bubble surface near the elastic boundary with the maximum value of 3.5 MPa, while it appears at the top of the bubble surface near the rigid boundary with the maximum value of 12.0 MPa. Although two bubbles have the same initial position and maximum radius, the flow structures around them are obviously different, even opposite. It is well known that a bubble near the rigid boundary is always affected by a pair of opposing forces, namely, the buoyancy and Bjerknes force reported by Blake et al. [

40]. As for the elastic boundary case, the high-pressure region appears at the bottom of the bubble, indicating that Bjerknes force is smaller than the buoyancy force. As for the rigid boundary case, the high-pressure region appears at the top of the bubble, indicating that Bjerknes force is much larger than the buoyancy force. Therefore, the resultant force of Bjerknes force and buoyancy force dominate the migration direction of the bubble.