# Morphological Analysis of a Collapsing Cavitation Bubble near a Solid Wall with Complex Geometry
## 4 Morphological Analysis Method Exerted on the Cavitation Bubble Collapse Model  
### Morphological Verification  
The phenomenon examined in this study belongs to multiphase flow. However, it is necessary to verify the validity and feasibility of the Min-MAM in quantitatively describing the evolution of collapsing bubble profiles. Therefore, Figure 8 depicts the time evolution  
Figure 7: Evolution of pressure and velocity fields of the collapsing bubble near a solid wall with complex geometry for \(\lambda=1.5\) and \(D=1.03476\): (**a**) \(t=0\), (**b**) \(t=550\) ts, (**c**) \(t=670\) ts, (**d**) \(t=705\) ts, (**e**) \(t=735\) ts, and (**f**) \(t=770\) ts.  
of the Minkowski measures for the collapsing cavitation bubble near a solid wall with a complex geometry. This figure also includes the density field at several representative moments and the extended dot-line diagrams of the Minkowski measures for the period before the second collapse.  
As shown in Figure 6, the collapsing bubble near the solid wall generally goes through the stages of inward collapse, top depression, micro-jet formation, first collapse, and second collapse. Therefore, as shown in Figure 8, the entire evolution process can be divided into the following parts: the first deformation from \(t=0\) to \(t=550\) ts, the second deformation before \(t=705\) ts, the first collapse at \(t=705\) ts, the circular bubble formation from \(t=705\) ts to \(t=770\) ts, and the second collapse at \(t=770\) ts. In the first deformation part, the initial spherical bubble deforms into an elliptical bubble under the pressure difference \(\Delta p\). The fractional area \(F\) and boundary length \(U\) decrease during this period, and their slopes increase with time \(t\). This indicates that the inward collapse velocity of the bubble increases gradually. In addition, the fluid velocity increases toward the solid wall, thereby forming an inceptive micro-jet. In the second deformation part, \(F\) and \(U\) continue to decrease, but their corresponding accelerations begin to increase. From \(t=550\) ts, the bubble top wall begins to sag downwards, which partly compensates for the effect of the inward collapse on the bubble fractional area and boundary length. When the upper and bottom bubble walls collide at \(t=705\) ts, the cavitation bubble splits into two small asymmetrical bubbles. After the first collapse, the visible bubble surface is composed of two asymmetrical bubbles in the 2D space and an annular bubble in the three-dimensional (3D) space. At this stage, the smaller bubble collapses earlier than the larger bubble, causing the slope of the \(U(t)\) curve to increase significantly. After the second collapse, the values of \(F\) and \(U\) simultaneously decrease to zero.  
As the third Minkowski measure, the Euler characteristic \(X\) can directly reflect the number of bubbles in the bubble collapse process. \(X\) remains at 1 before the first collapse. \(X\) increases to 2 at \(t=705\) ts and remains at 2 before the smaller bubble collapses. \(X\) decreases to 1 when the smaller bubble collapses. \(X\) decreases to 0 after \(t=770\) ts, indicating that the cavitation bubble completely collapses. In the curves for \(U(t)\) and \(X(t)\), the red circles in the figure can be described as \"the first collapse point\" and \"the second collapse point\".  
The results demonstrate that the three Minkowski measures can describe the morphological characteristics of the collapsing bubble in the vicinity of a solid wall with a complex geometry from different perspectives. The three curves can clearly distinguish between several typical moments in the cavitation bubble collapse process. Hence, we success  
Figure 8: Time evolution of Minkowski measures for the collapsing bubble near a solid wall with complex geometry. (**a**) the whole process from \(t=0\) to \(t=800\) ts, (**b**) the part process from \(t=680\) ts to \(t=780\) ts.  
fully verify the validity and feasibility of the Min-MAM in quantifying the morphological evolution of the collapsing bubble.