# Characteristics and Mechanisms of the Zigzag and Spiral Movement of Rising Bubbles in Still Water
## 3 Results and Discussion
### Zigzag Movement
#### 3.2.2 Movement Mechanism of Zigzag Movement  
The movement of the bubble is related to the surrounding fluid, and its acceleration can relatively reflect the force. Bubbles are mainly subject to buoyancy \(F_{n}\) and the tractive force \(F_{v}\) of the vortices at the tail of the bubble, as shown in Table 1. The edge velocity \(v\) of the vortex is about equal to the incoming velocity (bubble rising velocity). According to the Bernoulli equation, the pressure at the edge of the vortex can be obtained [25]. \[p=p_{0}-\frac{1}{2}\rho v^{2} \tag{4}\]  
Figure 6: Zigzag movement law of bubbles. (**a**) Changes in displacement and velocity over time. (**b**) Displacement and velocity frequency spectrum analysis diagram. (**c**) Changes in displacement and acceleration over time. (**d**) Displacement and acceleration frequency spectrum analysis diagram. The higher the incoming flow velocity, the lower the pressure at the vortex edge, and the greater the tractive force \(F_{v}\) on the bubble. The flow inside the vortex is not discussed here, because it cannot directly affect the movement of bubbles. The buoyancy \(F_{n}\) is related to the bubble volume. Air is compressible, and its compressibility is related to \(Ma\) (Mach number). When \(Ma\) < 0.3, air can be regarded as incompressible [26]. Therefore, the buoyancy is unchanged. It can be seen from Figure 6 that the bubble velocity oscillates within a range, without monotonic change, that is, the \(F_{n}\) and \(F_{v}\) of the bubble reach a dynamic balance. Table 1 shows the change in the trailing vortices in half a period of bubble movement (the movement trajectory is the bold part of the solid trajectory line in Table 1). The right side is the positive direction of the \(x\)-axis, and the upper side is the positive direction of the \(z\)-axis. The force analysis is shown in Table 1, where \(F_{R}\) represents the resultant force and \(v\) represents the resultant velocity. Throughout the whole process, the bubble always maintains a vertical upward velocity. During zigzag motion, the force on the bubble is in a dynamic equilibrium state. When the bubble moves to the midpoint of the oscillation, the buoyancy \(F_{n}\) and the traction force \(F_{v}\) of the two vortices on the bubble reaches equilibrium, and at this time, the bubble velocity is towards the upper right. Subsequently, the bubble moves to the right, causing the vortex on the right to fall off and \(F_{v2}\) to rapidly decrease. The bubble is subjected to a combined force towards the upper left. The horizontal velocity u decreases and the vertical velocity w increases. Because the vertical velocity determines the overall velocity of the bubble, the overall velocity q of the bubble gradually increases, and the vortex strength on the left side of the bubble increases. According to the Bernoulli equation, \(F_{v1}\) gradually increases, and the resultant force of the bubble rotates counterclockwise until it reaches the right endpoint. At this point, the direction of the resultant force is to the left, and the bubble velocity is at its maximum, and the direction is upward. Subsequently, the bubbles move to the left, the vortices on the right regenerate, \(F_{v2}\) increases, and the combined force of the bubbles faces the lower left. The horizontal velocity u increases, the vertical velocity w decreases, and the overall velocity q of the bubble decreases. The left vortex strength decreases. According to the Bernoulli equation, \(F_{v1}\) gradually decreases until the bubble returns to the midpoint of the oscillation, and the force is balanced. At this point, the bubble