# Numerical Study on the Rising Motion of Bubbles near the Wall
## 1 Introduction  
Two-phase flow is a very common phenomenon. As an important branch of fluid mechanics, it began to develop rapidly in the 1960s. It is mainly used to study the interaction between two phases and the changes in flow patterns and motion disciplines. Generally, it is divided into gas-liquid two-phase flow, solid-liquid two-phase flow and liquid-liquid two-phase flow according to the phase state of the substances constituting the flow system [1]. Gas-liquid two-phase flow is one of the most common types of multi-phase flow. It refers to the flow phenomenon of gas-phase fluid and liquid-phase fluid in the same flow system [2, 3]. The motion and change of the gas phase in the flow will affect the characteristics of the two-phase flow. Therefore, studying the motion discipline of the gas phase is the key to studying the characteristics of the two-phase flow. The rise of bubbles in water is a very complicated and unstable process. The bubbles continue to rise in the water. Due to the combined action of gravity, buoyancy and surface tension, bubbles will deform, burst and coalesce. Eventually, the rising path of the bubble will be unstable, usually spiral, \"Z\"-shaped motion, etc. [4, 5]. Bubbles deform and break into many \(\mu\)m-level micro-bubbles during their rising motion. The bubbles usually appear in the form of micro-bubble groups, so it is also very important to study the interaction between bubbles. In this article, the influence of the wall surface on the motion characteristics of single and double bubbles will be explored, and the discipline of motion of bubbles near the wall will be studied. Since the discipline of bubble motion in liquid is widely used in the chemical industry, bubble dynamics theory was established to study the bubble motion and the interaction between bubbles [6]. At present, many scholars have conducted researches on the motion discipline of a single bubble and the motion discipline of bubble groups. The American scientist [PERSON] et al. [7, 8] studied the shape change of bubbles in the liquid after consulting a large number of data in the literature. According to the dimensionless criterion, they drewthe bubble shape phase diagram, which is called the Grace diagram. Some scholars [9; 10; 11] also verified the bubble shape phase diagram obtained by [PERSON], and found that the result is consistent with the bubble phase diagram. [PERSON] [12] found that when the equivalent diameter of the bubble is less than 1.8 mm, because the bubble shape tends to be spherical, the rising path presents a straight line; when the equivalent diameter of the bubble exceeds 1.8 mm, the path of bubble will change from a straight line to a \"Z\" shape. The bubble-free surface interaction has been experimentally investigated by [PERSON] et al. [13], when bubbles are close together, their mutual interactions can influence the existence of free surfaces. These factors can influence the migratory behavior and lifetime of the bubbles. [PERSON] et al. [14] studied the near-wall motion of deformable bubbles in viscous fluids, and found that the bubbles have a significant lateral deviation in the wall domain. When the distance between the bubble and the wall changes, the bubble will bounce due to the pressure difference between the two sides and the lift between the wall, which is the \"Z\"-shaped motion path mentioned above. [PERSON] et al. [15] tracked the three-dimensional deformation of the two-phase interface based on the numerical simulation method of Front Tracking and compared the data with unbounded still water. They found that the bubbles around the wall would be suppressed by the wall, causing the bubbles to gradually deviate from the wall. [PERSON] et al. [16]further proposed a high-resolution numerical method based on the level set method to solve the motion and deformation of the gas-liquid interface. It is found that the rising bubbles will be affected by the wall surface and migrate laterally to the central domain when they are close to the wall surface.