## 1 Introduction  
The bearing chamber is an important part in the aero-engine lubrication system. The lubricating oil is shed into the bearing chamber in the form of oil droplet via under-race lubrication method [1]. After the high-speed moving oil droplets collide with the wall, some of them are deposited to form the oil film, and others are splashed to form a large number of secondary oil droplets. Then, the complex air-oil two phase flow and heat transfer state of air, oil droplet, and oil film coexist in the bearing chamber. Because the oil droplet size is very small, many secondary oil droplets are suspended in the bearing chamber under the action of the air phase flow field. Furthermore, the interaction between oil droplets induces many dynamic events, such as collision, coalescence, deposition, evaporation, breakup, etc. The coalescence and breakup of oil droplets will affect the size distribution of secondary oil droplets and further restrict the research of flow and heat transfer in the bearing chamber. Moreover, the oil droplet size distribution directly affects the mixture state of oil and gas. Previously the Rosin-Rammler (R-R) distribution is widely used for the calculation of oil droplet size distribution in bearing chamber. However, experimental research indicates that R-R distribution is not consistent with the actual condition. Therefore, the study of droplet diameter distribution of the secondary oil droplets, which considers the coalescence and breakup of oil droplets, is of great significance in the aero-engine lubrication design. At present, the research on coalescence and breakup of oil droplets mostly adopts the population balance model, which is earliest used in chemical process by [PERSON] et al. [2]. Since then it has become a general method to analyze the particle diameter distribution of dispersed phase in multiphase system. The population balance model describes the time and space distribution of discrete phase by tracing the change of number density function of discrete phase with convection, diffusion and the source term of coalescence and breakup in the flow field. This method has been widely used because it remains part of the size information of discrete phase, and uses the averaging method to describe the motion of continuous phase, which reduces computation greatly. The core of population balance model lies in the profound understanding of coalescence and breakup mechanism of oil droplets. The coalescence and breakup of oil droplets is affected by the flow field around the droplets and the physical parameters (such as surface tension) of the oil droplet, which involves a very complicated multi-level and multiscale physical mechanism. The mechanism description of oil droplet breakup can be divided into three categories: (1) the droplet breakup caused by turbulence, (2) the droplet breakup caused by viscous shear force, and (3) the droplet breakup caused by the instability of the liquid-liquid interface. The coalescence mechanism of oil droplet is more complex. Moreover, the probability model is used more often, as the coalescence rate of oil droplet is equal to the product of collision frequency and coalescence efficiency between oil droplet. The collision mechanism of oil droplet can be divided into four categories: (1) collision caused by random turbulence fluctuation, (2) collision caused by buoyancy, (3) collision caused by shear force, and (4) collision induced by wave vortex. The coalescence efficiency model can be divided into three categories: energy model, velocity model, and liquid membrane drainage model. During the past decades, many scholars have done many researches on the coalescence and breakup of bubbles/droplet and the size distribution of bubbles/droplet. [PERSON] and [PERSON] et al. [3,4] reviewed the coalescence and breakup model of liquid, and introduced the research process of oil droplet coalescence and breakup mechanism. [PERSON] et al.