## 1 Introduction
[21] compared biodegradable and low-toxicity ester-based oils with commercial mineral oils to evaluate the efficiency of gear transmission. Research shows that [PERSON] and [PERSON] [22] found that gear lubricants with an increased viscosity index can significantly reduce friction and agitation losses. [PERSON] et al. [23] proposed a TEHL simulation method aimed at studying the effects of different lubricants on gear load related losses. In addition, the experimental results of [PERSON] et al. [24] indicate that even if the viscosity grade of the lubricating oil is the same, its power loss may vary significantly due to differences in the type of base oil and the combination of additives. Therefore, lubrication conditions are identified as a key factor affecting the overall meshing efficiency of gears [25; 26; 27; 28]. The application of lean lubrication techniques is becoming increasingly widespread in modern mechanical tools and other equipment, especially in liquid and compressed air supply automation. Minimum Quantity Lubrication (MQL) and Air-Oil Lubrication (AOLS) are standard lean lubrication techniques. MQL is a lubrication technique used in metal cutting processes that uses a minimal amount of lubricant to reduce friction and heat while minimizing negative impacts on the environment and workpiece. AOLS is a lubrication method that combines lubricating oil and compressed air, making it especially suitable for mechanical equipment that requires long-distance transportation of lubricating oil [29]. There is a high correlation between the implementation of automated liquid and air supply control and injection parameter optimization in lean lubrication systems. Lean lubrication techniques, including Minimum Quantity Lubrication (MQL) and Air-Oil Lubrication (AOLS), are increasingly adopted in modern machinery, emphasizing automation in liquid and air supply systems. MQL minimizes environmental and workpiece impacts by using small amounts of lubricant to reduce friction and heat, while AOLS combines oil and compressed air for long-distance lubrication. The optimization of injection parameters in these systems is closely linked to automated control, enhancing efficiency and sustainability. In lubrication research, plant-derived base oils are gaining traction due to their biodegradability and low toxicity, though challenges exist in thermal stability and antioxidant properties, especially under high temperatures. Nano-additives improve thermal management, reduce friction, extend equipment life, and enhance wear resistance, making them crucial for advanced lubrication systems. Additionally, appropriate lubricants and cooling techniques significantly influence machine temperature distribution, enhancing efficiency and reliability. The integration of AI and IoT technology offers substantial benefits by enabling real-time monitoring and adaptive lubrication control, optimizing performance, reducing maintenance, and boosting productivity [30; 31; 32; 33]. Presently, numerous scholars have extensively employed various methods to explore the optimal structural and oil injection parameters for gears. However, in practical engineering applications, different gear models must be adapted to various mechanical systems. Therefore, achieving optimal lubrication operations for various gear models became the central research topic of this study. This paper particularly focuses on the significant pressures present around the oil injection nozzle, aiming to investigate the oil injection speed guide curves under different variable tooth thickness gear structural parameters. By integrating the VOF multiphase flow technique with the Fluent UDF program, this paper accurately calculates the lubricant volume fraction on the surface of variable tooth thickness gears. Furthermore, through fitting and analyzing the oil injection speed curves under different gear structural parameters, a systematic guidance scheme for optimizing the oil injection lubrication of variable tooth thickness gears is proposed.