## 1 Introduction  
Diesel engines find widespread applications in many industries, including transportation, agriculture, and power generation, among others. Soot and NOx emissions, which are products of fuel combustion in these engines, pose a threat to the environment and the health of living organisms. Over the years, strict global regulations have been set to reduce the negative impact of such emissions. These stringent legislations require manufacturers to design cleaner and more efficient engines [1, 2]. Many technologies, such as diesel particulate filters (DPFs) [1] and selective catalytic reduction (SCR) [3], have been developed to reduce emissions from diesel engines. In addition, controlling the combustion process (e.g., by using low-temperature combustion (LTC) [2], homogeneous charge compression ignition (HCCI) [4], reactivity-controlled compression ignition (RCC) [5], and premixed charge compression ignition (PCCI)) have also attracted significant research interest [2]. Research on the fuel mixing process for preparing diesel combustion components in cylinders through effective injection rate adjustment is an important consideration. In this study, we only consider injection rate adjustment in the fuel injection system without changing the air intake and other contexts of the system; thus, this study does not modify the existing injection system too greatly. This is due to the significant influence of the air-fuel mixing process on combustion efficiency and exhaust emissions in diesel engines. The latest developments in fuel injection strategy efficiency tend to focus on the injection rate to increase fuel mixing efficiency and reduce emissions. Notably, the combustion time will be shorter for cases with high initial injection rates. Generally, a high initial injection rate will result in better atomization and air entrainment. The duration of combustion decreases as the mixture of air and fuel improves, meaning that combustion occurs faster. Conversely, under a low initial injection rate, the initial atomization of the fuel and premixing are not good. The slow initial injection fuel droplets will combine with faster fuel droplets, which will result in a larger droplet size that yields a poor spray breakup. The combustion period will then be longer as more time is required to inject the fuel to atomize and evaporate it for combustion. In the past, researchers were interested in investigating the injection rates to improve fuel injection strategies. [PERSON] et al. [6] noted that increasing the injection rate after a previous injection is sufficient to increase the collision frequency and formation of large droplets, resulting in high momentum and greater penetration. [PERSON] et al. [7] found that a higher peak injection rate yielded a higher spray tip penetration, peak entrainment rate, and entrainment rate after the end of injection (EOI). In addition, [PERSON] et al. [8] suggested that the start of injection is the main parameter that affects the impingement phenomenon, whereas [PERSON] et al. [9] found that the start of combustion for each rate shape is different, although the injection duration and start of injection are same due to the start of pressure rise in each case. The peak in-cylinder pressures are higher when the start of combustion is advanced due to the injection rate shaping. The combustion duration will be shorter with a higher initial injection velocity. Apart from that, the results from numerical study on the effects of boot injection rate shapes by co-workers of [PERSON] and [PERSON] [10; 11; 12], showed that NOx decreased but large soot particles occur due to low injection velocity and narrow soot distribution when the main injection velocity is higher [11]. [PERSON] et al. [12] found that higher boot injection velocity and shorter boot injection duration resulted in shorter ignition delay and more fuel burning at the premixed combustion stage.