## 3 Results and Discussion
### Discussions  
According to the analysis results, the rotational speed significantly influences the axial and circumferential distribution of force chains. However, the helical transportation process of particles is intricate, and it is essential to delve deeper into the relationship between the force chain distribution and flow fluctuation. The quantitative characterization of the flow fluctuation of a worm-type distributor was conducted, following the test method provided by NY/T1003-2006 [37]. The simulation extraction time was set at 19 s-25 s, and the calculation formula is as follows:  
\[\tilde{\mathbf{m}}=\frac{\sum_{i=1}^{\mathbf{j}}\mathbf{m}_{i}}{\mathbf{j}} \tag{14}\]  
\[\mathbf{S}=\sqrt{\frac{\sum_{i=1}^{\mathbf{j}}(\mathbf{m}_{i}-\overline{ \mathbf{m}})}{\mathbf{j}-1}} \tag{15}\]  
\[\mathrm{CV}_{1}=\frac{\mathbf{S}}{\overline{\mathbf{m}}} \tag{16}\]  
where \(\overline{m}\) is the mean mass of samples in all intervals. \(m_{i}\) is the mass in the interval \(i\). \(S\) is the standard deviation of the mass of all intervals. \(\mathrm{CV}_{1}\) is the uniform coefficient of variation. The uniform coefficient of variation under different rotational speeds is depicted in Figure 19. Clearly, the \(\mathrm{CV}_{1}\) decreases with the increase in rotational speed. This directly demonstrates that the flow fluctuation decreases with the increase in rotational speed. It is crucial to note that this aligns with the axial distribution laws of force chains. Consequently, it is deduced that force chains mainly exhibit axial distribution in the worm-type distributor, influencing the uniformity of flow fluctuation. Figure 18: The change in particle circumferential distribution kinetic energy, speed of (**a**) 20 rpm, (**b**) 50 rpm, and (**c**) 80 rpm. To verify the above deduction, the distribution diagram of the axial speed and circumferential speed of fertilizer particles in the barrier zone under different rotational speeds is presented in Figure 20. Evidently, the axial speed exhibits a linear growth with the increase in rotational speed, while the circumferential speed remains almost the same. Moreover, the axial speed is significantly higher than the circumferential speed as the rotational speed increases. In this scenario, the axial translation of particles plays a dominant role. This observation aligns with the research results of [PERSON] and [PERSON] [2,10]. It is clear that the force chain transmission among particles is primarily axial due to axial translation. This is consistent with the previous conjecture. For a comprehensive analysis of the influence of the axial force chain distribution on flow fluctuation, the relationship between the CV of axial force chain distribution and the CV\({}_{1}\) of the flow fluctuation was established through function fitting, as shown in Figure 21. A mathematical model was constructed:  
\[\text{Y}=6.365\text{X}-2.181 \tag{17}\]  
where Y is the CV\({}_{1}\) of flow fluctuation, and X is the CV of the axial force chain distribution. In this mathematical model, R\({}^{2}\) values are 0.95, indicating that the fitting line has high accuracy. It can be seen from the mathematical model that the axial force chain distribution is positively related with the CV\({}_{1}\) under different rotational speeds. With respect to the chosen worm-type distributor model, the flow fluctuation reaches the minimum when the rotational speed is 80 rpm. Figure 19: Uniform coefficient of variation in different rotational speeds. Figure 20: The distribution of axial velocity and circumferential velocity at different rotational speeds. Furthermore, it is crucial to emphasize that an increasing rotational speed enhances the contact force among particles and stabilizes the network structure that is formed by particle contact. However, the centrifugal force increases with the rise in rotational speed, leading to a significant decline in the final transport efficiency and performance.