The influence of cut-off speed in a Zeeman slower on the loading of blue magneto-optical trap based on the distribution function theory is analyzed and the relationship between the cut-off speed and the number of atoms trapped by the blue magneto-optical trap is investigated. A comparison between the fitting results from the distribution function theory and that from the general theory indicates that the distribution function theory model can better analyze the relationship between the above two. The cold atomic sample for strontium optical clock is prepared by the usage of a multi-turns Zeeman slower for slowing down the thermal atomic beam entering the blue magneto-optical trap. In the optical clock system, the design of the Zeeman slower is optimally designed based on the distribution function theory and thus the work efficiency of the Zeeman slower is improved. This research lays the foundation to achieve a more efficient and compact Zeeman slower and the miniaturization of an optical clock.