## 3 Results and Discussion
### Effect of the Outlet Structure of the Ball Seat on the Flow Characteristics  
Increasing the diameter of the ball seat improved the flow status of the drilling fluid. However, due to the abrupt change of the diameter in the ball seat outlet, a severe vortex and turbulence occurred within the differential motion assembly. To reduce these behaviors, it is necessary to improve the outlet structure of the ball seat. To study the effect of the outlet structure of the ball seat on the flow characteristics, and considering machining and structural strength factors, three outlet structures of the ball seat--arc, cone, and original right-angled structure--were designed, as shown in Figure 12. Their effect on the velocity is shown in Figure 13. The internal velocity distribution of the differential motion assembly of the cone outlet at the ball seat was similar to that of the original right-angled outlet. Their maximum velocities in the differential motion assembly were both 36 m/s at the axis behind the ball seat outlet. Notably, the arc ball seat outlet structure greatly decreased the fluid velocity in the differential motion assembly, and the maximum velocity was only 34 m/s at the ball seat inlet. In addition, the axial length of the vortex area in the differential motion assembly with a cone outlet structure at the ball seat was 211 mm, which was only 9 mm less than that for the original right-angled outlet. However, the axial vortex length in the differential motion assembly with the arc outlet structure decreased to 0.7 times that with the original right-angled outlet. Figure 11: Effect of the ball seat diameter on the pressure distribution; the fluid viscosity was set as 100 mPa\(\cdot\)s.  
To quantitatively study the effect of the outlet structure of the ball seat on the velocity distribution in the differential motion assembly, we set two velocity monitoring lines (shown in black and white) along the axial and radial directions, respectively. The velocity distributions at the two monitoring lines for different outlet structures at the ball seat are shown in Figure 14. In Figure 14a, according to the structural characteristics of the differential motion assembly, the computational domain of the differential motion assembly was divided into the inlet, ball seat, and outlet sections. The change in the outlet structure at the ball seat had almost no effect on the axial velocity distribution of the inlet section. However, when the drilling fluid entered the ball seat and outlet sections, the velocity of the arc structure was significantly less than that of the cone structure and the original right-angled structure. Notably, in the ball seat section of the arc structure, the drilling fluid first rose to 33 m/s and then started to decrease at the arc of the ball seat section. However, for the cone and original right-angled structure, the fluid continued to accelerate in the ball seat section. After entering the outlet section, the fluid continued to advance at a velocity greater than 35 m/s for approximately 130 mm before the velocity began to gradually decrease. In addition, as shown in Figure 14b, the maximum velocity of the central jet zone of the arc structure was only 28 m/s, which was 0.8 times that of the cone and the right-angled structure, and the vortex velocity near the wall of the arc structure was only 0.4 times that of the cone and right-angled structures. Therefore, it can be concluded that the arc structure in the ball seat outlet effectively decreased the fluid velocity in the differential motion assembly and improved the severe vortex and turbulence behaviors  
Figure 12: Differential motion assembly with different ball seat outlet structures: (**a**) arc, (**b**) cone, and (**c**) right angle. Figure 13: Effect of the outlet structure of the ball seat on the flow velocity; the fluid viscosity was set as 100 mPa-s.  
in the differential motion assembly. Furthermore, the erosion of the inner surface of the differential motion assembly by the drilling fluid could be improved.