## 5 Experiment and Analysis of Carbon Particle Adhesion Under Different Flow Velocity
Therefore, the fluid closer to the solid surface will experience stronger obstruction and lower flow velocity. The velocity of each layer of fluid gradually increases from the solid surface upwards until it is equal to the free flow velocity. This phenomenon means that there is a velocity gradient in the normal direction of the solid surface, and the particles will inevitably deviate towards the side with higher velocity and lower pressure of the transformer oil. At this time, the particles will be subjected to a force along the velocity gradient direction of the flow field, called Saffman force \(F_{S}\)[26]. The force situation of particles in critical adhesion conditions under the flow of transformer oil is as shown in Figure 15.  
The expression for the Saffman force \(F_{S}\) exerted on particles is shown in Equation (14). \[F_{S}=-K_{S}\sqrt{\rho_{oil}}\mu R^{2}\sqrt{\left|\frac{\partial(u-u_{p})}{ \partial y}\right|}\left|u-u_{p}\right| \tag{14}\]  
Figure 15: Schematic diagram of the forces on carbon particles under the action of flowing transformer oil. Among them, \(K_{S}\) is the slip shear coefficient, generally taking an empirical value of 6.46, and \(u\) and \(u_{p}\) are the horizontal velocities of the transformer oil and particles, respectively. Since the particles adhere to the solid surface, the particle velocity \(u_{p}=0\). Comparing Figures 10 and 15, when analyzing the critical adhesion conditions under which adhesive particles are pulled apart by flowing transformer oil, \(F_{S}\) needs to be taken into account, and the corresponding critical external load \(N_{cf}\) is shown in Equation (15). \[N_{cf}=F_{E}+F_{G}+F_{S}=-\frac{3\pi\omega^{*}R}{2} \tag{15}\]  
Due to the fact that the \(F_{S}\) and \(F_{E}\) directions are the same, both tend to pull apart particles adhering to the solid surface. Therefore, compared to stationary transformer oil, the probability of particle adhesion in flowing transformer oil decreases, and its critical adhesion charge \(Q_{cf}\) decreases compared to \(Q_{cs}\) in Equation (9), as shown in Equation (16). \[Q_{cf}=Q_{cs}-\frac{K_{S}\sqrt{\rho_{oil}\mu R^{2}}\sqrt{\left|\frac{\partial u }{\partial y}\right|}\left|u\right|}{\lambda_{E}E} \tag{16}\]  
In this experiment, due to the distance \(h\) between the epoxy resin boards being much smaller than the width of the boards, the flow of transformer oil between the boards can be approximated as a two-dimensional Poiseuille flow, with the flow velocity gradient direction perpendicular to the boards, and its value is determined by the distance \(y\) from the reference point to the nearest solid surface, as shown in Equation (17). \[\frac{\partial u}{\partial y}=\frac{8 yu}{h^{2}} \tag{17}\]  
Combining Equations (16) and (17), it can be seen that \(Q_{cf}\) is negatively correlated with the flow velocity u between the boards. The higher the flow velocity of the transformer oil, the smaller the critical adhesion charge \(Q_{cf}\) of the particles, and the lower the probability of particle adhesion. In summary, the flow velocity of transformer oil affects the adhesion degree of carbon particles. When the transformer oil is in a specific low flow velocity range, a large number of carbon particles migrate directionally to the range of electric field action with the oil flow, equivalent to an increase in particle concentration. At this time, the number of particles colliding with the solid surface increases, resulting in an increase in particle adhesion degree compared to static conditions; when the flow velocity of transformer oil is high, the influence of oil flow on particle adhesion will gradually dominate. The fluid near the solid surface has a longitudinal velocity gradient due to viscous friction. Under the combined action of Saffman force and electric field force, particles adhering to the solid surface are more likely to break free from the adhesive force and be pulled apart.