# Pattern and Analysis of Corrugation-Sand Retaining Seals for
## 2 New Structure Sealing Theory and Model  
### Theoretical Basis  
When analyzing the leakage problem of the sealing ring, the inverse problem theory of hydrodynamic lubrication and the [PERSON] theory [26] are often used as the research basis. [PERSON] theory avoids the use of complex hydrodynamic calculations. Considering that the internal friction generated by pressure and fluid viscosity is far greater than the influence of all other factors, it is a greatly simplified theoretical method compared to the inverse problem of hydrodynamic lubrication. Through analysis, the corrugated main seal structure involved in this paper can use this theory as the theoretical basis. [PERSON] theory is as follows.  
1. Velocity distribution. The velocity distribution in a tiny gap can be expressed by the following formula: \[x=\frac{vy}{h}+\frac{\partial p}{\partial x}\cdot\frac{h^{2}}{2\eta}\bigg{[} \Big{(}\frac{y}{h}\Big{)}^{2}-\frac{y}{h}\bigg{]}\] (1) where \(v\) is the moving speed, \(h\) is the gap height, \(\eta\) is the dynamic viscosity, and \(\frac{\partial p}{\partial x}\) is the pressure gradient. As shown in Figure 2, there is a pressure difference \(\Delta P\) inside and outside the seal cavity, \(P_{1}\) represents the pressure in the seal cavity, \(P_{2}\) represents the pressure outside the seal cavity, when the seal ring does not slide, the flow speed of the lubricating medium in the seal gap is shown in Figure 2a, the flow velocity of the lubricating medium near the wall is close to zero due to its own viscosity and wall friction, the flow velocity of the area away from the wall is determined by the pressure difference and the internal friction of the fluid. When there is no pressure difference between the inside and outside of the seal cavity, and the seal ring is sliding, the flow speed of the lubricating medium is shown in Figure 2b. The lubricating medium close to the wall of the sliding seal ring moves with the seal ring due to the friction of the wall, and the velocity distribution process close to the wall decreases to zero with the viscosity gradient. When there is a pressure difference between the inside and outside of the seal cavity and the seal ring slides in the direction of \(p_{2}\), the flow velocity of the lubricating medium is shown in Figure 2c, in this case, the flow situation of the fluid in the gap is the positive superposition of Figure 2a,b. When there is a pressure difference between the inside and outside of the seal cavity, and the seal ring slides in the direction of \(p_{1}\), the flow velocity of the lubricating medium in the seal gap  
Figure 1: Corrugated wedge sand retaining seal structure diagram of roller cone bit: (**a**) Assembly drawing, and (**b**) Circumferential development view of main seal. 1—roller cone, 2—steel ball, 3—tooth palm, 4—new type of sealing ring.  
is shown in Figure 2d; in this case, the flow situation of the fluid in the gap is the reverse superposition of Figure 2a,b.  
2. Volume flow. \[Q=\frac{1}{2}vh-\frac{1}{12}\cdot\frac{\partial P}{\partial x}\cdot\frac{h^{3}}{\eta}\] (2) where the first term is the flow velocity generated by the wall motion, and the second term is the influence of the pressure difference. It can be seen that the sealing gap \(h\) has a great influence on the leakage.  
3. Fluid friction. \[\begin{split}&\text{The fluid shear stress is:}\\ &\tau=\frac{v\eta}{h}+\frac{\partial p}{\partial x}y-\frac{ \partial p}{\partial x}\frac{h}{2}\end{split}\] (3) The sealing friction is: \[F=\pi dL\frac{v\eta}{h}-\pi dL\frac{\partial p}{\partial x}\frac{h}{2}\] (4) where \(d\) is the shaft diameter and \(L\) is the contact length.  
When the friction generated by the fluid flow dominates:  
\[F=\pi dL\frac{v\eta}{h}\] (5) Oil film thickness: \[h_{1}=\pi dL\frac{V\eta}{F} \tag{6}\]