# Numerical Analysis via Mixed Inverse Hydrodynamic
## 1 Introduction
According to mixed lubrication theory, the normal load acting on the two surfaces in sliding relative motion is divided and supported by the contact force of the asperities on the rough surface and the hydraulic force increased from the oil film interposed between the two surfaces; ignoring the deformation of the rubber seal due to the tangential force, it is clear that the contact force in the FE model (occurring between the two surfaces) is the sum of these two forces. Here, ignoring the contact force of a rough surface becomes IHL theory. The IHL method has the advantage that the calculation time is much faster than that of the EHL method because the deformation of the rubber seal does not need to be additionally calculated after the contact pressure of the rubber seal surface is calculated through FE analysis. Owing to its convenience, IHL analysis has been used to analyze the behavior of various seals, such as O-rings [9; 11; 12], rectangular seals [13; 14; 15], and U-cup seals [16; 17]. However, IHL analysis that assumes perfect fluid lubrication, without considering rough surface asperity contact, was found to be very inaccurate in predicting the amount of leakage and frictional force. In particular, the frictional force was found to be significantly smaller than that of the experiment [10]. Thus, a mixed IHL study was conducted by combining the Greenwood-Williamson (GW) contact model with IHL theory to account for the effect of contact between the rough surfaces of the two plates [10; 18; 19]. The interface between the rubber seal and rod experiences a continuous sliding motion. In a state where a sufficiently thick oil film is not formed, such as mixed fluid lubrication and boundary lubrication, numerous asperities on each surface come into direct contact. As the asperity contact friction is much greater than the fluid friction, the friction force increases rapidly with increasing asperity contact ratio. Most of this mechanical friction energy is converted into frictional heat, which causes the temperature of the sealing zone to rise, and the lubrication state can be changed by increasing the temperature of the lubricating oil in the sealing zone. In a state where frictional heat generation is extremely serious, problems--such as breakage of the lubricating film, changes in material properties, and surface melting--may occur. Therefore, it is important to design a mechanical system with consideration for temperature. In this study, the sealing performance was numerically analyzed by applying it to a U-cup seal. The novelty of this study is the investigation of the effect of the mixed IHLmethod, considering the temperature increase due to frictional heat in the sealing zone. The static contact pressure between the seal and the rod was simulated using ABAQUS. The oil film distribution in the sealing zone was calculated using IHL theory and was modified to a mixed lubrication model by combining it with the GW model. The frictional heat generated by the fluid shear and asperity contact friction in the mixed lubrication state was calculated and the temperature rise of the sealing zone by this frictional heat was calculated via moving heat source theory. From the numerical analysis results, it was confirmed that when predicting the sealing performance of rubber seals, it is very important to consider the thermal effect under the operating conditions at high temperatures or with a high asperity contact ratio.