## 4 Mathematical Model of Air Film Load-Bearing Characteristics  
This apparatus leverages the load-bearing phenomenon facilitated by the air film to achieve non-destructive support for the high-speed rotation of the cylindrical roller. In the context of the axial load-bearing air film for the cylindrical roller, established principles in fluid mechanics dictate that a decrease in air film thickness corresponds to an augmentation in load-bearing capacity. However, due to inherent geometric imperfections present on the upper and lower end surfaces of both the cylindrical roller and the vacuum suction cup, operational forces may induce vertical oscillations. In severe instances, this oscillation may culminate in contact between the lower end surface of the cylindrical roller and the base of the air flotation sleeve, resulting in detrimental effects on the end surface of the cylindrical roller. As an industry-wide practice, the thickness of the load-bearing air film is typically maintained at several tens of microns, a dimension that ensures maximal stiffness of the air film [18]. Regarding the radial load-bearing air film supporting the cylindrical roller, as explicated in the preceding sections, the non-destructive dynamic unbalance detection method outlined in this paper necessitates the generation of a cyclic force exerted by the cylindrical roller onto the air flotation sleeve. To optimize both the lubrication and load-bearing functions of this radial load-bearing air film, it is imperative to maximize its rigidity. Figure 5 illustrates the configuration of the air flotation sleeve and the gas flow dynamics. Upon the introduction of external compressed air into the interstice between the air flotation sleeve and the cylindrical roller, it engenders the creation of an air film characterized by a pressure distribution adhering to a specific pattern, as detailed in the references [19,20]. The differential pressure between the air film and the ambient atmosphere confers upon it a designated load-bearing capacity. The pressure distribution within the gas film is typically characterized by the Reynolds equation, a derivation drawn from the fundamental gas equation of motion, the ideal gas equation of state, and the equation governing mass conservation, resolved collectively. In practical engineering, the encountered viscous flow entails intricate boundary conditions, with flow parameters within the field exhibiting temporal and spatial variation leading to mutual interactions. Hence, prior to the calculation, it becomes necessary to posit the following assumptions. 1. The flow pattern within the gas film is laminar flow and conforms to [PERSON]'s laws of motion;
2. The flow of the gas film is purely viscous and incompressible, with a constant coefficient of viscosity;
3. The pressure variation in the direction of the gas film thickness is neglected, and the effects of gravity, magnetism, and other bulk forces are ignored;
4. The thickness of the gas film is very small and it is a single-phase medium with continuous flow;
5. The gas film is isothermal in the process of flow. Based on the above assumptions, the N-S equation is obtained after neglecting the minutiae terms:  
\[ \begin{cases}\frac{\partial p}{\partial x}=\frac{\partial}{\partial y}\left( \mu\frac{\partial u}{\partial y}\right)\\ \frac{\partial p}{\partial y}=0\\ \frac{\partial p}{\partial x}=\frac{\partial y}{\partial y}\left(\mu\frac{ \partial w}{\partial y}\right)\end{cases}  \tag{3}\]  
where \(p\) is the gas film pressure; \(\mu\) is the gas viscosity coefficient; and \(u\) and \(w\) are the velocity of the fluid unit moving along the \(x\)-direction and \(z\)-direction, respectively. The gas continuity equation is given by:  
\[\frac{\partial(\rho u)}{\partial x}+\frac{\partial(\rho v)}{\partial y}+\frac {\partial(\rho w)}{\partial z}+\frac{\partial\rho}{\partial t}=0 \tag{4}\]  
where \(\rho\) is the gas density, \(v\) is the velocity of the fluid unit moving along the y-direction, and \(t\) is the time.