## 5 Finite Element Analysis of Air Film Load-Bearing Characteristics  
To validate the accuracy of the proposed mathematical model and to investigate the gas flow dynamics within the air flotation sleeve, a finite element analysis of the air film was conducted. A representative air flotation sleeve with specifications denoted as \(L=D=5\) mm was selected for this purpose. The 3D fluid domain was constructed using SolidWorks and subsequently imported into Fluent for mesh generation. The physical field reference type was designated as Watertight Geometry. Inlet ports were labeled as pressure inlets, while outlet ports were denoted as pressure outlets. The surface interfacing with the cylindrical roller, where the air film came into contact, was identified as wall-force; all other surfaces were categorized as walls. Given the minuscule thickness of the air film, ranging from a few microns to a few tens of microns, a boundary layer in the direction of the film thickness was incorporated, comprising three layers. The minimum radial air film thickness was set at 0.01 mm, with a corresponding minimum mesh size of \(7\times 10^{-3}\) mm. Subsequent refinements in mesh size did not yield substantial alterations in solution outcomes. The final mesh comprised 179,802 elements, with a minimum mesh mass of 0.285, meeting the solution criteria. For the minimum axial air film thickness of 0.005 mm, the minimum mesh size was established as \(5\times 10^{-3}\) mm. Similar reductions in mesh size did not induce noteworthy changes in the solution results. The mesh for this configuration consisted of 96,859 elements, with a minimum mesh mass of 0.262, in conformity with the solution prerequisites. The resulting model following mesh generation is depicted in Figure 6.  
After the meshing is completed, the parameters and boundary conditions of the fluid analysis were set. 1. The solver was set to pressure-based, the velocity formulation is set to absolute, and the time is set to steady. 2. K-epsilon (2 eqn) was selected for the model, standard was chosen for the k-epsilon model parameter, standard wall functions were set for near-wall treatment, and other control factors were defaulted. 3. The material of the fluid was set to ideal air; the boundary conditions of inlet pressure were set to 600,000 Pa oriented perpendicular to the inlet interface inward; and the boundary conditions of outlet pressure were set to 100,000 Pa, oriented perpendicular to the outlet cross-section outward. 4. A force report definition was created, and the force was a positive pressure (N) on the wall-force surface, that is, the load-bearing force of the air film on the cylindrical roller. Figure 6: Air film mesh model. (**a**) Axial load-bearing. (**b**) Radial load-bearing. (5) An initialization method was selected standard initialization. Execute the solution after initialization. Figure 7 depicts the pressure distribution within the air film. Observably, the high-pressure zone of the axial load-bearing air film was notably concentrated in proximity to the pressure-equalizing groove. Gradually, the pressure diminished from this region towards the periphery and center, eventually reaching atmospheric levels. As for the radial load-bearing air film, the elevated pressure region primarily occupied the central expanse of the cylinder. Analogously, the pressure gradually tapered off towards atmospheric levels from this central region towards the upper and lower extremities. The axial and radial air film thicknesses were systematically varied as parameters, enabling a comprehensive examination and computation of the corresponding air film load-bearing capacities across different thicknesses. Additionally, two alternative configurations of air flotation sleeves were introduced for comparative analysis. Employing Fluent, a suite of simulation procedures was executed in accordance with the aforementioned protocols.