## 1 Introduction  
In self-acting (aerodynamic) gas bearings, aerodynamic lift is generated by the reciprocal movement of two neighboring surfaces and the resulting overpressure created in the thin layer of gas that separates these surfaces. The beginning of air lubrication technology can be traced back to 1828, when [PERSON] conducted the first experimental investigation on airflow between two parallel surfaces [1]. In 1897, [PERSON] constructed a radial air bearing and experimentally investigated its load capacity [2]. In 1913, the first theory of air lubrication was published by [PERSON] [2]. Despite the earlier developed theoretical basis, gas lubrication of mating machine parts did not find wider application until the third and fourth decades of the 20 th century, which was related, among other things, to the ongoing armed conflicts in the world and the search for new solutions to achieve technical superiority. The growth of the nuclear and armaments industries created the need to develop a bearing system that allows precision machinery to operate for long periods at high temperatures, high speeds, and with low friction losses [1]. Such requirements could be met by gas bearings, which are additionally characterized by low noise levels, negligible heat generation, and no emission of pollutants. The biggest disadvantages of gas bearings are their low load capacity and limited range of stable operation. Bearing load capacity increases with larger diameter and length, higher rotational speed, and reduced radial clearance [3; 4]. The desired thin layer of gaseous lubricating film often requires the use of unusual, often expensive materials and necessitates high manufacturing precision. Since the 1960s, gas bearings have been widely used in equipment such as pumps, blowers, and compressors, as well as in turbine engines operating at temperatures of 500-800 \({}^{\circ}\)C and high rotational frequencies [1]. In the 1990s, industrial interest in ultra-precision machining technology increased significantly, making precision machine tools particularly important. This led to an increase in the requirements for gas (aerostatic) bearings, one of the key elements enabling precision machining at high speeds [1]. Today, the typical application areas for aerodynamic gas bearings include high-speed spindles, aircraft engines, turbochargers, gyroscopes, gas microturbines, and high-speed compressors and blowers [5]. Decades of development of aerodynamic gas bearings have refined this technology in terms of both design and materials. Research on such gas bearings in recent years has primarily focused on improving methods for modeling them. One trend in the development of such methods is to accurately represent the phenomena occurring in them with minimal computing power, aiming to significantly reduce calculation time. As early as the 1990s, attempts were made to develop simplified models of compressible gas flow in lubrication gaps [6]. The analyses carried out have shown that in some cases, using a simplified flow model can yield results close to those obtained with advanced numerical models. It was also noted that in cases of high flow velocity, it is important to consider the gas inertia [6]. It has been shown that, in some cases, the type of calculation scheme used (time-dependent or steady-state analysis) has no significant effect on the aerodynamic pressure distribution but can significantly affect the determined gas flow rate. In the work of [3], direct numerical analysis was performed for two variants of gas bearing characteristics using mass conservation at coupled boundaries. The first case corresponded to externally pressurized gas bearings with four feed holes. In the second case, aerodynamic (hydrodynamic) bearings were analyzed. The pressure distributions and load capacity of the radial and thrust bearings designed for a gas microturbine were determined as a function of aspect ratio, considering coupled boundary effects [3].