## 1 Introduction  
As it is known, the use of specific bearings in a machine is usually determined by the operating conditions of the machine, such as the weight of the shaft and its speed, the ecological nature of the lubricant used for the bearings, the operating temperature range of the machine, etc. One type of gas bearing is the foil bearing, best suited for low loads and high rotational speeds [1]. Some companies are trying to introduce devices that use foil bearings, as these are becoming better designed. What has contributed to this is the enormous work of scientists around the world who have performed numerous experiments and simulations to study the phenomena that occur in foil bearings. The main parameters in these types of bearings are the stiffness and damping of the foil assembly [2] (bump foil and top foil), which largely depend on the material they are made of [3], the shape of the bumps [4], as well as the friction pair selected for the journal and top foil [5,6]. These parameters and the correlations between them have already been well studied (although some aspects still need to be tested), and some numerical and theoretical models can predict them with a very good accuracy [7]. The ever-increasing computing power of computers has led to the creation of 3D numerical models that combine the deformation of the support structure caused by the pressure exerted by air in the bearing and, at the same time, the deformation of the lubricating film caused by the deformation of the foil assembly (via FSI--Fluid Structure Interaction) [8]. Another parameter that determines the proper operation of the foil bearing is the temperature of the gas film [9]. This important information has been neglected for some time because, in traditional gas bearings, small temperature fluctuations were assumed not to affect bearing operation, so isothermal models were used in the calculations [10]. It is important to note that a foil bearing is not an ordinary gas bearing as it has additional components between the journal and the sleeve that determine the operating performance of the bearing and its variable lubrication gap, which is associated with the temperature that is generated [11; 12]. There are experimental studies that confirm the non-isothermal nature of the operation of the foil bearing [13]. The very fact that the journal in a foil bearing begins to rotate under conditions of dry friction, then viscous friction and then is isolated by the air film formed, confirms the need to monitor the temperature of components that are at risk of damage (such as the top foil). In experimental studies, the number of temperature sensors used to monitor the proper operation of the bearing varies from one to several. In some studies, the temperature is specifically raised to several hundred degrees Celsius in order to investigate the capabilities of foil bearings and try to find ways how to cool them effectively [14]. All these studies require numerical and theoretical models capable of predicting the temperature value and, more importantly, its distribution on the top foil, with respect to the other operating parameters of the rotor [15]. Practically, the lubrication gap cannot be effectively cooled or flushed since we do not know how the cool air suction mechanism between the journal and the top foil works, and what the external and internal factors of the system that determine the temperature rise are. The authors undertook the difficult task of experimentally investigating the temperature distribution on the surface of the bump foil, something that has not been often covered in the literature to date. There are research papers that describe the effect of journal dynamics on the temperature in the bearing at the measuring points, but this has not been able to explain the temperature distribution on the top foil. There are many numerical works that show such a distribution on the top foil, but they have not yet been verified.