Modelagem matemática e computacional da dinâmica de mancais fluidodinâmicos usando teoria de filme fino em Coordenadas Curvilíneas Móveis

Abstract

The modeling of fluid dynamic bearings has historically been dominated by the classical Reynolds model, an effective tool for steady-state regimes and low Reynolds numbers, but limited by neglecting inertial and transient terms. This work proposes a new physical-mathematical approach for simulating the dynamics of shaft-bearing systems, based on thin film theory applied to balance equations for fluid dynamics in Generalized and Moving Curvilinear Coordinates. The developed model is one-dimensional in the radial direction, solving the average pressure and velocity fields, which drastically reduces the computational cost compared to complete 3D CFD simulations, while maintaining the ability to capture convective and transient inertia effects. Numerical discretization used the Finite Difference Method with pressure-velocity coupling via the fractional step method. Validation was performed by comparing equilibrium positions and dynamic coefficients (stiffness and damping) with literature data, obtaining excellent qualitative and quantitative agreement. Simulations demonstrated that, in regimes of higher density or velocity ($Re^\ast \approx 1$), the inclusion of inertial terms generates perceptible deviations from classical theory, justifying the need for the proposed model. The methodology proved to be robust for the analysis of transient trajectories and stability, offering an efficient intermediate tool between simplified lubrication models and complex computational fluid dynamics.