Summary: We study a moving free boundary problem related to the cavitation modelling in a lubricated device. After formulating the problem, we apply an implicit scheme for time discretization, and then we study the time-discretized problem. First, by means of regularization techniques, we prove the existence of solutions of the discretized problem, and then we state a monotonicity result for continuous solutions. As a consequence, we prove the convergence to the corresponding stationary solution.