A continuum model of the evolution of air ingestion and entrainment for open-ended squeeze film dampers is proposed in this paper. Hydrodynamic lubrication theory is extended to lubrication with mixture of a Newtonian liquid and an ideal gas. The solution to the universal Reynolds equation is determined numerically using a control volume method (Elrod algorithm) and the forth-order Range-Kutta method. This method conserves mass throughout the computational domain including air ingestion and entrainment. Excellent agreement is found with the experimental works of Diaz and San Andrès for the squeeze film damper [1, 2].