Coating Flow of a Rotating Cylinder in the Presence of an Irrotational Airflow

Coating flow of a thin film of viscous fluid on the outside of a uniformly rotating horizontal circular cylinder in the presence of a non-uniform pressure distribution due to an irrotational airflow with circulation is investigated. This problem is not only of interest in its own right, both as a rare example of an analytically tractable problem involving the interaction between an airflow and a fluid film and as a novel extension to the extensively studied classical coating-flow problem, but also as a paradigm for the wide range of practical situations in which a thin film of fluid on a moving solid substrate is subject to an airflow. In particular, it is shown that steady full-film solutions corresponding to a continuous film of fluid covering the entire cylinder are possible only when the flux and mass of fluid do not exceed critical values, which are determined in terms of the non-dimensional parameters representing the speed of the far-field airflow and the circulation of the airflow. In addition, when the circulation of the airflow is in the same direction as the rotation of the cylinder the maximum mass of fluid that can be supported on the cylinder is always less than that in classical coating flow, whereas when the circulation is in the opposite direction the maximum mass of fluid can be greater than that in classical coating flow. A long-time analysis of the unsteady problem reveals that the flow can be unconditionally stable.