Modeling and analysis of certain engineering questions will be taken up in this mathematical study of free boundaries arising in the lubrication theory. The work involves efforts to understand the dynamics of thin viscous fluid films confined between solid boundaries, such as fluid flow between two concentric or eccentric rotating cylinders - a journal bearing. An interesting and mechanically significant aspect of journal bearings is the occurrence of cavitation, a phenomenon in which the lubricant vaporizes due to its inability to sustain large vacuums. On the mathematical level, a representation of this phenomenon is given by a variational problem involving free boundaries. The primary goal of the project is to determine properties of the free boundary: regularity, shape and location. Two particular models will be analyzed. In both cases the object is to describe the pressure in the lubricant film, with vaporization pressure taken to be zero. The first is the solid or non-porous bearing of finite length in which the outer and inner cylinders are not concentric. For small eccentricity, the region of cavitation is known to have a single component. Whether this condition remain under more general conditions is yet to be resolved. The second model concerns the porous, concentric journal bearing of infinite length. This journal is composed of a porous matrix which permits continuous fluid flow between the matrix and the lubricant. The regularity of the pressure function is better understood in this case and work will concentrate on developing numerical approximations to the function by finite element methods, with special emphasis on the convergence of the discrete free boundaries.
