Fluid Flow Between a Roller and Absorbent Compressible Paper

The lubrication theory approximation is used to model the flow of a resin between a pair of rollers through which is fed a sheet of compressible porous paper. A simple model for the porosity and compressibility of the paper leads to a modified Reynolds equation which is non-linear, and has boundary conditions applied at the unknown position where a cavity develops. The solution depends on three non-dimensional parameters measuring the porosity k, the compressibility c and the ratio of the paper thickness to roller separation distance. Various asymptotic forms of the solution are found for small c and k for diflerent values of the paper thickness. Critical conditions are obtained at which the lubrication channel closes and the paper touches the roller.