Power-law creep of a material being compressed between parallel plates: a singular perturbation problem

Summary

The flow of a power-law creeping or viscous material which is being compressed in a narrow gap between parallel plates is studied. A perturbation scheme based on the small gap size is developed and the approximations which lead to classical lubrication theory are formally identified. The solution obtained from lubrication theory is shown to correspond to an outer solution which is not uniformly valid because it predicts infinite longitudinal stresses along both the centerline of the gap and across the entire gap on the line connecting the plate midpoints. The failure of lubrication theory to describe the flow in these regions is not due to an inherent failure of the power-law constitutive equation to model material behavior, but is due to a breakdown in the approximations made in lubrication theory. The solution is corrected by constructing inner solutions in the regions where lubrication theory fails and a uniformly valid solution for the stress field and velocity field is obtained.