The stretching of a thin viscous inclusion and the drawing of glass sheets

Abstract

A thin, two-dimensional inclusion of a highly viscous fluid is deformed by an external Stokes flow—a suggested mechanism by which the Earth’s oceanic crust is entrained and thinned by the mantle. The method of matched asymptotic expansions is applied with the small expansion parameter being the inverse aspect ratio of the inclusion. The kinematic condition and continuity of shear and normal stresses lead to boundary conditions for the biharmonic equation governing the outer flow in the two cases of stretching by a pure shear and by a simple shear flow. The evolution of the inclusion can then, in principle, be determined. The equations governing the drawing of glass sheets may be derived as a limiting case, and any asymmetry in the centerline of the sheet is deformed over a short time scale before stretching occurs. Several examples are computed, showing how the straightening of the sheet can be slowed down by constraints on the gradients at the two ends. It is shown that an inclusion with smooth initial data cannot break up in a finite time.