Numerical solution of the flow of viscous sheets under gravity and the inverse windscreen sagging problem

The slumping of a thin sheet of very viscous liquid glass is used in the manufacture of windscreens in the automotive industry. The governing equations for an asymptotically thin sheet with variable viscosity are derived in which the vertical coordinate forms the centre-line of the sheet. The time-dependant equations have been solved numerically using the backward Euler method to give results in both two and three dimensions. The flow of an initially flat sheet falls freely under gravity until it becomes curved and the flow becomes very slow in the slumped phase. Finally the sheet freefalls as the thickness becomes small at the boundaries. The inverse problem in which the viscosity profile is to be determined for a given shape can be solved as an embedding problem in which a search is made amongst the forward solutions. Possible shapes in the two-dimensional problem are very restrictive and are shown to be related to the sheet thickness. In three dimensions the range of shapes is much greater