Dynamics and shape instability of thin viscous sheets

An asymptotic theory for predicting the thickness distribution and geometry of the boundaries of a thin nearly planar fluid sheet and analyzing the stability of its shape is developed starting from the balance law for mass and momentum for a continuous medium. The resulting equations comprise a transient two-dimensional model where the sheet thickness and out of plane displacement are additional distributed parameters along with the continuum velocities. The analysis of the sheet motion in the transverse direction showed that the existence of compressive stresses inevitably leads to viscous sheet shape instability, while the equations describing the out of plane sheet displacement become of mixed type. As examples, two practical problems involving nonisoviscous sheets have been considered: a two-dimensional viscous sheet, which shape is unstable in the case when the compressive stress is applied at the exit end, and a three-dimensional problem of viscous sheet redraw (constant stretching), where the existence of the compressive stresses leads to the development of hyperbolic zones in the sheet, resulting in the sheet shape instability. (C) 2010 American Institute of Physics. [doi:10.1063/1.3286434]