Wrinkling and sagging of viscous sheets

Abstract

This thesis explores the wrinkling and sagging behavior of thin viscous Newtonian sheets and filaments motivated by analogous scenarios in elasticity. These problems involve dynamic free boundaries and geometric nonlinearities but use simple physics. The first problem examined concerns an annular viscous sheet subjected to torsional shearing which consequently develops spiral wrinkles. Examination of the behavior of this system leads to a scaling of the Stokes equations for zero Reynolds number flow resulting in a reduced order mathematical model for the evolution of the sheet that includes the effects of gravity and surface tension. Linear stability analysis yields the most unstable modes for wrinkling of the sheet and their associated growth rates at onset which agree with experimental observations. In the limit of a narrow annular gap, the problem reduces to that of a sheared rectilinear sheet. Interestingly, this Couette problem shows instabilities even in the zero Reynolds number limit. The second problem examined concerns the sagging of a horizontal viscida (fluid filament) under the influence of gravity. Resistance of the viscida to bending controls the initial phase of deformation, while resistance to stretching begins to play a principal role in later stages. At very late times the process resembles droplet break-off from two thin filaments.