Bending and buckling of a falling viscous thread

Abstract

This thesis analyses the behaviour of a slender viscous thread as it falls through air from a nozzle and lands on a solid surface, whereupon it bends and twists. If the diameter of the nozzle is much smaller than the height of fall, then numerical solution of a slender-thread model yields predictions in good agreement with experiment. Moreover, if the thread falls through a height large enough that extensiona.l forces are important, then bending forces are significant only in a small region near the point of impact and negligible in the remainder of the thread, which forms a 'tail'. In this thesis the effects of bending forces near the point of impact are examined by means of asymptotic analysis in the limit of a very slender thread and the key processes that govern the behaviour of such threads is elucidated. The analysis focuses on three particular physical problems that have recently been studied experimentally. The first problem concerns the steady motion of a thread as it lands on, and is dragged sideways by, a horizontally moving belt. It is shown herein that there are three distinct asymptotic regimes which correspond to the belt speed being faster than, slower than or roughly equal to a 'free-fall' speed at the bottom of the tail. Solutions are obtained for each regime, which provide good predictions of the shape and dragout distance of the thread. The second problem concerns the stability of such a steadily dragged thread to transverse meandering. It is shown that meandering is caused by bending forces near the impact point, which cause the thread to slump slowly sideways, and the restoring tension in the tail, which is pinned at the nozzle. The competition between these effects and the structure of the eigenmode is analysed, and quantitative asymptotic estimates are obtained for the onset of meandering and the frequency of meandering at onset. The third problem concerns steady coiling of a viscous thread as it lands on a stationary surface. Provided the fall height is not too small, steady coiling is known to fall into one of three distinct regimes, depending on the relative importance of gravity and inertia within the thread. The asymptotic structure of the thread near the contact point is determined in each regime, and the role of bending forces in this region is analysed. In particular, an analysis of the interaction between the region of bending forces and the 'tail' yields quantitative estimates for the coiling frequency and radius.