The asymptotic structure of a slender dragged viscous thread

The behaviour of a viscous thread as it falls onto a moving belt is analysed in the asymptotic limit of a slender thread. While the bending resistance of a slender thread is small, its effects are dynamically important near the contact point with the belt, where it changes the curvature and orientation of the thread. Steady flows are shown to fall into one of three distinct regimes, depending on whether the belt is moving faster than, slower than or close to the same speed as the free-fall velocity of the thread. The key dynamical balances in each regime are explained and the role of bending stresses is found to be qualitatively different. The asymptotic solutions exhibit the 'backward-facing heel' observed experimentally for low belt speeds, and provide the leading-order corrections to the stretching catenary in theory previously developed for high belt speeds. The asymptotic stability of the thread to the onset of meandering is also analysed. It is shown that the entire thread, rather than the bending boundary layer alone, governs the stability. A balance between the destabilising reaction forces near the belt and the restoring force of gravity on the remainder of the thread determines the onset of meandering, and an analytic estimate for the meandering frequency is thereby obtained. At leading order, neutral stability occurs with the belt moving a little more slowly than the free-fall velocity of the thread, not when the lower part of the thread begins to be under compression, but when the horizontal reaction force at the belt begins to be slightly against the direction of belt motion. The onset of meandering is the heel 'losing its balance'.