Asymptotic investigations into the `fluid mechanical sewing machine'

The fall of a slender viscous thread from a nozzle onto a moving horizontal belt exhibits a wide range of behaviour. Steady motion is observed above a critical belt speed. Below this speed the thread undergoes a buckling instability, and lays down on the belt a variety of stable, periodic patterns referred to as a `fluid mechanical sewing machine'. We expand on previous theoretical progress [1] by including the effects arising from the resistance of the thread to bending. While the bending resistance of a slender viscous thread is small, under certain circumstances it has a dominant effect. We work in the asymtotic limit of a slender thread, and investigate the full range of steady solutions. An asymptotic refinement to the estimate derived in [1] for the onset of buckling instability is presented, and the behaviour of the thread near onset is discussed. [1] S. Chiu-Webster \& J.R. Lister, {\it J. Fluid Mech.} {\bf 569}, 89-111.