Drop formation during coating of vertical fibres

Abstract

When the coating film around a vertical fibre exceeds a critical thickness hc, the interfacial disturbances triggered by Rayleigh instability can undergo accelerated growth such that localized drops much larger in dimension than the film thickness appear. We associate the initial period of this strongly nonlinear drop formation phenomenon with a self-similar intermediate asymptotic blow-up solution to the long-wave evolution equation which describes how static capillary forces drain fluid into the drop. Below hc, we show that strongly nonlinear coupling between the mean flow and axial curvature produces a finite-amplitude solitary wave solution which prevents local finite-time blow up and hence disallows further growth into drops. We thus estimate hc by determining the existence of solitary wave solutions. This is accomplished by a matched asymptotic analysis which joins the capillary outer region of a large solitary wave to the thin-film inner region. Our estimate of hc = 1.68R3H–2, where R is the fibre radius and H is the capillary length H = (σ/ρg)½, is favourably compared to experimental data.