Capillary drainage of an annular film: the dynamics of collars and lobes

This paper considers the capillary drainage of a thin annular film on the inside or outside of a circular cylinder of radius $a$. A film of uniform thickness and axial length greater than $\pi a$ suffers a Rayleigh instability and evolves to form an axisymmetric structure in which the film thickness varies with axial distance. The fluid is gathered into collars, having axial length $2\pi a$, and shorter lobes; the pressure within each collar or lobe is spatially uniform and adjacent collars and lobes are separated by thin necks. We examine numerically the evolution of this structure and demonstrate that, for sufficiently short cylinders, lobes drain into collars as described by Hammond (J. Fluid Mech. vol. 137, 1983, p. 363). For longer cylinder lengths we find that, in spite of the energetic advantage, neighbouring collars do not drain into one another, and that the neck region between adjacent collars is governed by a similarity solution of the thin-film equation having axial length that varies as $t^{-1/2}$ after time $t$, and film thickness that varies as $t^{-1}$, which is different from that found by Jones & Wilson (J. Fluid Mech. vol. 87, 1978, p. 263). We also find a new phenomenon: a collar can spontaneously and episodically translate back and forth along the cylinder, on each occasion consuming the lobe ahead and leaving a smaller daughter lobe behind. This motion takes place on several different timescales: the relatively rapid translation is governed by Landau–Levich equations; the collision with a neighbouring collar is governed by the similarity equation for the neck regions ahead and behind; and the delay between one episode of translation and the next is governed via the Landau–Levich equation by a slow peeling process. Asymptotic results for each of the processes of translation, collision and peeling are obtained and are compared with a full numerical solution. Each episode of translation reduces the thickness of the daughter lobe by a factor 0.115, and successive translations back and forth give rise to a lobe thickness that decays on average and on very long timescales like $t^{-1/2}$. A thin film of fluid trapped beneath a two-dimensional drop sedimenting towards a rigid horizontal plane is described by the same evolution equation, and analogous lobe and collar dynamics are found (Lister, Morrison & Rallison, J. Fluid Mech. vol. 552, 2006, p. 345).