The overflowing cylinder (OFC) is an experimental apparatus designed to generate
a controlled straining flow at a free surface, whose dynamic properties may then be
investigated. Surfactant solution is pumped up slowly through a vertical cylinder.
On reaching the top, the liquid forms a flat free surface which expands radially
before over flowing down the side of the cylinder. The velocity, surface tension and
surfactant concentration on the expanding free surface are measured using a variety
of non-invasive techniques.A mathematical model for the OFC has been previously derived by Breward et al.
(2001) and shown to give satisfactory agreement with experimental results. However, a
puzzling indeterminacy in the model renders it unable to predict one scalar parameter
(e.g. the surfactant concentration at the centre of the cylinder), which must be therefore
be taken from the experiments.In this paper we analyse the OFC model asymptotically and numerically. We show
that solutions typically develop one of two possible singularities. In the first, the
surface concentration of surfactant reaches zero a finite distance from the cylinder
axis, while the surface velocity tends to infinity there. In the second, the surfactant
concentration is exponentially large and a stagnation point forms just inside the rim
of the cylinder. We propose a criterion for selecting the free parameter, based on
the elimination of both singularities, and show that it leads to good agreement with
experimental results.
Mathematical modelling of the overflowing cylinder experiment