The Mathematical Institute, University of Oxford, Eprints Archive

Mathematical modelling of the overflowing cylinder experiment

Howell, P. D. and Breward, C. J. W. (2003) Mathematical modelling of the overflowing cylinder experiment. Journal of Fluid Mechanics, 474 . pp. 275-298.

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Abstract

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 overflowing 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, Darton, Howell and Ockendon 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.

Item Type:Article
Uncontrolled Keywords:Fluid Mechanics, Mathematical Modelling, Asymptotics, Numerical solution
Subjects:D - G > Fluid mechanics
Research Groups:Oxford Centre for Industrial and Applied Mathematics
ID Code:118
Deposited By:Chris J.W. Breward
Deposited On:31 Aug 2004
Last Modified:20 Jul 2009 14:18

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