Thin liquid films

Thin films of liquid occur widely in nature and in industrial processes. We are interested in developing and analysing mathematical models for the evolution of such films. The principal idea is to exploit the large aspect ratio (i.e. the fact that the film is thin), using techniques from asymptotic analysis and differential geometry.

There are two broad categories of films that behave in rather different ways. The first occurs when a thin layer of liquid flows over a substrate, for example a coating of paint or the tear film on the eye. Such films are often governed by degenerate nonlinear parabolic differential equations, whose properties are of wide current interest in the mathematical community. Rather different behaviour occurs in free liquid films, for example in soap films and foams. These tend to evolve more rapidly, since their surfaces are free and, without a substrate to which to conform, their geometry is unknown in advance. Current topics of interest include the following.

Inviscid thin-film flows

The dynamics of inviscid thin films are determined by the balance between inertia, gravity and, sometimes, surface tension (although this is more important in jets and sprays). A famous example is the shallow water model, and one topic of interest is the small-gravity limit of these equations. When gravity is neglected altogether, for very fast or rapidly-changing flows, an even simpler model can be used to analyse the free sheets that you see if you hold a spoon horizontally under a jet of water from a tap. The picture opposite shows a surf-skimmer, and all the features of the flow including the two splash wakes behind the board can be predicted using a model of this kind.

Inviscid small aspect ratio fluid flows also appear on much larger scales, as approximate descriptions of planetary atmospheres and oceans. The shallow water equations are commonly used to study phenomena like wave-vortex interactions that also arise in more complicated systems in geophysical fluid dynamics like the meteorological primitive equations. Higher order shallow water equations have also been developed, by assuming some approximate vertical structure and then averaging over the layer depth. More recently, shallow water equations with a horizontal magnetic field have been used to model the solar tachocline, the thin layer inside the Sun marking the transition from solid body rotation in the radiative interior to differential rotation in the outer convection zone.

Thin film behaviour under surface tension

In thin films, surface tension effects are localised at the edges where the curvature is large. For very viscous fluids, the whole film retracts and thickens, as in the figure opposite; when inertial effects are important, `blobs' of fluid collect at the edges as they retract.

Coating flow on curved substrates

In many industrial coating processes, the ability of the film to cover and/or conform to any irregularities in the substrate is crucial. Many biological systems (e.g. the lungs or the eye) also involve surface-tension-driven flow over topography. Substrate curvature variations introduce nonlinear forcing terms in the governing equations that may lead to the formation of sudden jumps in the film thickness and other singularities.

Lamella modelling

To understand the stability of foam, it is necessary to model the thinning and eventual rupture of the thin free films, called lamellae, that separate the gas bubbles. This involves incorporating various stabilising agents (e.g. surfactants) in the standard thin-film models and matching with the "Plateau borders" that border the lamellae and tend to suck fluid out of them. The figure below shows some typical flow profiles.

Curtain coating

Curtain coating is an industrial process, traditionally used to coat photographic film, which is now beginning to be used by the paper industry. A reservoir of coating mix, typically an aqueous solution containing 20-50% solids and surfactants, is formed into a curtain of fluid using either a slot or a slide as shown in the figure. This curtain falls under gravity until it hits the substrate or "web" to be coated, which is conveyed quickly underneath. Thus mathematically we have a free surface problem of a thin viscous film falling under gravity and impinging onto a moving surface below.The coated substrate is passed through driers to evaporate off the water and thus the finished paper is left with a, hopefully uniform, solid coating. In the paper industry this technology is run at much higher web speeds, and with fluids of very different rheologies, than in the photographic industry.

Selected references

1. P. D. Howell, 1999 The draining of a two-dimensional bubble, J. Engng Maths 35, 251-272.
2. S. D. Howison, J. R. Ockendon & J. M. Oliver, 2002 Deep- and shallow-water slamming at small and zero deadrise angles, J. Engng Maths 42, 373-388.
3. C. J. W. Breward & P. D. Howell, 2002 The drainage of a foam lamella, J. Fluid Mech. 458, 379-406.
4. T. G. Myers, J. P. F. Charpin & S. J. Chapman, 2002 The flow and solidification of a thin fluid film on an arbitrary three-dimensional surface, Phys. Fluids 14, 2788-2803.
5. P. J. Dellar, Common Hamiltonian structure of the shallow water equations with horizontal temperature gradients and magnetic fields, Phys. Fluids, 15 292-297.
6. P. J. Dellar, Dispersive shallow water magnetohydrodynamics, Phys. Plasmas, 10 581-590.
7. P. D. Howell, Surface-tension-driven flow on a moving curved surface, J. Engng Maths, 45, 283-308.

People working in this area within OCIAM are