Not only are thin fluid films of enormous importance in numerous practical applications, including painting, the manufacture of foodstuffs, and coating processes for products ranging from semi-conductors and magnetic tape to television screens, but they are also of great fundamental interest to mathematicians, physicists and engineers. Thin fluid films can exhibit a wealth of fascinating behaviour, including wave propagation, rupture, and transition to quasi-periodic or chaotic structures. More details of various aspects of thin-film flow can be found in the recent review articles by Oron, Davis & Bankoff (1997) and Myers (1998), and in the volumes edited by Kistler & Schweizer (1997) and Batchelor, Moffatt & Worster (2000).

When a solid plate is removed from a pool of fluid, a film of fluid is attached to the plate. There are two possible outcomes. The edge of the fluid may be raised through a finite distance, with the edge slipping on the plate. Alternatively, a continuous film of a certain thickness may be drawn up. For plates which have a small slope, it is shown that the first alternative holds when the speed of withdrawal is sufficiently small, and that, when a critical speed is exceeded, the height of the edge above the fluid level in the pool increases with time. A related problem concerns the shape of a receding meniscus in a channel. If the static contact angle is small, lubrication theory can be applied to the film of fluid adjacent to the wall of the channel, and the results of this part of the solution can be matched to the central part of the meniscus, which is controlled by capillarity and gravity, but for which lubrication theory does not apply. As in the draw-up problem, for small speeds the shape of the meniscus is time-independent, but above a critical speed, a tail of fluid of increasing length remains in the tube.

Long-wavelength models for van der Waals driven rupture of a free thin viscous sheet and for capillary pinchoff of a viscous fluid thread both give rise to families of first-type similarity solutions. The scaling exponents in these solutions are independent of the dimensionality of problem. However, the structure of the similarity solutions exhibits an intriguing geometric dependence on the dimensionality of the system: van der Waals driven sheet rupture proceeds symmetrically, whereas thread rupture is inherently asymmetric. To study the bifurcation of rupture from symmetric to asymmetric forms, we generalize the governing equations with the dimension serving as a control parameter. The bifurcation is governed by leading-order inviscid dynamics in which viscous effects are asymptotically small but nevertheless provide the selection mechanism.

In this paper two similarity solutions describing a steady, slender, symmetric dry patch in an infinitely wide liquid film draining under gravity down an inclined plane are obtained. The first solution, which predicts that the dry patch has a parabolic shape and that the transverse profile of the free surface always has a monotonically increasing shape, is appropriate for weak surface-tension effects and far from the apex of the dry patch. The second solution, which predicts that the dry patch has a quartic shape and that the transverse profile of the free surface has a capillary ridge near the contact line and decays in an oscillatory manner far from it, is appropriate for strong surface-tension effects (in particular, when the plane is nearly vertical) and near (but not too close) to the apex of the dry patch. With the average volume flux per unit width (or equivalently with the uniform height of the layer far from the dry patch) prescribed, both solutions contain a free parameter. For each value of this parameter there is a unique solution in the first case and either no solution or a one-parameter family of solutions in the second case. The solutions capture some of the qualitative features observed in experiments.

Recent studies of liquid films driven by competing forces due to surface tension gradients and gravity reveal that undercompressive travelling waves play an important role in the dynamics when the competing forces are comparable. In this paper, we provide a theoretical framework for assessing the spectral stability of compressive and undercompressive travelling waves in thin film models. Associated with the linear stability problem is an Evans function which vanishes precisely at eigenvalues of the linearized operator. The structure of an index related to the Evans function explains computational results for stability of compressive waves. A new formula for the index in the undercompressive case yields results consistent with stability. In considering stability of undercompressive waves to transverse perturbations, there is an apparent inconsistency between long-wave asymptotics of the largest eigenvalue and its actual behaviour. We show that this paradox is due to the unusual structure of the eigenfunctions and we construct a revised long-wave asymptotics. We conclude with numerical computations of the largest eigenvalue, comparisons with the asymptotic results, and several open problems associated with our findings.

We present a finite element scheme for nonlinear fourth-order diffusion equations that arise for example in lubrication theory for the time evolution of thin films of viscous fluids. The equations are in general fourth-order degenerate parabolic, but in addition singular terms of second order may occur which model the effects of intermolecular forces or thermocapillarity. Discretizing the arising nonlinearities in a subtle way allows us to establish discrete counterparts of the essential integral estimates found in the continuous setting. As a consequence, the algorithm is efficient, and results on convergence, nonnegativity or even strict positivity of discrete solutions follow in a natural way. Applying this scheme to the numerical simulation of different models shows various interesting qualitative effects, which turn out to be in good agreement with physical experiments.

A variety of mass preserving moving boundary problems for the thin film equation, ut = −(unuxxx)x, are derived (by formal asymptotics) from a number of regularisations, the case in which the substrate is covered by a very thin pre-wetting film being discussed in most detail. Some of the properties of the solutions selected in this fashion are described, and the full range of possible mass preserving non-negative solutions is outlined.

Free surface flows generated by a moving distribution of pressure are considered. The fluid consists of two superposed layers in a two-dimensional channel. The upper layer is inviscid and the lower layer, which is introduced as a damping mechanism, is modelled by the mathematically convenient lubrication equations. Numerical and analytical solutions are presented. Special attention is given to solutions for which there is a train of waves on each side of the distribution of pressure. It is shown that, depending on the values of the parameters, the short waves can appear on either side of the distribution of pressure.

We study the effect of van der Waals forces on globally energy minimizing profiles for liquid droplets which lie on a solid substrate in a vapour atmosphere and which are assumed to have a uniform cross-section. We prove that for repulsive van der Waals forces as well as for certain short range repulsive-long range attractive forces, there exists a unique globally minimizing profile. Although this profile necessarily contains vertical bounding segments, the height A of the vertical bounding segments can often be demonstrated to be order of magnitude smaller that the overall height B of the droplet. This is the case, in particular, when the droplet is sufficiently large, the Hamaker constant is sufficiently small, and the attractive forces are sufficiently mild. In the presence of repulsive forces only, A is on the order of angströms when B is on the order of millimeters, for realistic parameter values. Moreover, conditions are prescribed under which Young's law is satisfied to leading order despite the appearance of the vertical segments, when the contact angle is measured via an inscribed circle construction at a distance ξ0 from the edge of the droplet, where A [Lt ] ξ0 [Lt ] B.

In multiple-roll coaters thin liquid films are transferred from roll to roll by means of liquid ‘beads’ which occupy the small gaps between adjacent rolls. Double-Film-Fed (DFF) beads are those which feature two ingoing films instead of the usual one, and arise in the intermediate stages of certain types of roll coater. One of the ingoing films, h1, is supplied from the previous inter-roll gap while the other, h2, ‘returns’ from the subsequent gap. Such a flow is investigated here under the conditions of low flow rate, small capillary number and negligible gravity and inertia, using lubrication theory and finite element analysis. The thickness of film h1 is fixed independently, while that of h2 is specified as a fraction, ζ, of the film output on the same roll. This simple approach allows a degree of feedback between the output and input of the bead, and enables one to simulate different conditions in the subsequent gap. Predictions of outgoing film thicknesses made using the two models agree extremely well and show that, for each value of ζ < 1, one outgoing film thickness decreases monotonically with speed ratio, S, while the other features a maximum. Good agreement is also seen in the pressure profiles, which are entirely sub-ambient in keeping with the small capillary number conditions. The finite element solutions reveal that in the ‘zero-flux’ case (when ζ = 1) the flow structures are very similar to those seen in an idealized cavity problem. In the more general (ζ < 1) situation, as in single-film-fed meniscus roll coating, several liquid transfer-jets occur by which liquid is conveyed through the bead from one roll to the other. The lubrication model is used to calculate several critical flow rates at which the flow is transformed, and it is shown that when the total dimensionless flow rate through the bead exceeds 1/3, the downstream flow structure is independent of the relative sizes of the ingoing films.

Two-dimensional flow is considered in a fluid bead located in the gap between a pair of contra-rotating cylinders and bounded by two curved menisci. The stability of such bead flows with two inlet films, and hence no contact line, are analysed as the roll speed ratio S is increased. One of the inlet films can be regarded as an ‘input flux’ whilst the other is a ‘returning film’ whose thickness is specified as a fraction ζ of the outlet film on that roll. The flow is modelled via lubrication theory and for Ca [Lt ] 1, where Ca represents the capillary number, boundary conditions are formally developed that account for S ≠ 1 and the non-constant gap. It is shown that there is a qualitative difference in the results between the single and double inlet film models unless small correction terms to the pressure drops at the interfaces are taken into account. Futhermore, it is shown that the inclusion of these small terms produces an O(1) effect on the prediction of the critical value of S at which bead break occurs. When the limits of the returning film fraction are examined it is found that as ζ → 0 results are in good agreement with those for the single inlet film. Further it is shown for a fixed input flux that as ζ → 1 a transition from bead break to upstream flooding of the nip can occur and multiple two-dimensionally stable solutions exist. For a varying input flux and fixed and ‘sufficiently large’ values of ζ there is a critical input flux &λmacr;(ζ) such that as S is increased from zero:

(i) bead break occurs for λ < &λmacr;;

(ii) upstream flooding occurs for λ > &λmacr;;

(iii) when λ = &λmacr; the flow becomes neutrally stable at a specific value of S beyond which there exist two steady solutions (two-dimensionally stable) leading to bead break and upstream flooding, respectively.