to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.

Linear and nonlinear waves, by G. B. Whitham. Pp.636. £50. 1999. ISBN 0 471 35942 4 (Wiley).

Wetting phenomena are ubiquitous in nature and technology. A solid substrate exposed to the environment is almost invariably covered by a layer of fluid material. In this review, the surface forces that lead to wetting are considered, and the equilibrium surface coverage of a substrate in contact with a drop of liquid. Depending on the nature of the surface forces involved, different scenarios for wetting phase transitions are possible; recent progress allows us to relate the critical exponents directly to the nature of the surface forces which lead to the different wetting scenarios. Thermal fluctuation effects, which can be greatly enhanced for wetting of geometrically or chemically structured substrates, and are much stronger in colloidal suspensions, modify the adsorption singularities. Macroscopic descriptions and microscopic theories have been developed to understand and predict wetting behavior relevant to microfluidics and nanofluidics applications. Then the dynamics of wetting is examined. A drop, placed on a substrate which it wets, spreads out to form a film. Conversely, a nonwetted substrate previously covered by a film dewets upon an appropriate change of system parameters. The hydrodynamics of both wetting and dewetting is influenced by the presence of the three-phase contact line separating "wet" regions from those that are either dry or covered by a microscopic film only. Recent theoretical, experimental, and numerical progress in the description of moving contact line dynamics are reviewed, and its relation to the thermodynamics of wetting is explored. In addition, recent progress on rough surfaces is surveyed. The anchoring of contact lines and contact angle hysteresis are explored resulting from surface inhomogeneities. Further, new ways to mold wetting characteristics according to technological constraints are discussed, for example, the use of patterned surfaces, surfactants, or complex fluids.

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Wetting and Spreading

Extended tanh-function method and its applications to nonlinear equations

The (G' G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics

Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations

Laminar gravity-driven thin-film flow down a helically wound channel of rectangular cross-section with small torsion in which the fluid depth is small is considered. Neglecting the entrance and exit regions we obtain the steady-state solution that is independent of position along the axis of the channel, so that the flow, which comprises a primary flow in the direction of the axis of the channel and a secondary flow in the cross-sectional plane, depends only on position in the two-dimensional cross-section of the channel. A thin-film approximation yields explicit expressions for the fluid velocity and pressure in terms of the free-surface shape, the latter satisfying a nonlinear ordinary differential equation that has a simple exact solution in the special case of a channel of rectangular cross-section. The predictions of the thin-film model are shown to be in good agreement with much more computationally intensive solutions of the small-helix-torsion Navier–Stokes equations. The present work has particular relevance to spiral particle separators used in the mineral-processing industry. The validity of an assumption commonly used in modelling flow in spiral separators, namely, that the flow in the outer region of the separator cross-section is described by a free vortex, is shown to depend on the problem parameters.

/pdf/thin-film-flow-in-helically-wound-rectangular-channels-with-6qko0ymnib.pdf

Thin-film flow in helically wound rectangular channels with small torsion

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/pdf/comment-on-increased-evaporation-kinetics-of-sessile-59au0h6xir.pdf

Comment on "Increased evaporation kinetics of sessile droplets by using nanoparticles".

The behaviour of a steady thin sessile or pendent ridge of fluid on an inclined planar substrate which is strongly coupled to the external pressure gradient arising from an inviscid airflow parallel to the substrate far from the ridge is described. When the substrate is nearly horizontal, a very wide ridge can be supported against gravity by capillary and/or external pressure forces; otherwise, only a narrower (but still wide) ridge can be supported. Classical thin-aerofoil theory is adapted to obtain the governing singular integro-differential equation for the profile of the ridge in each case. Attention is focused mainly on the case of a very wide sessile ridge. The effect of strengthening the airflow is to push a pinned ridge down near to its edges and to pull it up near to its middle. At a critical airflow strength, the upslope contact angle reaches the receding contact angle at which the upslope contact line de-pins, and continuing to increase the airflow strength beyond this critical value results i...

/pdf/strongly-coupled-interaction-between-a-ridge-of-fluid-and-an-3e521v6sf2.pdf

Strongly coupled interaction between a ridge of fluid and an inviscid airflow

A detailed analysis of small-scale locally unidirectional gravity-driven rivulet flow with prescribed volume flux down an inclined slippery substrate for a rivulet with either constant width (i.e., pinned contact lines) or constant contact angle is undertaken. In particular, we determine the effect of varying the Navier slip length λ (i.e., the strength of the slip at the solid–fluid interface) on the rivulet. The present analysis shows that the shape and size of the rivulet and the velocity within it depend strongly on the value of λ. Increasing the value of λ reduces the viscous resistance at the substrate and, hence, leads to a larger velocity within the rivulet, and so the prescribed flux is achieved with a smaller rivulet. In particular, in the limit of strong slip, λ → ∞, for a rivulet of a perfectly wetting fluid and a rivulet with constant width, the velocity becomes large and plug-like like O(λ1/2) ≫ 1, and the rivulet becomes shallow like O(λ−1/2) ≪ 1, while for a rivulet with positive constant contact angle, the velocity becomes large and plug-like like O(λ2/3) ≫ 1, and the rivulet becomes narrow like O(λ−1/3) ≪ 1 and shallow like O(λ−1/3) ≪ 1.

/pdf/rivulet-flow-down-a-slippery-substrate-2dvjwjv697.pdf

Rivulet flow down a slippery substrate

We construct a fluid-dynamical model for the flow of a solution with a free surface at which surface tension acts. This model can describe both classical surfactants, which decrease the surface tension of the solution relative to that of the pure solvent, and ‘anti-surfactants’ (such as many salts when added to water, and small amounts of water when added to alcohol) which increase it. We demonstrate the utility of the model by considering the linear stability of an infinitely deep layer of initially quiescent fluid. In particular, we predict the occurrence of a novel instability driven by surface-tension gradients, which occurs for anti-surfactant, but not for surfactant, solutions.

Brian R. Duffy

Papers

Thin-film flow in helically wound rectangular channels with small torsion

Comment on "Increased evaporation kinetics of sessile droplets by using nanoparticles".

Strongly coupled interaction between a ridge of fluid and an inviscid airflow

Rivulet flow down a slippery substrate

A fluid-dynamical model for ‘anti-surfactants’