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

We show that the porous medium equation has a gradient flow structure which is both physically and mathematically natural. In order to convince the reader that it is mathematically natural, we show that the time asymptotic behavior can be easily understood in this framework. We use the intuition and the calculus of Riemannian geometry to quantify this asymptotic behavior.

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The geometry of dissipative evolution equations: the porous medium equation

The Heat Equation is one of the three classical linear partial differential equations of second order that form the basis of any elementary introduction to the area of PDEs, and only recently has it come to be fairly well understood. In this monograph, aimed at research students and academics in mathematics and engineering, as well as engineering specialists, Professor Vazquez provides a systematic and comprehensive presentation of the mathematical theory of the nonlinear heat equation usually called the Porous Medium Equation (PME). This equation appears in a number of physical applications, such as to describe processes involving fluid flow, heat transfer or diffusion. Other applications have been proposed in mathematical biology, lubrication, boundary layer theory, and other fields. Each chapter contains a detailed introduction and is supplied with a section of notes, providing comments, historical notes or recommended reading, and exercises for the reader.

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The Porous Medium Equation: Mathematical Theory

The viscous gravity current that results when fluid flows along a rigid horizontal surface below fluid of lesser density is analysed using a lubrication-theory approximation. It is shown that the effect on the gravity current of the motion in the upper fluid can be expressed as a condition of zero shear on the unknown upper surface of the gravity current. With the supposition that the volume of heavy fluid increases with time like F, where a is a constant, a similarity solution to the governing nonlinear partial differential equations is obtained, which describes the shape and rate of propagation of the current. The viscous theory is shown to be valid for t & t, when a -c a, and for t -4 t, when a > a,, where t, is the transition time at which the inertial and viscous forces are equal, with a, = $ for a two-dimensional current and a, = 3 for an axisymmetric current. The solutions confirm the functional forms for the spreading relationships determined for a = 1 in the preceding paper by Didden & Maxworthy (1982), as well as evaluating the multiplicative factors appearing in the relationships. The relationships compare very well with experimental measurements of the axisymmetric spreading of silicone oils into air for a = 0 and 1. There is also very good agreement between the theoretical predictions and the measurements of the axisymmetric spreading of salt water into fresh water reported by Didden & Maxworthy and by Britter (1979). The predicted multiplicative constant is within 10 Yo of that measured by Didden & Maxworthy for the spreading of salt water into fresh water in a channel.

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The propagation of two-dimensional and axisymmetric viscous gravity currents over a rigid horizontal surface

Preface to the Third Edition.- Preface.- Linear Partial Differential Equations.- Nonlinear Model Equations and Variational Principles.- First-Order, Quasi-Linear Equations and Method of Characteristics.- First-Order Nonlinear Equations and Their Applications.- Conservation Laws and Shock Waves.- Kinematic Waves and Real-World Nonlinear Problems.- Nonlinear Dispersive Waves and Whitham's Equations.- Nonlinear Diffusion-Reaction Phenomena.- Solitons and the Inverse Scattering Transform.- The Nonlinear Schroedinger Equation and Solitary Waves.- Nonlinear Klein--Gordon and Sine-Gordon Equations.- Asymptotic Methods and Nonlinear Evolution Equations.- Tables of Integral Transforms.- Answers and Hints to Selected Exercises.- Bibliography.- Index.

R. E. Pattle

Papers

Diffusion from an instantaneous point source with a concentration-dependent coefficient