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Long-scale evolution of thin liquid films

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TLDR
In this article, a unified mathematical theory is presented that takes advantage of the disparity of the length scales and is based on the asymptotic procedure of reduction of the full set of governing equations and boundary conditions to a simplified, highly nonlinear, evolution equation or to a set of equations.
Abstract
Macroscopic thin liquid films are entities that are important in biophysics, physics, and engineering, as well as in natural settings. They can be composed of common liquids such as water or oil, rheologically complex materials such as polymers solutions or melts, or complex mixtures of phases or components. When the films are subjected to the action of various mechanical, thermal, or structural factors, they display interesting dynamic phenomena such as wave propagation, wave steepening, and development of chaotic responses. Such films can display rupture phenomena creating holes, spreading of fronts, and the development of fingers. In this review a unified mathematical theory is presented that takes advantage of the disparity of the length scales and is based on the asymptotic procedure of reduction of the full set of governing equations and boundary conditions to a simplified, highly nonlinear, evolution equation or to a set of equations. As a result of this long-wave theory, a mathematical system is obtained that does not have the mathematical complexity of the original free-boundary problem but does preserve many of the important features of its physics. The basics of the long-wave theory are explained. If, in addition, the Reynolds number of the flow is not too large, the analogy with Reynolds's theory of lubrication can be drawn. A general nonlinear evolution equation or equations are then derived and various particular cases are considered. Each case contains a discussion of the linear stability properties of the base-state solutions and of the nonlinear spatiotemporal evolution of the interface (and other scalar variables, such as temperature or solute concentration). The cases reducing to a single highly nonlinear evolution equation are first examined. These include: (a) films with constant interfacial shear stress and constant surface tension, (b) films with constant surface tension and gravity only, (c) films with van der Waals (long-range molecular) forces and constant surface tension only, (d) films with thermocapillarity, surface tension, and body force only, (e) films with temperature-dependent physical properties, (f) evaporating/condensing films, (g) films on a thick substrate, (h) films on a horizontal cylinder, and (i) films on a rotating disc. The dynamics of the films with a spatial dependence of the base-state solution are then studied. These include the examples of nonuniform temperature or heat flux at liquid-solid boundaries. Problems which reduce to a set of nonlinear evolution equations are considered next. Those include (a) the dynamics of free liquid films, (b) bounded films with interfacial viscosity, and (c) dynamics of soluble and insoluble surfactants in bounded and free films. The spreading of drops on a solid surface and moving contact lines, including effects of heat and mass transport and van der Waals attractions, are then addressed. Several related topics such as falling films and sheets and Hele-Shaw flows are also briefly discussed. The results discussed give motivation for the development of careful experiments which can be used to test the theories and exhibit new phenomena.

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Citations
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Journal ArticleDOI

Molecular transport and flow past hard and soft surfaces: computer simulation of model systems

TL;DR: In this paper, the authors used the Navier boundary condition to describe the effect of the surface on the deformation at the three-phase contact line of a soft, deformable substrate like a polymer brush.
Journal ArticleDOI

Marangoni instability in a thin film heated from below: Effect of nonmonotonic dependence of surface tension on temperature.

TL;DR: It is demonstrated that for such a fluid layer upon heating from below, both monotonic and oscillatory instability can appear for a certain range of the dimensionless parameters, viz., Biot number (Bi), Galileo number (Ga), and inverse capillary number (Σ).
Journal ArticleDOI

Electric field induced instabilities of thin leaky bilayers: pathways to unique morphologies and miniaturization.

TL;DR: Nonlinear simulations uncover that the interfaces can develop unique morphologies when the spatiotemporal variation of the attractive or repulsive force at the charged interface act together or against the electrical stress due to the induced charge separation across the interface.
Journal ArticleDOI

Self-organization in a chromium thin film under laser irradiation

TL;DR: In this article, the beam of a nanosecond pulse laser tightly focused to a line was applied to the back-side ablation of the chromium thin film on a glass substrate.
Journal ArticleDOI

Thermocapillary effects on steadily evaporating contact line: A perturbative local analysis

TL;DR: In this article, a linear stability analysis of very thin very thin films (∼10nm) accounting for surface forces is presented. But the analysis shows no instability for (quasi-) flat-very-thin films, which is consistent with general knowledge.
References
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Book

Intermolecular and surface forces

TL;DR: The forces between atoms and molecules are discussed in detail in this article, including the van der Waals forces between surfaces, and the forces between particles and surfaces, as well as their interactions with other forces.
Book

Boundary layer theory

TL;DR: The flow laws of the actual flows at high Reynolds numbers differ considerably from those of the laminar flows treated in the preceding part, denoted as turbulence as discussed by the authors, and the actual flow is very different from that of the Poiseuille flow.
Book

Linear and Nonlinear Waves

G. B. Whitham
TL;DR: In this paper, a general overview of the nonlinear theory of water wave dynamics is presented, including the Wave Equation, the Wave Hierarchies, and the Variational Method of Wave Dispersion.
Journal ArticleDOI

Wetting: statics and dynamics

TL;DR: In this paper, the authors present an attempt towards a unified picture with special emphasis on certain features of "dry spreading": (a) the final state of a spreading droplet need not be a monomolecular film; (b) the spreading drop is surrounded by a precursor film, where most of the available free energy is spent; and (c) polymer melts may slip on the solid and belong to a separate dynamical class, conceptually related to the spreading of superfluids.