Microfabricated integrated circuits revolutionized computation by vastly reducing the space, labor, and time required for calculations. Microfluidic systems hold similar promise for the large-scale automation of chemistry and biology, suggesting the possibility of numerous experiments performed rapidly and in parallel, while consuming little reagent. While it is too early to tell whether such a vision will be realized, significant progress has been achieved, and various applications of significant scientific and practical interest have been developed. Here a review of the physics of small volumes (nanoliters) of fluids is presented, as parametrized by a series of dimensionless numbers expressing the relative importance of various physical phenomena. Specifically, this review explores the Reynolds number Re, addressing inertial effects; the Peclet number Pe, which concerns convective and diffusive transport; the capillary number Ca expressing the importance of interfacial tension; the Deborah, Weissenberg, and elasticity numbers De, Wi, and El, describing elastic effects due to deformable microstructural elements like polymers; the Grashof and Rayleigh numbers Gr and Ra, describing density-driven flows; and the Knudsen number, describing the importance of noncontinuum molecular effects. Furthermore, the long-range nature of viscous flows and the small device dimensions inherent in microfluidics mean that the influence of boundaries is typically significant. A variety of strategies have been developed to manipulate fluids by exploiting boundary effects; among these are electrokinetic effects, acoustic streaming, and fluid-structure interactions. The goal is to describe the physics behind the rich variety of fluid phenomena occurring on the nanoliter scale using simple scaling arguments, with the hopes of developing an intuitive sense for this occasionally counterintuitive world.

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Microfluidics: Fluid physics at the nanoliter scale

We review the development of diffuse-interface models of hydrodynamics and their application to a wide variety of interfacial phenomena. These models have been applied successfully to situations in which the physical phenomena of interest have a length scale commensurate with the thickness of the interfacial region (e.g. near-critical interfacial phenomena or small-scale flows such as those occurring near contact lines) and fluid flows involving large interface deformations and/or topological changes (e.g. breakup and coalescence events associated with fluid jets, droplets, and large-deformation waves). We discuss the issues involved in formulating diffuse-interface models for single-component and binary fluids. Recent applications and computations using these models are discussed in each case. Further, we address issues including sharp-interface analyses that relate these models to the classical free-boundary problem, computational approaches to describe interfacial phenomena, and models of fully miscible fluids.

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Diffuse-interface methods in fluid mechanics

Atmospheric and Oceanic Fluid Dynamics

The study of waves can be traced back to antiquity where philosophers, such as Pythagoras (c.560-480 BC), studied the relation of pitch and length of string in musical instruments. However, it was not until the work of Giovani Benedetti (1530-90), Isaac Beeckman (1588-1637) and Galileo (1564-1642) that the relationship between pitch and frequency was discovered. This started the science of acoustics, a term coined by Joseph Sauveur (1653-1716) who showed that strings can vibrate simultaneously at a fundamental frequency and at integral multiples that he called harmonics. Isaac Newton (1642-1727) was the first to calculate the speed of sound in his Principia. However, he assumed isothermal conditions so his value was too low compared with measured values. This discrepancy was resolved by Laplace (1749-1827) when he included adiabatic heating and cooling effects. The first analytical solution for a vibrating string was given by Brook Taylor (1685-1731). After this, advances were made by Daniel Bernoulli (1700-82), Leonard Euler (1707-83) and Jean d’Alembert (1717-83) who found the first solution to the linear wave equation, see section (3.2). Whilst others had shown that a wave can be represented as a sum of simple harmonic oscillations, it was Joseph Fourier (1768-1830) who conjectured that arbitrary functions can be represented by the superposition of an infinite sum of sines and cosines now known as the Fourier series. However, whilst his conjecture was controversial and not widely accepted at the time, Dirichlet subsequently provided a proof, in 1828, that all functions satisfying Dirichlet’s conditions (i.e. non-pathological piecewise continuous) could be represented by a convergent Fourier series. Finally, the subject of classical acoustics was laid down and presented as a coherent whole by John William Strutt (Lord Rayleigh, 1832-1901) in his treatise Theory of Sound. The science of modern acoustics has now moved into such diverse areas as sonar, auditoria, electronic amplifiers, etc.

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Linear and nonlinear waves

A complete analysis of coordinate transformations in elastic media by Milton et al has shown that, in general, the equations of motion are not form invariant and thus do not admit transformation-type solutions of the type discovered by Pendry et al for electromagnetics. However, in a two-dimensional (2D) geometry, the acoustic equations in a fluid are identical in form to the single polarization Maxwell equations via a variable exchange that also preserves boundary conditions. We confirm the existence of transformation-type solutions for the 2D acoustic equations with anisotropic mass via time harmonic simulations of acoustic cloaking. We discuss the possibilities of experimentally demonstrating acoustic cloaking and analyse why this special equivalence of acoustics and electromagnetics occurs only in 2D.

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One path to acoustic cloaking

Preface Prologue 1. Sound waves 2. One-dimensional waves in fluids 3. Water waves 4. Internal waves Epilogue Bibliography Notation list Author index Subject index.

Waves in fluids

Sir James Lighthill is one of the world's leading authorities on the subject of fluid dynamics. In this book he draws on his extensive teaching experience to produce an informal introduction to theoretical fluid mechanics for undergraduate mathematicians or engineers. The book will initiate students into the creative use of data from experimental studies and theoretical analyses to generate practically useful mathematical models (including manageable computer models) of a broad range of important fluid flows.

An Informal Introduction to Theoretical Fluid Mechanics

An informal introduction to theoretical fluid mechanics

This is a reissue in the Cambridge Mathematical Library series of a book first published in 1978. When it was first issued, it had enormous impact on all those working in fluid flows, and on waves in particular. I quote from the review by D C Pack, published in 1979 in the Journal of Fluid Mechanics (vol 90, pp 605-8): ``The book will become a standard work for reading and consultation by all interested in fluid flow. It will be invaluable to research students ... Even among the most experienced readers few, if any, will fail to find in it new insights ... comments and comparisons that illuminate the chapters.'' The same comments can be made today, because in spite of the two and a half decades that have now passed, it retains its immediacy and vitality. The book is designed to develop the fundamental concepts of waves in fluids through an in-depth analysis, in each of four chapters, of four important and representative examples of waves in fluids, namely, sound waves, one-dimensional waves, water waves and internal waves. A major highlight of the book is the thorough treatment of group velocity and ray-tracing ideas, developed throughout the text and with a climax in chapter four. As a potential textbook, or simply as an invaluable research aid, it remains indispensable. However, it does not cover such major developments in the last two decades as solitons, and the role of canonical equations such as the Korteweg-de Vries equation and the nonlinear Schrodinger equation. Hence it will need to be used in conjunction with more recent texts which cover these topics. R Grimshaw

James Lighthill

Papers

Waves in fluids

Waves in fluids

An Informal Introduction to Theoretical Fluid Mechanics

An informal introduction to theoretical fluid mechanics

Waves in Fluids