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

Matter is commonly found in the form of materials. Analytical mechanics turned its back upon this fact, creating the centrally useful but abstract concepts of the mass point and the rigid body, in which matter manifests itself only through its inertia, independent of its constitution; “modern” physics likewise turns its back, since it concerns solely the small particles of matter, declining to face the problem of how a specimen made up of such particles will behave in the typical circumstances in which we meet it. Materials, however, continue to furnish the masses of matter we see and use from day to day: air, water, earth, flesh, wood, stone, steel, concrete, glass, rubber, ... All are deformable. A theory aiming to describe their mechanical behavior must take heed of their deformability and represent the definite principles it obeys.

The non-linear field theories of mechanics

Jets, ie collimated streams of matter, occur from the microscale up to the large-scale structure of the universe Our focus will be mostly on surface tension effects, which result from the cohesive properties of liquids Paradoxically, cohesive forces promote the breakup of jets, widely encountered in nature, technology and basic science, for example in nuclear fission, DNA sampling, medical diagnostics, sprays, agricultural irrigation and jet engine technology Liquid jets thus serve as a paradigm for free-surface motion, hydrodynamic instability and singularity formation leading to drop breakup In addition to their practical usefulness, jets are an ideal probe for liquid properties, such as surface tension, viscosity or non-Newtonian rheology They also arise from the last but one topology change of liquid masses bursting into sprays Jet dynamics are sensitive to the turbulent or thermal excitation of the fluid, as well as to the surrounding gas or fluid medium The aim of this review is to provide a unified description of the fundamental and the technological aspects of these subjects

Physics of liquid jets

An account is given of the development of the idea of a yield stress for solids, soft solids and structured liquids from the beginning of this century to the present time. Originally, it was accepted that the yield stress of a solid was essentially the point at which, when the applied stress was increased, the deforming solid first began to show liquid-like behaviour, i.e. continual deformation. In the same way, the yield stress of a structured liquid was originally seen as the point at which, when decreasing the applied stress, solid-like behaviour was first noticed, i.e. no continual deformation. However as time went on, and experimental capabilities increased, it became clear, first for solids and lately for soft solids and structured liquids, that although there is usually a small range of stress over which the mechanical properties change dramatically (an apparent yield stress), these materials nevertheless show slow but continual steady deformation when stressed for a long time below this level, having shown an initial linear elastic response to the applied stress. At the lowest stresses, this creep behaviour for solids, soft solids and structured liquids can be described by a Newtonian-plateau viscosity. As the stress is increased the flow behaviour usually changes into a power-law dependence of steady-state shear rate on shear stress. For structured liquids and soft solids, this behaviour generally gives way to Newtonian behaviour at the highest stresses. For structured liquids this transition from very high (creep) viscosity (>106 Pa.s) to mobile liquid (

The yield stress—a review or ‘παντα ρει’—everything flows?

Viscoelastic instabilities are of practical importance, and are the subject of growing interest. Reviewed here are the fresh developments as well as earlier work in this area, organized into the following categories: instabilities in Taylor-Couette flows, instabilities in cone-and-plate and plate-and-plate flows, instabilities in parallel shear flows, extrudate distortions and fracture, instabilities in shear flows with interfaces, instabilities in extensional flows, instabilities in multidimensional flows, and thermohydrodynamic instabilities. Emphasized in the review are comparisons between theory and experiment and suggested directions for future work.

Instabilities in viscoelastic flows

A rheometrical study is reported on a series of Boger fluids (solutions of polyacrylamide in a water/maltose syrup base). Steady-shear and oscillatory-shear experiments lead to the now-accepted conclusion that Boger fluids appear to be more “elastic” in steady shear than they do in oscillatory shear. At the same time, the limiting continuum relations for the material functions at low shear rates and low frequencies are not violated. It is concluded that it is more appropriate to use an Oldroyd B model in the interpretation of the results than the upper-convected Maxwell model, in line with the suggestion of Prilutski et al. [9]. At relatively high shear rates, shear thickening is observed in steady-shear flow, associated with a time-dependent anti-thixotropic behaviour. The new Sangamo Spin-Line Rheometer is used to obtain an estimate of the Boger fluids' resistance to an extensional flow. Trouton ratios as high as 104 are obtained for one of the Boger fluids.

A rheometrical study of boger fluids

This paper contains theoretical and experimental work on the squeeze-film situation for Newtonian and non-Newtonian liquids. To facilitate the interpretation of experimental results, a theoretical analysis is developed for an inelastic liquid having an arbitrary viscosity/shear rate relationship. In the process, it is found necessary to relax the common assumption that material planes which are initially horizontal remain so during the subsequent deformation. It is also shown how the inertia of the moving plate can be accommodated in the analysis. The experimental work contains a description of how a commercial rheometer can be adapted with ease to perform in a squeeze-film mode. Experimental data on polymer solutions indicate that under light-loading conditions, the behaviour of the liquids is predictable from a knowledge of the shear dependent viscosity only. However, under conditions of heavy loading ( i.e. high Deborah number), viscoelastic effects are in evidence. Under these conditions, the liquids behave as better lubricants than one would predict from viscosity considerations.

Elastico-viscous squeeze films. part I

In this paper, we readdress the long-standing question as to whether the viscoelastic properties of multigrade oils can have a measurable effect on lubrication characteristics. To facilitate this, we first investigate the rheometrical properties of a number of multigrade oils with similar shear-viscosity responses. Specifically, we employ a Weissenberg Rheogoniometer and a Lodge Stressmeter to provide a measure of the normal stresses. We then investigate the behaviour of the oils in a journal bearing simulator. We find that viscoelasticity does indeed produce a measurable and beneficial effect on lubrication characteristics at the higher eccentricity ratios and that this effect correlates well with a characteristic relaxation time.

The viscoelastic properties of multigrade oils and their effect on journal-bearing characteristics

A flow visualization technique by means of an expanded laser beam and trace amounts of particulate additives is used to study the behaviour of Newtonian and non-Newtonian elastic liquids in a number of complex geometries. Particular attention is paid to the effect of fluid elasticity on the flow characteristics. Attempts are made to simulate numerically the observed flows by using finite-difference techniques. The agreement between theory and experiment is very satisfactory.

On Newtonian and Non-Newtonian Flow in Complex Geometries

A laser flow-visualization technique is used to study the behaviour of Newtonian and non-Newtonian elastic liquids in complex flows involving abrupt changes in geometry. Particular attention is paid to those situations where dramatic elastico-viscous effects are in evidence and the flow characteristics for non-Newtonian liquids are qualitatively different from those for Newtonian liquids. Consideration is also given to the effect of `rounding' sharp re-entrant corners. Two basic types of geometry are studied in detail. The first is the so-called combined mixing-and-separating flow geometry introduced in an earlier study (Cochrane et al. 1981), while the second includes both planar and circular contractions. Attempts to simulate numerically the observed flows are very satisfactory in the Newtonian case, but the elastico-viscous simulations can do no more than indicate trends in the right direction.

Kenneth Walters

Papers

A rheometrical study of boger fluids

Elastico-viscous squeeze films. part I

The viscoelastic properties of multigrade oils and their effect on journal-bearing characteristics

On Newtonian and Non-Newtonian Flow in Complex Geometries

On Dominating Elastico-Viscous Response in Some Complex Flows