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

The first part of this paper contains a reconsideration of the conventional squeeze-film situation. It is shown that the Part 1 concentration on the half-time t 1 2 sometimes hides interesting elastico-viscous effects, since these are most pronounced at short times after the load is applied. The bulk of the paper is concerned with the more general situation in which a rotational flow is superimposed on the basic squeezing flow. This is brought about by rotating the bottom plate. An approximate theoretical analysis is shown to lead to a satisfactory prediction of observed behaviour under conditions of light loading. The experiments show substantial transient stress-overshoot effects under some conditions and there is also a possible indication of long-range memory effects in some of the experiments.

Elastico-viscous squeeze films: Part 2. superimposed rotation

It is shown that the thrust, T , exerted by a jet on the tube from which it flows, and the corresponding die-swell ratio, D , are closely related and dependent on the axial velocity and stress profiles at the exit plane. Velocity-profile data, calculated by Tanner using a finite element method, have been used to demonstrate that for a Newtonian liquid the reduction in measured thrust from the expected value arises from a re-arranged, non-parabolic axial velocity profile and the related re-arranged non-zero axial stress profile at the exit plane. The axial stress re-arrangement is the major effect. Using the correction-curve thus derived to determine the normal stresses, ν 1 + 1 2 ν 2 aqueous and non-aqueous polymer solutions gives values that are higher than the “correct” results by a significant, substantial amount. The difference is not due to neglect of the second normal stress difference, ν 2 , nor to the neglect of the wall pressure at the exit plane, which is shown experimentally to be very small. It is suggested that the difference, which is a function only of the shear stress (or rate of shear) at the wall, may arise from a difference in the stress profile associated with the velocity re-arrangement at the exit between Newtonian liquids and elasticoviscous liquids for which the extensional viscosity may be high.

A critical re-appraisal of the jet-thrust technique for normal stresses, with particular reference to axial velocity and stress rearrangement at the exit plane

This numerical study focuses on regularised Bingham-type and viscoelastoplastic fluids, performing simulations for 4:1:4 contraction–expansion flow with a hybrid finite element–finite volume subcell scheme. The work explores the viscoplastic regime, via the Bingham–Papanastasiou model, and extends this into the viscoelastoplastic regime through the Papanastasiou–Oldroyd model. Our findings reveal the significant impact that elevation has in yield stress parameters, and in sharpening of the stress singularity from that of the Oldroyd/Newtonian models to the ideal Bingham form. Such aspects are covered in field response via vortex behaviour, pressure-drops, stress field structures and yielded–unyielded zones. With rising yield stress parameters, vortex trends reflect suppression in both upstream and downstream vortices. Viscoelastoplasticity, with its additional elasticity properties, tends to disturb upstream–downstream vortex symmetry balance, with knock-on effects according to solvent-fraction and level of elasticity. Yield fronts are traced with increasing yield stress influences, revealing locations where relatively unyielded material aggregates. Analysis of pressure drop data reveals significant increases in the viscoplastic Bingham–Papanastasiou case, O (12%) above the equivalent Newtonian fluid, that are reduced to 8% total contribution increase in the viscoelastoplastic Papanastasiou–Oldroyd case. This may be argued to be a consequence of strengthening in first normal stress effects.

Computations with viscoplastic and viscoelastoplastic fluids

A technique of flow visualization by means of an expanded laser beam and trace amounts of particulate additives is used to study the behaviour of newtonian and nonnewtonian elastic liquids in complex geometries. Differences in response are highlighted between the newtonian and non-newtonian fluids when these flow separately in certain contraction flows and also in a two-dimensional T geometry. This information is then used to interpret the behaviour when both types of fluid flow together in the same geometry with a well-defined interface between them. Of major interest is the observation that a newtonian liquid in the two-liquid situation (the other liquid being non-newtonian) can behave as if it were highly elastic. We are led to associate this behaviour with the boundary conditions existing at the interface between the newtonian and the non-newtonian liquids. Powerful finite-element numerical techniques are used in an attempt to simulate the observed flow characteristics. The techniques are able to meet the challenges posed by the two-liquid situation when both liquids are newtonian. They are also able to simulate the distinctive vortex structure observed when only one highly elastic liquid is used in the experiments. They are, as yet, unable to meet the difficult challenge where one of the two liquids is highly elastic. The experimental and theoretical work has an obvious potential application to the important practical problem of ‘coextrusion’.

Interfacial effects in the flow of viscous and elasticoviscous liquids

Consideration is given to the stability of the flow of an idealized elastico-viscous liquid in a narrow curved channel, the motion being due to a pressure gradient acting around the channel It is shown that the main effect of the elasticity of the liquid is to lower the value of the critical Reynolds number at which instability occurs

Kenneth Walters

Papers

Elastico-viscous squeeze films: Part 2. superimposed rotation

A critical re-appraisal of the jet-thrust technique for normal stresses, with particular reference to axial velocity and stress rearrangement at the exit plane

Computations with viscoplastic and viscoelastoplastic fluids

Interfacial effects in the flow of viscous and elasticoviscous liquids

The Stability of the Flow of an Elastico-Viscous Liquid in a Curved Channel