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 Laser Doppler technique is used to study flow characteristics in the vicinity of a re-entrant corner for both Newtonian and elastic liquids. An L-shaped channel is chosen as the test geometry and a constant-viscosity Boger fluid as the elastic liquid. It is found that near the re-entrant corner the flow characteristics are virtually independent of both fluid inertia and fluid elasticity. The observed influence of fluid inertia is in agreement with existing theoretical predictions, whilst that of fluid elasticity is consistent with the elastic liquid behaving asymptotically like a Newtonian viscous liquid near the corner, suggesting the use of Oldroyd- (rather than Maxwell-) type models in future simulation studies. Numerical predictions for the Newtonian cases using finite-difference techniques are in satisfactory agreement with experiment, whilst those for an Oldroyd model are in qualitative agreement with experiment results for the Boger fluid.

Newtonian and non-Newtonian flow near a re-entrant corner

The review begins with a critical assessment of existing work in continuum non-Newtonian fluid mechanics. To facilitate this, relevant papers are divided into three categories, these being associated with (i) slow flow or slightly elastic liquids, (ii) nearly viscometric flows and (iii) complex flows. A general argument is then developed which deprecates too much emphasis on one particular strand of research, whether this is associated with a preoccupation with “generality” or with the need to be “practically relevant”. By drawing attention to certain case histories, the review concludes that a unifying and helpful picture emerges when the best of the work associated with the various strands of research is taken into account.

Developments in non-Newtonian fluid mechanics — a personal view

Viscoelastic effects in pressure-driven flow through a curved pipe of circular cross-section are studied by perturbation methods. Relatively simple constitutive models are studied, including the Upper Convected Maxwell (UCM) fluid. It is shown that in the swirling flow generated by the curvature of the pipe, fluid inertia and viscoelasticity are complementary influences in determining the direction of swirl, in conflict with general expectation. Moreover, for creeping flow of the UCM model, drag reduction followed by drag enhancement is predicted as the pressure gradient generating the flow is systematically increased.

On viscoelastic effects in swirling flows

In this paper, we give detailed attention to a relatively recent method for the determination of the linear dynamic properties of viscoelastic systems, namely, the so-called oscillatory squeeze flow (OSF) technique. We provide a comprehensive theory for the OSF rheometer, which includes a full discussion of the influence of fluid inertia. In the process, it is argued that, fortuitously perhaps, fluid inertia is more easily accommodated in the OSF rheometer than in the corresponding torsional-flow techniques. A new version of the OSF rheometer is described and experimental results on a set of viscoelastic systems are used to demonstrate the versatility of the technique. In the process, the potential use of the instrument within an industrial quality control environment is stressed.

The oscillatory squeeze flow rheometer: comprehensive theory and a new experimental facility

A capillary rheometer is adapted to permit measurements of the rheological properties of lubricants at elevated pressures (up to 80 MPa) The pressure drop in the capillary section of the rheometer is used to provide shear viscosity data and the ‘entrance pressure loss’ in the contraction flow before the capillary section is used to estimate the extensional viscosity, employing ideas pioneered by Cogswell and Binding Experimental results on single grade oils (which are near-Newtonian in response) are used to validate procedures, and attention is then given to two multigrade oils in the 15W/40 category, both in a virgin and a degraded state A substantial pressure dependence of the Trouton ratio for the virgin oils is demonstrated, and this is shown to be greatly reduced in the degraded samples The relevance of the data to the performance of multigrade oils in practical journal-bearing situations is emphasised

Kenneth Walters

Papers

Newtonian and non-Newtonian flow near a re-entrant corner

Developments in non-Newtonian fluid mechanics — a personal view

On viscoelastic effects in swirling flows

The oscillatory squeeze flow rheometer: comprehensive theory and a new experimental facility

The rheology of multigrade oils at elevated pressures