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

We give theoretical and experimental consideration to the stress overshoot phenomenon and attempt to distinguish between (apparent) overshoot arising frominstrument effects and (real) overshoot which can be regarded as a property of the material. The experiments are performed on a Weissenberg Rheogoniometer and particular attention is paid to the effect of the natural frequency of the torquemeasuring system and also to edge effects. The experimental results for Newtonian liquids are found to be in excellent quantitative agreement with the theoretical predictions. In the case of elastic liquids, the experimental results are in qualitative agreement with the predictions for a simple elastico-viscous model. We do not find any significant evidence of aberrations due to edge effects. It is concluded that although stress overshoot can be significantly affected by instrument factors, it is nevertheless a true material property.

Stress overshoot: Real and apparent

Experimental results are presented for the case of a sphere falling under gravity along the axis of a cylindrical tube containing an elastic liquid. The radius of the container is twice that of the sphere. The work is a logical extension of the earlier experiments of Jones et al. [W.M. Jones, A.H. Price and K. Walters, J. Non-Newtonian Fluid Mech., 53 (1994) 175–196] and Degand and Walters [E. Degand and K. Walters, J. Non-Newtonian Fluid Mech., 53 (1995) 103–115], the new feature being the substantial shear thinning exhibited by the two polymer solutions investigated. It is found that, in general terms, shear thinning has a much stronger influence on the settling viscosity than the fluid elasticity, in line with the earlier conclusions of Mena et al. [B. Mena, O. Menero and L.G. Leal, J. Non-Newtonian Fluid Mech., 26 (1987) 247–275]. When an attempt is made to accommodate the shear thinning using a numerical simulation for an inelastic generalized Newtonian model, the generic behaviour of the normalized drag against Weissenberg number is not inconsistent with that postulated by Walters and Tanner [K. Walters and R.I. Tanner, in R.P. Chhabra and D. Dekee (Eds.), Transport Process in Bubbles, Drops and Particles, Hemisphere, 1991, pp. 73–86], i.e. a drag decrease at low Weissenberg numbers followed by a drag increase at high Weissenberg numbers.

A note on settling in shear-thinning polymer solutions

Consideration is given to the flow of Newtonian and non-Newtonian liquids through a “corrugated” pipe of circular cross section whose radius sinusoidally along its longitudinal axis. The flow is produced by a constant pressure gradient. The theory predicts that in the Newtonian case, the corrugated wall of the pipe causes a reduction in flow rate when compared with the value expected on the basis of a straight pipe of the same mean radius. This prediction is confirmed by experiment. In the case of elastico-viscous liquids, the theoretical analysis for a certain fluid model indicates that the reduction in flow rate may be amplified or reduced by the non-Newtonian properties of the liquid depending on the precise flow conditions. The qualitative predictions are again confirmed by experiment. The possible practical implications of the work are outlined.

On the flow of Newtonian and non-Newtonian liquids through corrugated pipes

This paper addresses the problem of matching experimental findings with numerical prediction for the extreme experimental levels of pressure-drops observed in the 4:1 sharp-corner contraction flows, as reported by Nigen and Walters [“Viscoelastic contraction flows: Comparison of axisymmetric and planar configurations,” J. Non- Newtonian Fluid Mech. 102, 343–359 (2002)]. In this connection, we report on significant success in achieving quantitative agreement between predictions and experiments. This has been made possible by using a new swanINNFM model, employing an additional dissipative function. Notably, one can observe that extremely large pressure-drops may be attained with a suitable selection of the extensional viscous time scale. In addition, and on vortex structure, the early and immediate vortex enhancement for Boger fluids in axisymmetric contractions has also been reproduced, which is shown to be absent in planar counterparts.

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Numerical vs experimental pressure drops for Boger fluids in sharp-corner contraction flow

Reasonable agreement is found between values of the first normal-stress differenceN
1 for samples of “D2”, a polyisobutylene/decalin solution, measured in steady shear flow using three different instruments: a Weissenberg Rheogoniometer (a cone-plate rotational rheometer), a Torsional Balance (plate-plate rotational) Rheometer, and a Stressmeter (a transverse-slot slit-die rheometer). Viscosity values are also in reasonable agreement. Ranges of variables common to at least two rheometers include values of shear stressσ up to 3,700 Pa and shear rate
 
$$\dot \gamma $$

 up to 20,700 s−1 near 25 °C. The agreement supports the approximate validity of the semi-empirical “HPBL equation” used to calculateN
1 from Stressmeter data over a range of shear rates up to 20,700 s−1 near 25 °C. Time-temperature superposition behavior exhibited by Stressmeter data at temperatures in the range 21 °C to 111 °C suggest that the range of validity of the Stressmeter method for determiningN
1 approximately may extend up to shear rates of 290,000 s−1 at 111 °C.

Kenneth Walters

Papers

Stress overshoot: Real and apparent

A note on settling in shear-thinning polymer solutions

On the flow of Newtonian and non-Newtonian liquids through corrugated pipes

Numerical vs experimental pressure drops for Boger fluids in sharp-corner contraction flow

Measurement of the first normal-stress difference at high shear rates for a polyisobutylene/decalin solution “D2”