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.

/pdf/microfluidics-fluid-physics-at-the-nanoliter-scale-4bpahtd8qa.pdf

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

Aqueous solutions of polyacrylamide are investigated in various planar-contraction flows in an attempt to answer some unresolved questions from an earlier paper of the present authors. It is concluded that a lip-vortex mechanism can be responsible for vortex enhancement for any planar-contraction ratio, provided the polymer concentration is chosen carefully or, alternatively, the contraction angle is varied.

Further remarks on the lip-vortex mechanism of vortex enhancement in planar-contraction flows

This is a theoretical paper which attempts to study for the first time the effect of high elasticity in flow situations involving elastico-viscous liquids and abrupt changes in geometry. It is argued that implicit rheological models are essential in this exercise and, accordingly, the numerical method of solution is forced to recognise the equations of continuity, the stress equations of motion and the rheological equations as separate equations involving velocity, pressure and stress variables with appropriate boundary conditions on these variables. The present paper is concerned with L-shaped and T-shaped geometries, and the effect of elasticity is assessed by comparing the numerical predictions for an elastic liquid with those for an inelastic liquid with the same “viscosity” behaviour. This comparison is facilitated by a simple limiting procedure outlined in Section 2. The main conclusions from the work are that, in general terms, elasticity works against inertia, reducing the pressure drop caused by the abrupt change in geometry and reducing the area of influence of the bend (for finite Reynolds numbers). So far as the stress fields are concerned most interest centres on the corner region, as one would expect, but there is also a region of normal-stress activity, which is generated by “stretching” rather than “shearing”. In an appendix, some consideration is given to the entry-length and exit-length problems. It is concluded that the overall problem is a complex one, since it depends to a large measure on the criterion one uses for “fully-developed” flow. If a fairly crude criterion is used, fluid elasticity is found to decrease the entry-length and increase the exit-length.

Long-range memory effects in flows involving abrupt changes in geometry

We consider the behaviour of non-Newtonian liquids as they are made to flow through straight pipes of circular cross section under the action of a pressure -gradient which oscillates sinusoidally about a non-zero mean. The theory for such a situation is developed in detail and certain predictions, some quantitative some qualitative, are made. Quite a substantial increase in mean flow rate due to the fluctuation in the pressure-gradient is predicted under some conditions. The theoretical predictions are shown to be in good agreement with experimental results and a method for locating the “optimum conditions” is outlined which makes use of the experimentally determined apparent viscosity of the fluids only. The possible relevance of the work to blood flow in the human body is discussed.

On pulsatile flow of non-Newtonian liquids

The Cox-Merz empirical relationship between the linear (oscillatory) and nonlinear (steady-state) viscosities has been shown to be valid for many polymeric systems. Here, we present an equivalent expression to relate the linear (G′) and nonlinear (N
1) elastic properties of viscoelastic systems. Like the Cox-Merz relationship, it uses a combination of elastic and viscous parameters. The modified form of the storage modulus is then equivalent to the Cox-Merz complex viscosity. It can be used to correlate with (half) the normal force at numerically equal circular frequency and shear rate, respectively. This new expression and the Cox-Merz rule are tested for a range of polymeric and colloidal systems. It is found that both expression work for the polymeric systems considered, but fail for the colloidal systems. In the latter, the steady state values of viscosity and elasticity are consistently low, and replacing them by the complex viscosity and our new elastic expression only makes matters worse. For polymer systems, we suggest this is a general but not universal observation, since we are aware of exceptions to the rule that polymeric systems obey the Cox-Merz rule for viscosity and our rule for elasticity. For colloidal systems we find that neither rule is obeyed for any of our systems.

The relationship between the linear (oscillatory) and nonlinear (steady-state) flow properties of a series of polymer and colloidal systems

A capillary rheometer has been modified, by the addition of a second chamber and valve arrangement below the main die, in order to measure the pressure drops associated with the capillary and entry flows of a number of polymer melts as a function of pressure. The five polymer melts investigated are high- and low-density polyethylene, polypropylene, polymethyl-methacrylate and polystyrene, each of which is tested at three temperatures within the normal processing range, at apparent shear rates between 50 and 2500 s−1 and at mean pressures ranging from atmospheric up to 70 MPa. The capillary pressure drop data are used to obtain shear viscosity functions using conventional capillary rheometry expressions, whilst extensional viscosities are estimated from orifice pressure drop data via the Cogswell–Binding analysis. Both the shear and extensional viscosity curves for all of the polymers are seen to exhibit an exponential pressure dependence that can be characterised by pressure coefficients that are found to be independent of temperature. Trouton ratios for the polymers can be specified by an expression with separable strain rate and pressure dependence terms, the latter of which is again exponential. The pressure coefficients of the Trouton ratio terms then orders the pressure dependence: PS>PMMA>PP>HDPE>LDPE. Our major conclusion is that the Trouton ratio for some of the polymer melts can be a strong function of the pressure, indicating that the variation of extensional properties with pressure can be greater than that of the shear properties.

Kenneth Walters

Papers

Further remarks on the lip-vortex mechanism of vortex enhancement in planar-contraction flows

Long-range memory effects in flows involving abrupt changes in geometry

On pulsatile flow of non-Newtonian liquids

The relationship between the linear (oscillatory) and nonlinear (steady-state) flow properties of a series of polymer and colloidal systems

The pressure dependence of the shear and elongational properties of polymer melts