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

It has often been remarked that turbulence is a subject of great scientific and technological importance, and yet one of the least understood (e.g. McComb 1990). To an outsider this may seem strange, since the basic physical laws of fluid mechanics are well established, an excellent mathematical model is available in the Navier-Stokes equations, and the results of well over a century of increasingly sophisticated experiments are at our disposal. One major difficulty, of course, is that the governing equations are nonlinear and little is known about their solutions at high Reynolds number, even in simple geometries. Even mathematical questions as basic as existence and uniqueness are unsettled in three spatial dimensions (cf Temam 1988). A second problem, more important from the physical viewpoint, is that experiments and the available mathematical evidence all indicate that turbulence involves the interaction of many degrees of freedom over broad ranges of spatial and temporal scales. One of the problems of turbulence is to derive this complex picture from the simple laws of mass and momentum balance enshrined in the NavierStokes equations. It was to this that Ruelle & Takens (1971) contributed with their suggestion that turbulence might be a manifestation in physical

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The Proper Orthogonal Decomposition in the Analysis of Turbulent Flows

Microfluidic devices for manipulating fluids are widespread and finding uses in many scientific and industrial contexts. Their design often requires unusual geometries and the interplay of multiple physical effects such as pressure gradients, electrokinetics, and capillarity. These circumstances lead to interesting variants of well-studied fluid dynamical problems and some new fluid responses. We provide an overview of flows in microdevices with focus on electrokinetics, mixing and dispersion, and multiphase flows. We highlight topics important for the description of the fluid dynamics: driving forces, geometry, and the chemical characteristics of surfaces.

Engineering flows in small devices

It is difficult to mix solutions in microchannels. Under typical operating conditions, flows in these channels are laminar—the spontaneous fluctuations of velocity that tend to homogenize fluids in turbulent flows are absent, and molecular diffusion across the channels is slow. We present a passive method for mixing streams of steady pressure-driven flows in microchannels at low Reynolds number. Using this method, the length of the channel required for mixing grows only logarithmically with the Pe «clet number, and hydrodynamic dispersion along the channel is reduced relative to that in a simple, smooth channel. This method uses bas-relief structures on the floor of the channel that are easily fabricated with commonly used methods of planar lithography.

https://www.princeton.edu/%7Estonelab/Publications/pdfs/From%20Howard/ScienceNaturePNAS/Science/abe-mixer-science.pdf

Chaotic Mixer for Microchannels

A new subgrid scale model is proposed for Large Eddy Simulations in complex geometries. This model which is based on the square of the velocity gradient tensor accounts for the effects of both the strain and the rotation rate of the smallest resolved turbulent fluctuations. Moreover it recovers the proper y
3 near-wall scaling for the eddy viscosity without requiring dynamic procedure. It is also shown from a periodic turbulent pipe flow computation that the model can handle transition.

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Subgrid-scale stress modelling based on the square of the velocity gradient tensor

An important achievement of modern experimental fluid mechanics is the invention and development of techniques for the measurement of whole, instantaneous fields of scalars and vectors. These techniques include tomographic interferometry (Hesselink 1988) and planar laser-induced fluorescence for scalars (Hassa et al 1987), and nuclear-magnetic-resonance imaging (Lee et al 1987), planar laser-induced fluorescence, laser-speckle velocimetry, particle-tracking velocimetry, molecular-tracking velocimetry (Miles et al 1989), and particle-image velocimetry for velocity fields. Reviews of these methods can be found in articles by Lauterborn & Vogel (1984), Adrian (1986a), Hesselink (1988), and Dudderar et al (1988), in books written by Merzkirch (1987) and edited by Chiang & Reid (1988) and Gad-el-Hak (1989).

Particle-Imaging Techniques for Experimental Fluid Mechanics

The evolution of a single hairpin vortex-like structure in the mean turbulent field of a low-Reynolds-number channel flow is studied by direct numerical simulation. The structure of the initial three-dimensional vortex is extracted from the two-point spatial correlation of the velocity field by linear stochastic estimation given a second-quadrant ejection event vector. Initial vortices having vorticity that is weak relative to the mean vorticity evolve gradually into omega-shaped vortices that persist for long times and decay slowly. As reported in Zhou, Adrian & Balachandar (1996), initial vortices that exceed a threshold strength relative to the mean flow generate new hairpin vortices upstream of the primary vortex. The detailed mechanisms for this upstream process are determined, and they are generally similar to the mechanisms proposed by Smith et al. (1991), with some notable differences in the details. It has also been found that new hairpins generate downstream of the primary hairpin, thereby forming, together with the upstream hairpins, a coherent packet of hairpins that propagate coherently. This is consistent with the experimental observations of Meinhart & Adrian (1995). The possibility of autogeneration above a critical threshold implies that hairpin vortices in fully turbulent fields may occur singly, but they more often occur in packets. The hairpins also generate quasi-streamwise vortices to the side of the primary hairpin legs. This mechanism bears many similarities to the mechanisms found by Brooke & Hanratty (1993) and Bernard, Thomas & Handler (1993). It provides a means by which new quasi-streamwise vortices, and, subsequently, new hairpin vortices can populate the near-wall layer.

Mechanisms for generating coherent packets of hairpin vortices in channel flow

The structure of energy-containing turbulence in the outer region of a zero-pressure-
 gradient boundary layer has been studied using particle image velocimetry (PIV) to measure the instantaneous velocity fields in a streamwise-wall-normal plane. Experiments performed at three Reynolds numbers in the range 930 0) that occur on a locus inclined at 30–60° to the wall.In the outer layer, hairpin vortices occur in streamwise-aligned packets that propagate with small velocity dispersion. Packets that begin in or slightly above the buffer layer are very similar to the packets created by the autogeneration mechanism (Zhou, Adrian & Balachandar 1996). Individual packets grow upwards in the streamwise direction at a mean angle of approximately 12°, and the hairpins in packets are typically spaced several hundred viscous lengthscales apart in the streamwise direction. Within the interior of the envelope the spatial coherence between the velocity fields induced by the individual vortices leads to strongly retarded streamwise momentum, explaining the zones of uniform momentum observed by Meinhart & Adrian (1995). The packets are an important type of organized structure in the wall layer in which relatively small structural units in the form of three-dimensional vortical structures are arranged coherently, i.e. with correlated spatial relationships, to form much longer structures. The formation of packets explains the occurrence of multiple VITA events in turbulent ‘bursts’, and the creation of Townsend's (1958) large-scale inactive motions. These packets share many features of the hairpin models proposed by Smith (1984) and co-workers for the near-wall layer, and by Bandyopadhyay (1980), but they are shown to occur in a hierarchy of scales across most of the boundary layer.In the logarithmic layer, the coherent vortex packets that originate close to the wall frequently occur within larger, faster moving zones of uniform momentum, which may extend up to the middle of the boundary layer. These larger zones are the induced interior flow of older packets of coherent hairpin vortices that originate upstream and over-run the younger, more recently generated packets. The occurence of small hairpin packets in the environment of larger hairpin packets is a prominent feature of the logarithmic layer. With increasing Reynolds number, the number of hairpins in a packet increases.

Vortex organization in the outer region of the turbulent boundary layer

The development of the method of particle image velocimetry (PIV) is traced by describing some of the milestones that have enabled new and/or better measurements to be made. The current status of PIV is summarized, and some goals for future advances are addressed.

Twenty years of particle image velocimetry

A three-dimensional serpentine microchannel design with a "C shaped" repeating unit is presented in this paper as a means of implementing chaotic advection to passively enhance fluid mixing. The device is fabricated in a silicon wafer using a double-sided KOH wet-etching technique to realize a three-dimensional channel geometry. Experiments using phenolphthalein and sodium hydroxide solutions demonstrate the ability of flow in this channel to mix faster and more uniformly than either pure molecular diffusion or flow in a "square-wave" channel for Reynolds numbers from 6 to 70. The mixing capability of the channel increases with increasing Reynolds number. At least 98% of the maximum intensity of reacted phenolphthalein is observed in the channel after five mixing segments for Reynolds numbers greater than 25. At a Reynolds number of 70, the serpentine channel produces 16 times more reacted phenolphthalein than a straight channel and 1.6 times more than the square-wave channel. Mixing rates in the serpentine channel at the higher Reynolds numbers are consistent with the occurrence of chaotic advection. Visualization of the interface formed in the channel between streams of water and ethyl alcohol indicates that the mixing is due to both diffusion and fluid stirring.

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Ronald J. Adrian

Papers

Particle-Imaging Techniques for Experimental Fluid Mechanics

Mechanisms for generating coherent packets of hairpin vortices in channel flow

Vortex organization in the outer region of the turbulent boundary layer

Twenty years of particle image velocimetry

Passive mixing in a three-dimensional serpentine microchannel