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

Small-scale turbulence has been an area of especially active research in the recent past, and several useful research directions have been pursued. Here, we selectively review this work. The emphasis is on scaling phenomenology and kinematics of small-scale structure. After providing a brief introduction to the classical notions of universality due to Kolmogorov and others, we survey the existing work on intermittency, refined similarity hypotheses, anomalous scaling exponents, derivative statistics, intermittency models, and the structure and kinematics of small-scale structure—the latter aspect coming largely from the direct numerical simulation of homogeneous turbulence in a periodic box.

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The phenomenology of small-scale turbulence

For several centuries fluid dynamics studies have relied upon the assumption that when a liquid flows over a solid surface, the liquid molecules adjacent to the solid are stationary relative to the solid. This no-slip boundary condition (BC) has been applied successfully to model many macroscopic experiments, but has no microscopic justification. In recent years there has been an increased interest in determining the appropriate BCs for the flow of Newtonian liquids in confined geometries, partly due to exciting developments in the fields of microfluidic and microelectromechanical devices and partly because new and more sophisticated measurement techniques are now available. An increasing number of research groups now dedicate great attention to the study of the flow of liquids at solid interfaces, and as a result a large number of experimental, computational and theoretical studies have appeared in the literature. We provide here a review of experimental studies regarding the phenomenon of slip of Newtonian liquids at solid interfaces. We dedicate particular attention to the effects that factors such as surface roughness, wettability and the presence of gaseous layers might have on the measured interfacial slip. We also discuss how future studies might improve our understanding of hydrodynamic BCs and enable us to actively control liquid slip.

https://iopscience.iop.org/article/10.1088/0034-4885/68/12/R05/pdf

Boundary slip in Newtonian liquids: a review of experimental studies

The Lagrangian description of turbulence is characterized by a unique conceptual simplicity and by an immediate connection with the physics of dispersion and mixing. In this article, we report some motivations behind the Lagrangian description of turbulence and focus on the statistical properties of particles when advected by fully developed turbulent flows. By means of a detailed comparison between experimental and numerical results, we review the physics of particle acceleration, Lagrangian velocity structure functions, and pairs and shapes evolution. Recent results for nonideal particles are discussed, providing an outlook on future directions.

Lagrangian Properties of Particles in Turbulence

The properties of the structure functions and other small-scale quantities in turbulent Rayleigh-Benard convection are reviewed, from an experimental, theoretical, and numerical point of view. In particular, we address the question of whether, and if so where in the flow, the so-called Bolgiano-Obukhov scaling exists, i.e., Sθ(r) ∼ r2/5 for the second-order temperature structure function and Su(r) ∼ r6/5 for the second-order velocity structure function. Apart from the anisotropy and inhomogeneity of the flow, insufficiently high Rayleigh numbers, and intermittency corrections (which all hinder the identification of such a potential regime), there are also reasons, as a matter of principle, why such a scaling regime may be limited to at most a decade, namely the lack of clear scale separation between the Bolgiano length scale LB and the height of the cell.

Small-Scale Properties of Turbulent Rayleigh-Bénard Convection

We perform gas flow experiments in a shallow microchannel, 1.14±0.02 μm deep, 200 μm wide, etched in glass and covered by an atomically flat silicon wafer. The dimensions of the channel are accurately measured by using profilometry, optical microscopy and interferometric optical microscopy. Flow-rate and pressure drop measurements are performed for helium and nitrogen, in a range of averaged Knudsen numbers extending up to 0.8 for helium and 0.6 for nitrogen. This represents an extension, by a factor of 3 or so, of previous studies. We emphasize the importance of the averaged Knudsen number which is identified as the basic control parameter of the problem. From the measurements, we estimate the accommodation factor for helium to be equal to 0.91±0.03 and that for nitrogen equal to 0.87±0.06. We provide estimates for second-order effects, and compare them with theoretical expectations. We estimate the upper limit of the slip flow regime, in terms of the averaged Knudsen number, to be 0.3±0.1, for the two gases.

Second-order slip laws in microchannels for helium and nitrogen

We investigate flows of helium IV driven by two counter-rotating disks, in a range of temperatures varying between 1.4 and 2.3 K. The local pressure fluctuations obtained on a small total-head tube are analyzed. Above Tλ, the sensor allows to measure the local velocity fluctuations, and below Tλ, it determines the local fluctuations of a linear combination of the normal and superfluid flow components. Above and below Tλ, Kolmogorov spectra are clearly obtained, with similar Kolmogorov constants. Evidence for persistence of inertial range intermittency in the superfluid region is presented. At all temperatures below Tλ, the structure function exponents are found indistinguishable from those currently observed in normal fluid turbulence.

https://iopscience.iop.org/article/10.1209/epl/i1998-00314-9/pdf

Local investigation of superfluid turbulence

A summary of experimental results on structure functions obtained using extended self-similarity in various flow configurations (jet, grid, mixing layer, duct flow, cylinder) at Reynolds numbers ranging between 30 and 5000 is presented.

http://iopscience.iop.org/article/10.1209/epl/i1996-00472-2/pdf

Structure functions in turbulence, in various flow configurations, at Reynolds number between 30 and 5000, using extended self-similarity

Probability density functions (PDF) of longitudinal velocity increments are measured in an experiment using low temperature helium gas, in which a large domain of microscale Reynolds numbers ${\mathit{R}}_{\ensuremath{\lambda}}$ (from 150 to 5040) is explored. The technique of measurement, which is essentially hot wire anemometry, but operating in nonstandard conditions, is described. In the inertial range of scales, and for a given scale of separation, the PDF are found to be independent of the Reynolds number at large ${\mathit{R}}_{\ensuremath{\lambda}}$. Measurements of the skewness and the flatness of the velocity derivatives are performed in a range of ${\mathit{R}}_{\ensuremath{\lambda}}$ lying between 150 and 3200. The surprising result that we find is that these quantities first increase with the Reynolds number up to ${\mathit{R}}_{\ensuremath{\lambda}}$\ensuremath{\approxeq}700 and then cease to increase further, indicating a transitional behavior. \textcopyright{} 1996 The American Physical Society.

Probability density functions, skewness, and flatness in large Reynolds number turbulence

We report measurements of the probability distribution function of the velocity derivatives, and the corresponding hyperflatness factors, up to order 6, as a function of the microscale Reynolds number Rλ. The measurements are performed in a flow produced between counter-rotating disks, using low-temperature helium gas as the working fluid, in a range of microscale Reynolds numbers lying between 150 and 2300. Consistently with previous studies, a transitional behavior is found around Rλ≈700. We determine a simple scaling law, in terms of Rλ, which allows the collapse of the tails of the pdf of the velocity derivatives onto a single curve, below the transition. We find well-defined relative power laws for the hyperflatness factors Hp and Hp*, throughout the entire range of variation of Rλ: H4=F=(0.99±0.05)H60.376±0.015 and H5*=(0.95±0.05)H60.67±0.022. These results are compared to those of previous investigators and to various theoretical approaches both statistical (multifractal model) and structural (i.e., based on a model of fine scales).

J. Maurer

Papers

Second-order slip laws in microchannels for helium and nitrogen

Local investigation of superfluid turbulence

Structure functions in turbulence, in various flow configurations, at Reynolds number between 30 and 5000, using extended self-similarity

Probability density functions, skewness, and flatness in large Reynolds number turbulence

Velocity gradient distributions in fully developed turbulence : an experimental study