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

A finite-difference formation is employed to study the use of integral rheological models in the numerical simulation of complex flows A contra-variant Maxwell model is used in the study In the method, the physical variables are determined numerically at grid points and no interpolation procedure is required The equations for the displacement functions are solved using a Lax−Wendroff scheme and the complete set of governing equations is solved iteratively using preconditioned conjugate gradient methods The numerical data obtained for the integral model are in good agreement with existing results for the corresponding differential model

Long-range memory effects in flows involving abrupt changes in geometry Part 4: Numerical simulation using integral rheological models

The contents of this lecture are a part of a general research programme which seeks to investigate flows for which elastico-viscous effects are a dominating influence. The basic experimental work is supported by attempts to simulate numerically the observed flow characteristics using finite-difference techniques (see, for example, Cochrane, Walters and Webster [1] Walters and Webster [2]).

On some contraction flows for Boger fluids

A numerical solution is provided for the three-dimensional flow of an upper-convected Maxwell model in a journal bearing, operating under static loading conditions. Realistic values are taken for the material parameters and the geometrical and flow variables. It is concluded that a relaxation time of the order of 10−4 s is required before viscoelasticity results in a practically-important increase in load-bearing capacity. This conclusion is unchanged if fluid inertia is included in the analysis; it is also effectively independent of the L/D ratio of the bearing.

On viscoelastic effects in journal-bearing lubrication

Further consideration is given to the use of implicit rheological equations in the solution of complex flow problems A finite difference solution is obtained for the problem generated by the presence of a protruberance in a basic two-dimensional Couette flow New features of the work include (i) a refinement of the numerical method used in Parts 1 and 2, (ii) the applicability (or otherwise) of explicit rheological models of the “second-order” type, (iii) the handling of the differential boundary conditions on the stress components brought about by the moving boundary in Couette flow

Long range memory effects in flows involving abrupt changes in geometry: Part 3: moving boundaries

The numerical simulation of some contraction flows of highly elastic liquids and their impact on the relevance of the Couette correction in extensional rheology

Kenneth Walters

Papers

Long-range memory effects in flows involving abrupt changes in geometry Part 4: Numerical simulation using integral rheological models

On some contraction flows for Boger fluids

On viscoelastic effects in journal-bearing lubrication

Long range memory effects in flows involving abrupt changes in geometry: Part 3: moving boundaries

The numerical simulation of some contraction flows of highly elastic liquids and their impact on the relevance of the Couette correction in extensional rheology