Julio M. Ottino

Introduction: mixing in microfluidics

In this paper we briefly review the main issues associated with mixing at the microscale and introduce the papers comprising the Theme Issue.

Foundations of chaotic mixing

The simplest mixing problem corresponds to the mixing of a fluid with itself; this case provides a foundation on which the subject rests. The objective here is to study mixing independently of the mechanisms used to create the motion and review elements of theory focusing mostly on mathematical foundations and minimal models. The flows under consideration will be of two types: two–dimensional (2D) ‘blinking flows’, or three–dimensional (3D) duct flows. Given that mixing in continuous 3D duct flows depends critically on cross–sectional mixing, and that many microfluidic applications involve continuous flows, we focus on the essential aspects of mixing in 2D flows, as they provide a foundation from which to base our understanding of more complex cases. The bakerstransformation is taken as the centrepiece for describing the dynamical systems framework. In particular, a hierarchy of characterizations of mixing exist, Bernoulli→mixing→ergodic, ordered according to the quality of mixing (the strongest first). Most importantly for the design process, we show how the so–called linked twist maps function as a minimal picture of mixing, provide a mathematical structure for understanding the type of 2D flows that arise in many micromixers already built, and give conditions guaranteeing the best quality mixing. Extensions of these concepts lead to first–principle–based designs without resorting to lengthy computations.

Laminar mixing and chaotic mixing in several cavity flows

The objective of this work is an experimental study of laminar mixing in several kinds of two-dimensional cavity flows by means of material line and blob deformation in a new experimental system consisting of two sets of roller pairs connected by belts. The apparatus can be adjusted to produce a range of aspect ratios (0.067–10), Reynolds numbers (0.1–100), and various kinds of flow fields with one or two moving boundaries. Flow visualization is conducted by marking underneath the free surface of the flow with a tracer solution of low diffusivity and of approximately the same density and viscosity as the flowing fluid. The effects of the initial location of the material blob, relative motion of the two bands, and minor changes in the geometry of the flow region are investigated experimentally. The alternate periodic motion of two bands in a cavity flow is an example of a laminar flow which might lead to chaotic mixing. The governing parameter is the dimensionless frequency of oscillation of the walls f which, under the proper conditions, is able to produce horseshoe functions of various types. The deformation of blobs is central to the understanding of mixing and can be studied to identify horseshoe functions. It is found that the efficiency of mixing depends strongly on the value of f and that there exists an optimal value of f that produces the best mixing in a given time.

Satellite and subsatellite formation in capillary breakup

An investigation of the interfacial-tension-driven fragmentation of a very long fluid filament in a quiescent viscous fluid is presented. Experiments covering almost three orders of magnitude in viscosity ratio reveal as many as 19 satellite droplets in between the largest droplets; complementary boundary-integral calculations are used to study numerically the evolution of the filament as a function of the viscosity ratio of the fluids and the initial wavenumber of the interface perturbation. Satellite drops are generated owing to multiple breakup sequences around the neck region of a highly deformed filament. In low-viscosity ratio systems, p O (0.1), the breakup mechanism is self-repeating in the sense that every pinch-off is always associated with the formation of a neck, the neck undergoes pinch-off, and the process repeats. In general the agreement between computations and experiments is excellent; both indicate that the initial wavenumber of the disturbance is important in the quantitative details of the generated drop size distributions. However, these details are insignificant when compared with the large variations produced in the drop size distributions owing to variation in the viscosity ratio.

A comparative computational and experimental study of chaotic mixing of viscous fluids

The objective of this work is to develop techniques to predict the results of experiments involving chaotic dispersion of passive tracers in two-dimensional low Reynolds number flows. We present the design of a flow apparatus which allows the unobstructed observation of the entire flow region. Whenever possible we compare the experimental results with those of computations. Conventional tracking of the boundaries of the tracer is inefficient and works well only for low stretches (order 10 2 at most). However, most mixing experiments involve extremely large perturbations from steady flow since this is where the best mixing occurs. The best prediction of widespread mixing and large stretching (order 10 4 –10 5 ) is provided by lineal stretching plots; surprisingly the technique also works for relatively low numbers of periods (as low as 2 or 3). The second best prediction is provided by a combination of low-order unstable manifolds – which indicate where the tracer goes, especially for short times – and the eigendirections of low-order hyperbolic periodic points – which indicate the alignment of striations in the flow. On the other hand, Poincare sections provide only a gross picture of the spreading: they can be used primarily to detect what regions are inaccessible to the dye. Comparison of computations and experiments invariably reveals that bifurcations within islands have little impact in the mixing process.

Experiments on mixing due to chaotic advection in a cavity

We present a versatile cavity flow apparatus, capable of producing a variety of two-dimensional velocity fields, and use it to conduct a detailed experimental study of mixing in low-Reynolds-number flows. Since the goal is detailed understanding, only two time-periodic co-rotating flows induced by wall motions are considered: one continuous and the other discontinuous. Both types of flows produce exponential growth of intermaterial area, as expected from chaotic flows, and a mixture of islands and chaotic regions

TRANSVERSE FLOW AND MIXING OF GRANULAR MATERIALS IN A ROTATING CYLINDER

The focus of this work is analysis of mixing in a rotating cylinder—a prototype system for mixing of granular materials—with the objective of understanding and highlighting the role of flow on the dynamics of the process. The analysis is restricted to low speeds of rotation, when the free surface of the granular solids is nearly flat, and when particles are identical so that segregation is unimportant. The flow is divided into two regions: a rapid flow region of the cascading layer at the free surface, and a fixed bed of particles rotating at the angular speed of the cylinder. A continuum model, in which averages are taken across the layer, is used to analyze the flow in the layer. Good agreement is obtained between the predictions of the flow model for the layer thickness profile and experimental results obtained by digital image analysis. The dynamics of the mixing process are studied by advecting tracer particles by the flow and allowing for particle diffusion in the cascading layer. The mixing model predictions for distribution of tracer particles and mixing rates are compared qualitatively and quantitatively to experimental data. Optimal operating conditions, at which mixing rates are maximum, are determined.

Chaos, Symmetry, and Self-Similarity: Exploiting Order and Disorder in Mixing Processes

Fluid mixing is a successful application of chaos. Theory anticipates the coexistence of order and disorder—symmetry and chaos—as well as self-similarity and multifractality arising from repeated stretching and folding. Experiments and computations, in turn, provide a point of confluence and a visual analog for chaotic behavior, multiplicative processes, and scaling behavior. All these concepts have conceptual engineering counterparts: examples arise in the context of flow classification, design of mixing devices, enhancement of transport processes, and controlled structure formation in two-phase systems.

Experiments on mixing in continuous chaotic flows

We present the design and operation of a flow apparatus for investigations of mixing in time-periodic and spatially periodic chaotic flows. Uses are illustrated in terms of two devices operating in the Stokes regime: the partitioned-pipe mixer , a spatially periodic system consisting of sequences of flows in semicircular ducts, and the eccentric helical annular mixer , a time-periodic velocity field between eccentric cylinders with a superposed Poiseuille flow; other mixing flows can be implemented with relative ease. Fundamental differences between spatially periodic and time-periodic duct flows are readily apparent. Steady spatially periodic systems show segregated KAM-tubes coexisting with chaotic advection; such tubes are remarkably stable under a variety of experimental conditions. Time-periodic duct flows lead to complex streakline structures; since regular regions in the cross-sectional flow move through space, a streakline can find itself injected in a regular domain for some time then be trapped in a chaotic region, and so on, leading to ‘intermittent’ behaviour.

Experimental and computational studies of mixing in complex Stokes flows: the vortex mixing flow and multicellular cavity flows

A complex Stokes flow has several cells, is subject to bifurcation, and its velocity field is, with rare exceptions, only available from numcrical computations. We present experimental and computational studies of two new complex Stokes flows: a vortex mixing flow and multicell flows in slender cavities. We develop topological relations between the geometry of the flow domain and the family of physically realizable flows; we study bifurcations and symmetries, in particular to reveal how the forcing protocol’s phase hides or reveals symmetries. Using a variety of dynamical tools, comparisons of boundary integral equation numerical computations to dye advection experiments are made throughout. Several findings challenge commonly accepted wisdom. For example, we show that higher-order periodic points can be more important than period-one points in establishing the advection template and extended regions of large stretching. We demonstrate also that a broad class of forcing functions produces the same qualitative mixing patterns. We experimentally verify the existence of potential mixing zones for adiabatic forcing and investigate the crossover from adiabatic to non-adiabatic behaviour. Finally, we use the entire array of tools to address an optimization problem for a complex flow. We conclude that none of the dynamical tools alone can successfully fulfil the role of a merit function; however, the collection of tools can be applied successively as a dynamical sieve to uncover a global optimum.