Transport and Mixing in Laminar Flows

From Microfluidics to Oceanic Currents

Roman Grigoriev, Heinz Georg Schuster, Roman Grigoriev

• Publisher: Wiley-VCH
• Year: 2012
• ISBN:
  9783527639755
• Edition: 1
• Topic: Technology & Engineering
• Subtopic: Microelectronics
• Index: Technology & Engineering

This book provides readers from academia and industry with an up-to-date overview of important advances in the field, dealing with such fundamental fluid mechanics problems as nonlinear transport phenomena and optimal control of mixing at the micro- and nanoscale.
The editors provide both in-depth knowledge of the topic as well as vast experience in guiding an expert team of authors. The review style articles offer a coherent view of the micromixing methods, resulting in a much-needed synopsis of the theoretical models needed to direct experimental research and establish engineering principles for future applications.
Since these processes are governed by nonlinear phenomena, this book will appeal to readers from both communities: fluid mechanics and nonlinear dynamics.

Chapter 1

Resonances and Mixing in Near-Integrable Volume-Preserving Systems

Dmitri Vainchtein

1.1 Introduction

Many laminar flows are often characterized by a high degree of symmetry due to the confining effect of surface tension (for free-surface flows, e.g., in microdroplets) and/or device geometry (e.g., for flows in microchannels). Designing a flow with good mixing properties is particularly difficult in the presence of symmetries. Symmetry leads to the existence of (flow) invariants [1, 2], which are functions of coordinates that are constant along streamlines of the flow. The level sets of one invariant define surfaces on which the (three-dimensional) flow is effectively two-dimensional. An additional invariant further reduces the flow dimensionality: a flow with two invariants is effectively one-dimensional. Since the flow cannot cross invariant surfaces, the existence of invariants is highly undesirable in the mixing problem as their presence inhibits complete stirring of the full fluid volume by advection. Neither is chaotic advection per se sufficient for good mixing, as time-dependent flows [3, 4] can have chaotic streamlines restricted to two-dimensional surfaces in the presence of an invariant. Thus, the key to achieving effective chaotic mixing in any laminar flow is to ensure that all flow invariants are destroyed.

In this section we will focus on the class of laminar flows characterized by small deviations from exact symmetries. Not only are such flows common in various applications of microfluidics, this is the only class of flows that generically affords a quantitative analytical treatment. The description of the weakly perturbed flow in terms of the action and angle variables allows quantitative analytical treatment using perturbation theory. Indeed, if the symmetries are broken weakly, the invariants (or actions) of the unperturbed flow become slowly varying functions of time (start to drift, in the more technical language) for the perturbed flow, while the angle variable remains quickly varying. Such perturbed flows are referred to as near-integrable, in contrast to the flows with exact symmetries which are integrable, that is, possess an exact analytical solution. Near-integrable systems play a prominent role in many areas of science. Often they arise naturally when there is a large separation of scale...