Transport and Mixing in Laminar Flows E-Book

• Herausgeber: Wiley-VCH
• Kategorie: Wissenschaft und neue Technologien
• Serie: Reviews of Nonlinear Dynamics and Complexity
• Sprache: Englisch

Seitenzahl: 368

Veröffentlichungsjahr: 2012

Beschreibung

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.

Leseprobe

Contents

Cover

Related Titles

Title Page

Copyright

List of Contributors

Mixing in Laminar Fluid Flows: From Microfluidics to Oceanic Currents

Introduction

Chapter 1: Resonances and Mixing in Near-Integrable Volume-Preserving Systems

1.1 Introduction

1.2 General Properties of Near-Integrable Flows and Different Types of the Resonance Surfaces

1.3 Separatrix Crossings in Volume-Preserving Systems

1.4 Passages Through Resonances in Autonomous Flows

1.5 Passages Through Resonances in Nonautonomous Flows

Acknowledgments

References

Chapter 2: Fluid Stirring in a Tilted Rotating Tank

2.1 Introduction and Background Information

2.2 Tilted-Rotating Tank Analysis

2.3 Experiments

2.4 Conclusion

Acknowledgments

References

Chapter 3: Lagrangian Coherent Structures

3.2 Background

3.3 Global Approach

3.4 Computational Strategy

3.5 Robustness

3.6 Applications

3.7 Conclusions

Acknowledgments

References

Chapter 4: Interfacial Transfer from Stirred Laminar Flows

4.1 Introduction

4.2 Phenomena and Definitions

4.3 Experimental Methods

4.4 Modeling Approaches

4.5 Conclusions

References

Chapter 5: The Effects of Laminar Mixing on Reaction Fronts and Patterns

5.1 Introduction

5.2 Background

5.3 Advection–Reaction–Diffusion: General Principles

5.4 Local Behavior of ARD Systems

5.5 Synchronization of Oscillating Reactions

5.6 Front Propagation in ARD Systems

5.7 Additional Comments

References

Chapter 6: Microfluidic Flows of Viscoelastic Fluids

6.1 Introduction

6.2 Mixing in Microfluidics

6.3 Non-Newtonian Viscoelastic Fluids

6.4 Governing Equations

6.5 Passive Mixing for Viscoelastic Fluids: Purely Elastic Flow Instabilities

6.6 Other Forcing Methods - Applications

6.7 Conclusions and Perspectives

Acknowledgments

References

Index

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List of Contributors

Manuel A. Alves Universidade do Porto Faculdade de Engenharia Centro de Estudos de Fenómenos de Transporte Departamento de Engenharia Química Rua Dr. Roberto Frias s/n 4200-465 Porto Portugal

Roman O. Grigoriev Georgia Institute of Technology School of Physics 837 State St. Atlanta, GA 30332-0430 USA

Joseph D. Kirtland Dordt College Department of Physics Sioux Center, IA 51250 USA

Mónica S. N. Oliveira Universidade do Porto Faculdade de Engenharia Centro de Estudos de Fenómenos de Transporte Departamento de Engenharia Química Rua Dr. Roberto Frias s/n 4200-465 Porto Portugal

Fernando T. Pinho Universidade do Porto Faculdade de Engenharia Centro de Estudos de Fenómenos de Transporte Departamento de Engenharia Mecânica Rua Dr. Roberto Frias s/n 4200-465 Porto Portugal

Shawn C. Shadden Illinois Institute of Technology Mechanical, Materials & Aerospace Engineering 10 West 32nd Street 243 Engineering 1 Building Chicago, IL 60616-3793 USA

Tom Solomon Bucknell University Department of Physics & Astronomy Lewisburg, PA 17837 USA

Abraham D. Stroock Cornell University School of Chemical and Biomolecular Engineering 120 Olin Hall Ithaca, NY 14853 USA

Thomas Ward North Carolina State University Department of Mechanical and Aerospace Engineering 911 Oval Dr. Raleigh, NC 27695-7910 USA

Dmitri Vainchtein Temple University Department of Mechanical Engineering 1947 N 12th St. Philadelphia, PA 19122 USA

Mixing in Laminar Fluid Flows: From Microfluidics to Oceanic Currents

Roman O. Grigoriev

Introduction

Transport properties of laminar fluid flows have attracted a lot of attention over the past decade. To a large extent, this increased interest has been driven by the rapid development of microfluidic technologies which revolutionized practical molecular biology by enabling high-throughput sequencing. Speeding up the scaled-down versions of chemical essays, such as polymerase chain reaction, demanded by applications in genomics, proteomics, and clinical pathology requires quick and thorough mixing of several components (e.g., DNA fragments and fluorescent markers) inside microscopic liquid volumes. Although molecular diffusion becomes an effective mixing mechanism at small scales, it is not fast enough at the scale of a typical microfluidic device. Advection by a moving fluid provides an alternative, and much quicker, mixing mechanism. This conclusion follows directly from the advection–diffusion equation

(1)

which describes the evolution of solute concentration with molecular diffusion constant in the fluid solvent moving with velocity . Nondimensionalizing this equation we discover that, for a fluid flow with the characteristic length scale and velocity scale , molecular diffusion described by the term is dominant when the nondimensional Peclet number is small, while advection described by the term dominates in the opposite limit. Typical microfluidic devices happen to be characterized by , placing them in the diffusion-dominated regime.

We should also point out that the evolution equation (1), where the concentration is replaced by the temperature of the fluid and the molecular diffusivity is replaced by the heat diffusivity , describes heat transport in the fluid, with numerous modern applications in microelectronics cooling. At the opposite end of the size scale, the same equation has been used to describe the transport properties of atmospheric and oceanic currents. Some of the high-profile applications include problems describing the spreading of environmental pollutants such as oil and radioactive elements released as a result of accidents at offshore drilling platforms and nuclear plants.

A mathematically equivalent, but more analytically tractable, description of advective transport in the limit of negligible diffusion is provided by replacing partial differential equation (1) with an ordinary differential equation

(2)

which defines the streamlines of the flow, but also describes the motion of infinitesimal fluid elements characterized by a constant value of concentration (or temperature ) on time scales on which diffusion can be neglected. The Jacobian of the system (2) describes the local shape dynamics of the fluid elements in the local comoving reference frame, as they are stretched in some directions and compressed in others, as explained in Chapter 3. This process is usually referred to as Lagrangian transport, reflecting the choice of the Lagrangian reference frame comoving with the flow, as opposed to the static global Eulerian reference frame used in (1).

While this reduction in the transport problem affords tremendous simplification, the system (2) defines a set of three coupled nonlinear ordinary differential equations, which in general can only be solved numerically. One exception is the special limit of a steady integrable flow which possesses one or two conserved quantities, referred to as integrals of motion, . These integrals arise when the flow possesses a symmetry. At the microscale, symmetries are pervasive and usually reflect the geometry of the flow (e.g., spherical shape of microdroplets, round or rectangular shape of microchannels, etc.). The flows with one invariant, also known as action–angle–angle flows, are effectively two-dimensional (with quasi-periodic streamlines), while the flows with two invariants, of the action–action–angle type are one-dimensional (with time-periodic streamlines). Integrable flows possess extremely poor mixing properties because level sets of each invariant represent a transport barrier – a two-dimensional surface which cannot be crossed by any streamline of the flow.

Both types of integrable flows allow exact analytical solutions to be found, although rarely in explicit form. Solutions to nonintegrable flows are chaotic and cannot be found in analytic form. However, approximate semianalytical short-term solutions can be computed, for both steady and time-periodic flows, using perturbation theory described in Chapter 1 when the velocity field is near-integrable, that is, , where is an integrable flow and . The perturbative analysis shows that, somewhat surprisingly, the simpler action–action–angle flows possess dramatically better mixing properties than the more complex action–angle–angle flows in the presence of a generic perturbation. This is illustrated in Chapter 2 which compares theoretical predictions with experimental observations for a perturbed action–action–angle flow arising in a tilted rotated tank.

While the perturbative description of advective transport is unique in that it allows one to describe the mixing process quantitatively, it breaks down in the nonperturbative regime. There are two leading alternatives which have been used to characterize mixing in the general case; both providing a qualitative description. One, analytical, is based on the topological analysis of the streamlines of the flow in the Eulerian reference frame. Another, numerical, is based on finite-time Lyapunov exponents which measure the rate of stretching or contraction of infinitesimal fluid elements in the Lagrangian reference frame. As Chapter 3 explains, the Lagrangian coherent structures defining regions of the flow characterized by the largest or the smallest values of Lyapunov exponents tend to organize the transport. Specifically, the largest exponents define the locations of the transport barriers and the smallest – the regions of the flow where mixing is most efficient.

Advective transport discussed in Chapters 1--3 describes only the initial stage of mixing. For a flow with chaotic streamlines, the fluid elements are stretched in some directions and, at the same time, compressed in others exponentially fast. The associated sharpening of the concentration gradients leads to a greatly enhanced molecular diffusion (or heat conduction). Therefore, on longer time scales, one inevitably has to revert to using the advection–diffusion equation (1). Moreover, diffusive transport might be non-negligible even at earlier times, either for sufficiently small volumes or in the regions of the flow where the velocity of the fluid becomes very small (e.g., near solid walls). The latter situation is discussed in Chapter 4, which considers transport across interfaces, either between fluids (characterized by stress-free boundary conditions) or between a fluid and a solid (no-slip boundary conditions). Understanding transport across interfaces is essential in problems involving, for example, heat exchange between fluids and solids or chemical reactions localized to the walls.

Chapter 5 is devoted to the interplay between mixing and chemical reactions taking place in the bulk. Indeed, chemical reaction rates depend on the local concentrations of reactants and products, and hence transport phenomena are expected to play an important role. The most general models of chemical reactors are formulated in terms of reaction–advection–diffusion equations which generalize (1):

(3)

where the nonlinear functions describe the reaction rates, are the concentrations of different chemicals, and – the corresponding diffusion constants. While the field of reaction–diffusion systems is well established – the Turing mechanism for pattern formation has been proposed in 1952 – no comprehensive theory of reaction–advection–diffusion system currently exists. Experiments described in Chapter 5 show that the fluid flow, especially when it is time-dependent, has a highly nontrivial effect on the dynamics of reaction fronts, illustrating the importance of both transport mechanisms – diffusion and advection – on the chemical reaction patterns arising in fluids. In particular, transport barriers are determined to play a crucial role in the organization of chemical patterns.

Quite generally, and especially at the microscale, steady flows are found to have inferior mixing properties, compared with their time-dependent counterparts. This is quite logical since mixing is caused by stretching and compression of fluid elements advected along chaotic streamlines. According to dynamical systems theory, in order for chaos to arise, the flow has to be at least three-dimensional. On the other hand, both symmetries and spatial confinement tend to reduce the effective dimensionality of the flow, making it two- or even one-dimensional and hence precluding chaotic streamlines. The effective dimensionality of the flow can be increased either by removing spatial confinement and symmetry (forcing the fluid through a twisted microchannel or a microchannel with a variable cross-section are good examples) or by introducing time-dependence.

Microscale flows of Newtonian fluids such as water, however, tend to be steady. Indeed, the flow of Newtonian fluids is governed by the Navier–Stokes equation

(4)

where is the density of the fluid, is its dynamic viscosity, is the pressure, and is the body force density. At the microscale, the Reynolds number is usually much smaller than unity, so (4) reduces to the Stokes equation

(5)

Hence, the direction of the driving force has to be time-dependent in order to generate an unsteady flow , which can be quite hard to achieve in practice.

An alternative approach, described in Chapter 6, is to use non-Newtonian – or viscoelastic – fluids, which are characterized by both viscous and elastic stresses. Elastic stresses are represented by additional terms on the right-hand-side of (4) which can remain non-negligible even at the typical length scales of microfluidic devices. As a result, in certain geometries the flow becomes susceptible to viscoelastic instabilities which introduce time-dependence, dramatically enhancing mixing. Understanding transport in non-Newtonian fluids is interesting not only from the fundamental perspective, but also due to numerous emerging applications such as clinical pathology which involve handling of polymeric fluids, for example, whole blood or protein solutions.

While this book does not pretend to provide complete coverage of the subject of transport and mixing in laminar fluid flows, it provides a review of the main research thrusts in this active field. It is our hope that the book will appeal to both scientists working on the fundamental aspects of the transport problem and the engineers looking to use the existing knowledge base in the design of a new generation of microfluidic devices.

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 scales and, hence, of the associated forces, for example, as in many problems in celestial mechanics, where the gravitational interaction with the Sun dominates all other forces which can be considered small perturbations [5, 6]. Similarly, for weakly perturbed action–action–angle fluid flows there is a large separation of timescales on which the actions and the angle change.

The space of the integrable unperturbed system is foliated into invariant tori and the motion on these tori is quasi-periodic or periodic. If there are two independent integrals, the tori are invariant closed curves. In general, the integrability requires the existence of at least one conserved quantity (or action or invariant), so all flows of interest belong to one of two classes: action–action–angle or action–angle–angle [1]. Transport in the perturbed action–angle–angle flows is severely restricted by KAM tori (it was illustrated in [7]), while the effective degeneracy of the action–action–angle flows opens the possibility of global transport and mixing. We will, therefore, focus our attention on action–action–angle flows and possible mechanisms leading to chaotic advection.

Exact analytic solutions for near-integrable dynamics cannot be obtained. Direct brute-force numerical simulation of such systems is possible, but usually very challenging precisely due to a big separation of timescales. Approximate analytical tools represent an important alternative for studying such systems. Specifically, the assumption of a weak perturbation allows one to use a collection of perturbation theory methods to describe the dynamics quantitatively. In particular, by averaging the evolution equations for the actions and over a period of the fast motion described by the angle one finds that although the original exact invariants are destroyed, the averaged system of equations itself possesses an invariant . Since the averaged equations are an approximation, in the exact perturbed system is only conserved in the adiabatic sense: its value undergoes small oscillations with period close to that of but the average value of remains the same on much longer time intervals [8]. Therefore, this approximate invariant is referred to as an adiabatic invariant (AI).

As it turns out, the existence of an AI enables a greatly simplified description of the mixing dynamics in near-integrable flows. Of course, if the AI were conserved everywhere, mixing would be restricted to the two-dimensional level sets of the AI (usually tori) defined by , which is often indeed the case (i.e., nonintegrability does not lead to mixing). Mixing requires the breakdown of the adiabatic invariance, which can occur in the flows possessing certain types of singular manifolds, where the fast subsystem slows down and the separation of scales disappears. A very elegant description of the dynamics can be obtained by separating the evolution into regular advection along the level sets of the AI between singular manifolds, and fast passages through the singular manifolds. During the motion near the singular manifolds, the value of the AI typically experiences a change that is much larger than the magnitude of the oscillations of the AI during the motion far from the singular manifolds. As the time of the passage near the singular manifolds is much shorter than the characteristic time of motion between them, the changes in the AI can be treated as instantaneous jumps in describing the evolution of the AI. For every set of initial conditions, the magnitude of the jump in the AI can be calculated exactly. However, a small change in the initial conditions produces in general a large change in the jump magnitude [9, 10]. Hence, for weak perturbations, in computing the statistical properties of many consequent jumps, it is possible to treat the jump magnitude as a random variable with statistical properties obtained from the dependence of the jump magnitude on initial conditions. The dynamics in the vicinity of singular manifolds (separatrices or resonance surfaces) can be described using a different perturbation expansion, where the small parameters are not just the perturbation strength but also the distance to the singular manifold. For resonant surfaces, this approach was first introduced in [11, 12] and further developed in [10, 13] in the context of Hamiltonian systems and subsequently applied to volume-preserving autonomous (such as flows of incompressible fluids) in [14] and nonautonomous [4] systems. A theory for systems with separatrix crossings was proposed in [15, 16] and later developed in [17–19] for Hamiltonian systems and in [9, 20] for volume-preserving autonomous systems.

If allowed by the geometry of the system, the streamline keeps coming to the singular surface(s) again and again and the process of jumps repeats itself. Accumulation of jumps at multiple crossings results in destruction of the adiabatic invariance (i.e., the AI changes by a value of the order 1) and leads to chaotic dynamics in the system. Therefore, the physical space becomes partitioned into the domains of chaotic and regular dynamics filled, respectively, by the streamlines that do, or do not, cross the singular manifold(s). In the chaotic domain, the jumps of the AI associated with separatrix or resonance crossings lead to the destruction of adiabatic invariance and transport across the level sets of the AI. Over long timescales, accumulation of small jumps coupled with the divergence of initially close streamlines lead to effective diffusion of the AI and mixing. This dynamical picture allows one to compute the size and shape of the chaotic and regular domains and to estimate the rate of mixing.

The theory of long-time transport in volume-preserving flows in the presence of chaotic advection and regular diffusion is by no means limited to mixing in fluid flows [21–25]. Such a description of long-time transport in near-integrable Hamiltonian and volume-preserving systems is crucial for long-term predictions of, for example, the dynamics of comets and asteroids [26–28], customizing transport to achieve selective segregation in electromagnetic diverters in plasma confinement devices [29–31], energy exchange between coupled oscillators [32–34], chaotic billiards [35], arrays of Josephson junctions [36, 37], and the drift of charged particles in the Earth magnetosphere [38–42].

In this chapter, we describe destruction of AIs at separatrices and resonances and use several examples studied earlier [4, 14, 43, 44] to illustrate different aspects of the complete picture. We refer the reader to the corresponding chapter for details of derivations and additional discussions. General properties are discussed in Section 1.2. Separatrix crossings are discussed in Section 1.3, and passages through resonances in autonomous and nonautonomous flows are considered in Sections 1.4 and 1.5, respectively.

1.2 General Properties of Near-Integrable Flows and Different Types of the Resonance Surfaces

The motion of passive tracers advected by the flow can be described by a volume-preserving system of ODE in depending on a small parameter, :

(1.1)

Velocity field in (1.1) defines an unperturbed (base) flow; is a perturbation and is supposed to be a smooth function of . We restrict our discussion to autonomous base flows, while the perturbation may be autonomous or nonautonomous. System (1.1) at corresponds to the unperturbed system. In a sense, passive tracers in the flows are equivalent to phase points in generic dynamical systems.

The effects of the small perturbation in (1.1) start manifesting themselves on time intervals of order at least . A function of phase variables is called an AI if its value along a phase trajectory of (1.1) has only small (with ) variations on time intervals of such length. In other words, an AI is an approximate first integral of the system. Perpetual conservation of AI presents a barrier for complete mixing.

Let unperturbed system (1.1) be integrable and of the action–action–angle type. Then, almost the entire phase space is filled with closed streamlines. Denote the two independent integrals of motion as and . Every joint level of the two integrals defines a closed unperturbed phase trajectory . Introduce on an angular variable changing at a constant rate in the unperturbed motion.

The perturbation in (1.1) causes the values of and to change at a rate of order in the motion along a perturbed streamline. In terms of the variables , perturbed system (1.1) can be written as

(1.2)

The functions are -periodic in . In (1.2), the variables are “slow,” and the variable is “fast.” Define the averaged system:

(1.3)

where functions and are obtained by averaging and , respectively, over :

(1.4)

In (1.4), is calculated at , parentheses denote the scalar product, and the integration is performed along . Far from the singular surfaces (described below), solutions of the averaged system describe variations of in complete system (1.1) with the accuracy of order on time intervals of order [45, 46].

Let be the flux of the perturbation through a surface spanning . Due to the preservation of the volume, the value of does not depend on a particular choice of the surface. A remarkable fact is that averaged system (1.3) is Hamiltonian, and is a Hamiltonian function (see, e.g., [9, 20]):

(1.5)

where is a certain function of and , determined by the base flow and is the period of the unperturbed motion along . In almost all the systems we studied recently, . Moreover, it is always the case when the base flow has axial symmetry and the two invariants are the streamfunction and the azimuthal angle. It follows from (1.5) that is an integral of the averaged system. Standard assertions about the accuracy of the averaging method (see, e.g., [8, 45, 46]) imply that is an approximate integral of the motion in exact system (1.1), that is, is an adiabatic invariant (see Figure 1.1a below).

Figure 1.1 Adiabatic invariants (a) away from the singular surfaces and (b) when a streamline crosses them. The model from [14].

However, the averaging method breaks down in a neighborhood of singular surfaces. These surfaces can be of one of three types:

1.Separatrix surfaces containing nondegenerated hyperbolic fixed points of the unperturbed system and filled by heteroclinic trajectories connecting them. A systems of this type is considered in Section 1.3.

2.Separatrix surfaces containing a line of degenerate singular points (this case occurs, in particular, in 1 d.o.f. Hamiltonian systems depending on a slowly varying parameter or a no-autonomous flows with axial symmetry, and it is considered elsewhere [15, 16, 47]).

3.Resonance surfaces, corresponding to closed curves filling a surface. In autonomous systems with one angle variable, the function in (1.2) is identically zero everywhere on such a surface. A system of this type is considered in Section 1.4. In nonautonomous systems in which time appears as an additional fast phase with a frequency , on resonance surface is a rational number (see Section 1.5). The major difference between separatrix and resonance surfaces is that near separatrix surfaces the base flow slows down only in the immediate vicinity of fixed points of the base system. In comparison, a linear combination (with integer coefficients) of the phase of the base flow and the time variable slows down everywhere near a resonance surface.

Let us just briefly note that the problem of jumps of AIs at separatrix crossings in volume-preserving systems cannot be reduced to similar problems in Hamiltonian systems depending on a slowly varying parameter [15, 16, 47], or in slow–fast Hamiltonian systems [48]. Although ideologically close to them, this problem needs an independent study. Similar phenomena were also observed in 3D volume-preserving maps [7].

A complete description of chaotic advection in these problems starts with a description of a single crossing of a resonance or a separatrix surface. Let a phase point (passive tracer) closely follow a trajectory of the averaged system. The quantity along the streamline oscillates with an amplitude of order around a certain constant value, say, . When the streamline crosses a small neighborhood of a singular surface, changes by a value , , which is in general much greater than . In the case of scatterings on resonance and almost all the separatrix crossings, . After this neighborhood is crossed, the value of along the trajectory oscillates near a new constant value, . As the main change occurs in a narrow neighborhood of a singular surface, we shall call this change a jump of the AI. In every particular problem, an asymptotic formula (in the limit of small values of ) for this change of the AI can be obtained following a standard procedure which was reported in several publications [4, 9, 20, 43, 44, 49]. An example of such a dynamics is illustrated in (see Figure 1.1b). In the case of crossing a resonance when there is the possibility of capture into resonance, the captured dynamics can be also described.

The magnitude of a jump turns out to be very sensitive to variations of initial conditions. Therefore, the jump is in a sense random. If allowed by the geometry of the system, the streamline comes to the separatrix again and the process repeats itself. Accumulation of jumps at multiple crossings results in destruction of the adiabatic invariance (i.e., the AI changes by a value of the order 1) and leads to chaotic dynamics in the system. Based on the equations for a single passage, we can describe statistical properties of jumps and use them to study the long-time dynamics on time intervals that include many crossings.

1.2.1 Metrics of Mixing

Two different (and generally unrelated) metrics should be used to describe chaotic advection in a bounded flow such as the one considered here: the size of the chaotic domain and the characteristic rate of mixing inside the chaotic domain.

The first of these two metrics is the volume of the chaotic domain, . We define the volume and the dimensionality of mixing as the properties of the domain occupied (after a long time) by the tracers that originate in a small ball (say, size ). For , the whole interior is almost completely regular for any kind of perturbation. Actually, any however minuscule perturbation (e.g., molecular diffusion) leads to mixing: the dependence of in (1.2) on the values of and/or results in stretching of the original ball along . This happens over order times. In perturbed systems without singular surfaces, the tracers stay forever in the vicinity of the original surface of constant AI (see Figure 1.1a), eventually covering the whole surface . Thus we say that the adiabatic invariancy leads to mixing.

In the presence of singular surfaces, a chaotic domain of a finite size appears as soon as becomes nonzero. This is due to the fact that while the resonance phenomena are themselves local events (they are only affected by the dynamics in the vicinity of a corresponding surface), their effect is global, extending the chaotic domain to the scale of the entire flow. As a result, depends on parameters of the base flow but not on the magnitude of perturbation, (for infinitesimal ). Depending on parameters of the base flow (e.g., in Section 1.3), the flow domain can be completely regular, partially regular and partially chaotic, or completely chaotic. The size of the chaotic domain is, to leading order, determined by the shape of its boundary – the torus tangential to the singular surface – which is independent of . However, for any finite , the boundary between the two domains is more complex. The chaotic domain penetrates inside adding a small layer (most often with thickness of order , see [50] and Section 1.4 below for details). Further, small islands of stability may appear inside the mixing domain. As a result there are small corrections to .

The second metric, the rate of mixing , on the other hand, strongly depends on . Assuming statistical independence of consecutive crossings (see below), we can describe the evolution of the AI by a random walk with a characteristic step size of order , . Hence, after crossings, the value of the AI changes by a quantity of the order . The mixing can be considered complete when a typical chaotic streamline samples the entire chaotic domain. The difference between the values of the AI that bound the chaotic domain in our problem is of order unity. Therefore, it takes on the order of separatrix crossings for diffusion to cover the whole domain. As the typical time between successive crossings is of the order , we find the characteristic time for mixing to be . This characteristic time diverges for , so the rate of mixing, defined as , vanishes for .

For infinitesimal , the consecutive jumps can be considered statistically independent for most of the streamlines, so the accumulation of jumps can be described as a random walk without memory, leading to the standard Fokker–Planck equation for the probability density function (PDF) of the AI

(1.6)

where is defined as the probability that at time the tracer resides between the surfaces and . The drift velocity and the diffusion coefficient are determined by the first and the second moments of the distribution of the jumps, respectively [51]. Since the jump magnitude distribution is a function of , so are its moments. Moreover, since the dynamics between the jumps takes place on the surfaces of constant AI, the time between the jumps is also a function of [43]. Therefore, both and should be computed by taking this dependence into account.

1.2.2 Correlations of Successive Jumps and Ergodicity

Quantitative properties of the diffusion of the AI, in particular the validity of (1.6), depend on whether consecutive crossings are statistically dependent or independent. In volume-preserving systems, statistical independence (and, thus, the hypothesis of ergodicity, at least up to a residual of a small measure) for vanishing value of can be deduced from the divergence of resonance phases (denoted by in the following sections) along streamlines. A similar problem for the Hamiltonian system was discussed in [10, 52, 53]. For finite values of the consecutive jumps become somewhat correlated, especially near the boundaries of the system. We discuss this in more details in Section 1.3.

Consider statistical properties of the jumps in along one-phase trajectory of the system. Let two successive separatrix crossings be characterized by values and . A small variation in produces a variation of the jump in by . In the period of time before the next crossing, the value obtains a variation . Thus . Therefore, it is natural to suppose that and are statistically independent and the successive jumps in are not correlated.

For many flows, it was verified numerically that inside the chaotic domain one does indeed find a positive Lyapunov exponent for , confirming the divergence of nearby streamlines. Furthermore, for small , the flow possesses good ergodic properties inside the mixing domain, as the Poincaré sections illustrate, indicating very thorough mixing. Indeed, the chaotic domain is essentially devoid of regular islands, so a single streamline densely fills the whole chaotic domain. For decreasing , the regular islands (of size ) are expected to gradually disappear, resulting in perfect mixing.

1.3 Separatrix Crossings in Volume-Preserving Systems

In this section, we consider a flow where the presence of separatrix crossings results in the destruction of adiabatic invariance to illustrate different aspects of the evolution. The following problem was studied in details in [43].

Consider a microdroplet suspended at the free surface of a liquid substrate and driven using the thermocapillary effect with a constant speed in a straight line along the substrate surface [3, 54]. Experiments found the mixing to be very poor in this regime [54]. However, a numerical study of the simplified model of the flow constructed in [3] shows that the mixing efficiency can be improved dramatically by appropriately choosing the parameters such as the magnitude of the temperature coefficients of surface tension at different fluid interfaces, the ratio of the fluid viscosities inside and outside the droplet, and the curvature of the temperature field driving the flow.

To simplify the mathematical description of the problem, we follow [3, 43] assuming that the droplet is suspended below the free surface of the liquid substrate and consider the limit of small capillary numbers such that the droplet can be considered spherical. Under these assumptions, in nondimensional units (with distances scaled by the droplet radius and the origin located at the center of the drop), there are three component flows: the dipole flow

(1.7)

the quadrupole flow

(1.8)

and the Taylor flow

(1.9)

where , (the value is used in all numerical calculations), the axis points in the direction of the thermal gradient, and the axis is vertical. The components and are caused by the thermocapillary effect at the droplet surface, while arises due to the thermocapillary effect at the surface of the liquid substrate. The complete flow inside the droplet can be written as a linear superposition of the dipole, Taylor, and quadrupole flows

(1.10)

The parameters and determine the relative strengths of the three components which depend on the temperature coefficients of surface tension at the droplet surface and the free surface of the substrate fluid and on the nonuniformity of the imposed temperature gradient [3]. As the dipole component is present in almost any setting, it is convenient to set its magnitude to unity by an appropriate choice of the timescale. In what follows, we will restrict our attention to the case of and . This will allow us to describe the mixing process quantitatively using perturbation theory [8].

Flows (1.7)–(1.10) are volume preserving and bounded by the droplet surface , which represents an invariant set. Moreover, the plane is an invariant set for each flow. Since the flow for is a mirror image of the flow for and there is no transport across the plane, we will restrict our attention to the flow inside the hemisphere characterized by positive values of .

For , flow (1.10) reduces to a superposition of the dipole and quadrupole flows and possesses two invariants: the azimuthal angle (around the axis) and the streamfunction :

where we have defined . The flow structure of the unperturbed system depends on the value of . Note that the dynamics for is the same as for up to the reflection with respect to the plane . There is always a pair of hyperbolic fixed points at the poles , and, for , two circles of degenerate elliptic fixed points, accompanied by a hyperbolic fixed point , and a circle of degenerate hyperbolic fixed points on the surface at , . The plane (denoted below) is a separatrix (see Figure 1.2). Away from the separatrix, the axis, and the surface of the sphere, each joint level of the two integrals and defines a closed unperturbed phase trajectory . The motion on is periodic with frequency . Note that if the base flow possesses axial symmetry, its frequency is naturally independent on azimuthal angle.

Figure 1.2 A sketch of the unperturbed flow for .

1.3.1 Flow Structure

For , system (1.10) is no longer integrable. Integrals and are not preserved. Streamlines are not closed and cross the separatrix . Figure 1.3a represents a result of long integration of one perturbed-phase trajectory.

Figure 1.3 Dynamics of the perturbed system over one long period. The parameters are and . (a) A perturbed streamline. (b) Phase portrait of the averaged system. The thick line in (b) shows the boundary of chaotic domain, .

The structure of the phase portrait on the slow plane depends on the values of . For , there is no separatrix, so the averaging procedure is valid everywhere and hence the AI is constant, if one ignores small bounded oscillations with amplitude of order . Therefore, the entire drop is a regular domain: all streamlines reside on the tori that are levels sets of the AI.

For , the separatrix plane defined by in the physical space (or in the slow plane) appears inside the drop. For , the interior of the drop is divided between the regular domain and the chaotic domain. Numerically, we find . The regular domain corresponds to streamlines lying on the level sets of that do not cross the separatrix (see Figure 1.3b), whereas the rest of the streamlines belong to the chaotic domain. To the leading order in , the boundary between the regular and the chaotic domain is a torus tangential to . On the plane, corresponds to a closed curve passing through the origin (see Figure 1.3b). Thus, we conclude that it is the level set of the AI that serves as the boundary between the regular and the chaotic domains. Note that the level set is not a sharp boundary: for any finite there are (although very few) regular trajectories inside the chaotic domain and vice versa. We will return to the discussion of the boundary between the domains in Section 1.4.

1.3.2 Dynamics Near the Separatrix Surface

For streamlines that cross the separatrix, the value of may change significantly. It is shown in [43] that the jump of AI during a single passage of the exact system through the vicinity of is

(1.11)

where describes the dependence of the jump magnitude on the distance between the crossing point and the axis, parameterized by variable , is the normalized value of the AI. We refer the reader to [43] for the explicit definitions of , , and . The values of and (and hence ) can be calculated exactly for any initial condition. However, a small change of order in the initial conditions produces, in general, a large (order 1) change in . Hence, for small it is possible to treat as a random variable uniformly distributed on the unit interval [10].

Equation (1.11) was verified numerically for various values of parameters , , and . A typical plot of is presented in Figure 1.4a. The function – and hence – has singularities at both and . Thus, there is a possibility (albeit quite small) of large changes in associated with a separatrix crossing. By direct calculation, the ensemble average of can be shown to vanish regardless of the value of :

(1.12)

Figure 1.4 Jumps of AI: (a) the plot of . The solid line was obtained using analytical result (1.11) and the dots show the values obtained numerically from (1.10) for , , . (b) The distribution of the sizes of the jumps versus the values of before the crossings.

1.3.3 Finite Perturbations

In most of the studies of resonance-induced chaotic diffusion, only infinitesimal perturbations were considered. To the best of our knowledge, the only papers that address, in any significant detail, the case of finite are [43, 53]. However, the dynamics in the presence of small but finite perturbations differs in several important ways from that with an infinitesimally small perturbation.

First, the very applicability of the method of averaging for larger is somewhat questionable as the ratio of the characteristic frequencies (e.g., and ) may not be very large. Numerical simulations, however, indicate that the main result of the averaging method, that changes most significantly near the separatrix, holds for a wide range of . Moreover, this problem can be somewhat addressed by implementing the improved adiabatic invariants, see Section 1.4 below.

The second effect is that for finite values of , there is a finite probability that the jump size can become comparable to the range of the AI. Consequently, the boundaries of the system start playing an important role in the statistics of the jumps. While magnitudes of most of the jumps are still given by (1.11) and satisfy the zero-average statement, the distribution of large jumps differs from the original prediction. Indeed, (1.11) breaks down when the value of either before or after the separatrix crossing is close to one of the domain boundaries. Take, for example, and . Then, approximately of the jumps feel the presence of the boundaries. While the influence of the boundaries on the properties and statistics of single crossings (albeit for a different system) was discussed in details in [53], here we are interested in necessary modifications to the long-time dynamics of the system and, in particular, the rate of mixing.

Figure 1.4b presents the distribution of the sizes of the jumps versus the values of before the corresponding crossing. There are three types of jumps. Most of the jumps are small and concentrate near the axis (the densely covered region). These jumps are well described by (1.11). In particular, the average value of these jumps is zero. The second type are jumps corresponding to points that lie between the lines , but outside of the densely covered region. These jumps happen when streamlines pass through the vicinities of the singularities of , given by (1.11). Such jumps were studied in detail in [53]. Finally, there are jumps that lie on either of the lines and . They were called “axis crossings” in [55].

1.4 Passages Through Resonances in Autonomous Flows