Volume 413, Issues 2–3, July 2005, Pages 91–196

Chemical and biological activity in open flows: A dynamical system approach

  • a Institute for Theoretical Physics, Eötvös University, P.O. Box 32, H-1518, Budapest, Hungary
  • b Instituto de Física, Universidade de São Paulo, Caixa Postal 66318, 05315-970, São Paulo, SP, Brazil
  • c Max-Plank-Institute for the Physics of Complex Systems, Nöthnitzer Str. 38, D-01187 Dresden, Germany
  • d Center for Applied Mathematics and Computational Physics, and Department of Structural Mechanics, Budapest University of Technology and Economics, Műegyetem rkp. 3, H-1521, Budapest, Hungary
Referred to by

Abstract

Chemical and biological processes often take place in fluid flows. Many of them, like environmental or microfluidical ones, generate filamentary patterns which have a fractal structure, due to the presence of chaos in the underlying advection dynamics. In such cases, hydrodynamical stirring strongly couples to the reactivity of the advected species: the outcome of the reaction is then typically different from that of the same reaction taking place in a well-mixed environment. Here we review recent progress in this field, which became possible due to the application of methods taken from dynamical system theory. We place special emphasis on the derivation of effective rate equations which contain singular terms expressing the fact that the reaction takes place on a moving fractal catalyst, on the unstable foliation of the reaction free advection dynamics.

PACS

  • 05.45.-a;
  • 47.52.+j;
  • 47.53.+n;
  • 47.70.Fw

Keywords

  • Reaction;
  • Chaotic advection;
  • Diffusion;
  • Filamentary fractal;
  • Advection–reaction–diffusion;
  • Population dynamics;
  • Inertial effect;
  • Non-hyperbolic effect

1. Introduction

1.1. Motivation

Several chemical and biological processes occur in fluid flows, both in natural systems and in engineering. Such processes are especially important in environmental systems, such as plankton blooming in the oceans [2], [3], [25], [116], [117], [118], [142], [173] and [210], and the formation of the ozone hole in the stratosphere [41], [190] and [58]. They are, however, relevant in a broad range of other fields as well, including chemistry [125], [43], [123], [124] and [5], population dynamics [179], [181] and [86], geophysical sciences [62], [202] and [186], microfluids [197], [121], [23], [150], [216] and [134] and combustion [219], [54], [89] and [90]. In these systems, particles are carried by the motion of the fluid, and they change due to their chemical and biological interactions. We say that these particles are active, in the sense that they are not just passively advected by the flow, but they follow dynamical processes of their own. In other words, they do something. For example, plankton ‘particles’ (cells) reproduce and die, their number changes; the various chemicals involved in ozone depletion transform into each other, and lead to an overall decay of ozone. Fig. 1 illustrates the idea, showing a particle that is advected for some time by the flow, and then undergoes a multiplicative process. For simplicity, we refer to such systems as active flows, by which we mean flows carrying active particles. We emphasize, however, that this kind of activity is assumed to have no feedback on the fluid flow, which is a realistic assumption in many applications. An unusual feature of many of these reactions, as shown e.g. in Fig. 2, Fig. 3, Fig. 4 and Fig. 5, is that they take place along filamentary spatial structures.

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Fig. 1. Illustration of a kind of active process taking place in a flow.

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Fig. 2. (Color online.) Concentration distribution of a passive scalar (red: full concentration, dark blue: zero concentration) in a micromixer. The flow in the main channel is stationary and is manipulated by time-periodic flows in the secondary channels. The frequency of these perturbations can be used to enhance mixing. Picture by S.D. Müller, I. Mezić, J.H. Walther and P. Koumoutsakos, with their kind permission.

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Fig. 3. Shape of a dye (flouresceine) droplet (of initial diameter about 1 cm) after stirring on the surface of a thin layer of glycerol in a Petri dish. The stirring protocol is that of the experiment by Villermaux and Duplat [214]: a number of parallel cuts is made by a rod through the fluid in two direction in an alternating manner. Experiment carried out by I.M. Jánosi, K.G. Szabó, T. Tél, and M. Wells at the von Kármán Laboratory of Eötvös University, Budapest.

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Fig. 4. (Color online.) Filamentation in a phytoplankton bloom in the Norwegian Sea. Provided by the SeaWiFS Project, NASA/Goddard Space Flight Center, and ORBIMAGE, URL: http://visibleearth.nasa.gov/cgi-bin/viewrecord?5278.

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Fig. 5. (Color online.) Ozone distribution (in mixing ratio measured in parts per million (ppm)) above the South Pole region at about 18 km altitude on September 24, 2002. The chemical reactions leading to ozone depletion are simulated over 23 days in the flow given by meteorological wind analyses (from Grooß, Konopka, and Müller, Ozone chemistry during the 2002 Antarctic vortex split [58], with permission) during a very rare event when the so-called polar vortex splits into two parts.

The main fields of phenomena where chemical or biological reactivity plays a role can be grouped into the following categories, according to their length scale (see Table 1):

Microfluidics: Recent technological advancement has made the fabrication of microchannels of a few hundred micrometers in cross-section possible [23] and [134] (Fig. 2). These are used e.g. in printers and in bio-medical equipment. At the microfluidic scale, viscosity dominates and turbulence cannot be present; also, diffusion can usually be neglected. The only effective source of mixing is then chaotic advection.

Laboratory scale flows: This covers a broad range of phenomena on the human scale, including chemical industry applications, flames and combustion, as well as laboratory experiments (see Fig. 3).

Flows in oceans or large lakes: The most typical reactive process on this scale is plankton dynamics (see Fig. 4). Phytoplankton is a key ingredient in the carbon exchange with the atmosphere and plays thus a role in regulating the greenhouse effect. The population dynamics of other micro-organisms and the spread of chemically reacting pollutants also belongs here.

Atmospheric flows: The prototype process is ozone depletion in the stratosphere (see Fig. 5), influencing essentially the greenhouse effect. Several other reactions of atmospheric chemistry are also of interest. Outside the planetary boundary layer, flows are practically two-dimensional (as well as in the oceans), due to the dominance of the Coriolis effect and density stratification.

Table 1. Typical length and velocity scales of important flows, as detailed in the main text

MicrofluidsLaboratoryOceanAtmosphere
L (m)5×10-41105106
U (m/s)10-210-210-110
Full-size table

In studying active flows, it is of particular importance to relate the dynamics of the reactive system to the underlying advection dynamics of the flow. In other words, how does the advection dynamics affect total reaction productivity, or population dynamics of different species in the flow? The Lagrangian formulation of the equations of motion for the flow is the relevant framework in this context, because we are interested in the behavior of the particles advected by the fluid. It is well-known (see e.g. [7], [153], [33], [35], [78], [229], [174], [87], [221], [114], [128] and [225]) that even very simple flows may have chaotic advection dynamics, characterized by an extreme sensitivity of the motion to initial conditions. Lagrangian chaos is a very general feature, found in most real flows.

In this work, we review recent results which show that the character of a reaction can drastically change if it takes place in a time-dependent chaotic flow. A reaction can lead to a pattern formation of a new type: the product is asymptotically distributed around a filamentary fractal which moves in a rhythm corresponding to the time dependence of the flow. In a periodic case, the total amount of product is thus oscillating about a mean: a kind of limit-cycle behavior sets in even if the original reaction kinetics leads to a time-independent stationary state. This pattern formation is due to the interplay of the chaotic particle motion produced by hydrodynamics and the production of the new particles by the reaction.

Flows can be grouped into two main classes: closed and open ones. A flow is closed if its motion is confined to a bounded domain (see Fig. 6a). A flow is open if there is a net current flowing through the region of observation. We also assume that far from the region of observation the flow is simple (nearly homogeneous). A typical example is the flow around an obstacle (see Fig. 6b). In open flows, most trajectories are unbounded, and particles escape the observation region in a finite time. Even flows that are actually closed can in many cases be considered open, if the time it takes for typical particles to return to the observation region is much longer than the relevant time-scale of observation. As an example, the ocean is of course a closed fluid system, but if we are looking at a small region surrounding an island, the average return time might be of the order of hundreds of years. The fluid flow in the vicinity of the island can, for all practical purposes, be considered open. There are many other important flow systems which can be effectively open, especially environmental and microfluidic flows (see e.g. Fig. 2).

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Fig. 6. Illustrations of (a) closed, and (b) open flows.

Both closed and open flows are important, and can of course carry some kind of activity. The dynamical system approach to the active problem turns out, however, to be more relevant for the case of open flows: it is in this class where the effective rate equation to be derived explicitly contains the chaos parameters of the advection dynamics. Our presentation is therefore biased towards open flows, and touches only some aspects of closed ones.

1.2. Relevance of open flows

The question of the interplay between chaotic advection and activity was first addressed by Metcalfe and Ottino in the context of closed flows [125] and [43]. The asymptotic state (reached as t→∞) is typically a homogeneous distribution of the components in the fluid, and there is no enhancement of activity in this state.

In open flows, however, chaos takes a different form [204], because typical advected particle orbits escape the observation region in a finite time. There are, however, non-escaping orbits which are bounded within a finite region, located e.g. in the vicinity of the wake in the case of a flow around an obstacle. These non-escaping orbits form a chaotic set of the dynamical system associated with advection, which is known as a chaotic saddle. Even though the orbits in the chaotic saddle are unstable and in many cases make up a set of measure zero, they control the long-term dynamics of the system. Chaotic saddles have a complex, fractal geometric structure, and they give rise to extreme sensitivity to initial conditions of particle trajectories. This sensitivity is related to the fractal dimension of the chaotic saddle, and characterizes its geometry. The chaotic dynamics of open flows is a special case of transient chaos[204], i.e., chaos of finite lifetime, as opposed to persistent motion on attractors, found in dissipative systems. It is worth mentioning that due to the openness of a flow even the spectrum of passive scalars changes: it becomes steeper than the Batchelor spectrum, and the shift of the spectral exponent can be expressed in terms of transient chaos properties [137].

In this work, we present a general theoretical framework for the understanding of the reactive dynamics of particles advected by open chaotic flows. This area has attracted great interest since the appearance of the first papers by Toroczkai, Károlyi, Péntek, Tél and Grebogi [209], [83] and [205], and there have been many new and interesting results since then. Due to space constraints, this review is necessarily limited in its scope. However, we try to present the main results in the area as well as the theoretical concepts that unify them.

The basic picture that emerges from the study of active dynamics in open chaotic flows is that in all cases most of the reactivity is concentrated along a fractal set[204], forming clearly visible filamentary patterns in space. This fractal set is associated with the chaotic saddle, being its unstable manifold. The concentration of reactivity along a filamentary fractal results in a singular enhancement of the productivity associated with the reaction (as compared to the productivity in non-chaotic flows). As we will show, this enhancement is reflected by a singular production term in the reaction rate equation, which is absent if the flow is not chaotic. This singular term is related in a well-defined way to the fractal dimension of the chaotic saddle [204] (in fact, to the fractal dimension of its unstable manifold), and therefore it is a consequence of the fractal geometry of the set of non-escaping orbits. This is because the surface (or perimeter, in the case of two-dimensional flows) of the fractal filaments of the unstable manifold diverges with refining resolution. As it is a consequence of a basic geometric property of fractals, the enhancement of activity is a very general and pervading phenomenon, and is largely independent of the particulars of the active process, or of the details of the flow, so long as it is open and its dynamics is chaotic. We will illustrate this through many examples and applications. We stress that this phenomenon is not a result of artificial assumptions of particular models, but is expected to be a general feature of real active open flows. Hence, the theory we present here has potentially broad applicability, and could be a valuable tool in areas where reactivity in flows plays an important role, like chemistry, biology, oceanography, atmospheric sciences, engineering, and others.

1.3. Organization of the paper

We start with a presentation of the phenomena in simple systems (two-dimensional incompressible flows with non-inertial particles displaying autocatalytic activity), and then proceed to more complex situations, including three-dimensional flows, transport barriers (non-hyperbolic motion), chaotically time-dependent or random flows, inertial tracers, and more complex chemical or biological activity. Throughout the presentation, we try to emphasize how the active dynamics in all these cases can be understood within a single unifying theoretical framework.

In Section 2, we review the basic results related to chaotic advection, focusing on applications to open flows. We put emphasis on those aspects that are of direct relevance to the understanding of active dynamics, including the notions of invariant chaotic sets, stable and unstable manifolds, and fractal dimension. We also introduce some important examples of two-dimensional incompressible open flows that display Lagrangian chaos: the von Kármán vortex street, the blinking vortex–sink systems and the case of four point vortices.

In Section 3, we present some simple models of activity, corresponding to different ways by which the advected particles can be active. We introduce both a discrete kind of modeling, where individual particles are considered, and a continuous modeling, where concentrations are considered. We use the autocatalytic reaction (A+B⟶2B) as an example because it is both simple and important, and it underlies many real reactive processes.

Section 4 brings together the motion of the flow (Section 2) and the activity of the advected particles (Section 3), and presents the basic theory of the dynamics of active processes in open flows. We expose the theory in the context of 2D incompressible time-periodic flows. This allows us to present the basic ideas of the theory as clearly as possible. We argue that our approach and its results are valid for any situation in which a well-defined reaction front exists, independently of the details of the flow motion or of the particular kind of reaction. We derive an effective rate equation, and we emphasize the new features in the reaction dynamics brought about by chaos, especially the enhancement of productivity.