In this chapter, we will discuss barriers to purely advective transport in velocity fields that may have complex spatial features but a simple (recurrent) temporal structure: steady, periodic or quasiperiodic. Such velocity fields can be integrated for all times on bounded domains and hence their trajectories can be interrogated over infinite time intervals. While such exact recurrence is atypical in nature, mixing processes with precisely repeating stirring protocols are abundant in technological applications. Here, we survey classic results on temporally recurrentvelocity fields partly for motivation, partly for historical completeness and partly because their predictions in distinguished (recurrent) frames coincide with the predictions of Lagrangian coherent structure (LCS) methods to be discussed in the next chapter. For this reason, recurrent velocity fields are ideal benchmarks for LCS techniques because their transport barriers can be unambiguously identified. There are also a number of technological mixing processes in which the velocity field is engineered to be spatially recurrent, and hence the techniques discussed here apply directly to them.

Here, we take our first step to discover barriers to transport outside the idealized setting of temporally recurrent (steady, periodic or quasiperiodic) velocity fields. While we can no longer hope for even approximately recurring material surfaces in this general setting, we can certainly look for material surfaces that remain coherent. We perceive a material surface to be coherent if it preserves the spatial integrity without developing smaller scales. Those smaller scales would manifest themselves as protrusions from either side of the material surface without a break-up of that surface. In other words, using the terminology of the Introduction, we seek advective transport barriers in nonrecurrent flows as Lagrangian coherent structures (LCS). We will refer to this instantaneous limit of LCSs as objective Eulerian coherent structures (OECSs). These Eulerian structures act as LCSs over infinitesimally short time scales and hence their time-evolution is not material. Despite being nonmaterial, OECSs have advantages and important applications in unsteady flow analysis, as we will discuss separately.

Flow separation is the ejection of fluid particles from a small neighborhood of a solid boundary. Such a breakaway from the boundary is often due to the detachment of a boundary layer, but it also occurs in highly viscous flows where the boundary layer description is inapplicable. Accordingly, we will treat separation here as a purely kinematic phenomenon: the formation of a material spike from a flow boundary.Such material spikes form along attracting LCSs, as we have already seen inthe previous chapter. We consider LCSs acting as separation or attachment profiles here separately because their contact points with the boundary and their local shapes near the boundary can be located from a purely Eulerian analysis along the boundary. Since the attachment points of material separation profiles cannot move under no-slip boundary conditions, such profiles necessarily create fixed separation. In contrast, material spikes emanating from off-boundary points generally result in moving separation in unsteady flows. We will discuss how both fixed and moving separation can be described via material barriers to transport.

Classical continuum mechanics focuses on the deformation field of moving continua. This deformation field is composed of the trajectories of all material elements, labeled by their initial positions. This initial-condition-based, material description is what we mean here by the Lagrangian description of a fluid motion. In contrast to typical solid-body deformations, however, fluid deformation may be orders of magnitude larger than the net displacement of the total fluid mass. The difficulty of tracking individual fluid elements has traditionally shifted the focus in fluid mechanics from individual trajectories to the instantaneous velocity field and quantities derived from it. These quantities constitutethe Eulerian description of fluids. This chapter surveys the fundamentals of both the Lagrangian and the Eulerian approaches. We also cover notions and results from differential equations and dynamical systems theory that are typically omitted from fluid mechanics textbooks, yet are heavily used in later chapters ofthis book.

While the transport of concentration fields arising in nature and technology is often predominantly advective, it invariably has at least a small diffusive component as well. The inclusion of diffusivity in transport studies increases their complexity significantly, as we will see.At the same time, introducing the diffusivity creates an opportunity to settle on a broadly agreeable definition for a transport barrier. Indeed, diffusive transport through a material surface is a uniquely defined, fundamental physical quantity, whose extremizing surfaces can be defined without reliance on any special notion of coherence. In the limit of zero diffusivity, the results we describe in this chapter also give a unique, physical definition of purely advective LCSs as material surfaces that will block transport most efficiently under the addition of the slightest diffusion or uncertainty to the velocity field.

Transport barriers offer a simplified global template for the redistribution ofsubstances without the need to simulate or observe numerous different initial distributions in detail. Because of their simplifying role, transport barriers are broadly invoked as explanations for observations in several physical disciplines, including geophysical flows,fluid dynamics,plasma fusion, reactive flowsand molecular dynamics. Despite their frequent conceptual use, however, transport barriers are rarely defined precisely or extracted systematically from data. The purpose of this book is to survey effective and mathematically grounded methods for defining, locating and leveraging transport barriers in numerical simulations, laboratory experiments, technological processes and nature. In the rest of this Introduction, we briefly survey the main topics that we will be covering in later chapters.

In this chapter, we will be concerned with barriers to the transport of inertial (i.e., small but finite-size) particles in a carrier fluid. As a general rule, the more the density of inertial particles diverts from the carrier fluid density, the more they tend to depart from fluid trajectories. Specifically, while small enough neutrally buoyant particles often remain close to fluid motion, the same is not true for heavy particles (aerosols) and light particles (bubbles). Practical flow problems involving inertial particles tend to be temporally aperiodic and hence the machinery of LCSs discussed in earlier chaptersis also highly relevant for inertial particles. By inertial LCSs (or iLCSs, for short), we mean coherent structures composed of distinguished inertial particles that govern inertial transport patterns. In contrast, LCSs (composed of distinguished fluid particles) govern fluid transport patterns. The purpose of this chapter is to examine how iLCSs differ from LCSs of the carrier fluid.

In the preceding chapters, we have discussed definitions and identification techniques for observed material barriers to the transport of fluid particles, inertial particles and passive scalar fields. All these barriers are directly observable in flow visualizations based on their impact on tracers carried by the flow. However, the transport of several important physical quantities, such as the energy, momentum, angular momentum, vorticity and enstrophy, is also broadly studied but allows no direct experimental visualization. These important scalar and vector quantities are dynamically active fields, i.e., functions of the velocity field and its derivatives.We will collectively refer to barriers to the transport of such fieldsas dynamically active transport barriers. This chapter will be devoted to the development ofan objective notion of active barriers in 3D unsteady velocity data. Additionally, 2D velocity fields can also be handled via this approach by treating them as 3D flows with a symmetry.

Here, we elaborate in more detail on a few technical notions that we have used throughout this book. Among other things, the subjects include the implicit function theorem, the notion of a ridge, classic Lyapunov exponents, differentiable manifolds, Hamiltonian systems, the AVISO data set, surfaces normal to a vector field, calculus of variations, Beltrami flows and the Reynolds transport theorem.

If we accept that transport barriers should be material features for experimental verifiability, we must also remember a fundamental axiom of mechanics: material response of any moving continuum, including fluids, must be frame-indifferent.This means that the conclusions of different observers regarding material behavior must transform into each other by exactly the same rigid-body transformation that transforms the frames of the observers into each other. This requirement of the frame-indifference of material response is called objectivity in classical continuum mechanics. Its significance in fluid mechanics is often overlooked or forgotten, which prompts us to devote a whole chapter to this important physical axiom. We clarify some common misunderstandings of the principle of objectivity in fluid mechanics and discuss in detail the mathematical requirements imposed by objectivity on scalars, vectors and tensors to be used in describing transport barriers.