This work reviews the present position of and surveys future perspectives in the physics of chaotic advection: the field that emerged three decades ago at the intersection of fluid mechanics and nonlinear dynamics, which encompasses a range of applications with length scales ranging from micrometers to hundreds of kilometers, including systems as diverse as mixing and thermal processing of viscous fluids, microfluidics, biological flows, and oceanographic and atmospheric flows.

Three-dimensional advection of passive tracers in non-inertial flows is studied in a finite cylinder confined by two parallel endwalls by means of numerical simulations and laboratory experiments. The fluid is set in motion through steady or time-periodic forcing by in-plane motion of the endwalls via a given forcing protocol. The numerical analysis centres on a dynamical-systems approach and concerns symmetry-based identification of coherent structures in the web of tracer paths (collectively defining the flow topology) for a number of archetypal flow configurations. The role of the flow topology in the process of tracer transport is investigated by numerical tracking of finite-size material objects released at strategic locations in the flow. Experimental validation of key aspects of the numerical results has been carried out in laboratory experiments by flow visualization with dye and flow measurement via three-dimensional particle tracking velocimetry.

Mixing under laminar flow conditions is key to a wide variety of industrial fluid systems of size extending from micrometres to metres. Profound insight into three-dimensional laminar mixing mechanisms is essential for better understanding of the behaviour of such systems and is in fact imperative for further advancement of (in particular, microscopic) mixing technology. This insight remains limited to date, however. The present study concentrates on a fundamental transport phenomenon relevant to laminar mixing: the formation and interaction of coherent structures in the web of three-dimensional paths of passive tracers due to fluid inertia. Such coherent structures geometrically determine the transport properties of the flow and thus their formation and topological structure are essential to three-dimensional mixing phenomena. The formation of coherent structures, its universal character and its impact upon three-dimensional transport properties is demonstrated by way of experimentally realizable time-periodic model flows. Key result is that fluid inertia induces partial disintegration of coherent structures of the non-inertial limit into chaotic regions and merger of surviving parts into intricate three-dimensional structures. This response to inertial perturbations, though exhibiting great diversity, follows a universal scenario and is therefore believed to reflect an essentially three-dimensional route to chaos. Furthermore, a first outlook towards experimental validation and investigation of the observed dynamics is made.

Tracer advection of non-Newtonian fluids in reoriented duct flows is investigated in terms of coherent structures in the web of tracer paths that determine transport properties geometrically. Reoriented duct flows are an idealization of in-line mixers, encompassing many micro and industrial continuous mixers. The topology of the tracer dynamics of reoriented duct flows is Hamiltonian. As the stretching per reorientation increases from zero, we show that the qualitative route from the integrable state to global chaos and good mixing does not depend on fluid rheology. This is due to a universal symmetry of reoriented duct flows, which we derive, controlling the topology of the tracer web. Symmetry determines where in parameter space global chaos first occurs, while increasing non-Newtonian effects delays the quantitative value of onset. Theory is demonstrated computationally for a representative duct flow, the rotated arc mixing flow.

We consider a relatively simple model for pool boiling processes. This model involves only the temperature distribution within the heater and describes the heat exchange with the boiling fluid via a nonlinear boundary condition imposed on the fluid-heater interface. This results in a standard heat equation with a nonlinear Neumann boundary condition on part of the boundary. In this paper we analyse the qualitative structure of steady-state solutions of this heat equation. It turns out that the model allows both multiple homogeneous and multiple heterogeneous solutions in certain regimes of the parameter space. The latter solutions originate from bifurcations on a certain branch of homogeneous solutions. We present a bifurcation analysis that reveals the multiple-solution structure in this mathematical model. In the numerical analysis a continuation algorithm is combined with the method of separation-of-variables and a Fourier collocation technique. For both the continuous and discrete problem a fundamental symmetry property is derived that implies multiplicity of heterogeneous solutions. Numerical simulations of this model problem predict phenomena that are consistent with laboratory observations for pool boiling processes.

Kinematic features of three-dimensional mixing by advection of passive particles in time-periodic flows are the primary subject of this study. A classification of periodic points, providing important information about the mixing properties of a flow, is presented, and the dynamics of the Poincaré map in the vicinity of periodic points is analysed for all identified types. Three examples of Stokes flow in a finite cylindrical cavity with discontinuous periodic motion of its end walls are used to illustrate the determination of both periodic lines and isolated periodic points in the flow domain. The stable and unstable manifolds of points on the periodic lines create two surfaces in the flow. A numerical technique based on tracking of a material surface is presented to study the manifold surfaces and their intersections. It is illustrated with numerical examples that flows with periodic lines possess only quasi-two-dimensional mechanisms of chaotic advection.

In this study, we explore the spectral properties of the distribution matrices of the mapping method and its relation to the distributive mixing of passive scalars. The spectral (or eigenvector-eigenvalue) decomposition of these matrices constitutes discrete approximations to the eigenmodes of the continuous advection operator in periodic flows. The eigenvalue spectrum always lies within the unit circle and due to mass conservation, always accommodates an eigenvalue equal to one with trivial (uniform) eigenvector. The asymptotic state of a fully chaotic mixing flow is dominated by the eigenmode corresponding with the eigenvalue closest to the unit circle (“dominant eigenmode”). This eigenvalue determines the decay rate; its eigenvector determines the asymptotic mixing pattern. The closer this eigenvalue value is to the origin, the faster is the homogenization by the chaotic mixing. Hence, its magnitude can be used as a quantitative mixing measure for comparison of different mixing protocols. In nonchaotic cases, the presence of islands results in eigenvalues on the unit circle and associated eigenvectors demarcating the location of these islands. Eigenvalues on the unit circle thus are qualitative indicators of inefficient mixing; the properties of its eigenvectors enable isolation of the nonmixing zones. Thus important fundamental aspects of mixing processes can be inferred from the eigenmode analysis of the mapping matrix. This is elaborated in the present paper and demonstrated by way of two different prototypical mixing flows: the time-periodic sine flow and the spatially periodic partitioned-pipe mixer.

Inertia-induced changes in transport properties of an incompressible viscous time-periodic flow due to fluid inertia ͑nonzero Reynolds numbers Re͒ are studied in terms of the topological properties of volume-preserving maps. In the noninertial Stokes limit ͑vanishing Re͒, the flow relates to a so-called one-action map. However, the corresponding invariant surfaces are topologically equivalent to spheres rather than the common case of tori. This has fundamental ramifications for the response to small departures from the noninertial limit and leads to a new type of response scenario: resonance-induced merger of coherent structures. Thus several coexisting families of two-dimensional coherent structures are formed that make up two classes: fully closed structures and leaky structures. Fully closed structures restrict motion as in a one-action map; leaky structures have open boundaries that connect with a locally chaotic region through which exchange of material with other leaky structures occurs. For large departures from the noninertial limit the above structures vanish and the topology becomes determined by isolated periodic points and associated manifolds. This results in unrestricted chaotic motion.