Stefano Cerbelli

Laminar dispersion at high Péclet numbers in finite-length channels: Effects of the near-wall velocity profile and connection with the generalized Leveque problem

This article develops the theory of laminar dispersion in finite-length channel flows at high Peclet numbers, completing the classical Taylor–Aris theory which applies for long-term, long-distance properties. It is shown, by means of scaling analysis and invariant reformulation of the moment equations, that solute dispersion in finite length channels is characterized by the occurrence of a new regime, referred to as the convection-dominated transport. In this regime, the properties of the dispersion boundary layer and the values of the scaling exponents controlling the dependence of the moment hierarchy on the Peclet number are determined by the local near-wall behavior of the axial velocity. Specifically, different scaling laws in the behavior of the moment hierarchy occur, depending whether the cross-sectional boundary is smooth or nonsmooth (e.g., presenting corner points or cusps). This phenomenon marks the difference between the dispersion boundary layer and the thermal boundary layer in the classical Leveque problem. Analytical and numerical results are presented for typical channel cross sections in the Stokes regime.

Fractional diffusion equation and relaxation in complex viscoelastic materials

A fractional equation describing relaxation phenomena in complex viscoelastic materials is derived by employing a formal analogy between linear viscoelasticity and difusion in a disordered structure. From this analogy, a power-law relaxation follows which is in agreement with experimental results obtained in many complex viscoelastic materials.

The intermaterial area density generated by time- and spatially periodic 2D chaotic flows

Abstract This paper explores in some detail the spatial structure and the statistical properties of partially mixed structures evolving under the effects of a time-periodic chaotic flow. Numerical simulations are used to examine the evolution of the interface between two fluids, which grows exponentially with a rate equal to the topological entropy of the flow. Such growth is much faster than predicted by the Lyapunov exponent of the flow. As time increases, the partially mixed system develops into a self-similar structure. Frequency distributions of interface density corresponding to different times collapse onto an invariant curve by a simple homogeneous scaling. This scaling behavior is a direct consequence of the generic asymptotic directionality property characteristic of 2D time-periodic flows. Striation thickness distributions (STDs) also acquire a time-invariant shape after a few (∼5–10) periods of the flow and are collapsed onto a single curve by standardization. It is also shown that STDs can be accurately predicted from distributions of stretching values, thus providing an effective method for calculation of STDs in complex flows.

The geometry of mixing in time-periodic chaotic flows.: I. asymptotic directionality in physically realizable flows and global invariant properties

Abstract This paper demonstrates that the geometry and topology of material lines in 2D time-periodic chaotic flows is controlled by a global geometric property referred to as asymptotic directionality . This property implies the existence of local asymptotic orientations at each point within the chaotic region, determined by the unstable eigendirections of the Jacobian matrix of the n th iterative of the Poincare map associated with the flow. Asymptotic directionality also determines the geometry of the invariant unstable manifolds, which are everywhere tangent to the field of asymptotic eigendirections. This fact is used to derive simple non-perturbative methods for reconstructing the global invariant manifolds to any desired level of detail. The geometric approach associated with the existence of a field of invariant unstable subspaces permits us to introduce the concept of a geometric global unstable manifold as an intrinsic property of a Poincare map of the flow, defined as a class of equivalence of integral manifolds belonging to the invariant unstable foliation. The connection between the geometric global unstable manifold and the global unstable manifold of hyperbolic periodic points is also addressed. Since material lines evolved by a chaotic flow are asymptotically attracted to the geometric global unstable manifold of the Poincare map, in a sense that will be made clear in the article, the reconstruction of unstable integral manifolds can be used to obtain a quantitative characterization of the topological and statistical properties of partially mixed structures. Two physically realizable systems are analyzed: closed cavity flow and flow between eccentric cylinders. Asymptotic directionality provides evidence of a global self-organizing structure characterizing chaotic flow which is analogous to that of Anosov diffeomorphisms, which turns out to be the basic prototype of mixing systems. In this framework, we show how partially mixed structures can be quantitatively characterized by a nonuniform stationary measure (different from the ergodic measure) associated with the dynamical system generated by the field of asymptotic unstable eigenvectors.

Universality and imaginary potentials in advection-diffusion equations in closed flows

We address the scaling and spectral properties of the advection-diffusion equation in closed two-dimensional steady flows. We show that homogenization dynamics in simple model flows is equivalent to a Schrodinger eigenvalue problem in the presence of an imaginary potential. Several properties follow from this formulation: spectral invariance, eigenfunction localization, and a universal scaling of the dominant eigenvalue with respect to the Peclet number Pe. The latter property means that, in the high-Pe range (in practice Pe≥10 2 -10 3 ), the scaling exponent controlling the behaviour of the dominant eigenvalue with the Peclet number depends on the local behaviour of the potential near the critical points (local maxima/minima). A kinematic interpretation of this result is also addressed

A spectral approach to reaction/diffusion kinetics in chaotic flows

A classical spectral approach based on the set of eigenfunctions of the Laplacian operator is proposed for the numerical solution of advection/diffusion/reaction equations for reactive mixing in 2-D laminar chaotic flows. This approach overcomes numerical diffusion problems and provides accurate spatiotemporal concentration fields in reasonable computer time up to very high values of Pe, such as Pe=105 and higher. Moreover, a pseudo-spectral approach, combining spectral expansion with an FFT algorithm, provides an efficient computational strategy for both polynomial and non-polynomial nonlinearities such as those arising in non-isothermal reactive mixing problems with Arrhenius dependence of kinetic rates on temperature.

The evolution of material lines curvature in deterministic chaotic flows

Abstract It is by now well established that curvature plays a fundamental role in the description of the topology emerging from the partially mixed structures advected by chaotic flows. This article focuses on the dynamics of curvature in volume-preserving time-periodic flows. Previous work on the subject dealt with the evolution of curvature in the time-continuous framework. Here we derive the dynamical equations for the time-discrete dynamical system associated with the Poincare return map of the flow. We show that this approach allows one to gain more insight into understanding the mechanisms of folding of material lines as they are passively stirred by the mixing process. By exploiting the incompressibility assumption, we analyze dependence on initial conditions (i.e. on the initial curvature and tangent vectors), and discuss under which circumstances the dependence on the initial curvature vector becomes immaterial as time increases. This analysis is closely connected with the properties of an invariant geometric structure referred to as the global unstable manifold associated with the flow system. Direct numerical simulations for physically realizable systems are used to provide concrete examples of the results that arise from theoretical considerations. The impact of this information on the prediction of the behavior of diffusing-reacting mixing processes (e.g. pattern formation and generation of lamellar structures) is also addressed.

Enhanced diffusion regimes in bounded chaotic flows

We analyze the exponent characterizing the decay towards the equilibrium distribution of a generic diffusing scalar advected by a nonlinear flow on the two-torus. When the kinematics of the stirring field is predominantly regular (e.g., autonomous flows or protocols possessing large quasiperiodic islands) a purely diffusive scaling of the dominant exponent Λ as a function of the diffusivity D, Λ(D) ∼ D, coexists with a convection-enhanced diffusion regime, with an apparent exponent λ that scales as √ D. For globally chaotic conditions we find Λ(D) → const as D → 0. We provide physical arguments to explain this new

Non-uniform stationary measure properties of chaotic area-preserving dynamical systems

This article shows the existence of a non-uniform stationary measure (referred to as the w-invariant measure) associated with the space-filling properties of the unstable manifold and characterizing some statistical properties of chaotic two-dimensional area-preserving systems. The w-invariant measure, which differs from the ergodic measure and is non-uniform in general, plays a central role in the statistical characterization of chaotic fluid mixing systems, since several properties of partially mixed structures can be expressed as ensemble averages over the w-invariant measure. A closed-form expression for the w-invariant density is obtained for a class of mixing systems topologically conjugate with the linear toral automorphism. The physical implications in the theory of fluid mixing, and in the statistical characterization of chaotic Hamiltonian systems, are discussed.

A Continuous Archetype of Nonuniform Chaos in Area-Preserving Dynamical Systems

We propose a piecewise linear, area-preserving, continuous map of the two-torus as a prototype of nonlinear two-dimensional mixing transformations that preserve a smooth measure (e.g., the Lebesgue measure). The model lends itself to a closed-form analysis of both statistical and geometric properties. We show that the proposed model shares typical features that characterize chaotic dynamics associated with area-preserving nonlinear maps, namely, strict inequality between the line-stretching exponent and the Lyapunov exponent, the dispersive behavior of stretch-factor statistics, the singular spatial distribution of expanding and contracting fibers, and the sign-alternating property of cocycle dynamics. The closed-form knowledge of statistical and geometric properties (in particular of the invariant contracting and dilating directions) makes the proposed model a useful tool for investigating the relationship between stretching and folding in bounded chaotic systems, with potential applications in the fields of chaotic advection, fast dynamo, and quantum chaos theory.