Massimiliano Giona

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 for transport phenomena in random media

A differential equation for diffusion in isotropic and homogeneous fractal structures is derived within the context of fractional calculus. It generalizes the fractional diffusion equation valid in Euclidean systems. The asymptotic behavior of the probability density function is obtained exactly and coincides with the accepted asymptotic form obtained using scaling argument and exact enumeration calculations on large percolation clusters at criticality. The asymptotic frequency dependence of the scattering function is derived exactly from the present approach, which can be studied by X-ray and neutron scattering experiments on fractals.

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.

Fast Distributed Average Consensus Algorithms Based on Advection-Diffusion Processes

Distributed consensus algorithms have recently gained large interest in sensor networks as a way to achieve globally optimal decisions in a totally decentralized way, that is, without the need of sending all the data collected by the sensors to a fusion center. However, distributed algorithms are typically iterative and they suffer from convergence time and energy consumption. In this paper, we show that introducing appropriate asymmetric interaction mechanisms, with time-varying weights on each edge, it is possible to provide a substantial increase of convergence rate with respect to the symmetric time-invariant case. The basic idea underlying our approach comes from modeling the average consensus algorithm as an advection-diffusion process governing the homogenization of fluid mixtures. Exploiting such a conceptual link, we show how introducing interaction mechanisms among nearby nodes, mimicking suitable advection processes, yields a substantial increase of convergence rate. Moreover, we show that the homogenization enhancement induced by the advection term produces a qualitatively different scaling law of the convergence rate versus the network size with respect to the symmetric case.

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 THEORY OF TRANSPORT PHENOMENA IN DISORDERED SYSTEMS

Abstract Starting from the fractional calculus formulation of diffusion equations in euclidean media, a fractional equation is proposed for describing transport in fractal and disordered structures. The solution of this equation, obtained analytically, is consistent with known results for diffusion on fractals and should also apply to systems showing more general anomalous transport behaviour. Analytical expressions for the Fourier transform of the concentration profile are predicted which can be studied by X-ray- and neutron-scattering experiments.

Exact solution of linear transport equations in fractal media - I. Renormalization analysis and general theory

We develop in detail a renormalization analysis of transport equations on fractals by considering regular model structures represented by means of families of graphs G(n), each of which is characterized by its adjacency matrix. Particular attention is paid to the correct representation of boundary conditions relevant to specific transport problems. The extension theory for the solution of generic transport problems defined by positive functions in the algebra of a given adjacency matrix is also developed.

Two-layer shrinking-core model: parameter estimation for the reaction order in leaching processes

This article develops a modified version of the shrinking-core model accounting for the surface heterogeneity of solid particles. This model can be primarily used as a shortcut method for estimating the dependence of the kinetic rates on the concentration of fluid reactants and specifically the reaction order in the presence of a polydisperse solid mixture. This approach is tested numerically for model reactions involving polydisperse systems and is applied to the dissolution of MnO2 in sulphuric acid solutions containing glucose as the reductant agent.