Advection diffusion in nonchaotic closed flows: non-Hermitian operators, universality, and localization

Chebyshev-polynomial expansion of the localization length of Hermitian and non-Hermitian random chains

We study Chebyshev-polynomial expansion of the inverse localization length of Hermitian and non-Hermitian random chains as a function of energy. For Hermitian models, the expansion produces this energy-dependent function numerically in one run of the algorithm. This is in strong contrast to the standard transfer-matrix method, which produces the inverse localization length for a fixed energy in each run. For non-Hermitian models, as in the transfer-matrix method, our algorithm computes the inverse localization length for a fixed (complex) energy. We also find a formula of the Chebyshev-polynomial expansion of the density of states of non-Hermitian models. As explained in detail, our algorithm for non-Hermitian models may be the only available efficient algorithm for finding the density of states of models with interactions.

Scaling of the density of state of the weighted Laplacian in the presence of fractal boundaries

Spectral properties of the weighted Laplace operator in the presence of fractal boundaries are numerically investigated for both Neumann and Dirichlet boundary conditions. This corresponds to the characterization of heat and mass transport in microchannels with irregular and rough surfaces induced by the microfabrication process. The axial velocity field with no-slip boundary conditions, representing the weighting function of the Laplace operator, influences the localization properties of the eigenfunctions and the scaling of the integrated density of state (IDOS) N(lambda). The results indicate that N(lambda) deviates from the form given by the modified Weyl-Berry-Lapidus conjecture as it shows a correction of DeltaN(lambda) approximately lambda(D(f)/4) to the leading-order Weil term. Numerical results are presented for Koch and Koch snowflake fractal boundaries. The role of slip or no-slip boundary conditions of the velocity field on the IDOS is also investigated.

Localization and spectral phase transition in an open advecting-diffusing three-dimensional Stokes flow

We study the steady-state characterization of an advecting-diffusing three-dimensional flow defined in the annular region between coaxial cylinders of finite length that can rotate independently. A phase transition occurs when the cylinder velocities vary, which controls the spectral properties of the dominant eigenvalue and eigenfunction of the advection-diffusion equation for high Péclet numbers. The localization abscissa of the dominant eigenfunction can be used as the order parameter of the transition, and is a continuous function of the wall velocity. Conversely, the exponent characterizing the scaling of the real part of the dominant eigenvalue displays a discontinuous behavior at the critical point. Theoretical arguments support the localization properties observed numerically and provide a simple explanation of this phenomenon.

Topology of advective-diffusive scalar transport in laminar flows

The present study proposes a unified Lagrangian transport template for topological description of advective fluid transport and advective-diffusive scalar transport in laminar flows. The key to this unified description is the expression of scalar transport as purely advective transport by the total scalar flux. This admits generalization of the concept of transport topologies known from laminar mixing studies to scalar transport. The study is restricted to two-dimensional systems and the fluid and scalar transport topologies, as a consequence, prove to be Hamiltonian. The unified Lagrangian transport template is demonstrated and investigated for a heat-transfer problem with nonadiabatic boundaries, representing generic scalar transport with permeable boundaries. The fluid and thermal transport topologies under steady conditions both accommodate islands (constituting isolated flow and thermal regions) that undergo Hamiltonian disintegration into chaotic seas upon introducing time periodicity. The thermal transport topology invariably comprises transport conduits that connect the nonadiabatic boundaries and facilitate wall-wall and wall-fluid heat transfer. For steady conditions these transport conduits are regular; for time-periodic conditions these conduits may, depending on degree of diffusion, be regular or chaotic. Regular conduits connect nonadiabatic walls only with specific flow regions; chaotic heat conduits establish connection (and thus heat exchange) of the nonadiabatic walls with the entire flow domain.

Does a fast mixer really exist?

In this paper, we are concerned about the limit behavior of the decay rate of variance of a passive and diffusive scalar in a flow field as the diffusivity of the scalar goes to zero. Motivated by the concept of the fast dynamo in the dynamo theory, we term a flow as fast mixer if the decay rate remains away from zero as the diffusivity goes to zero. We first repeat numerical simulations with flow maps and velocity fields used in the existing literature, including the lattice map, the 1D baker's map, and the sinusoidal shear flow. Our simulations shows that, in all cases, the decay rate tends to zero as the diffusivity goes to zero. For the closed flows in a bounded domain, we then theoretically proved this result under certain plausible conditions on the flows. For the open flows in the whole space, we show that the effective diffusivity matrix tends to zero in the limit without the conditions for the closed flow. In conclusion, although a fast mixer might exist, it could be very difficult to find one.