Steady stokes flow with long-range correlations, fractal fourier spectrum, and anomalous transport

Subdiffusive and superdiffusive transport in plane steady viscous flows

Deterministic transport of passive tracers in steady laminar plane flows of incompressible viscous fluids through lattices of solid bodies or arrays of steady vortices can be anomalous. Motion along regular patterns of streamlines is often aperiodic: Repeated slow passages near stagnation points and/or solid surfaces serve for eventual decorrelation. Singularities of passage times near the obstacles, dictated by the boundary conditions, affect the character of transport anomalies: Flows past arrays of vortices are subdiffusive whereas tracers advected through lattices of solid obstacles can feature superdiffusion. We calculate the transport characteristics with the help of the simple and computationally efficient model: the special flow.

Critical dispersion of advecting-diffusing tracers in periodic landscapes of hard-wall symmetric potentials

Large-scale, time-asymptotic dispersion properties of diffusing tracers dragged by a uniform drive through a two-dimensional periodic lattice of hard-wall symmetric potentials are investigated. Dispersion is quantified by a typically anisotropic effective diffusivity tensor D, whose eigenvalues and eigenvectors depend on the dimensionless bare diffusivity 1/Pe for each given lattice geometry. Attention is focused on critical lattice geometries yielding sustained macroscale dispersion D([perpendicular]) along the direction orthogonal to the uniform drive in the limit where Pe→∞. A simple one-dimensional model is proposed, which predicts the anomalous scaling D([perpendicular])~1/[A(1)+A(2)log(Pe)].

Hydrodynamics of fractal continuum flow

A model of fractal continuum flow employing local fractional differential operators is suggested. The generalizations of the Green-Gauss divergence and Reynolds transport theorems for a fractal continuum are suggested. The fundamental conservation laws and hydrodynamic equations for an anisotropic fractal continuum flow are derived. Some physical implications of the long-range correlations in the fractal continuum flow are briefly discussed. It is noteworthy to point out that the fractal (quasi)metric defined in this paper implies that the flow of an isotropic fractal continuum obeying the Mandelbrot rule of thumb for intersection is governed by conventional hydrodynamic equations.

Porosity and tortuosity relations as revealed by a mathematical model of biofilm structure

A biofilm is assumed to be submerged in a fluid with given viscosity and low Reynolds number. The interaction between fluid and bacteria is modeled through streamlines. We use finite-difference and boundary element numerical schemes to predict streamlines within the biofilm. The results show that this approach can provide information about prior distribution and geometry of the biofilm structure. Theoretical values of porosity and tortuosity are computed and compared with published data.