We study the permeability of quasi two-dimensional porous structures of randomly placed overlapping monodisperse circular and elliptical grains. Measurements in microfluidic devices and lattice Boltzmann simulations demonstrate that the permeability is determined by the Euler characteristic of the conducting phase. We obtain an expression for the permeability that is independent of the percolation threshold and shows agreement with experimental and simulated data over a wide range of porosities. Our approach suggests that the permeability explicitly depends on the overlapping probability of grains rather than their shape.

-We experimentally investigate hydrodynamic dispersion in elastic turbulent flows of a semi-dilute aqueous polymer solution within a periodic porous structure at ultra-low Reynolds numbers < 10 −3 by particle tracking velocimetry. Our results indicate that elastic turbulence can be characterized by an effective dispersion coefficient which exceeds that of Newtonian liquids by several orders of magnitude and grows non-linearly with the Weissenberg number Wi. Contrary to laminar flow conditions, the velocity field, and thus the shear rate, is not proportional to the flow rate and becomes asymmetric at high Wi.Introduction. -Turbulent flow is characterized by unsteady velocity fields which suddenly vary in space and time. For Newtonian liquids, this regime is reached for high Reynolds numbers Re, where inertial effects dominate over viscous forces. In contrast, for viscoelastic fluids, turbulent flow at arbitrarily small Re numbers, usually referred to as elastic turbulence [1], can be observed. Such fluids are characterized by an elastic response, e.g. due to the entanglement and the dynamics of polymer chains [2]. The degree of elastic effects can be described by the dimensionless Weissenberg number Wi = λ ·γ, which quantifies the anisotropic polymer alignment in the presence of shear [3]. Here, λ is the equilibrium polymer relaxation time andγ the shear rate of the flow. Accordingly, for Wi → 0 the response of the fluid to a sudden stress is purely viscous while for Wi > 0 an elastic short-time response is obtained. Since the polymer alignment not only depends on the instantaneous but also on the previous shear, this explains why a highly non-linear flow dependence on the shear strength is observed [2]. It has been experimentally demonstrated that above a critical Weissenberg number Wi c 1, elastic turbulence occurs [4]. Similar to inertial turbulence, this regime is characterized by an enhanced flow resistance and a power-law decay of the spectral power density [1,5,6]. This effect can be exploited to dramatically increase the mixing efficiency at small length scales and small Reynolds numbers (Re < 10 −4 ) in microfluidic devices [7,8].

We study the relation of permeability and morphology for porous structures composed of randomly placed overlapping circular or elliptical grains, so-called Boolean models. Microfluidic experiments and lattice Boltzmann simulations allow us to evaluate a power-law relation between the Euler characteristic of the conducting phase and its permeability. Moreover, this relation is so far only directly applicable to structures composed of overlapping grains where the grain density is known a priori. We develop a generalization to arbitrary structures modeled by Boolean models and characterized by Minkowski functionals. This generalization works well for the permeability of the void phase in systems with overlapping grains, but systematic deviations are found if the grain phase is transporting the fluid. In the latter case our analysis reveals a significant dependence on the spatial discretization of the porous structure, in particular the occurrence of single isolated pixels. To link the results to percolation theory we performed Monte-Carlo simulations of the Euler characteristic of the open cluster, which reveals different regimes of applicability for our permeability-morphology relations close to and far away from the percolation threshold.

We present two methods how the permeability in porous microstructures can be experimentally obtained from particle tracking velocimetry of finite-sized colloidal particles suspended in a liquid. The first method employs additional unpatterned reference channels where the liquid flow can be calculated theoretically and a relationship between the velocity of the particles and the liquid is obtained. The second method takes advantage of a timedependent pressure drop that leads to an exponential decrease in the particle velocity inside a porous structure. From the corresponding decay time, the permeability can be calculated independently of the particle size. Both methods lead to results comparable with permeabilities derived from numerical simulations.

Using a combined experimental-numerical approach, we study the first-passage time distributions (FPTD) of small particles in two-dimensional porous materials. The distributions in low-porosity structures show persistent long-time tails, which are independent of the Péclet number and therefore cannot be explained by the advectiondiffusion equation. Instead, our results suggest that these tails are caused by stagnant, i.e., quiescent areas where particles are trapped for some time. Comparison of measured FPTD with an analytical expression for the residence time of particles, which diffuse in confined regions and are able to escape through a small pore, yields good agreement with our data.