Self-similarity and structure of DLA and viscous fingering clusters

Dendritic patterns from shear-enhanced anisotropy in nematic liquid crystals

Controlling the growth morphology of fluid instabilities is challenging because of their self-amplified and nonlinear growth. The viscous fingering instability, which arises when a less viscous fluid displaces a more viscous one, transitions from exhibiting dense-branching growth characterized by repeated tip splitting of the growing fingers to dendritic growth characterized by stable tips in the presence of anisotropy. We controllably induce such a morphology transition by shear-enhancing the anisotropy of nematic liquid crystal solutions. For fast enough flow induced by the finger growth, the intrinsic tumbling behavior of lyotropic chromonic liquid crystals can be suppressed, which results in a flow alignment of the material. This microscopic change in the director field occurs as the viscous torque from the shear flow becomes dominant over the elastic torque from the nematic potential and macroscopically enhances the liquid crystal anisotropy to induce the transition to dendritic growth.

Geometry-Driven Detection, Tracking and Visual Analysis of Viscous and Gravitational Fingers

Viscous and gravitational flow instabilities cause a displacement front to break up into finger-like fluids. The detection and evolutionary analysis of these fingering instabilities are critical in multiple scientific disciplines such as fluid mechanics and hydrogeology. However, previous detection methods of the viscous and gravitational fingers are based on density thresholding, which provides limited geometric information of the fingers. The geometric structures of fingers and their evolution are important yet little studied in the literature. In this article, we explore the geometric detection and evolution of the fingers in detail to elucidate the dynamics of the instability. We propose a ridge voxel detection method to guide the extraction of finger cores from three-dimensional (3D) scalar fields. After skeletonizing finger cores into skeletons, we design a spanning tree based approach to capture how fingers branch spatially from the finger skeletons. Finally, we devise a novel geometric-glyph augmented tracking graph to study how the fingers and their branches grow, merge, and split over time. Feedback from earth scientists demonstrates the usefulness of our approach to performing spatio-temporal geometric analyses of fingers.

Solvability of the p-adic Analogue of Navier–Stokes Equation via the Wavelet Theory

P-adic numbers serve as the simplest ultrametric model for the tree-like structures arising in various physical and biological phenomena. Recently p-adic dynamical equations started to be applied to geophysics, to model propagation of fluids (oil, water, and oil-in-water and water-in-oil emulsion) in capillary networks in porous random media. In particular, a p-adic analog of the Navier–Stokes equation was derived starting with a system of differential equations respecting the hierarchic structure of a capillary tree. In this paper, using the Schauder fixed point theorem together with the wavelet functions, we extend the study of the solvability of a p-adic field analog of the Navier–Stokes equation derived from a system of hierarchic equations for fluid flow in a capillary network in porous medium. This equation describes propagation of fluid’s flow through Geo-conduits, consisting of the mixture of fractures (as well as fracture’s corridors) and capillary networks, detected by seismic as joint wave/mass conducts. Furthermore, applying the Adomian decomposition method we formulate the solution of the p-adic analog of the Navier–Stokes equation in term of series in general form. This solution may help researchers to come closer and find more facts, taking into consideration the scaling, hierarchies, and formal derivations, imprinted from the analogous aspects of the real world phenomena.

Multifractal analysis of the branch structure of diffusion-limited aggregates

We examine the branch structure of radial diffusion-limited aggregation (DLA) clusters for evidence of multifractality. The lacunarity of DLA clusters is measured and the generalized dimensions D(q) of their mass distribution is estimated using the sandbox method. We find that the global n-fold symmetry of the aggregates can induce anomalous scaling behavior into these measurements. However, negating the effects of this symmetry, standard scaling is recovered.

Coupled air/granular flow in a linear Hele-Shaw cell

We investigate experimentally the pattern formation process during injection of air in a noncohesive granular material confined in a linear Hele-Shaw cell. We characterize the features and dynamics of this pattern formation on the basis of fast image analysis and sensitive pressure measurements. Behaviors are classified using two parameters--injection pressure and plate opening--and four hydrodynamic regimes are defined. For some regions of the parameter space, flows of air and grains are shown to be strongly coupled and instable, and lead to channelization within the granular material with obvious large-scale permeability variations.

Pattern formation during air injection into granular materials confined in a circular Hele-Shaw cell

We investigate the dynamics of granular materials confined in a radial Hele-Shaw cell, during central air injection. The behavior of this granular system, driven by its interstitial fluid, is studied both experimentally and numerically. This allows us to explore the associated pattern formation process, characterize its features and dynamics. We classify different hydrodynamic regimes as function of the injection pressure. The numerical model takes into account the interactions between the granular material and the interstitial fluid, as well as the solid-solid interactions between the grains and the confining plates. Numerical and experimental results are comparable, both to reproduce the hydrodynamical regimes experimentally observed, as well as the dynamical features associated to fingering and compacting.

Power-law tail probabilities of drainage areas in river basins

We examine the appearance of power-law behavior in rooted tree graphs in the context of river networks. It has long been observed that the tails of statistical distributions of upstream areas in river networks, measured above every link, obey a power-law relationship over a range of scales. We examine this behavior by considering a subset of all links, defined as those links which drain complete Strahler basins, where the Strahler order defines a discrete measure of scale, for self-similar networks with both deterministic and random topologies. We find an excellent power-law structure in the tail probabilities for complete Strahler basin areas, over many ranges of scale. We show analytically that the tail probabilities converge to a power law under the assumptions of (1) simple scaling of the distributions of complete Strahler basin areas and (2) application of Horton's law of stream numbers. The convergence to a power law does not occur for all underlying distributions, but for a large class of statistical distributions which have specific limiting properties. For example, underlying distributions which are exponential and gamma distributed, while not power-law scaling, produce power laws in the tail probabilities when rescaled and sampled according to Horton's law of stream numbers. The power-law exponent is given by the expression phi=ln(R(b))/ln(R(A)), where R(b) is the bifurcation ratio and R(A) is the Horton area ratio. It is commonly observed that R(b) approximately equal R(A) in many river basins, implying that the tail probability exponent for complete Strahler basins is close to 1.0.

Interface scaling in a two-dimensional porous medium under combined viscous, gravity, and capillary effects

We have investigated experimentally the competition between viscous, capillary, and gravity forces during drainage in a two-dimensional synthetic porous medium. The displacement of a mixture of glycerol and water by air at constant withdrawal rate has been studied. The setup can be tilted to tune gravity, and pressure is recorded at the outlet of the model. Viscous forces tend to destabilize the displacement front into narrow fingers against the stabilizing effect of gravity. Subsequently, a viscous instability is observed for sufficiently large withdrawal speeds or sufficiently low gravity components on the model. We predict the scaling of the front width for stable situations and characterize it experimentally through analyses of the invasion front geometry and pressure recordings. The front width under stable displacement and the threshold for the instability are shown, both experimentally and theoretically, to be controlled by a dimensionless number F which is defined as the ratio of the effective fluid pressure drop (i.e., average hydrostatic pressure drop minus viscous pressure drop) at pore scale to the width of the fluctuations in the threshold capillary pressures.

Mean-field diffusion-limited aggregation: a "density" model for viscous fingering phenomena

We explore a universal "density" formalism to describe nonequilibrium growth processes, specifically, the immiscible viscous fingering in Hele-Shaw cells (usually referred to as the Saffman-Taylor problem). For that we develop an alternative approach to the viscous fingering phenomena, whose basic concepts have been recently published in a Rapid Communication [Phys. Rev. E 63, 045305(R) (2001)]. This approach uses the diffusion-limited aggregation (DLA) paradigm as a core: we introduce a mean-field DLA generalization in stochastic and deterministic formulations. The stochastic model, a quasicontinuum DLA, simulates Monte Carlo patterns, which demonstrate a striking resemblance to natural Hele-Shaw fingers and, for steady-state growth regimes, follow precisely the Saffman-Taylor analytical solutions in channel and sector configurations. The relevant deterministic theory, a complete set of differential equations for a time development of density fields, is derived from that stochastic model. As a principal conclusion, we prove an asymptotic equivalency of both the stochastic and deterministic mean-field DLA formulations to the classic Saffman-Taylor hydrodynamics in terms of an interface evolution.

Hyperbranched Polymers and Aggregates: Distribution Kinetics of Dendrimer Growth

We present an analytic solution for growth of branched aggregates or polymers with distributed cluster (dendrimer) size. Monomer addition to each branch follows first-order polymerization kinetics leading to a distribution of branch lengths. The rate constant for monomer addition is considered diffusion-dependent. Deterministic branching occurs so that at prescribed times t(j) (j >/= 1), p branches emanate from the tip of a branch that began to grow at t(j-1). When the ratio of average branch lengths is constant, L(j)/L(j-1) = a, the fractal dimension of branches is ln(p)/ ||ln(a) ||. Closed expressions for cluster mass moments show unbounded growth with time unless ap < 1. The number of clusters (zeroth moment) is constant during growth and equal to the number of initiating buds. Expressions for the number of branches and clusters, average cluster mass, volume, density, and viscosity in solution are functions of p and a. Density and molecular weight show features similar to observed behavior of dendrimers and hyperbranched polymers. Copyright 1999 Academic Press.