Maze solvers demystified and some other thoughts

There is a growing interest towards implementation of maze solving in spatially-extended physical, chemical and living systems. Several reports of prototypes attracted great publicity, e.g. maze solving with slime mould and epithelial cells, maze navigating droplets. We show that most prototypes utilise one of two phenomena: a shortest path in a maze is a path of the least resistance for fluid and current flow, and a shortest path is a path of the steepest gradient of chemoattractants. We discuss that substrates with so-called maze-solving capabilities simply trace flow currents or chemical diffusion gradients. We illustrate our thoughts with a model of flow and experiments with slime mould. The chapter ends with a discussion of experiments on maze solving with plant roots and leeches which show limitations of the chemical diffusion maze-solving approach.Comment: This is a preliminary version of the chapter to be published in Adamatzky A. (Ed.) Shortest path solvers. From software to wetware. Springer, 201

FLUID MAPPERS AS VISUAL ANALOGS FOR POTENTIAL FIELDS

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/73340/1/j.1749-6632.1955.tb40080.x.pd

Particle dynamics of a cartoon dune

The spatio-temporal evolution of a downsized model for a desert dune is observed experimentally in a narrow water flow channel. A particle tracking method reveals that the migration speed of the model dune is one order of magnitude smaller than that of individual grains. In particular, the erosion rate consists of comparable contributions from creeping (low energy) and saltating (high energy) particles. The saltation flow rate is slightly larger, whereas the number of saltating particles is one order of magnitude lower than that of the creeping ones. The velocity field of the saltating particles is comparable to the velocity field of the driving fluid. It can be observed that the spatial profile of the shear stress reaches its maximum value upstream of the crest, while its minimum lies at the downstream foot of the dune. The particle tracking method reveals that the deposition of entrained particles occurs primarily in the region between these two extrema of the shear stress. Moreover, it is demonstrated that the initial triangular heap evolves to a steady state with constant mass, shape, velocity, and packing fraction after one turnover time has elapsed. Within that time the mean distance between particles initially in contact reaches a value of approximately one quarter of the dune basis length

Breakdown of smoothness for the Muskat problem

In this paper we show that there exist analytic initial data in the stable regime for the Muskat problem such that the solution turns to the unstable regime and later breaks down i.e. no longer belongs to $C^4$.Comment: 93 pages, 10 figures (6 added

Cavitation-induced force transition in confined viscous liquids under traction

We perform traction experiments on simple liquids highly confined between parallel plates. At small separation rates, we observe a simple response corresponding to a convergent Poiseuille flow. Dramatic changes in the force response occur at high separation rates, with the appearance of a force plateau followed by an abrupt drop. By direct observation in the course of the experiment, we show that cavitation accounts for these features which are reminiscent of the utmost complex behavior of adhesive films under traction. Surprisingly enough, this is observed here in purely viscous fluids.Comment: Submitted to Physical Review Letters on May 31, 2002. Related informations on http://www.crpp.u-bordeaux.fr/tack.htm

Rayleigh-Taylor breakdown for the Muskat problem with applications to water waves

The Muskat problem models the evolution of the interface between two different fluids in porous media. The Rayleigh-Taylor condition is natural to reach linear stability of the Muskat problem. We show that the Rayleigh-Taylor condition may hold initially but break down in finite time. As a consequence of the method used, we prove the existence of water waves turning.Ministerio de Ciencia e InnovaciónEuropean Research CouncilNational Science FoundationOffice of Naval Researc

Generic critical points of normal matrix ensembles

The evolution of the degenerate complex curve associated with the ensemble at a generic critical point is related to the finite time singularities of Laplacian Growth. It is shown that the scaling behavior at a critical point of singular geometry $x^3 \sim y^2$ is described by the first Painlev\'e transcendent. The regularization of the curve resulting from discretization is discussed.Comment: Based on a talk given at the conference on Random Matrices, Random Processes and Integrable Systems, CRM Montreal, June 200

Gas migration regimes and outgassing in particle-rich suspensions

Understanding how gasses escape from particle-rich suspensions has important applications in nature and industry. Motivated by applications such as outgassing of crystal-rich magmas, we map gas migration patterns in experiments where we vary (1) particle fractions and liquid viscosity (10–500 Pa s), (2) container shape (horizontal parallel plates and upright cylinders), and (3) methods of bubble generation (single bubble injections, and multiple bubble generation with chemical reactions). We identify two successive changes in gas migration behavior that are determined by the normalized particle fraction (relative to random close packing), and are insensitive to liquid viscosity, bubble growth rate or container shape within the explored ranges. The first occurs at the random loose packing, when gas bubbles begin to deform; the second occurs near the random close packing, and is characterized by gas migration in a fracture-like manner. We suggest that changes in gas migration behavior are caused by dilation of the granular network, which locally resists bubble growth. The resulting bubble deformation increases the likelihood of bubble coalescence, and promotes the development of permeable pathways at low porosities. This behavior may explain the efficient loss of volatiles from viscous slurries such as crystal-rich magmas

Hele-Shaw beach creation by breaking waves: a mathematics-inspired experiment

Fundamentals of nonlinear wave-particle interactions are studied experimentally in a Hele-Shaw configuration with wave breaking and a dynamic bed. To design this configuration, we determine, mathematically, the gap width which allows inertial flows to survive the viscous damping due to the side walls. Damped wave sloshing experiments compared with simulations confirm that width-averaged potential-flow models with linear momentum damping are adequately capturing the large scale nonlinear wave motion. Subsequently, we show that the four types of wave breaking observed at real-world beaches also emerge on Hele-Shaw laboratory beaches, albeit in idealized forms. Finally, an experimental parameter study is undertaken to quantify the formation of quasi-steady beach morphologies due to nonlinear, breaking waves: berm or dune, beach and bar formation are all classified. Our research reveals that the Hele-Shaw beach configuration allows a wealth of experimental and modelling extensions, including benchmarking of forecast models used in the coastal engineering practice, especially for shingle beaches

Shocks and finite-time singularities in Hele-Shaw flow

Hele-Shaw flow at vanishing surface tension is ill-defined. In finite time, the flow develops cusp-like singularities. We show that the ill-defined problem admits a weak {\it dispersive} solution when singularities give rise to a graph of shock waves propagating in the viscous fluid. The graph of shocks grows and branches. Velocity and pressure jump across the shock. We formulate a few simple physical principles which single out the dispersive solution and interpret shocks as lines of decompressed fluid. We also formulate the dispersive weak solution in algebro-geometrical terms as an evolution of the Krichever-Boutroux complex curve. We study in detail the most generic (2,3) cusp singularity, which gives rise to an elementary branching event. This solution is self-similar and expressed in terms of elliptic functions.Comment: 24 pages, 11 figures; references added; figures change