scholarly journals Patterns formed by chains of magnetic beads

2021 ◽  
Vol 249 ◽  
pp. 15004
Author(s):  
Danilo S. Borges ◽  
Hans J. Herrmann ◽  
Humberto A. Carmona ◽  
José Soares Andrade ◽  
Ascânio D. Araújo
Keyword(s):  
Fractal Dimension ◽  
Numerical Simulations ◽  
Magnetic Beads ◽  
Power Laws ◽  
Packing Fraction ◽  
Self Similar ◽  
A Chain ◽  
Hele Shaw Cell ◽  
Injection Procedure

Magnetic beads attract each other forming rather stable chains. We consider such chains formed by magnetic beads and push them into a Hele-Shaw cell either from the boundary or from the center. When such a chain is pushed into a cavity, it bends and folds spontaneously forming interesting unreported patterns. These patterns are self-similar and an effective fractal dimension can be defined. As found experimentally and with numerical simulations, the numbers of beads, loops and contacts follow power laws as a function of packing fraction and, depending on the injection procedure, even energetically less favorable triangular configurations can be stabilized.

Fractal and Convolutional Analysis for Deep Atmospheric Turbulence Using Machine Learning

2021 ◽  
Author(s):  
Nicholas Dudu ◽  
Arturo Rodriguez ◽  
Gael Moran ◽  
Jose Terrazas ◽  
Richard Adansi ◽  
...  
Keyword(s):  
Fractal Dimension ◽  
Atmospheric Turbulence ◽  
Isotropic Turbulence ◽  
Passive Scalar ◽  
Counting Method ◽  
Data Set ◽  
Box Counting ◽  
Box Counting Dimension ◽  
Self Similar ◽  
Box Counting Method

Abstract Atmospheric turbulence studies indicate the presence of self-similar scaling structures over a range of scales from the inertial outer scale to the dissipative inner scale. A measure of this self-similar structure has been obtained by computing the fractal dimension of images visualizing the turbulence using the widely used box-counting method. If applied blindly, the box-counting method can lead to misleading results in which the edges of the scaling range, corresponding to the upper and lower length scales referred to above are incorporated in an incorrect way. Furthermore, certain structures arising in turbulent flows that are not self-similar can deliver spurious contributions to the box-counting dimension. An appropriately trained Convolutional Neural Network can take account of both the above features in an appropriate way, using as inputs more detailed information than just the number of boxes covering the putative fractal set. To give a particular example, how the shape of clusters of covering boxes covering the object changes with box size could be analyzed. We will create a data set of decaying isotropic turbulence scenarios for atmospheric turbulence using Large-Eddy Simulations (LES) and analyze characteristic structures arising from these. These could include contours of velocity magnitude, as well as of levels of a passive scalar introduced into the simulated flows. We will then identify features of the structures that can be used to train the networks to obtain the most appropriate fractal dimension describing the scaling range, even when this range is of limited extent, down to a minimum of one order of magnitude.

scholarly journals Thermal and Wet Comfort of Fabrics Based on Fractal Dimension of Silicone Coating

2018 ◽  
Vol 13 (1) ◽  
pp. 155892501801300
Author(s):  
Yunlong Shi ◽  
Liang Wang ◽  
Wenhuan Zhang ◽  
Xiaoming Qian
Keyword(s):  
Fractal Dimension ◽  
Thermal Insulation ◽  
Coating Layer ◽  
Fractal Theory ◽  
Barrier Effect ◽  
Evaporative Resistance ◽  
Silicone Coating ◽  
Self Similar ◽  
Warmth Retention ◽  
Coated Fabrics

In this paper, thermal and wet comforts of silicone coated windbreaker shell jacket fabrics were studied. Both thermal insulation and evaporative resistance of fabric increased with an increase in coating area due to the barrier effect of the silicone coating layer. Moreover, the coated fabrics with self-similar structures showed different thermal insulation and evaporative resistance under the same total coating area. Fractal theory was used to explain this phenomenon. Optimal thermal-wet comfort properties were obtained when the fractal dimension (D=1.599) was close to the Golden Mean (1.618). When the fractal dimension of coating was lower than 1.599, fabric warmth retention was not high enough. In contrast, fabric evaporative resistance was beyond the value at which people would feel comfortable when the fractal dimension was greater than 1.599.

Local structure at a ferrofluid surface: chain-like aggregates revealed by non-specular X-ray reflection

2003 ◽  
Vol 36 (2) ◽  
pp. 244-248 ◽  
Author(s):  
I. Takahashi ◽  
N. Tanaka ◽  
S. Doi
Keyword(s):  
Surface Tension ◽  
Fractal Dimension ◽  
Local Structure ◽  
Fine Particles ◽  
Capillary Waves ◽  
Aggregation Kinetics ◽  
X Ray ◽  
Surface Fluctuations ◽  
Irreversible Aggregation ◽  
A Chain

The surface structure of a ferrofluid was investigated by means of non-specular X-ray reflection. Strong intensity that is impossible to explain by surface fluctuations due to capillary waves was observed. It can be related to lateral correlation within aggregates of super-paramagnetic fine particles in the vicinity of the specimen surface. The fractal dimension of these surface-induced aggregates and the surface-tension coefficient of the ferrofluid were simultaneously determined. The fractal dimension was found to be around 1.1, indicating a chain-like character of the aggregates that have few branches. Strong and anisotropic interaction among the particles, as well as irreversible aggregation kinetics must be the origin of such a high-density and low-fractal-dimension system of dipolar 10 nm sized particles. The temperature variation of the fractal dimension indicated that the fractal aggregates stabilize themselves by losing their branches at increasing temperatures.

scholarly journals Splash wave and crown breakup after disc impact on a liquid surface

2013 ◽  
Vol 724 ◽  
pp. 553-580 ◽  
Author(s):  
Ivo R. Peters ◽  
Devaraj van der Meer ◽  
J. M. Gordillo
Keyword(s):  
Numerical Simulations ◽  
Weber Number ◽  
Threshold Value ◽  
Bond Number ◽  
Radius Of Curvature ◽  
Time Varying ◽  
Self Similar Solution ◽  
Air Cushion ◽  
Self Similar ◽  
The Impact

AbstractIn this paper we analyse the impact of a circular disc on a free surface using experiments, potential flow numerical simulations and theory. We focus our attention both on the study of the generation and possible breakup of the splash wave created after the impact and on the calculation of the force on the disc. We have experimentally found that drops are only ejected from the rim located at the top part of the splash – giving rise to what is known as the crown splash – if the impact Weber number exceeds a threshold value ${\mathit{We}}_{crit} \simeq 140$. We explain this threshold by defining a local Bond number $B{o}_{\mathit{tip}} $ based on the rim deceleration and its radius of curvature, with which we show using both numerical simulations and experiments that a crown splash only occurs when $B{o}_{\mathit{tip}} \gtrsim 1$, revealing that the rim disrupts due to a Rayleigh–Taylor instability. Neglecting the effect of air, we show that the flow in the region close to the disc edge possesses a Weber-number-dependent self-similar structure for every Weber number. From this we demonstrate that ${\mathit{Bo}}_{\mathit{tip}} \propto \mathit{We}$, explaining both why the transition to crown splash can be characterized in terms of the impact Weber number and why this transition occurs for $W{e}_{crit} \simeq 140$. Next, including the effect of air, we have developed a theory which predicts the time-varying thickness of the very thin air cushion that is entrapped between the impacting solid and the liquid. Our analysis reveals that gas critically affects the velocity of propagation of the splash wave as well as the time-varying force on the disc, ${F}_{D} $. The existence of the air layer also limits the range of times in which the self-similar solution is valid and, accordingly, the maximum deceleration experienced by the liquid rim, that sets the length scale of the splash drops ejected when $We\gt {\mathit{We}}_{crit} $.

ESTIMATING THE FRACTAL DIMENSION OF PLANTS USING THE TWO-SURFACE METHOD: AN ANALYSIS BASED ON 3D-DIGITIZED TREE FOLIAGE

Fractals ◽  
2006 ◽  
Vol 14 (03) ◽  
pp. 149-163 ◽  
Author(s):  
FRÉDÉRIC BOUDON ◽  
CHRISTOPHE GODIN ◽  
CHRISTOPHE PRADAL ◽  
OLIVIER PUECH ◽  
HERVÉ SINOQUET
Keyword(s):  
Fractal Dimension ◽  
Field Measurements ◽  
Three Dimensions ◽  
Total Leaf Area ◽  
Topological Information ◽  
Surface Method ◽  
Plant Foliage ◽  
Self Similar ◽  
Branching System ◽  
Peach Trees

In this paper, we present a method to estimate the fractal dimension of plant foliage in three dimensions (3D). This method is derived from the two-surface method introduced in the 90s to estimate the fractal dimension of tree species from field measurements on collections of trees. Here we adapted the method to individual plants. The multiscale topology and geometry of the plant must first be digitized in 3D. Then leafy branching systems of different sizes are constructed from the plant database, using the topological information. 3D convex envelops are then computed for each leafy branching system. The fractal dimension of the plant is finally estimated by comparing the total leaf area and the convex envelop area of these leafy modules. The method was assessed on a set of four peach trees entirely digitized at shoot scale. Results show that the peach trees have a marked self-similar foliage with fractal dimension close to 2.4.

scholarly journals A laboratory study of sediment deformation: stress heterogeneity and grain-size evolution

1996 ◽  
Vol 22 ◽  
pp. 167-175 ◽  
Author(s):  
Neal R. Iverson ◽  
Thomas S. Hooyer ◽  
Roger Leb. Hooke
Keyword(s):  
Grain Size ◽  
Fractal Dimension ◽  
Normal Stress ◽  
Size Distribution ◽  
Grain Size Distribution ◽  
Local Stress ◽  
Large Strains ◽  
Self Similar ◽  
Deformation Stress ◽  
Stress Heterogeneity

In shearing sediment beneath glaciers, networks of grains may transiently support shear and normal stresses that are larger than spatial averages. Consistent with studies of fault-gouge genesis, we hypothesize that crushing of grains in such networks is responsible for surrounding larger grains with smaller grains. At sufficiently large strains, this should minimize stress heterogeneity, favor intergranular sliding and abrasion rather than crushing, and result in a self-similar grain-size distribution.This hypothesis is tested with a ring-shear device that slowly shears a large annular sediment sample to high strains. Shearing and comminution of weak equigranular (2.0–3.3 mm) sediment resulted in a self-similar grain-size distribution with a fractal dimension that increased with shear strain toward a steady value of 2.85. This value is significantly larger than that of gouges produced purely by crushing, 2.6, but it is comparable to values for tilts thought to be deforming beneath modern glaciers, 2.8 to nearly 3.0. At low strains, under a steady mean normal stress of 84 kPa, variations in normal stress measured locally ranged in amplitude from 50 to 300 kPa with wavelengths that were 100 times larger than the initial grain diameter. Crushing of grains, observed through the transparent walls of the device, apparently caused the failure of grain networks. At shearing displacements ranging from 0.7 to 1.0 m, the amplitude of local stress fluctuations decreased abruptly. This change is attributed to fine sediment that distributed stresses more uniformly and caused grain networks to fail primarily by intergranular sliding rather than by crushing of grains. Sliding between grains apparently produced silt by abrasion and resulted in a fractal dimension that was higher than if there had been only crushing.A size distribution with a fractal dimension greater than 2.6 is probably a necessary but not sufficient condition for determining whether a basal till has been highly deformed. Stress heterogeneity in subglacial sediment that is shearing through its full thickness should contribute to the erosion of underlying rock.

scholarly journals Large resistivity in numerical simulations of radially self-similar outflows

2014 ◽  
Vol 442 (2) ◽  
pp. 1133-1141 ◽  
Author(s):  
M. Čemeljić ◽  
N. Vlahakis ◽  
K. Tsinganos
Keyword(s):  
Numerical Simulations ◽  
Self Similar

scholarly journals Influence of fault roughness on surface displacement: from numerical simulations to coseismic slip distributions

2019 ◽  
Vol 220 (3) ◽  
pp. 1857-1877 ◽  
Author(s):  
Lucile Bruhat ◽  
Yann Klinger ◽  
Amaury Vallage ◽  
Eric M Dunham
Keyword(s):  
Shear Stress ◽  
Numerical Simulations ◽  
Spectral Characteristics ◽  
Field Studies ◽  
Dynamic Rupture ◽  
Coseismic Slip ◽  
Fault Roughness ◽  
Critical Wavelength ◽  
Self Similar ◽  
Background Shear

SUMMARY Field studies have characterized natural faults as rough, non-planar surfaces at all scales. Fault roughness induces local stress perturbations during slip, which dramatically affect rupture behaviour, resulting in slip heterogeneity. However, the relation between fault roughness and slip heterogeneity remains a key knowledge gap between current numerical and field studies. In this study, we analyse numerical simulations of earthquake rupture to determine how roughness influences final slip. Using a rupture catalogue containing thousands of dynamic rupture simulations on band-limited self-similar fractal fault profiles with varying roughness and background shear stress levels, we quantify how fault roughness affects the spectral characteristics of the resulting slip distribution. We find that slip distributions become increasingly more self-affine, that is, containing more short wavelength fluctuations as compared to the self-similar fault profiles, as roughness increases. We also find that, at very short wavelengths (<1 km), the fractal dimension of the slip distributions dramatically changes with increasing roughness, background shear stress, and rupture speed (sub-Rayleigh versus supershear). The existence of a critical wavelength around 1 km, under which more short wavelengths are either preserved or created, suggests the role of rupture process and dynamic effects, together with fault geometry, in controlling the final slip distributions. The same spectral analysis is performed on high-resolution coseismic surface slip distributions from a catalogue of real strike-slip earthquakes. Compared to numerical simulations, all earthquakes feature slip distributions that are much more self-affine than the slip distributions from numerical simulations. A different critical wavelength, here around 5–6 km, appears, potentially informing about a critical asperity length. While we show here that the relation between fault roughness and slip is much more complex than expected, this study is a first attempt at using statistical analyses of numerical simulations on rough faults to investigate observed coseismic slip distributions.

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