scholarly journals Self-Similarity in Abrasiveness of Hard Carbon-Containing Coatings

2002 ◽  
Vol 125 (1) ◽  
pp. 1-7 ◽  
Author(s):  
Feodor M. Borodich ◽  
Leon M. Keer ◽  
Stephen J. Harris
Keyword(s):  
Power Law ◽  
Contact Region ◽  
Self Similarity ◽  
Hard Carbon ◽  
Flat Disk ◽  
Abrasion Rate ◽  
Flat Steel ◽  
Self Similar ◽  
Number Of Cycles ◽  
Experimental Scatter

The abrasiveness of hard carbon-containing thin films such as diamond-like carbon (DLC) and boron carbide (nominally B4C) towards steel is considered here. First, a remarkably simple experimentally observed power-law relationship between the abrasion rate of the coatings and the number of cycles is described. This relationship remains valid over at least 4 orders of magnitude of the number of cycles, with very little experimental scatter. Then possible models of wear are discussed. It is assumed that the dominant mechanism of steel wear is its mechanical abrasion by nano-scale asperities on the coating that have relatively large attack angles, i.e. by the so-called sharp asperities. Wear of coating is assumed to be mainly due to physical/chemical processes. Finally, models of the abrasion process for two basic cases are presented, namely a coated ball on a flat steel disk and a steel ball on a coated flat disk. The nominal contact region can be considered as constant in the former case, while in the latter case, the size of the region may be enlarged due to wear of the steel. These models of the abrasion process are based on the assumption of self-similar changes of the distribution function characterizing the statistical properties of patterns of scattered surface sharp asperities. It is shown that the power-law relationship for abrasion rate follows from the models.

THE SCALE-FREE AND SMALL-WORLD PROPERTIES OF COMPLEX NETWORKS ON SIERPINSKI-TYPE HEXAGON

Fractals ◽  
2020 ◽  
Vol 28 (03) ◽  
pp. 2050054
Author(s):  
KUN CHENG ◽  
DIRONG CHEN ◽  
YUMEI XUE ◽  
QIAN ZHANG
Keyword(s):  
Power Law ◽  
Path Length ◽  
Small World ◽  
The Self ◽  
Renewal Theorem ◽  
Self Similarity ◽  
Scale Free ◽  
Power Law Exponent ◽  
Self Similar ◽  
Encoding Method

In this paper, a network is generated from a Sierpinski-type hexagon by applying the encoding method in fractal. The criterion of neighbor is established to quantify the relationships among the nodes in the network. Based on the self-similar structures, we verify the scale-free and small-world effects. The power-law exponent on degree distribution is derived to be [Formula: see text] and the average clustering coefficients are shown to be larger than [Formula: see text]. Moreover, we give the bounds of the average path length of our proposed network from the renewal theorem and self-similarity.

SCALING LAWS IN A TURBULENT BAROCLINIC INSTABILITY

Fractals ◽  
1995 ◽  
Vol 03 (02) ◽  
pp. 297-314 ◽  
Author(s):  
C. SERIO ◽  
V. TRAMUTOLI
Keyword(s):  
Power Law ◽  
Scaling Laws ◽  
Spatial Scales ◽  
Baroclinic Instability ◽  
Satellite Image ◽  
Horizontal Resolution ◽  
Turbulent Motion ◽  
Self Similarity ◽  
Wavelet Transform Analysis ◽  
Self Similar

This work provides an empirical investigation of scaling laws in a cloud system generated and advected by a strong baroclinic instability. An infrared satellite image with a spatial (horizontal) resolution of about 1 km has been analyzed. The presence of two sizeable and unmistakable scaling regions, one extending from 1 to 15 km and characterized by a power law with an exponent close to 1, the other stretching from 20 km up to 100 km and characterized by a power law with exponent close to 1/3, have been revealed by variogram analysis. These two scaling laws are in agreement with the idea of scale invariance of the turbulent motion and also suggest the presence of a self-similar structure. To explore this possibility, wavelet transform analysis at different spatial scales has been used. Our findings are that self-similarity is present at the smallest scales, but this universal characteristic may be masked by non-universal effects which influence the homogeneity of the underlying turbulent motion. The implications of the two scaling exponents, 1 and 1/3, are also discussed.

Experimental study on radial gravity currents flowing in a vegetated channel

2022 ◽  
Vol 933 ◽  
Author(s):  
D. Petrolo ◽  
M. Ungarish ◽  
L. Chiapponi ◽  
S. Longo
Keyword(s):  
Experimental Study ◽  
Theoretical Model ◽  
Power Law ◽  
Gravity Currents ◽  
Tuning Parameter ◽  
Ambient Fluid ◽  
Self Similarity ◽  
Circular Sector ◽  
Power Law Function ◽  
Self Similar

We present an experimental study of gravity currents in a cylindrical geometry, in the presence of vegetation. Forty tests were performed with a brine advancing in a fresh water ambient fluid, in lock release, and with a constant and time-varying flow rate. The tank is a circular sector of angle $30^\circ$ with radius equal to 180 cm. Two different densities of the vegetation were simulated by vertical plastic rods with diameter $D=1.6\ \textrm{cm}$ . We marked the height of the current as a function of radius and time and the position of the front as a function of time. The results indicate a self-similar structure, with lateral profiles that after an initial adjustment collapse to a single curve in scaled variables. The propagation of the front is well described by a power law function of time. The existence of self-similarity on an experimental basis corroborates a simple theoretical model with the following assumptions: (i) the dominant balance is between buoyancy and drag, parameterized by a power law of the current velocity $\sim |u|^{\lambda-1}u$ ; (ii) the current advances in shallow-water conditions; and (iii) ambient-fluid dynamics is negligible. In order to evaluate the value of ${\lambda}$ (the only tuning parameter of the theoretical model), we performed two additional series of measurements. We found that $\lambda$ increased from 1 to 2 while the Reynolds number increased from 100 to approximately $6\times10^3$ , and the drag coefficient and the transition from $\lambda=1$ to $\lambda=2$ are quantitatively affected by D, but the structure of the model is not.

scholarly journals SELF-SIMILARITY IN THE TAXONOMIC CLASSIFICATION OF HUMAN LANGUAGES

2001 ◽  
Vol 04 (02n03) ◽  
pp. 281-286 ◽  
Author(s):  
DAMIAN H. ZANETTE
Keyword(s):  
Long Range ◽  
Power Law ◽  
Biological Species ◽  
Statistical Properties ◽  
Taxonomic Classification ◽  
Self Similarity ◽  
Self Similar ◽  
Evolutionary Foundation ◽  
Language Classification

Statistical properties of the taxonomic classification of human languages are studied. It is shown that, at the highest levels of the taxonomic hierarchy, the frequency of taxon members as a function of the number of languages belonging to each member decays as a power law. This feature reveals that a self-similar structure underlies the taxonomy of languages, exactly as observed in the taxonomic classification of biological species. Such an analogy is a clue to the evolutionary foundation of language classification based on long-range comparison.

GENERATION AND PREDICTION OF SELF-SIMILAR PROCESSES BY SURROGATES

Fractals ◽  
2006 ◽  
Vol 14 (01) ◽  
pp. 17-26 ◽  
Author(s):  
D. CHAKRABORTY ◽  
T. K. ROY
Keyword(s):  
Time Series ◽  
Power Spectrum ◽  
Power Law ◽  
Rank Order ◽  
Surrogate Data ◽  
Similar Process ◽  
Self Similarity ◽  
Random Data ◽  
Similarity Parameter ◽  
Self Similar

A self-similar process has power spectrum with power law depending on its self-similarity parameter H and we use this property for its generation by the method of surrogate data. The surrogates are a set of random data with the same distribution as that of the increments of the process. These are iteratively rearranged according to the rank order of a time series obtained with the same power law spectrum. The method is fast, and reliable as shown by the characteristics reproduced. It is also possible to extend this method to prediction of a self-similar process, the success of which will depend on the number of available data.

scholarly journals Extended power-law scaling of air permeabilities measured on a block of tuff

2012 ◽  
Vol 16 (1) ◽  
pp. 29-42 ◽  
Author(s):  
M. Siena ◽  
A. Guadagnini ◽  
M. Riva ◽  
S. P. Neuman
Keyword(s):  
Power Law ◽  
Structure Functions ◽  
Self Similarity ◽  
Scaling Exponents ◽  
Multi Scale ◽  
Sample Structure ◽  
Heavy Tailed ◽  
Non Gaussian ◽  
Nonlinear Fashion ◽  
Extended Power

Abstract. We use three methods to identify power-law scaling of multi-scale log air permeability data collected by Tidwell and Wilson on the faces of a laboratory-scale block of Topopah Spring tuff: method of moments (M), Extended Self-Similarity (ESS) and a generalized version thereof (G-ESS). All three methods focus on q-th-order sample structure functions of absolute increments. Most such functions exhibit power-law scaling at best over a limited midrange of experimental separation scales, or lags, which are sometimes difficult to identify unambiguously by means of M. ESS and G-ESS extend this range in a way that renders power-law scaling easier to characterize. Our analysis confirms the superiority of ESS and G-ESS over M in identifying the scaling exponents, ξ(q), of corresponding structure functions of orders q, suggesting further that ESS is more reliable than G-ESS. The exponents vary in a nonlinear fashion with q as is typical of real or apparent multifractals. Our estimates of the Hurst scaling coefficient increase with support scale, implying a reduction in roughness (anti-persistence) of the log permeability field with measurement volume. The finding by Tidwell and Wilson that log permeabilities associated with all tip sizes can be characterized by stationary variogram models, coupled with our findings that log permeability increments associated with the smallest tip size are approximately Gaussian and those associated with all tip sizes scale show nonlinear variations in ξ(q) with q, are consistent with a view of these data as a sample from a truncated version (tfBm) of self-affine fractional Brownian motion (fBm). Since in theory the scaling exponents, ξ(q), of tfBm vary linearly with q we conclude that nonlinear scaling in our case is not an indication of multifractality but an artifact of sampling from tfBm. This allows us to explain theoretically how power-law scaling of our data, as well as of non-Gaussian heavy-tailed signals subordinated to tfBm, are extended by ESS. It further allows us to identify the functional form and estimate all parameters of the corresponding tfBm based on sample structure functions of first and second orders.

scholarly journals Onset of the Mach Reflection of Zel’dovich–von Neumann–Döring Detonations

Entropy ◽  
2021 ◽  
Vol 23 (3) ◽  
pp. 314
Author(s):  
Tianyu Jing ◽  
Huilan Ren ◽  
Jian Li
Keyword(s):  
Hot Spot ◽  
Mach Reflection ◽  
Near Field ◽  
The Self ◽  
Small Scale ◽  
Mach Stem ◽  
Self Similarity ◽  
Von Neumann ◽  
Similarity Problem ◽  
Self Similar

The present study investigates the similarity problem associated with the onset of the Mach reflection of Zel’dovich–von Neumann–Döring (ZND) detonations in the near field. The results reveal that the self-similarity in the frozen-limit regime is strictly valid only within a small scale, i.e., of the order of the induction length. The Mach reflection becomes non-self-similar during the transition of the Mach stem from “frozen” to “reactive” by coupling with the reaction zone. The triple-point trajectory first rises from the self-similar result due to compressive waves generated by the “hot spot”, and then decays after establishment of the reactive Mach stem. It is also found, by removing the restriction, that the frozen limit can be extended to a much larger distance than expected. The obtained results elucidate the physical origin of the onset of Mach reflection with chemical reactions, which has previously been observed in both experiments and numerical simulations.

scholarly journals Depressurization-Induced Nucleation in the “Polylactide-Carbon Dioxide” System: Self-Similarity of the Bubble Embryos Expansion

Polymers ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1115
Author(s):  
Dmitry Zimnyakov ◽  
Marina Alonova ◽  
Ekaterina Ushakova
Keyword(s):  
Initial Rate ◽  
Initial Pressure ◽  
External Pressure ◽  
Self Similarity ◽  
Embryo Formation ◽  
Linear Behavior ◽  
Joint Influence ◽  
External Pressures ◽  
Adiabatic Cooling ◽  
Self Similar

Self-similar expansion of bubble embryos in a plasticized polymer under quasi-isothermal depressurization is examined using the experimental data on expansion rates of embryos in the CO2-plasticized d,l-polylactide and modeling the results. The CO2 initial pressure varied from 5 to 14 MPa, and the depressurization rate was 5 × 10−3 MPa/s. The constant temperature in experiments was in a range from 310 to 338 K. The initial rate of embryos expansion varied from ≈0.1 to ≈10 µm/s, with a decrease in the current external pressure. While modeling, a non-linear behavior of CO2 isotherms near the critical point was taken into account. The modeled data agree satisfactorily with the experimental results. The effect of a remarkable increase in the expansion rate at a decreasing external pressure is interpreted in terms of competing effects, including a decrease in the internal pressure, an increase in the polymer viscosity, and an increase in the embryo radius at the time of embryo formation. The vanishing probability of finding the steadily expanding embryos for external pressures around the CO2 critical pressure is interpreted in terms of a joint influence of the quasi-adiabatic cooling and high compressibility of CO2 in the embryos.

scholarly journals Self-similar resistive circuits as fractal-like structures

2017 ◽  
Vol 40 (1) ◽  
Author(s):  
Claudio Xavier Mendes dos Santos ◽  
Carlos Molina Mendes ◽  
Marcelo Ventura Freire
Keyword(s):  
Scale Invariance ◽  
Self Similarity ◽  
General Properties ◽  
Modern Physics ◽  
Resistive Networks ◽  
Self Similar ◽  
And Mathematics ◽  
Definition Of ◽  
The Individual

Fractals play a central role in several areas of modern physics and mathematics. In the present work we explore resistive circuits where the individual resistors are arranged in fractal-like patterns. These circuits have some of the characteristics typically found in geometric fractals, namely self-similarity and scale invariance. Considering resistive circuits as graphs, we propose a definition of self-similar circuits which mimics a self-similar fractal. General properties of the resistive circuits generated by this approach are investigated, and interesting examples are commented in detail. Specifically, we consider self-similar resistive series, tree-like resistive networks and Sierpinski’s configurations with resistors.

TUBE FORMULAS FOR GRAPH-DIRECTED FRACTALS

Fractals ◽  
2010 ◽  
Vol 18 (03) ◽  
pp. 349-361 ◽  
Author(s):  
BÜNYAMIN DEMÍR ◽  
ALI DENÍZ ◽  
ŞAHIN KOÇAK ◽  
A. ERSIN ÜREYEN
Keyword(s):  
The Self ◽  
Self Similarity ◽  
Complex Dimensions ◽  
Tubular Neighborhoods ◽  
Self Similar ◽  
Tube Formulas

Lapidus and Pearse proved recently an interesting formula about the volume of tubular neighborhoods of fractal sprays, including the self-similar fractals. We consider the graph-directed fractals in the sense of graph self-similarity of Mauldin-Williams within this framework of Lapidus-Pearse. Extending the notion of complex dimensions to the graph-directed fractals we compute the volumes of tubular neighborhoods of their associated tilings and give a simplified and pointwise proof of a version of Lapidus-Pearse formula, which can be applied to both self-similar and graph-directed fractals.

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