scholarly journals The Fractal Calculus for Fractal Materials

2019 ◽  
Vol 3 (1) ◽  
pp. 8 ◽  
Author(s):  
Fakhri Jafari ◽  
Mohammad Asgari ◽  
Amir Pishkoo
Keyword(s):  
Fractal Structure ◽  
Differential Forms ◽  
Mass Function ◽  
Ideal Gas ◽  
Fractal Structures ◽  
Volume Expansivity ◽  
Fractal Calculus ◽  
Fractal Materials ◽  
Physical Quantities ◽  
Dimensionality Of Space

The major problem in the process of mixing fluids (for instance liquid-liquid mixers) is turbulence, which is the outcome of the function of the equipment (engine). Fractal mixing is an alternative method that has symmetry and is predictable. Therefore, fractal structures and fractal reactors find importance. Using F α -fractal calculus, in this paper, we derive exact F α -differential forms of an ideal gas. Depending on the dimensionality of space, we should first obtain the integral staircase function and mass function of our geometry. When gases expand inside the fractal structure because of changes from the i + 1 iteration to the i iteration, in fact, we are faced with fluid mixing inside our fractal structure, which can be described by physical quantities P, V, and T. Finally, for the ideal gas equation, we calculate volume expansivity and isothermal compressibility.

MATHEMATICAL MODELS OF GEOFILTRRATION AND GEOMIGRATION IN POROUS MEDIA WITH FRACTAL STRUCTURE

2020 ◽  
Vol 6 (3) ◽  
pp. 21-27
Author(s):  
R.A. Yusupov ◽  
◽  
Sh.S. Axrolov ◽  
N.M. Mirzanova ◽  
A.N. Nasiriddinov ◽  
...  
Keyword(s):  
Porous Media ◽  
Mathematical Models ◽  
Linear Models ◽  
Fractal Structure ◽  
Fractal Structures

In this study 2-D linear models are coming from generalised, Boussinesq eqution describing geofiltration in soils with fractal structures are presented. In this study are presented too mathematical models geomigration of contaminations with groundwater in classical way and in soils with fractal structures.

Visualizing Coherent and Fractal Structures in Numerical Experiments on Chaotic Advection in Fluids

2005 ◽  
Author(s):  
M. V. Budyansky ◽  
S. V. Prants
Keyword(s):  
Numerical Experiments ◽  
Fractal Structure ◽  
Point Vortex ◽  
Chaotic Advection ◽  
Mixing Zone ◽  
Background Current ◽  
Fractal Structures ◽  
Chaotic Scattering ◽  
And Topology ◽  
Fractal Properties

We investigate typical mixing and fractal properties of chaotic scattering of passive particles in open hydrodynamic flows taking as an example a model two-dimensional incompressible flow composed of a fixed point vortex and a background current with a periodic component, the model inspired by the phenomenon of topographic eddies over mountains in the ocean and atmosphere. We have found, described and visualized a non-attracting invariant chaotic set defining chaotic scattering, fractality, and trapping of incoming particles. Geometry and topology of chaotic scattering have been studied and visualized. Scattering functions in the mixing zone have been found to have a fractal structure with a complicated hierarchy that has been described in terms of strophes and epistrophes. Mixing, trapping, and fractal properties of passive particles have been studied under the influence of a white noise with different amplitudes and frequency ranges. A new effect of clustering the particles in a noised flow has been demonstrated in numerical experiments.

Multi-Scale Regular-Fractal Topography Characterization and Modeling

2014 ◽  
Author(s):  
Jinya Liu ◽  
Vijaya Chalivendra ◽  
Charles L. Goldsmith ◽  
Wenzhen Huang
Keyword(s):  
Fractal Structure ◽  
Fractal Model ◽  
Rf Switch ◽  
Fractal Structures ◽  
Force Microscopy ◽  
Multi Scale ◽  
Atomic Force ◽  
Model Components ◽  
Sample Allocation ◽  
Random Irregularity

Regular-fractal topography on RF-switch MEMS surface is reported over different scale ranges. Surface topography is crucial in understanding underling physics associated with the surface contacts, switch working performance, and reliability. The complexity of these structures requires new techniques to characterize topography and then replicate the multi-scale regular-fractal structure for analysis. Topography on RF-switch contacting surfaces are scanned by atomic force microscopy (AFM) at different length scales (e.g. 1×1, 10×10 and 60×60 μm2). A sample allocation plan is designed to maximize the spatial representative of the AFM scanning patches with different resolutions and uniformly distributed sample patches. The scanning data are used for characterizing and model estimation. Hexagonal patterns are found on at coarser scales (e.g. 10×10 and 60×60 μm2). They were formed by the remnant (polymer) of etching process. Random irregularity is observed and the fractal structure at finer scales (e.g. 1×1 μm2) is identified. A regular-fractal model is proposed to decompose and characterize the regular and fractal structures with two model components: one for the regular geometric pattern and the other for fractal irregularity. The former uses a 2D cosine functions to characterize dominant modes in the regular (larger scale) patterns. The later summarizes random irregularity in finer scales with a statistical fractal model estimated from the data on the scattered sample patches. The model validation is made through the comparisons of topography and conventional roughness parameters between the results of simulation from the proposed model and that derived from AFM scanned data.

Investigation of the influence of boundary conditions in the numerical solution of a vortex tube model

2019 ◽  
Vol 14 (2) ◽  
pp. 89-100
Author(s):  
M.R. Minibaev ◽  
C.I. Mikhaylenko
Keyword(s):  
Boundary Conditions ◽  
Turbulent Flows ◽  
Gas Dynamics ◽  
Vortex Tube ◽  
Computational Simulation ◽  
Ideal Gas ◽  
Tube Model ◽  
Gas Dynamics Problems ◽  
Dynamics Problems ◽  
Physical Quantities

The applicability of various boundary conditions in the computational simulation of a Ranque–Hilsch vortex tube is investigated. A review of existing works on the effect of geometry and various thermodynamic parameters on the efficiency of the pipe is made. The substantiation of the possibility of introducing additional computational domains when moving the boundaries to study the influence of boundary conditions when modeling gas dynamics problems is given. To simulate the dynamics of a gas in a vortex tube, a mathematical model is written that includes the Navier–Stokes system of equations describing a compressible viscous fluid, which is closed by the equation of state of an ideal gas. Existing methods for calculating turbulent flows are considered. The applicability of various semi-empirical models of turbulence for modeling a vortex tube is described. The possibility of using the selected k−ε model and its description is argued. The boundary conditions characteristic of the vortex tube model are described, and the boundary conditions most combined in the simulation of gas dynamics problems are also shown. Presents a grid that takes into account the area formed by the removal of boundaries. The solution is based on the sonicFoam algorithm in the OpenFOAM package. Utilities of the postprocessor are used when preparing the model for calculations on a high-performance cluster and utilities for averaging the obtained physical quantities. The simulation results for different combinations of boundary conditions and models with remote boundaries are given. Comparison of the results obtained. It is shown that the geometrical dimensions have a strong influence on the operation of the pipe; the correct choice of boundary conditions makes it possible to obtain the values of physical quantities that are closest to the known experimental ones. Moving the boundaries away from direct exits provides an opportunity to more accurately estimate the effects that arise near the real boundaries of the vortex tube, especially affecting the magnitude of the Ranque–Hilsch effect.

MATHEMATICAL MODELS OF GEOFILTRRATION AND GEOMIGRATION IN POROUS MEDIA WITH FRACTAL STRUCTURE

2020 ◽  
Vol 5 (3) ◽  
pp. 39-45
Author(s):  
R.A. Yusupov ◽  
◽  
Sh.S. Axrolov ◽  
N.M. Mirzanova ◽  
A.N. Nasiriddinov
Keyword(s):  
Porous Media ◽  
Mathematical Models ◽  
Linear Models ◽  
Fractal Structure ◽  
Fractal Structures

In this study 2-D linear models are coming from generalised, Boussinesq eqution describing geofiltration in soils with fractal structures are presented. In this study are presented too mathematical models geomigration of contaminations with groundwater in classical way and in soils with fractal structures

Dual-band metamaterial absorber with stable absorption performance based on fractal structure

2021 ◽  
Author(s):  
Shuguang Fang ◽  
Lianwen Deng ◽  
Pin Zhang ◽  
Lei-Lei Qiu ◽  
Haipeng Xie ◽  
...  
Keyword(s):  
Fractal Structure ◽  
Low Cost ◽  
Impedance Matching ◽  
Metamaterial Absorber ◽  
Dual Band ◽  
Fractal Structures ◽  
Absorption Bands ◽  
Electromagnetic Resonance ◽  
Absorption Performance ◽  
Multi Band

Abstract In this paper, two kinds of dual-band metamaterial absorbers (MMAs) with stable absorption performance based on fractal structures are proposed. As the key feature, with the increase in fractal order, the fractal MMAs can reduce the weight while keeping the absorption performance. The multi-band absorption property is analyzed by multiple L-C resonances generated by the fractal structure. By virtue of good impedance matching characteristics and the synergy of the circuit and electromagnetic resonance, effective and stable microwave absorption is readily achieved. Finally, two prototypes are fabricated for demonstration, and the measurement result is consistent well with the simulation one. As expected, the proposed fractal MMAs have the advantage of low-cost, light-weight, and dual-effective absorption bands, and have great potential in the application of multi-band radar stealth.

scholarly journals The bivariate luminosity and mass functions of the local HRS galaxy sample

2018 ◽  
Vol 617 ◽  
pp. A33 ◽  
Author(s):  
P. Andreani ◽  
A. Boselli ◽  
L. Ciesla ◽  
R. Vio ◽  
L. Cortese ◽  
...  
Keyword(s):  
Statistical Analysis ◽  
Star Formation ◽  
Star Formation Rate ◽  
Formation Rate ◽  
Far Infrared ◽  
Mass Function ◽  
Stellar Mass ◽  
The Relationship ◽  
Mass Functions ◽  
Physical Quantities

Aims.We discuss the results of the relationships between theK-band and stellar mass, FIR luminosities, star formation rate, and the masses of the dust and gas of nearby galaxies computing the bivariateK-band-luminosity function (BLF) and bivariateK-band-mass function (BMF) of theHerschelReference Survey (HRS), a volume-limited sample with full wavelength coverage.Methods.We derive the BLFs and BMFs from theK-band and stellar mass, FIR luminosities, star formation rate, dust and gas masses cumulative distributions using a copula method, which is outlined in detail. The use of the computed bivariate taking into account the upper limits allows us to derive a more solid statistical ground for the relationship between the observed physical quantities.Results.The analysis shows that the behaviour of the morphological (optically selected) subsamples is quite different. A statistically meaningful result can be obtained over the whole HRS sample only from the relationship between theK-band and the stellar mass, while for the remaining physical quantities (dust and gas masses, far-infrared luminosity, and star formation rate), the analysis is distinct for late-type (LT) and early-type galaxies (ETG). However, the number of ETGs is small to perform a robust statistical analysis, and in most of the case results are discussed only for the LTG subsample. The luminosity and mass functions (LFs, MFs) of LTGs are generally dependent on theK-band and the various dependencies are discussed in detail. We are able to derive the corresponding LFs and MFs and compare them with those computed with other samples. Our statistical analysis allows us to characterise the HRS which, although non-homogeneously selected and partially biased towards low IR luminosities, may be considered as representative of the local LT galaxy population.

SQUARE FRACTAL ALGORITHM

Fractals ◽  
2005 ◽  
Vol 13 (01) ◽  
pp. 43-55 ◽  
Author(s):  
EYTAN H. SUCHARD
Keyword(s):  
Image Processing ◽  
Planar Graph ◽  
Fractal Structure ◽  
Cost Effective ◽  
Industrial Applications ◽  
Fractal Structures ◽  
Equivalent Problem ◽  
Fractal Properties ◽  
The Way

Spanning a planar graph the way D. Hilbert's curve does has various image processing and industrial applications. Spanning a planar graph by two disjoint curves with fractal properties has even more scientific and industrial uses. For example, given two liquids and an active osmosis through membrane between them, we would like to both cool the liquids and to find a cost-effective structure for the osmosis to occur. Another equivalent problem is to expose two liquids to light that passes through a transparent slab as the osmosis between them occurs. Two disjoint curves can be the answer for the required structure. Differences of lengths between the curves can also be useful. A fractal structure is obvious in the lungs, where osmosis of oxygen is vital. Fractal structures are often found in organic osmotic processes in Nature. In this article, a method for spanning a planar graph by two disjoint curves will be presented.

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