Fractal aggregates in simulated reduced gravity

We have developed a study of fractal aggregates in the laboratory geared towards their final production in a microgravity environment, on board of space shuttles and stations. Floating particles in gases and fluids have allowed simulation of conditions in reduced gravity. Production methods and dimensional analysis are described.     

Fractal cosmogony: similarity of the early universe to microscopic fractal aggregates

Images of fractal aggregates condensed at the micrometer scale exhibit structures strikingly similar to the early universe, the latter as pictured by a model due to Barkana and Loeb, interpreting the data of the Sloan Digital Sky Survey and an earlier one by Hernquist, Weinberg and Katz. The condensation of matter at very small scale is thus similar to that of the early universe as far as it can be reasonably established.     

Non-linear elastic behavior of filler aggregates

A filled rubber network consists of polymer chains which are suspended between filler aggregates. In this contribution, the nonlinear elastic behavior of the aggregated filler particles inside the rubber matrix is investigated. Previously, by using scaling theory, the influence of initial length and fractal dimension of aggregates on the elastic response of aggregated structures was studied. Here we additionally take into account a deformation induced evolution of the aggregate structure. To this end, the directional topology of the aggregate structure is represented by its backbone chain. Thus, the analytical approach proposed describes not only the geometrically but also the physically non-linear behavior of aggregates. Our solution can further be generalized for colloidal structures as for example granular materials or suspended solid structures. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modelling the impact of two different flocculants on the performance of a thickener feedwell

The performance of a thickener feedwell depends not only on its ability to generate large-sized aggregates from feed particles but also on aggregate density. The performance of the flocculant BASF Rheomax® DR 1050 has been previously compared to a conventional anionic flocculant in turbulent pipe flocculation of mineral suspension, suggesting that the flocculant can generate denser aggregates (i.e. larger effective fractal dimension). Such aggregates are generally stronger and reduce the need for solids dilution, with both factors favouring faster settling velocity at the feedwell exit. To investigate the impact of the internal aggregate structure on the flocculation performance of a feedwell, Computational Fluid Dynamics (CFD) simulations of a basic open feedwell with shelf design were carried out for both flocculants. A calcite with a fine particle size (Omyacarb 5) was modelled to emphasise the impact of the flocculation process on flow fields at the feedwell exit. Simulations were conducted using CFX-4.4 two-phase flow formulation incorporating equations for a population balance model of the flocculation process. The impact of the fractal dimension on the effectiveness of the aggregation process is presented for low and high solids concentrations. Comparison of the performance of the flocculants is presented in terms of both predicted mean aggregate size and settling flux.     

具有无界变差的连续函数研究进展

讨论了具有无界变差的连续函数的结构.首先按照局部结构和分形维数对连续函数进行了分类,给出了相应的例子.对这些具有无界变差的函数的性质进行了初步的讨论.对于新定义的奇异连续函数,给出了一个等价判别定理.基于奇异连续函数,又给出了局部分形函数和分形函数的定义.同时,分形函数又由奇异分形函数、非正则分形函数和正则分形函数组成.相应于不连续函数的情形也进行了简单的讨论.     

模糊分形及其在级配骨料研究中的应用

结合模糊集理论和分形理论提出了模糊分形集的概念,将分形论的应用范围从清晰现象推广到了模糊现象领域。模糊分形系统大量存在于自然界中,例如海岸线、混凝土的断裂表面、级配骨料的粒度分布等。级配骨料的分形特征具有模糊性,模糊分维是其粒度分布复杂程度的度量。文中给出模糊分形集在级配骨料研究中的应用实例。     

Construction of fractal surfaces by recurrent fractal interpolation curves

A method to construct fractal surfaces by recurrent fractal curves is provided. First we construct fractal interpolation curves using a recurrent iterated functions system (RIFS) with function scaling factors and estimate their box-counting dimension. Then we present a method of construction of wider class of fractal surfaces by fractal curves and Lipschitz functions and calculate the box-counting dimension of the constructed surfaces. Finally, we combine both methods to have more flexible constructions of fractal surfaces.     

Fractality in subatomic systems

Subatomic systems have unit components in permanent motion. Images in configuration space calculated for such systems are shown to have fractal characteristics. The fractal dimension is an invariant of the motion. The plausibility of extending fractal concepts to subnucleon structures is discussed.     

Temporal variation of velocity components in a turbulent open channel flow: Identification of fractal dimensions

Fractals are objects which have similar appearances when viewed at different scales. Such objects have details at arbitrarily small scales, making them too complex to be represented by Euclidian space; hence, they are assigned a non-integer dimension. Some natural phenomena have been modeled as fractals with success; examples include geologic deposits, topographic surfaces and seismic activities. In particular, time series have been represented as a curve with fractal dimensions between one and two. There are different ways to define fractal dimension, most being equivalent in the continuous domain. However, when applied in practice to discrete data sets, different ways lead to different results. In this study, three methods for estimating fractal dimension are described and two standard algorithms, Hurst’s rescaled range analysis and box-counting method (BC), are compared with the recently introduced variation method (VM). It was confirmed that the last method offers a superior efficiency and accuracy, and hence may be recommended for fractal dimension calculations for time series data. All methods were applied to the measured temporal variation of velocity components in turbulent flows in an open channel in Shiraz University laboratory. The analyses were applied to 2500 measurements at different Reynold’s numbers and it was concluded that a certain degree of randomness may be associated with the velocity in all directions which is a unique character of the flow independent of the Reynold’s number. Results also suggest that the rigid lateral confinement of flow to the fixed channel width allows for designation of a more-or-less constant fractal dimension for the spanwise velocity component. On the contrary, in vertical and streamwise directions more freedom of movements for fluid particles sets more room for variation in fractal dimension at different Reynold’s numbers.     

Fractal analysis of time series in oil and gas production

The peculiarities of fractal characteristics’ calculations for time series are described in this article. An algorithm for calculation of fractal dimension is suggested. It has been proved that the suggested method possesses high accuracy and the rapidity of convergence on the limited number of measurements compared to the methods of covering.The criteria of early diagnosis for changes in the condition of hydrodynamic processes, which do not vary by fractal dimension, have been recommended.The presented method is applicable for practical engineering calculations with self-affine, chaotic data, usually with relatively limited number of measurements. It is quite a simple method for calculation of fractal dimension, algorithm can be easily realized and it should be useful for engineers.The applicability of the proposed algorithm for fractal dimension calculation and early diagnosis criteria of qualitative changes in the behaviour of various dynamic systems has been tested both on simulated as well as practical examples of oil and gas production.     

Fractal structure and fractal dimension determination at nanometer scale

Three-dimensional fractures of different fractal dimensions have been constructed with successive random addition algorithm, the applicability of various dimension determination methods at nanometer scale has been studied. As to the metallic fractures, owing to the limited number of slit islands in a slit plane or limited datum number at nanometer scale, it is difficult to use the area-perimeter method or power spectrum method to determine the fractal dimension. Simulation indicates that box-counting method can be used to determine the fractal dimension at nanometer scale. The dimensions of fractures of valve steel 5Cr21Mn9Ni4N have been determined with STM. Results confirmed that fractal dimension varies with direction at nanometer scale. Our study revealed that, as to theoretical profiles, the dependence of frsctal dimension with direction is simply owing to the limited data set number, i.e. the effect of boundaries. However, the dependence of fractal dimension with direction at nanometer scale in real metallic fractures is correlated to the intrinsic characteristics of the materials in addition to the effect of boundaries. The relationship of fractal dimensions with the mechanical properties of materials at macrometer scale also exists at nanometer scale. Project supported by the National Natural Science Foundation of China (Grant Nos. 59771050 and 59872004) and the Foundation Fund of Ministry of Metallurgical Industry.     

Silica fractal atomic clusters saturated with OH

We constructed regular fractal SiOH atomic clusters which pending bonds are saturated with OH molecules. We calculated the binding energies of these clusters as well as for sp² hybridization as for sp³ hybridizations. The result are the following: for the two hybridizations, the total binding energies have a linear dependence on the size of the fractal cluster, which comes directly from the scaling law of the fractal characteristic of the building of the cluster. We related by a scaling law, the number of electronic bonds and the total bonding energy.     

Can mobile shapes of fractals cause rate enhancement?

The notion of mobile shapes of a fractal object has been introduced to explain rate enhancements beyond the limits set by three-dimensional diffusion control. Allowing for mobilities of fractal shapes in the practical range, theoretical simulations for a model system show several hundred-fold variation in rates. The results of the analysis can have wide implications in understanding several biochemical as well as physical and chemical systems.     

RESEARCH ANNOUNCEMENTS：On the Fractional Calculus of a Type of Weierstrass Functions

I Introduction In recent years, fractals have shown important applications in many fields. [1, 2] and [3] havedone some excellent initial and conclusion work on fractal and it's mathematical foundations.However, a fractal function: a type of Weierstrass functions defined bybecause of it's special fractal properties, [1,2, 4, 5] have given some detailed discussion about it'sgraph, fractal dimension, etc.     

On the arithmetic of fractal dimension using hyperhelices

A hyperhelix is a fractal curve generated by coiling a helix around a rect line, then another helix around the first one, a third around the second… an infinite number of times. A way to generate hyperhelices with any desired fractal dimension is presented, leading to the result that they have embedded an algebraic structure that allows making arithmetic with fractal dimensions and to the idea of an infinitesimal of fractal dimension.     

Kinetic and Structural Study of Interfacial Aggregation of Microparticles at Liquid-Air Interfaces

Aggregation of glass spheres (75±5 μm diameter) having different — medium and high — hydrophobicities was investigated at liquid-air interfaces (water-air, aqueous surfactant solution-air and aqueous glycerol solution-air) from morphological and kinetic point of view. The physical parameters of aggregation were calculated by computerized image analysis. The cluster structures were characterized by the fractal dimension calculated from the scaling law between radius of gyration and surface coverage of aggregates (growth function). The primary growth in every investigated case was accompanied by restructuring which makes the clusters denser and more compact. The kinetic order of aggregation was analyzed in terms of determining the number of clusters as a function of aggregation time. In the early stage of the aggregation the kinetics was second order in every investigated case but beyond a given time (and cluster size) the kinetics changed in the most cases.     

Analytical solutions for vibrating fractal composite rods and beams

Fractals have the potential to describe complex microstructures but presently no solution methodologies exist for the prediction of deformation on transiently deforming fractal structures. This is achieved in this paper with the development of analytical solutions on vibrating composite rods and beams. The fractals considered are necessarily deterministic and relatively simple in form to demonstrate the solution methodology. The solutions are limited to beams and rods constructed from an idealised-composite material consisting of relatively large rigid particles embedded in an infinitely thin pliable matrix. Although, as a result, the fractal composite system is not representative of a realistic physical system the methodologies presented do serve to highlight the practical difficulties in using fractals in structural dynamics. Static loading is restricted to spatially invariant axial forces and bending moments as solutions on a unified state of continuum stress are sought which then serve as initial conditions for the vibratory problem. It is demonstrated that measurable displacement is possible on a fractal structure and that finite measures of total, kinetic and strain energy are simultaneously achievable. The approach involves the use of modal analysis to determine modes at natural frequencies that satisfy boundary conditions. These are combined to provide a free vibration solution on a fractal that satisfies the initial conditions in the form of a fractal displacement field.     

Truncated fractal basin boundaries in the pendulum with nonperiodic forcing

Summary It is well known that oscillators such as the pendulum can have fractal basin boundaries when they are periodically forced with the consequence that the long term behavior of the system may be unpredictable. In engineering and physical applications, the forcing is often nonperiodic and eventually decays to zero, and simulation of the pendulum with decaying forcing (M. Varghese, J. S. Thorp, Physical Review Letters, vol. 60, no. 8, pp. 665–668, Feb. 1988) exhibits truncated fractal basin boundaries which also limit the system predictability. We develop a coordinate change for the pendulum with decaying forcing that allows us to apply standard qualitative methods to study the basin boundaries. We prove that the basin boundaries cannot be fractal and show by example how the extreme stretching and folding leading to a truncated fractal basin boundary may arise.     

Design and analysis of fractal detector for high resolution radars

Recently, fractal geometry has been used as a tool for improving the detection of targets in radar systems. The fractal dimension is utilized as a feature to distinguish between target and clutter in fractal detectors. In this paper, a general model is proposed for the target and clutter signals in high resolution radar (HRR). The fractal dimensions of the clutter and the target plus clutter are evaluated. Performing statistical tests on the distribution of the fractal dimension, it is proved that a gaussian distribution can approximately model the distribution of the fractal dimension for HRR signals. The fractal detector is designed based on the gaussian distribution of the fractal dimension and its performance is compared with a semi-optimum detector. It is demonstrated that the fractal detector has great capabilities in the rejection of colored clutter. Moreover, we show that the fractal detector is CFAR, i.e., the probability of false alarm remains approximately constant in different interference powers.     

Fractal functions with continuous Weil-type derivatives of variable order in control of distributed systems

Properties of fractal functions which are not differentiable in the classical sense but have continuous Weil-type derivatives of variable order at each point are studied. It is shown that the Weierstrass, Takagi, and Besicovitch classical fractal functions have such derivatives. An example of an oscillatory system controlling which requires constructing a fractal control function having a Weil-type derivative of variable order at each point is considered.