Fractals

We continue the program of Bedient et al.¹ by investigating some of the ways of embedding IFS with 1-step memory into IFS with 2-step memory, and 1- and 2-step memory into IFS with 3-step memory. This reveals a hierarchy of attractors of m-step memory IFS as subsets of attractors of n-step memory IFS.

A fractal model for nucleate pool boiling of nanofluids at high heat flux and critical heat flux (CHF) is developed based on the fractal distribution of nanoparticles and nucleation sites on boiling surfaces in this paper. The formula of calculating high heat flux and CHF for nanofluids in nucleate pool boiling is given by taking into account heat convection between nanoparticles and liquids due to the Brownian motion of nanoparticles in fluids. The proposed model is expressed as a function of temperature of nanofluids, the effective thermal conductivity of nanofluids, the average size of nanoparticles, the fractal dimension of nanoparticles and nucleation sites, the nanoparticles volume fraction of suspension, and physical properties of fluids. No additional/new empirical constant is introduced in this fractal model. An agreement between the proposed model predictions and experimental data is found. The validity of the fractal model for nucleate pool boiling of nanofluids at high heat flux and CHF is thus verified.

The macroscopic transport properties of porous media have received steadily attention in science and engineering areas in the past decades. It has been shown that the maximum pore size in a porous medium plays the crucial role in determination of transport properties such as flow resistance, permeability, thermal conductivity and electrical conductivity, etc. In this study, two models for predicting the maximum pore size in porous media based on fractal geometry are presented. The present analytical expressions may be used to calculate the maximum pore size from porosity and permeability data, as well as from liquid properties, structure parameters of media and imbibition coefficient data, respectively. Predicted maximum pore sizes by the proposed models show good agreement with the available experimental results.

The EEG patterns for normal and epileptic rats were studied in the framework of multifractal detrended fluctuation analysis (MFDFA). The patterns were found to be multifractal for normal rats. For epileptic rats the pattern was found to be monofractal in some cases and multifractal in others. The degree of multifractality was found to be higher in normal sample compared to the epileptic sample.

We describe the relationships among several notions of isoperimetry on a class of pre-fractal Sierpinski Carpets that satisfy a set of geometric assumptions as well as on the graphs corresponding to these carpets. We compute the L¹-isoperimetric profile at scale h, and we use this computation to determine both the classical isoperimetric profile and a profile computed using Jordan curve boundaries.

We studied the invariant fractals in convex N-gons given by N identical pure contractions at its vertices with non-overlapping images, which we called uniform Sierpinski fractals. We provided the explicit formulae for the maximal contraction ratio R_N, and the maximal Hausdorff dimension h_N, for the uniform Sierpinski fractals. We used maximal N-grams and principal crossing points as the main tools.

Historically, fractal analysis has been remarkably successful in describing wide ranging kinetic processes on (idealized) scale invariant objects in terms of elegantly simple universal scaling laws. However, as nanostructured materials find increasing applications in energy storage, energy conversion, healthcare, etc., one must reexamine the premise of traditional fractal scaling laws as it only applies to physically unrealistic infinite systems, while all natural/engineered systems are necessarily finite. In this article, we address the consequences of the 'finite-size' problem in the context of time dependent diffusion towards fractal surfaces via the novel technique of Cantor-transforms to (i) illustrate how finiteness modifies its classical scaling exponents; (ii) establish that for finite systems, the diffusion-limited reaction is decelerated below a critical dimension and accelerated above it; and (iii) to identify the crossover size-limits beyond which a finite system can be considered (practically) infinite and redefine the very notion of 'finiteness' of fractals in terms of its kinetic response. Our results have broad implications regarding dynamics of systems defined by the same fractal dimension, but differentiated by degree of scaling iteration or morphogenesis, e.g. variation in lung capacity between a child and adult.

The macroscopic branching and shape of a metal, electrodeposited in quasi-two dimensions, can be represented by a Lindenmayer System of multiple iterated function systems. The motif of a central branch that continues and two side branches is shown to allow modeling of the observed shapes. The dilation (or contraction) in length, and the angle of a branch with respect to the central stem are the only parameters needed to create realistic simulations of such ramified deposits. Values for these quantities together with standard deviations measured from a number of electrodeposited copper fractals have been used to generate simulations. Measurement of the parameters of the motif, the average linear dilation factor from one generation to the next, and the angles between branches, show a 5% relative standard deviation of these factors across one growth. The values of these parameters can also indicate the transition from an open fractal form to the more directed dendritic form. The results are compared with other approaches to describing these systems.

Suppose that the set of real n × n matrices has joint spectral radius less than 1. Then for any digit set D = {d₁, …, d_q} ⊂ ℝⁿ, there exists a unique non-empty compact set satisfying , which is typically a fractal set. We use the infinite digit expansions of the points of F to give simple necessary and sufficient conditions for the convex hull of F to be a polytope. Additionally, we present a technique to determine the vertices of such polytopes. These answer some of the related questions of Strichartz and Wang, and also enable us to approximate the Lebesgue measure of such self-affine sets. To show the use of our results, we also give several examples including the Levy dragon and the Heighway dragon.

The prevalence of Patent Foramen Ovale (PFO) is an increasing risk factor for paradoxical cerebral embolism. Decompression Sickness (DCS) results from gas coming out of solution in the bodily fluids and tissues when a diver ascends too quickly. This occurs because decreasing pressure lowers the solubility of gas in liquid. Also, the expansion of gas in the lungs may lead to alveolar rupture which may, in turn, result in Arterial Gas Embolism (AGE). The amount of nitrogen that diffuses into the bodily tissues during a dive depends on the depth of the dive and the duration of exposure. PFO acquires significance in various congenital heart diseases leading to right to left shunt and thus to paradoxical embolism. The presence of asymptomatic brain lesion like "spots" has been investigated in a randomized population of diving individuals. The partial pressure of the gas can be calculated by Henry's Law and the pressure of the blood stream can be calculated by Poiseuille's Law. The statistical comparison of the fractal dimension was obtained by means of Box Counting Method with slope analysis. We observe that PFO plays a significant role in the occurrence of unexplained cerebral DCS and white spots in the brain.