粉末注射成形坯孔隙度的分形模型

粉末注射成形坯是一种具有分形特性的典型的多孔介质,借助于多孔介质孔隙结构的分形理论,对粉末注射成形坯孔隙率的分形模型进行推导。首先分析了粉末注射成形坯孔隙结构的双重分形特性,介绍了粉末注射成形坯孔隙分布分形维数和孔隙迂曲分形维数,然后推导出粉末注射成形坯孔隙度的分形模型。     

Determining the limits of geometrical tortuosity from seepage flow calculations in porous media.

Recent investigations have found a distinct correlation of effective properties of porous media to sigmoidal functions, where one axis is the Reynolds number Re and the other is the effective property dependent of Re, Θ_i = S_i(Re). One of these properties is tortuosity. At very low Re (seepage flow), there is a characteristic value of tortuosity, and it is the upper horizontal asymptote of the sigmoidal function. With higher values of Re (transient flow) the tortuosity value decreases, until a lower asymptote is reached (turbulent flow). Estimations of this parameter have been limited to the low Reynolds regime in the study of porous media. The current state of the art presents different numerical measurements of tortuosity, such as skeletization, centroid binding, and arc length of streamlines. These are solutions for the low Re regime. So far, for high Re, only the arc length of stream lines has been used to calculate tortuosity. The present approach involves the simulation of fluid flow in large domains and high Re, which requires numerous resources, and often presents convergence problems. In response to this, we propose a geometrical method to estimate the limit of tortuosity of porous media at Re → ∞, from the streamlines calculated at low Re limit. We test our method by calculating the tortuosity limits in a fibrous porous media, and comparing the estimated values with numerical benchmark results. Ongoing work includes the geometric estimation of different intrinsic properties of porous media. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

One-dimensional drug release from finite Menger sponges: In silico simulation

The purpose of this work was to evaluate the consequences of the spatial distribution of components in pharmaceutical matrices type Menger sponge on the drug release kinetic from this kind of platforms by means of Monte Carlo computer simulation. First, six kinds of Menger sponges (porous fractal structures) with the same fractal dimension, d_f=2.727, but with different random walk dimension, d_w[2.149,3.183], were constructed as models of drug release device. Later, Monte Carlo simulation was used to describe drug release from these structures as a diffusion-controlled process. The obtained results show that drug release from Menger sponges is characterized by an anomalous behavior: there are important effects of the microstructure anisotropy, and porous structures with the same fractal dimension but with different topology produce different release profiles. Moreover, the drug release kinetic from heteromorphic structures depends on the axis used to transport the material to the external medium. Finally, it was shown that the number of releasing sites on the matrix surface has a significant impact on drug release behavior and it can be described quantitatively by the Weibull function.     

Experimental evaluation of phase velocities and tortuosity in fluid saturated highly porous media

In the current contribution, we present a novel method for the determination of the high frequency tortuosity parameter, α_∞ in high porous media. Therefore, time-domain measurements of ultrasonic signals are performed with a transmission technique. Aluminium foams with different pore fluids will be under the scope of experimental investigation. Finally, the experimental results are compared with analytical wave propagation tests. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

产业集群的分形结构—基于演化的观点

以宏观的视角来研究企业的地理分布,并在地理空间与社会网络结构的基础上建立一个产业集群模型;该模型显示出产业集群的分形结构进而揭示出产业集群是一种自组织系统,即在生产交易过程中自发形成有序的结构或状态的现象.根据该模型,运输成本或者禀赋只是形成产业集群的所有充分条件中的一个;影响产业集群的最重要指标是关系网络空间的分形维度,它显示了经济系统的层次结构性.网络密度很大的集群的关系网络可能是接近紊乱的,即分形维度接近于零;而紊乱将会导致这个集群效益下降,甚至促使集群崩溃.     

Describing some characters of serine proteinase using fractal analysis

In this paper we calculated the fractal dimensions of four proteins, chymotrypsin, elastase, trypsin and subtilisin, which are made up of about 220–275 amino acids and belong to the family of serine proteinase by using three definitions of fractal dimension i.e. the chain fractal dimension (D_L), the mass fractal dimension (D_m) and the correlation fractal dimension (D_c). We also analyzed the relationship between fractal dimension and space structure or secondary structure contents of proteins. The results showed that the values of fractal dimensions are almost same for the global mammalian enzymes (chymotrypsin, elastase and trypsin), but different for the global subtilisin. This demonstrated that the more similar structures, the more equal fractal dimensions, and if the fractal dimensions of proteins are different from each other, the three dimensional structures should not be similar. On the other hand, the detailed structures and fractal dimensions of the active sites of four enzymes are extraordinarily similar. Therefore, the fractal method can be applied to the elucidation of the proteins evolution.     

Micromechanics model for tortuosity and homogenized diffusion properties of porous materials with distributed micro-cracks

Continuum micromechanics [2,3] is used to model tortuosity within the context of ion transport to obtain the homogenized diffusivity of intact and micro-cracked porous materials. A novel cascade homogenization technique to model ion diffusion in porous materials is proposed. A REV^uc that represents the porous material is modelled as a spherical pore-space inclusion in a recursively updated matrix with an initial diffusivity without any tortuosity. A REV^c representing the micro-cracked material is modelled with prolate ellipsoidal inclusions representing micro-cracks embedded in a porous matrix [4]. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On turbulence in fractal porous media

We examine three fundamental equations governing turbulence of an incompressible Newtonian fluid in a fractal porous medium: continuity, linear momentum balance and energy balance. We find that the Reynolds stress is modified when a local, rather than an integral, balance law is considered. The heat flux is modified from its classical form when either the integral or local form of the energy density balance law is studied, but the energy density is always unchanged. The modifications of Reynolds stress and heat flux are expressed directly in terms of the resolution length scale, the fractal dimension of mass distribution and the fractal dimension of a fractal’s surface. When both fractal dimensions become integer (respectively 3 and 2), classical equations are recovered.      

Surface investigation by electrochemical methods and application of chaos theory and fractal geometry

The present study has considered the application of the noise analysis and fractal geometry as a promising dynamic method for exploiting the corrosion mechanism of the stainless steel 304 that is immersed in different concentrations of FeCl₃. The fractal dimension calculated from the electrochemical noise technique has a good correlation with the surface fractal dimension obtained by electrochemical impedance spectroscopy and scanning electron microscopy results. The complexity of system increases by divergence of Electrochemical Potential noise fractal dimension from 1.5 value and also the roughness of surface increases by an increase in surface fractal dimension. As the concentration of FeCl₃ increases (0.001 M, 0.01 M and 0.1 M) the value of Electrochemical Potential noise fractal dimension diverges from 1.5 value (1.57, 1.33 and 1.01 respectively) and the value of surface fractal dimension increases (2.107, 2.425 and 2.756 for impedance results and 2.073, 2.425 and 2.672 for scanning electron microscopy images). These results show that the complexity of system and roughness of the surface increases by an increase in concentration of FeCl₃. The present study has shown that chaos and noise analysis are effective methods for the study of the mechanism of the corrosion process.     

Homogenizing the acoustic properties of a porous matrix containing an incompressible inviscid fluid

We undertake a rigorous derivation of the Biot's law for a porous elastic solid containing an inviscid fluid. We consider small displacements of a linear elastic solid being itself a connected periodic skeleton containing a pore structure of the characteristic size ε. It is completely saturated by an incompressible inviscid fluid. The model is described by the equations of the linear elasticity coupled with the linearized incompressible Euler system. We study the homogenization limit when the pore size εtends to zero. The main difficulty is obtaining an a priori estimate for the gradient of the fluid velocity in the pore structure. Under the assumption that the solid part is connected and using results on the first order elliptic systems, we obtain the required estimate. It allows us to apply appropriate results from the 2‐scale convergence. Then it is proved that the microscopic displacements and the fluid pressure converge in 2‐scales towards a linear hyperbolic system for an effective displacement and an effective pressure field. Using correctors, we also give a strong convergence result. The obtained system is then compared with the Biot's law. It is found that there is a constitutive relation linking the effective pressure with the divergences of the effective fluid and solid displacements. Then we prove that the homogenized model coincides with the Biot's equations but with the added mass ρ_a being a matrix, which is calculated through an auxiliary problem in the periodic cell for the tortuosity. Furthermore, we get formulas for the matricial coefficients in the Biot's effective stress–strain relations. Finally, we consider the degenerate case when the fluid part is not connected and obtain Biot's model with the relative fluid displacement equal to zero. Copyright © 2003 John Wiley & Sons, Ltd.     

Fractal dimension of zeolite surfaces by calculation

A theoretical method for the estimation of the fractal dimensions of the pore surfaces of zeolites is proposed. The method is an analogy to the commonly employed box-counting method and uses imaginary meshes of various sizes (s) to trace the pore surfaces determined by the frameworks of crystalline zeolites. The surfaces formed by the geometrical shapes of the secondary building units of zeolites are taken into account for the calculations performed. The characteristics of the framework structures of the zeolites 13X, 5A and silicalite are determined by the help of the solid models of these zeolites and the total numbers of grid boxes intersecting the surfaces are estimated by using equations proposed in this study. As a result, the fractal dimension values of the zeolites 13X, 5A and silicalite are generally observed to vary in significant amounts with the range of mesh size used, especially for the relatively larger mesh sizes that are close to the sizes of real adsorbates. For these relatively larger mesh sizes, the fractal dimension of silicalite falls below 1.60 while the fractal dimension values of zeolite 13X and 5A tend to rise above 2. The fractal dimension values obtained by the proposed method seem to be consistent with those determined by using experimental adsorption data in their relative magnitudes while the absolute magnitudes may differ due to the different size ranges employed. The results of this study show that fractal dimension values much different from 2 (both higher and lower than 2) may be obtained for crystalline adsorbents, such as zeolites, in ranges of size that are close to those of real adsorbates.     

多孔介质热弥散系数的分形模型北大核心CSCD

热弥散系数是与流体的物性和多孔介质结构有关的,表征多孔介质传热传质强弱的重要参数.该文建立了分形多孔介质的孔喉结构模型,研究了在孔喉结构处流体由湍流状态变为层流状态的局部水头损失和速度弥散效应,在考虑微观孔喉结构和速度弥散效应的影响下,推导了热弥散系数关系式.研究表明,热弥散系数与孔喉比、孔喉结构个数和迂曲分形维数成正比,与孔隙率和面积分形维数成反比.进一步研究发现,孔喉比在1~150范围内对速度弥散效应有显著影响,流体在孔喉结构处存在局部水头损失,导致速度弥散效应增强,热弥散系数增大.     

Description of the Structure of the Polymer Matrix of Particulate-Filled Polymer Composites as a Thermal Cluster: the Critical Indices

The adequacy of the model of a thermal cluster for describing the structure of the polymer matrix of particulate-filled composites is shown. The equality of the critical indices _T = of thermal and percolation clusters is realized in the composites at a nonzero molecular mobility, which is characterized by the fractal dimension D of the chain fragment between entanglements. The variation interval of D for the polymer matrix of a composite is smaller than for the initial polymer because of the influence of the filler on its structure.     

Finite dimensionality of global attractors for a non-classical reaction-diffusion equation with memory

In this paper, we consider a periodic boundary value problem for a non-classical reaction-diffusion equation with memory. In other paper, we use the ω-limit compactness of the solution semigroup {S(t)}_t≥0 to get the existence of a global attractor. The main goal here is to give an estimate of the fractal dimension of the global attractor. By the fractal dimension theorem given by A.O. Celebi et al., we obtain that the fractal dimension of the global attractor for the problem is finite; this makes the results for the non-classical reaction-diffusion equations more substantial and perfect.     

Brownian motion on a homogeneous random fractal

Summary We introduce a simple random fractal based on the Sierpinski gasket and construct a Brownian motion upon the fractal. The properties of the process on the Sierpinski gasket are modified by the random environment. A sample path construction of the process via time truncation is used, which is a direct construction of the process on the fractal from the associated Dirichlet forms. We obtain estimates on the resolvent and transition density for the process and hence a value for the spectral dimension which satisfiesd _s=2d _f/d_w. A branching process in a random environment can be used to deduce some of the sample path properties of the process.     

Backbone fractal dimension and fractal hybrid orbital of protein structure

Fractal geometry analysis provides a useful and desirable tool to characterize the configuration and structure of proteins. In this paper we examined the fractal properties of 750 folded proteins from four different structural classes, namely (1) the α-class (dominated by α-helices), (2) the β-class (dominated by β-pleated sheets), (3) the (α/β)-class (α-helices and β-sheets alternately mixed) and (4) the (α + β)-class (α-helices and β-sheets largely segregated) by using two fractal dimension methods, i.e. “the local fractal dimension” and “the backbone fractal dimension” (a new and useful quantitative parameter). The results showed that the protein molecules exhibit a fractal behavior in the range of 1 ? N ? 15 (N is the number of the interval between two adjacent amino acid residues), and the value of backbone fractal dimension is distinctly greater than that of local fractal dimension for the same protein. The average value of two fractal dimensions decreased in order of α > α/β > α + β > β. Moreover, the mathematical formula for the hybrid orbital model of protein based on the concept of backbone fractal dimension is in good coincidence with that of the similarity dimension. So it is a very accurate and simple method to analyze the hybrid orbital model of protein by using the backbone fractal dimension.     

A set of formulae on fractal dimension relations and its application to urban form

The area-perimeter scaling can be employed to evaluate the fractal dimension of urban boundaries. However, the formula in common use seems to be not correct. By means of mathematical method, a new formula of calculating the boundary dimension of cities is derived from the idea of box-counting measurement and the principle of dimensional consistency in this paper. Thus, several practical results are obtained as follows. First, I derive the hyperbolic relation between the boundary dimension and form dimension of cities. Using the relation, we can estimate the form dimension through the boundary dimension and vice versa. Second, I derive the proper scales of fractal dimension: the form dimension comes between 1.5 and 2, and the boundary dimension comes between 1 and 1.5. Third, I derive three form dimension values with special geometric meanings. The first is 4/3, the second is 3/2, and the third is 1 + 2^1/2/2 ≈ 1.7071. The fractal dimension relation formulae are applied to China’s cities and the cities of the United Kingdom, and the computations are consistent with the theoretical expectation. The formulae are useful in the fractal dimension estimation of urban form, and the findings about the fractal parameters are revealing for future city planning and the spatial optimization of cities.     

On the Coefficients of Pore Tortuosity in the Effective Biot Model

In order to justify the effective Biot model of a porous medium, a precise expression for the density of kinetic energy is established and studied. In investigating this expression, the displacements and their velocities are averaged and the dispersion of normalized velocities in the fluid phase is introduced. The coefficients of pore tortuosity occurring in the Biot equations are expressed in terms of the dispersion, rendering the Biot equations of a porous continuous medium justified. Bibliography: 6 titles.     

Fractal dimension and complexity in the long-term dynamics of a monomolecular layer

The formation of three-dimensional domains in monomolecular layers (Nucleation Dynamics, ND) of four fatty acids, stearic, arachidic, behenic and lignoceric acids, containing the same carboxylic (–COOH) head and an alkyl chain with 18, 20, 22 and 24 carbon atoms, respectively, on water surface, has been studied through Specific Molecular Area (A) versus time (t) studies from Surface Pressure (π)-A isotherms and Brewster Angle Microscopy (BAM). To investigate the fractal nature, the gray-scale Brewster angle micrographs are converted to binary images containing only two pixel values – 0 for 2D phase and 255 for 3D phase before box-counting method is employed to compute the fractal dimension. The 3D phase in the background of 2D phase is found to be fractal in nature. In fact, 3D phase is an interpenetration of two fractal structures with two different fractal dimensions – one corresponding to smaller (intra-domain) structures and other corresponding to larger (inter-domain) structures. These fractal dimensions are seen to evolve with coverage as 3D phase grows. The two fractal dimensions and their evolution in ND dynamics are identical for the longest-tailed lignoceric acid which has a porous 3D phase.     

Transfer functions for a one-dimensional fluid-poroelastic system subject to an ultrasonic pulse

A one-dimensional model of an in vitro experiment, in which a specimen of cancellous bone is immersed in water and insonified by an ultrasonic pulse, is considered. The modification of the poroelastic model of Biot due to Johnson et al. [D.L. Johnson, J. Koplik, R. Dashen, Theory of dynamic permeability and tortuosity in fluid-saturated porous media, J. Fluid Mech. 176 (1987) 379-402] is used for the cancellous bone segment. By working with series expansions of the Laplace transform in terms of travel-time exponentials, a series of transfer functions for the reflection and transmission of fast and slow waves at the fluid-poroelastic interfaces are derived. The approach obviates numerical solution beyond the discretization involved in the use of the fast Fourier transform.