Darcy-Forchheimer flow by rotating disk with partial slip

The viscous dissipation and heat transfer in the Darcy-Forchheimer flow by a rotating disk are examined. The partial slip conditions are invoked. The optimal series solutions are computed via the optimal homotopic analysis method(OHAM). The thermophoresis and Brownian motions are studied. The Darcy-Forchheimer relation characterizes the porous space. The roles of influential variables on the physical quantities are graphically examined. A reduction in the local Nusselt number is observed through thermophoresis and thermal slip parameters. The local Sherwood number depicts an increasing trend for the higher Brownian motion and concentration slip parameters.     

A non-Fickian,particle-tracking diffusion model based on fractional Brownian motion

The work is motivated by the recent discovery that ocean surface drifter trajectories contain fractal properties. This suggests that the dispersion of pollutants in coastal waters may also be described using fractal statistics. The paper describes the development of a fractional Brownian motion model for simulating pollutant dispersion using particle tracking. Numerical test cases are used to compare this new model with the results obtained from a traditional Gaussian particle-tracking model. The results seems to be significantly different, which may have implications for pollution modelling in the coastal zone. © 1997 John Wiley & Sons, Ltd.     

Fractal dimensions of terrain profiles

Analysing terrain profiles of fields, roads, and other terrains, it was determined that terrain profiles are random and non-periodical. Mandelbrot has defined non-scaling, self-similar figures as fractals, and many investigators have tried to characterize natural forms and structures using fractal geometry. The work here investigates whether terrain profiles can be defined as fractals. Fractal dimensions of profiles were calculated. These were compared with a locus of Brownian motion further to investigate characteristics of terrain profiles. Fractals are defined to be self-similar and irregular. Measuring and analysing terrain profiles, it was established that the statistical characteristics of any part of a terrain profile are similar and that the statistical characteristics of profiles of any kind of terrain are similar irrespective of roughness. This means that terrain profiles are self-similar, and irregular. From these results, it was determined that terrain profiles are fractals. The fractal dimensions were calculated with a coarse-graining method and by Power Spectral Densities (PSD), and fractal dimensions by Scaling were between 1.1 and 1.8 and by PSD between 1.3 and 1.5. Using the locus of Brownian motion, fractal dimensions were 1.5 or slightly larger than those of the terrain profiles. Fractal dimensions for the locus of smoothed Brownian motion were nearly equal to terrain profiles. Therefore terrain profiles could be artificially generated from the locus of smoothed Brownian motion. It appears that terrain roughness is formed by random and non-periodical force.     

Excluded volume effects and fractional viscoelasticity in polymers

Meccanica - The excluded volume effect is added to a fractional viscoelastic model for modeling fractal polymers. This reveals a physical connection between the fractional time derivative, fractal...     

Viscoelasticity: Fractal parameters studied on mammalian erythrocytes under shear stress

In this work, experimental determinations are carried out using a home-made device called an erythrodeformeter, which has been developed and constructed for rheological measurements on red blood cells subjected to definite fluid shear stress. A numerical method formulated on the basis of the fractal approximation for ordinary and fractionary Brownian motion¹ is proposed to evaluate the viscoelastic behavior of mammalian erythrocyte membranes. The diffraction pattern, which is circular when the mammalian erythrocyte membranes are at rest, becomes elliptical when the cells undergo shear stress. Photometric readings of light intensity variation along the major axis of the elliptical diffraction pattern are recorded during the creep and recovery process. These data series are used to calculate, fractal rheological parameters of self-affine Brownian motion on the erythrocytes, averaged over several millions of cells. Three different parameters over the time dependent process could be obtained, which are: correlation coefficient <C(t)>, correlation integral, andK ₂-entropy, and very different results were obtained.     

Nonlocal-in-time kinetic energy in nonconservative fractional systems,disordered dynamics,jerk and snap and oscillatory motions in the rotating fluid tube

In this article we generalize the basic theoretical properties of nonlocal-in-time kinetic energy approach introduced in the framework of nonlocal classical Newtonian mechanics for the case of fractional dynamical systems explored in the context of the fractional actionlike variational approach. Two independent fractionally Lagrangians weights are considered independently: the Riemann-Liouville fractional weight and the extended exponentially fractional weight. For each weight, the corresponding nonlocal fractional Newton's law of motion is derived. Three main physical applications were discussed in details: free particles, oscillators and dynamics of particles in a rotating tube with earth frame. A number of differential equations depending on fractional and nonlocal-in-time parameters were obtained and their solutions are discussed accordingly. For specific parameters and particular initial conditions, it was observed that the dynamics exhibit a kind of strange phase plot trajectories that indicate the presence of disordered motions. However one of the main results concerns the physics of particles in the rotating tube which display, for specific values of fractional and nonlocal-in-time parameters, oscillatory motions controlled by the nonlocal-in-time parameter.     

Impact Dynamics in Milling of Thin-Walled Structures

The development of reliable high-speed spindles and motioncontrol systems has led to an increase in the industrial use ofhigh-speed milling. One of the primary applications of this newtechnology is the manufacture of thin-walled aluminum components foraircraft. The flexibility of the tools and workpieces, the high spindlefrequencies, and the inherent impact nonlinearities in the millingprocess can lead to complicated dynamic tool-workpieceinteractions. An experiment was constructed to study the vibrations ofa thin-walled part during milling. Time series, power spectra,autocorrelations, auto-bispectra, and phase portraits were examined.From this data, it is inferred that stiffness and damping nonlinearitiesdue to the intermittent cutting action have a pronounced effect on thedynamics of the workpiece. Delay space reconstructions and pointwisedimension calculations show that the associated motions arecharacterized by a fractal geometry. The auto-bispectra suggestquadratic phase coupling among the spectral peaks associated with thecutter frequency. A mechanics-based model with impact-nonlinearities wasdeveloped to explain the observed results. The predicted results agreewell with the experimental observations. The model predictions indicatethat aperiodic motions are possible over a large range ofcontrol-parameter values. These analytical and experimental results haveimplications for the prediction and control of vibrations in milling.     

Two-phase flows: Constitutive equations for lift and Brownian motion and some basic flows

Constitutive relations for the lift force on the particulate phase and the effect of Brownian motion are presented. These constitutive relations are derived subject to three new principles of constitutive equations. The effects of lift and Brownian motion in basic parallel flows are considered in order to determine the importance and the consequences of these effects. The relation of the Brownian motion model involving momentum balance to the diffusive model of particle motions is studied. Dimensional and scaling arguments are given.     

单自由度体系地震动力响应的混沌特性分析

引入非线性动力学理论和混沌时间序列分析方法考察地震动作用下单自由度体系动力响应的混沌特性。输入典型近断层地震动记录,定量计算了代表性周期的单自由度弹性和非弹性体系加速度响应时程的非线性特性参数。计算表明,这些加速度响应的关联维数为分数维,最大Lyapunov指数大于0;地震动激励下单自由度体系的地震动力响应具有混沌特性,不是完全的随机信号,为理解结构地震动力响应的不规则性与复杂性提供了新思路和新视角。     

Fractional derivative reconstruction of forced oscillators

Fractional derivatives are applied in the reconstruction from a single observable of the dynamics of a Duffing oscillator and a two-well experiment. The fractional derivatives of time series data are obtained in the frequency domain. The derivative fraction is evaluated using the average mutual information between the observable and its fractional derivative. The ability of this reconstruction method to unfold the data is assessed by the method of global false nearest neighbors. The reconstructed data is used to compute recurrences and fractal dimensions. The reconstruction is compared to the true phase space and the delay reconstruction in order to assess the reconstruction parameters and the quality of results.     

Gelation of polymeric nanoparticles

We review how, starting from polymeric nanoparticles, to generate clusters of fractal morphology and to expand the entire space and interconnect to form gels, through either Brownian motion or intense shear-induced aggregation. In the case of Brownian motion-induced gelation, specific techniques developed to obtain uniform structure of gels under both reaction-limited and diffusion-limited cluster aggregation conditions have been described. In the case of intense shear-induced gelation as a newly developed technique, our focus is on its principle, theoretical development and advantages with respect to Brownian motion-induced gelation in practical applications. We consider gelation of both rigid and soft particles. As a physical process, the bonding between the particles within gels is owed to van der Waals attractions, thus being easily broken. However, in the case of soft particles that can coalesce upon contact, the coalescence can allow the particles to stick together forming permanent gels. In this case, the gel structure can be controlled by controlling the degree of coalescence. Techniques used to control the degree of coalescence have also been described.     

Gelation of polymeric nanoparticles

We review how, starting from polymeric nanoparticles, to generate clusters of fractal morphology and to expand the entire space and interconnect to form gels, through either Brownian motion or intense shear-induced aggregation. In the case of Brownian motion-induced gelation, specific techniques developed to obtain uniform structure of gels under both reaction-limited and diffusion-limited cluster aggregation conditions have been described. In the case of intense shear-induced gelation as a newly developed technique, our focus is on its principle, theoretical development and advantages with respect to Brownian motion-induced gelation in practical applications. We consider gelation of both rigid and soft particles. As a physical process, the bonding between the particles within gels is owed to van der Waals attractions, thus being easily broken. However, in the case of soft particles that can coalesce upon contact, the coalescence can allow the particles to stick together forming permanent gels. In this case, the gel structure can be controlled by controlling the degree of coalescence. Techniques used to control the degree of coalescence have also been described.     

Analysis of fractional anomalous diffusion caused by an instantaneous point source in disordered fractal media

A theoretical analysis of fractional anomalous diffusion caused by an instantaneous point source in disordered fractal media is studied. Using the method of symmetry group of scaling transformations and the H-function, the analytical solutions of concentration distribution are given. At the same time we derive the expressions of scattering function spectrum.The result shows that the scattering function spectra still have the properties of scaling function. The scattering functions of point source, line source and area source in regular Euclidean space can be regarded as particular cases of this paper and are included in this paper. At the end of the paper we discuss the asymptotic behaviors of the solution in detail. The results of this paper can be taken to be the fundamental solutions for every kind of boundary value problems of fractional anomalous diffusion in disordered fractal media.     

Fish and Plankton Interplay Determines Both Plankton Spatio-Temporal Pattern Formation and Fish School Walks: A Theoretical Study

The fascinating variety of spatio-temporal patterns in aquatic ecosystems and the understanding of the governing mechanisms of its generation and further dynamics requires ongoing experimental and theoretical studies. After introducing a certain hybrid mathematical model, this paper makes an attempt to demonstrate that the predation of a mobile planktivorous fish school on zooplankton can initiate both plankton pattern formation and fish school walks. Nonlinear interactions in the model of a fish-zooplankton-algae trophic chain prevent a simple intuitive understanding of the system dynamics. It is shown that the fish school predation and motion can give rise to plankton spiral waves. In the course of the spiral wave formation, the amplitudes of the spatially averaged plankton density oscillations are decreasing dramatically. Fish school walks are shown to resemble a fractional Brownian motions with a Hurst exponent depending on the fish predation rate.     

Characterizing the creep of viscoelastic materials by fractal derivative models

In this paper, we make the first attempt to apply the fractal derivative to modeling viscoelastic behavior. The methodology of scaling transformation is utilized to obtain the creep modulus and relaxation compliance for the proposed fractal Maxwell and Kelvin models. Comparing with the fractional derivatives reported in the literature, the fractal derivative as a local operator has lower calculation costs and memory storage requirements. Moreover, numerical results show that the proposed fractal models require fewer parameters, have simpler mathematical expression and result in higher accuracy than the classical integer-order derivative models. Results further confirm that the proposed fractal models can characterize the creep behavior of viscoelastic materials.     

Structural changes of silica particles in elongated rubber by two-dimensional small-angle X-ray scattering and extended reverse Monte Carlo analysis

Two-dimensional pattern reverse Monte Carlo method is applied to investigate structural changes of silica particle filler in uniaxially elongat ed rubber. We calculated three-dimensional confor mations of spherical silica particles using a series of two-dimensional patterns of structure factors during elongation. The structure factors we used are inter polated from experimentally observed structure factors for the elongation ratio ε?=?0.0, 0.5, 1.0, and 1.5. To make a continuous motion of particles, we set the elongation ratio with small interval Δε?=?0.001. It is found that the structural change of the particles seem to be reliable at least for small ε regions because spurious motions like the Brownian motions are not observed.     

完全发展的非对称槽道湍流的分形维数研究

湍流的分形维数刻画了湍流的统计正则性 ,本文利用正交子波变换对完全发展的非对称槽道湍流脉动速度的实验数据进行了分析研究。文中首先将脉动速度信号分解到各个尺度 ,然后利用正则法和盒子法两种方法计算了不同尺度信号的分形维数 ,着重于考虑分形维数的变化趋势。研究结果表明 :(1 )随着信号分解尺度的增加 ,速度信号的低频部分和高频部分的维数都逐渐减小 ;(2 )逆输运现象对速度信号的分形维数没有本质的影响 ;(3)正则法计算得到的分形维数要略大于盒子法得到的分形维数     

On fractional difference logistic maps: Dynamic analysis and synchronous control

This paper investigates a logistic map derived from a difference equation in the framework of discrete fractional calculus. Through the Poincaré plots and Julia sets, the map’s chaotic and fractal characteristics are studied comparing with those of a quadratic map to be proposed. The memory effect of fractional difference maps is reflected in these dynamics, and some reasonable explanations are given by combining with quantitative analysis. A coupled controller is designed to realize synchronization between fractional difference logistic map and fractional difference quadratic map.

     

A Fractional Linear System View of the Fractional Brownian Motion

A definition of the fractional Brownian motion based on the fractional differintegrator characteristics is proposed and studied. It is shown that the model enjoys the usually required properties. A discrete-time version based in the backward difference and in the bilinear transformation is considered. Some results are presented.     

A Fractional Linear System View of the Fractional Brownian Motion

A definition of the fractional Brownian motion based on the fractional differintegrator characteristics is proposed and studied. It is shown that the model enjoys the usually required properties. A discrete-time version based in the backward difference and in the bilinear transformation is considered. Some results are presented.