On the Chaotic Instability of a Nonsliding Liquid-Filled Top with a Small Spheroidal Base via Melnikov-Holmes-Marsden Integrals

Chaotic orientations of a top containing a fluid filled cavity are investigated analytically and numerically under small perturbations. The top spins and rolls in nonsliding contact with a rough horizontal plane and the fluid in the ellipsoidal shaped cavity is considered to be ideal and describable by finite degrees of freedom. A Hamiltonian structure is established to facilitate the application of Melnikov-Holmes-Marsden (MHM) integrals. In particular, chaotic motion of the liquid-filled top is identified to be arisen from the transversal intersections between the stable and unstable manifolds of an approximated, disturbed flow of the liquid-filled top via the MHM integrals. The developed analytical criteria are crosschecked with numerical simulations via the 4th Runge-Kutta algorithms with adaptive time steps.     

Numbering homoclinic points of diffeomorphisms in the plane

In this paper we introduce a numerical conjugacy invariant for planar maps with homoclinic points. This invariant can be estimated based on partial information about the location of compact pieces of the stable and unstable manifolds of the system. The invariant is also related to topological entropy, and we indicate a method by which good bounds on the entropy of a system can be obtained.     

Identification of regions of fastest mixing in a system of point vortices

This paper describes a numerical method for efficiently identifying the regions of fastest mixing of a passive dye in a flow due to a system of point vortices. Results obtained from computations are presented for systems of three and four point vortices, both in the unbounded domain and inside a circular cylinder. The flow is two‐dimensional and the fluid is incompressible. The regions where mixing is possible are found by studying the largest Lagrangian Lyapunov exponent distribution with respect to various initial positions of tracer particles. The regions of fastest mixing are then identified from the Lyapunov exponent distribution at small times. The results of the method are verified by quantifying the mixing by using a traditional box counting technique. The technique is then applied to several different initial configurations of vortices and some interesting results are obtained. Some numerical findings about the nature of the exponents computed are also discussed. Copyright © 2002 John Wiley Sons, Ltd.     

Chaos in the Convective Flow of a Fluid Mixture in a Porous Medium

The results of the study of the global behaviour of the convective flow of a binary mixture in a porous medium are presented. Bifurcation diagram, fixed points, periodic, chaotic solutions, stable and unstable manifolds, and basins of attraction have been calculated. Different behaviours (chaos, undecidable behaviour, etc.) have been found.     

DEM investigation of mixing indices in a ribbon mixer

Mixing index is an important parameter to understand and assess the mixing state in various mixers including ribbon mixers, the typical food processing devices. Many mixing indices based on either sample variance methods or non-sample variance methods have been proposed and used in the past, however, they were not well compared in the literature to evaluate their accuracy of assessing the final mixing state. In this study, discrete element method (DEM) modelling is used to investigate and compare the accuracy of these mixing indices for mixing of uniform particles in a horizontal cylindrical ribbon mixer. The sample variance methods for mixing indices are first compared both at particle- and macro-scale levels. In addition, non-sample variance methods, namely entropy and non-sampling indices are compared against the results from the sample variance methods. The simulation results indicate that, among the indices considered in this study, Lacey index shows the most accurate results. The Lacey index is regarded to be the most suitable mixing index to evaluate the steady-state mixing state of the ribbon mixer in the real-time (or without stopping the impeller) at both the particle- and macro-scale levels. The study is useful for the selection of a proper mixing index for a specific mixture in a given mixer.     

Numerical simulation of 3d viscous fluid flow and convective mixing in a static mixer

Summary The paper deals with a hybrid method to simulate mixing processes of highly viscous single-phase shear-thinning liquids. A static mixer of SMX type is chosen for demonstration. In the first place, low Reynolds number flow is computed at different values of rheological parameters, considering periodicity and several geometrical and physical symmetries of the flow problem. The dynamical system of particle motion is treated, which turns out to be chaotic. With a large number of streak lines, Poincaré sections at different positions along the mixing element are made up, which gives a detailed insight into the mixing mechanism. Statistical quantities are extracted, which are suitable for a rating of the mixing quality. Received 12 April 1999; accepted for publication 30 May 1999     

圆板振子超谐分岔和混沌运动的实验研究

设计了非线性圆板混沌实验振动台,就轴对称圆板在简谐载荷作用下的非线性动力学行为进行了较为系统的实验研究,理论分析和数值计算,对基础作简谐运动,周边固支圆板,进行了实验。通过测量时间－中心挠度的加速度曲线,进行快速傅立叶变换（FFT）分析,实验发现了对称破缺,超谐分岔,调幅调相等复杂现象;对基础作简谐运动,周边固支,中心加质量块的圆板,进行了实验,通过测量时间－周边应变曲线,进行FFT分析,实验发现了混沌,对称破缺和恢复及超谐分岔等复杂动力学行为。     

Discrete element method study of parameter optimization and particle mixing behaviour in a soil mixer

Soil mixing is an emerging research in the field of construction resource recovery. In this study, the mixing behaviour of soil particles in a mixer is numerically simulated by the discrete element method (DEM). A four-factor, three-level orthogonal experiment is designed to optimize the mixer design by selecting the fly-cutter speed, spindle speed, number of blades and fly-cutter diameter, using Lacey mixing index and power consumption as evaluation indicators. Then, the impact of soil cohesion and type on the mixing behaviour is investigated. The results show that the optimal parameter combination of this experiment is 280 rpm fly-cutter speed, 40 rpm spindle speed, 4 blades and 250 mm fly-cutter diameter. This optimal combination reaches a comparatively uniform state mix in 5.9 s with an average power consumption of 704.11 W. In addition, the wear and tear of the mixer increases as soil cohesion increases, while the mixing quality of materials declines, resulting in a “shaft hugging” phenomenon. The mixing efficiency varies greatly among different soil types, but the radial and tangential velocities have a similar law. This work can provide some guidance for the optimization design of a mixer and study of soil mixing.     

Analysis of liquid circulation and mixing in a partitioned electrolytic tank

The performance of an electrochemical process depends critically on the mobility of the reacting species or ions towards the electrode surface. In this work, a partitioned electrolytic cell is studied. Here the fluid flow is induced by gases which evolve at the electrode surface. The liquid circulation induced by the rising bubbles is primarily responsible for mixing. In this study, the liquid circulation in a cell where an alkaline solution of water is electrolyzed using different Nickel designs of electrodes is investigated using PIV. For each electrode, the optimum operating conditions such as voltage and concentration of electrolyte which resulted in good mixing are found. The flow-field is quantified by calculating time averaged velocity profiles along the horizontal line and by analyzing the temporal variation of liquid velocity at a point. It is found that there are differences in the circulation and hence vorticity in the two compartments, anode and cathode. The effect of gas evolution on mixing between the two chambers is studied by taking uric acid in the cathode half and NaOH in the anode half. The flow induced by the evolved gas bubbles leads to convective mixing in the two chambers. The mixing time is calculated by measuring the variation of current with time under potentiostatic conditions. This is verified by measuring the pH in anode and cathode compartments during the electrolysis.     

超声速平面混合层小激波的形成与演变

为了揭示超声速混合层中小激波形成机理及其与涡相互作用的演变过程,本文基于大涡模拟（LES）方法,结合五阶精度混合TCD／WENO格式,对超声速平面混合层在对流马赫数为Mc=0．65条件下的流场结构进行了数值模拟,数值结果详细描述了超声速混合层中小激波的形成过程。研究了小激波形成后,随涡运动而产生的变形、脱落及发展过程。同时,对混合层双涡合并过程中,小激波与相邻涡相互作用所产生的变形与演变过程进行了讨论。     

Effects of dual jets distance on mixing characteristics and flow path within a cavity in supersonic crossflow

A rectangular open cavity with upstream dual injectors at a freestream Mach number of 1.9 was investigated experimentally. To evaluate the effect of the distance between the jets, the flow characteristics were investigated using the high-speed schlieren photography, particle image velocimetry, and surface oil flow techniques. The dual jet distances of 18 and 54 mm were used. Unstable flow occurs over the cavity in all cases and is not improved by changing the distance between the dual jets. Although the distance between the dual jets does not influence the flow stability, the flow field varies decidedly depending on the dual jets distance. The enhancement of air mixing depends on the distance between the jets. A long dual jets distance was found to yield better mixing characteristics within the cavity than a short one. When the jets are further apart, the mainstream between two counter-rotating vortex pairs behind the jets flows strongly into the cavity because of the increased blow-down occurring between the vortex pairs. Additionally, a counterflow with a low velocity magnitude occurs behind the jets. Hence, mixing is enhanced within the cavity by effects of the opposed flow. When the jet pairs are closer to each other, the counter-rotating vortex pairs are in contact; as a result, the blow-down effect does not occur between them. The flow drawn into the cavity from the mainstream is supplied from the sides of the test section into the cavity.     

Chaotic attitude motion of a magnetic rigid spacecraft in an elliptic orbit and its control

This paper deals with the chaotic attitude motion of a magnetic rigid spacecraft with internal damping in an elliptic orbit. The dynamical model of the spacecraft is established. The Melnikov analysis is carried out to prove the existence of a complicated nonwandering Cantor set. The dynamical behaviors are numerically investigated by means of time history, Poincaré map, Lyapunov exponents and power spectrum. Numerical simulations demonstrate the chaotic motion of the system. The input-output feedback linearization method and its modified version are applied, respectively, to control the chaotic attitude motions to the given fixed point or periodic motion. The project supported by the National Natural Science Foundation of Chine (10082003)     

Mixing and segregation assessment of bi-disperse solid particles in a double paddle mixer

A double paddle blender's flow patterns and mixing mechanisms were analyzed using discrete element method (DEM) and experiments. The mixing performance of this type of the blender containing bi-disperse particles has been rarely studied in the literature. Plackett-Burman design of experiments (DoE) methodology was used to calibrate the DEM input parameters. Subsequently, the impact of the particle number ratio, vessel fill level, and paddle rotational speed on mixing performance was investigated using the calibrated DEM model. The mixing performance was assessed using relative standard deviation and segregation intensity. Mixing performance was significantly affected by the paddle rotational speed and particle number ratio. Moreover, the Peclet number and diffusivity coefficient were used to evaluate the mixing mechanism in the blender. Results revealed that the diffusion was the predominant mixing mechanism, and the best mixing performance was observed when the diffusivity coefficients of 3 mm and 5 mm particles were almost equal.     

Large eddy simulation of hot and cold fluids mixing in a T-junction for predicting thermal fluctuations

Temperature fluctuations in a mixing T-junction have been simulated on the FLUENT platform using the large eddy simulation （LES） turbulent flow model and a sub-grid scale Smagorinsky-Lilly model. The normalized mean and root mean square temperatures for describing time-averaged temperature and temperature fluctuation intensity, and the velocity are obtained. The power spectrum densities of temperature fluctuations, which are key parameters for thermal fatigue analysis and lifetime evaluation, are analyzed. Simulation results are consistent with experimental data published in the literature, showing that the LES is reliable. Several mixing processes under different conditions are simulated in order to analyze the effects of varying Reynolds number and Richardson number on the mixing course and thermal fluctuations.     

Distribution of eddy viscosity and mixing length in a non-Newtonian flow of a solid particle-gas suspension

The present work concerns the study of the radial distribution of eddy viscosity and mixing length and their dependence upon the Reynolds number and the concentration of the solid phase in a non-Newtonian flow of a suspension of solid particles in a gas. The investigated systems have a pseudoplastic character and the deviation from Newtonian behaviour increases with an increase in the concentration of the dispersed phase. Relations are presented for the eddy viscosity and mixing length in the flow of pseudoplastic fluids. From the analysis of results it follows that the mixing length and eddy viscosity increases with an increase in the Reynolds number. In contrast, an increase in the concentration of the solid phase and consequently of the pseudoplasticity causes a decrease in the investigated quantities. The radial distribution of the mixing length and the eddy viscosity is characterized by a maximum, after which the investigated quantities vary only slightly. This enables the area of the core of the turbulent flow to be defined. E Non-dimensional eddy viscosity - K fluid consistency defined by Ostwald-De Waele's formula (power law) - K fluid consistency, eq. (12) - L mixing length - L _t non-dimensional mixing length - N ⁺ position parameter, eq. (3) - n power-law index - n ⁺ Reichardt's position parameter, eq. (4) - n slope of the dependence ln _w = f[ln (8w/D)] - R pipe radius - r radial distance from the pipe axis - Re=D W /µ Reynolds number - U ⁺ non-dimensional local mean axial velocity, eq. (2) - u ^* = w (/8)^0.5 friction velocity - non-dimensional local mean axial velocity - local mean axial velocity - u turbulence velocity component - mean axial velocity at pipe axis - w average velocity over cross-section of pipe - loading ratio of solid to air, i.e. ratio of mass flow rates of solids to air - Y _m ⁺ =Rⁿ u^2–n /K 8^n–1 non-dimensional distance of pipe centre from the wall - y distance from pipe wall - y ⁺ non-dimensional distance from pipe wall, eq. (5) - friction factor - µ _L laminar part of viscosity - µ _t eddy viscosity - density - shear stress - _w shear stress on the wall     

Three-dimensional multi-phase simulation of the mixing and segregation of binary particle mixtures in a two-jet spout fluidized bed

This study presents a three-dimensional numerical study of the mixing and segregation of binary particle mixtures in a two-jet spout fluidized bed based on an Eulerian–Eulerian three-fluid model. Initially, the particle mixtures were premixed and packed in a rectangular fluidized bed. As the calculation began, the gas stream was injected into the bed from the distributor and jet nozzles. The model was validated by comparing the simulated jet penetration depths with corresponding experimental data. The main features of the complex gas–solid flow behaviors and the mechanism of mixing and segregation of the binary mixtures were analyzed. Moreover, further simulations were carried out to evaluate the effects of operating conditions on the mixing and segregation of binary particle mixtures. The results illustrate that mixing can be enhanced by increasing the jet velocity or enlarging the difference of initial proportions of binary particle mixtures.     

Linear feedback and parametric controls of vibrations and chaotic escape in a Φ potential

The control of vibration amplitude and chaotic escape of an harmonically excited particle in a single well Φ⁶ potential is considered. The linear feedback and parametric control strategies are used. The control efficiency on amplitude is found by analysing the behaviour of the amplitude of the controlled system as compared to that of the uncontrolled system. The conditions for inhibition of the chaotic escape are obtained by means of the Melnikov method.