On centrifugal separation of particles of two different sizes

We consider flow in a centrifugal force field of a non-dilute suspension with particles or droplets of two sizes. The volume fraction and the velocity fields are determined assuming small convection and shear terms. The resulting flow field is quite different from that in a gravitational settling of a similar mixture. In particular, the volume fraction is a function of time and radius in the sectors separated by kinematic shocks and the settling velocity is a non-monotonic function of the particle size.     

Migration of a solid particle in the vicinity of a plane fluid–fluid interface

The slow migration of a small and solid particle in the vicinity of a gas–liquid, fluid–fluid or solid–fluid plane boundary when subject to a gravity or an external flow field is addressed. By contrast with previous works, the advocated approach holds for arbitrarily shaped particles and arbitrary external Stokes flow fields complying with the conditions on the boundary. It appeals to a few theoretically established and numerically solved boundary-integral equations on the particle’s surface. This integral formulation of the problem allows us to provide asymptotic approximations for a distant boundary and also, implementing a boundary element technique, accurate numerical results for arbitrary locations of the boundary. The results obtained for spheroids, both settling or immersed in external pure shear and straining flows, reveal that the rigid-body motion experienced by a particle deeply depends upon its shape and also upon the boundary location and properties.     

On the centrifugal separation of a bulk mixture

The centrifugal separation of a mixture of particles and fluid in an axisymmetric container is examined. The flow consists of three distinct regions—mixture, sediment and purified fluid—with Ekman boundary layers at the interfaces and walls. In the settling process, the mixture and pure fluid acquire retrograde and prograde rotations relative to the tank. This flow pattern, and the shape and locus of the interface which are easily determined, provide another simple means to compare mixture theory and experiment. It is shown that when the Coriolis force is important, the pure fluid layer on the “outwardly” inclined wall is not thin. Moreover the interface between the mixture and the pure fluid is not perpendicular to the centrifugal force. Both features contrast those of the gravitational Boycott effect. As a consequence, there is no obvious enhancement of settling due to geometrical configuration.     

The effect of unsteady and history forces in the gravity convection of suspensions

The effect of unsteady and history forces on the motion of a particle is studied in the case of gravitational sedimentation (rise) of a spherical particle in the harmonic velocity field of a viscous carrier phase. An algorithm is proposed to calculate the history Basset-Boussinesq force. A parameter range is found for the case when the consideration of unsteady and history forces is required to correctly describe the mesoscale motions in a settling (rising) suspension.     

Liquid bridge force between two unequal-sized spheres or a sphere and a plane

Liquid bridge force acting between wet particles is an important property in particle characterization. This paper deals with liquid bridge force between either two unequal-sized spherical particles or a sphere and a flat plate under conditions where gravitational effect arising from bridge distortion is negligible. In order to calculate the force of the liquid bridge efficiently and accurately, expressions of liquid configuration and liquid bridge force were derived by building a mechanical model, which assumes the liquid bridge to be circular in shape between either two unequal-sized spheres or a sphere and a plane. To assess the accuracy of the numerical results of the calculated liquid bridge forces, they were compared to the published experimental data.     

Effect of baffles on sloshing modulated fluid force and torque fluctuations on the dewar driven by gravity gradient acceleration in microgravity

The dynamical behavior of fluids affected by the asymmetric gravity gradient acceleration is studied. In particular the effect of surface tension on partially liquid filled rotating fluids applicable to a full-scale Gravity Probe-B Spacecraft dewar tank with and without a baffle have been investigated. Results of slosh wave excitation along the liquid-vapor interface induced by gravity gradient acceleration indicate that the gravity gradient acceleration is equivalent to the combined effect of a twisting force and torsional moment acting on the spacecraft. As viscous force (between liquid and solid interface), and surface tension force (between liquid-vapor-solid interface) greatly contribute to the damping effect of slosh wave excitation, a rotating dewar with baffle provides more areas of liquid-solid and liquid-vapor-solid interfaces than that of a rotating dewar without baffle. Results show that the damping effect provided by the baffle reduces the amplitude of slosh wave excitation, lowers the fluid force, torque, and the moment arm of fluid torque fluctuations than that without baffle, and also lowers the degree of asymmetry in the liquid-vapor distribution.     

ACTUATION OF SLOSHING MODULATED FORCE AND MOMENT ON LIQUID CONTAINER DRIVEN BY JITTER ACCELERATIONS ASSOCIATED WITH SIEW MOTION IN MICROGRAVITY

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A numerical study of the occurrence of convection in a liquid layer under the influence of periodic external forces

The development of convection in a horizontal liquid layer located in a periodically modulated gravitational field (or with periodically varying temperature gradient) is examined. The effect of modulation frequency on stability is studied. Modulation stabilizes equilibrium if the direction of the gravitational force remains constant at all times. In the opposite case, stabilization occurs only at sufficiently high frequencies. In [1] the dependence of the critical Rayleigh number on modulation amplitude of the external force for several fixed frequencies was examined. In all cases examined in [1], the modulation proves to have a stabilizing influence.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 81–86, May–June, 1972.     

Dynamic simulation of sedimentation of solid particles in an Oldroyd-B fluid

In this paper we present a two-dimensional numerical study of the viscoelastic effects on the sedimentation of particles in the presence of solid walls or another particle. The Navier-Stokes equations coupled with an Oldroyd-B model are solved using a finite-element method with the EVSS formalism, and the particles are moved according to their equations of motion. In a vertical channel filled with a viscoelastic fluid, a particle settling very close to one side wall experiences a repulsion from the wall; a particle farther away from the wall is attracted toward it. Thus a settling particle will approach an eccentric equilibrium position, which depends on the Reynolds and Deborah numbers. Two particles settling one on top of the other attract and form a doublet if their initial separation is not too large. Two particles settling side by side approach each other and the doublet also rotates till the line of centers is aligned with the direction of sedimentation. The particle-particle interactions are in qualitative agreement with experimental observations, while the wall repulsion has not been documented in experiments. The driving force for lateral migrations is shown to correlate with the pressure distribution on the particle's surface. As a rule, viscoelasticity affects the motion of particles by modifying the pressure distribution on their surface. The direct contribution of viscoelastic normal stresses to the force and torque is not important.     

明渠湍流边界层中颗粒的运动与分布

利用直接数值模拟、点球浸入边界法和颗粒离散元法相结合的方法, 模拟了颗粒在明渠湍流边界层中的运动, 并对颗粒的瞬时位置进行了Voronoi 分析, 定量研究了颗粒在湍流边界层中的运动和分布规律. 研究发现:颗粒的输运对湍流的统计特征有影响, 其运动与近壁区湍流拟序结构密切相关, 在"喷发"结构作用下被带离壁面, 在"扫掠" 结构和自身重力作用下回到壁面; 在湍流边界层中, 颗粒倾向于聚集在低流速带, 呈条带状分布;颗粒在大部分时间处于"簇"状态, 偶尔跳跃到"空" 状态, 但能够很快回到邻近低速区域.      

A numerical study of droplet motion/departure on condensation of mixture vapor using lattice Boltzmann method

Droplet motion/departure, which is governed by external force acceleration coefficient, droplet radius and surface wettability on solid surfaces under external forces such as gravitational force, play a significant role in characterizing condensation heat transfer, especially when high fractional non-condensable gases (NCG) present. However, due to the challenge in visualizing the vapor/steam velocity field imposed by droplet motion/departure, the detailed mechanism of droplet motion/departure on condensing surfaces has not been completely investigated experimentally. In this study, droplet motion/departures on solid surfaces under external forces and their interactions with steam flow are simulated using two dimensional (2D) multiphase lattice Boltzmann method (LBM). Large external force acceleration coefficient, droplet radius and contact angle, lead to large droplet deformation and high motion/departure velocity, which significantly shortens the droplet residual time on the solid surface. Our simulation shows that steam vortices (lateral velocity) induced by droplet motion/departure can greatly disturb the vapor flow and would be intensified by increasing external force acceleration coefficient, droplet radius, and contact angle. In addition, the location of vortex center shifts in the ascending direction with increase of these factors. The average lateral velocities induced by droplet motion/departure at various conditions are obtained. The mass transfer resistance is substantially reduced owing to the droplet motion/departure, leading to an enhanced heat flux. The experimental results are compared to validate the influence of droplet motion/departure on condensation heat transfer performance, especially for steam–air mixture with the presence of high fractional NCG.     

Thermo-acceleration waves and shock formation in Extended Thermodynamics of gravitational gases

The possibility of shock formation as degeneration of acceleration waves in a thermoviscous gravitational ideal gas is studied by exploiting the hyperbolic system of Extended Thermodynamics. The mathematical aspects of this problem are discussed by considering the different contributions of gravity and dissipative effects. In particular, we evaluate the critical time (i.e. the instant in which a shock wave starts) proving that it exists, in the usual physical situations, only for a sufficiently large critical initial amplitude of the acceleration jump. We show that an acceleration wave can never degenerate into a shock wave except in some limiting cases and so, since gravity force is overcome by dissipative effects, our results do not differ, qualitatively, from the case without gravity: this result implies the asymptotic stability (in the sense of [1]) of the static isothermal solutions.     

紊动流场中悬浮颗粒分布的随机理论

通过分析固体颗粒在紊动流场中的随机运动,建立了二维流场中垂直于时均流动的方向上颗粒随机位移的概率密度分布函数所满足的方程。由该方程解出的分布函数在一定条件下即相当于颗粒浓度分布函数。运用这一方法研究了[1]、[2]中报道的壁面附近颗粒浓度降低的现象。     

Slow motions of a circular cylinder experiencing slip near a plane wall

A theoretical study is presented for the two-dimensional creeping flow caused by a long circular cylindrical particle translating and rotating in a viscous fluid near a large plane wall parallel to its axis. The fluid is allowed to slip at the surface of the particle. The Stokes equations for the fluid velocity field are solved in the quasi-steady limit using cylindrical bipolar coordinates. Semi-analytical solutions for the drag force and torque acting on the particle by the fluid are obtained for various values of the slip coefficient associated with the particle surface and of the relative separation distance between the particle and the wall. The results indicate that the translation and rotation of the confined cylinder are not coupled with each other. For the motion of a no-slip cylinder near a plane wall, our hydrodynamic drag force and torque results reduce to the closed-form solutions available in the literature. The boundary-corrected drag force and torque acting on the particle decrease with an increase in the slip coefficient for an otherwise specified condition. The plane wall exerts the greatest drag on the particle when its migration occurs normal to it, and the least in the case of motion parallel to it. The enhancement in the hydrodynamic drag force and torque on a translating and rotating particle caused by a nearby plane wall is much more significant for a cylinder than for a sphere.     

Note on buble motion in a rotating liquid under residual gravity

The motion of a small gas bubble, presumed to retain its geometrical shape and contained in a rotating liquid, has been investigated. The fluid system liquid-gas is subject to the influence of a reduced gravitational field. It is demonstrated that under certain conditions (spin axis and direction of gravity are perpendicular to each other) the bubble travels on a circular path about the axis of rotation, as seen from an observer moving with the bulk of the liquid.     

HYDRODYNAMIC RESISTANCE EFFECT OF FLUID LAYER BETWEEN TWO IMMERSED APPROACHING PARTICLES

1. Introduction The mechanisms of impact and rebound of solid parti- cles in particulate flow systems are of interest over a wide range of application areas such as fluidized beds, pneu- matic transport, filtration processes, erosion and pollution control of suspended particles. In many cases, the colli- sions of particles against themselves and against walls may affect the properties of the mixture. Efforts have been made to describe the fundamental mechanics of particle collisions. The conta…     

The distribution of solid particles suspended in a turbulent flow: A stochastic approach

Basic fluid mechanics and stochastic theories are applied to show that the concentration distribution of suspended solid particles in a direction normal to the mean streamlines of a two-dimensional turbulent flow is greatly influenced by the lift force exerted on them in the vicinity of the wall. Analytic solution shows that, when the direction of the mean flow is horizontal, the probability density functionp (y, t) for random displacements of the particles will have a maximum value at a point from the wall where the perpendicular component of the lift force precisely balances particle gravity. Interpretation of experimental observations is presented using this theory.     

Development and validation of a reduced order history force model

The history force model accounts for temporal development in fluid gradients in the viscous region surrounding a particle in point particle methods. The calculation of the history force typically requires storing and using relative velocity information during the life time of the particle. For a large number of particles integrated over large times, history force calculation can become prohibitively expensive. The current work presents a new modeling approach to calculate the history force in which a decay function is applied to a stored cumulative value of the history force. The proposed formulation is equivalent to applying the same function obtained from a constant acceleration assumption to a running average of the acceleration within the memory time of the particle. The new force model is validated with experimental measurements of settling spheres at Reynolds numbers ranging from around one to a few hundreds and at density ratios from 1.2 to about 9.32. More validation work was carried-out with experimental measurements of oscillating spheres at different frequencies and amplitudes, as well as bouncing spheres at different Reynolds numbers and density ratios. The model shows very good agreement with the experiments of settling spheres and reasonable/good agreement with oscillating and bouncing sphere experiments. The proposed model significantly reduces the computational resources required to calculate the history force especially when large number of particles need to be integrated over long times.     

On the influence of buoyancy on the surface tension driven flow around a bubble on a heated wall

The surface tension driven flow in the liquid vicinity of gas bubbles on a heated solid wall has been investigated both, in a reduced gravity environment aboard a sounding rocket, and in an earth-bound experiment. Both experiments deal with temperature gradients within the liquid surrounding of a bubble which cause variations of the surface tension. These, in turn, lead to a liquid flow around the bubble periphery termed thermocapillary or thermal Marangoni-convection. On Earth, this phenomenon is widely masked by buoyancy. We therefore carried out an experiment under reduced gravitational acceleration. In order to simultaneously observe and record the flow field and the temperature field liquid crystal tracers have been applied. These particles offer the feature of selectively reflecting certain wavelengths of incident white light depending on the crystals temperature. Although the bubble injection system did not perform nominally during the flight experiment, some interesting flow characteristics could be observed. Comparison of results obtained in microgravity to data measured on Earth reveal that due to the interaction of thermocapillarity and buoyancy a very compact vortex flow results on ground, while in microgravity the influence on the surface tension driven flow penetrates much deeper into the bulk. This result is of special interest regarding the production of materials in space. Dedicated to Professor Dr. Julius Siekmann on the occasion of his 70th birthday The work described herein was supported by the German space agency DARA (Deutsche Agentur für Raumfahrtangelegenheiten GmbH) through DARA Grant 50 WM 9434. The authors thank the European Space Agency (ESA) for the opportunity to conduct the TEXUS 33 sounding rocket experiment. The flight hardware has been partly built by Daimler-Benz-Aerospace which is gratefully acknowledged. Also, the authors are indebted to Mr. H.-H. Wolf for his careful evaluation of the particle images