Effect of the Hydrodynamic Interaction of a Large Number of Particles on Their Sedimentation Rate in a Viscous Fluid

The gravity-induced sedimentation of a large number of identical spherical particles in a viscous fluid is considered. The hydrodynamic interaction of all the particles is taken into account. The problem is modeled numerically for random locations of 1–100 particles forming different configurations. A dependence of the mean sedimentation rate on the particle number and concentration in the cloud is obtained and compared with previous results.     

Oscillatory Motion of a Viscous Fluid in Contact with a Flat Layer of a Porous Medium

Analytical solutions are obtained for two problems of transverse internal waves in a viscous fluid contacting with a flat layer of a fixed porous medium. In the first problem, the waves are considered which are caused by the motion of an infinite flat plate located on the fluid surface and performing harmonic oscillations in its plane. In the second problem, the waves are caused by periodic shear stresses applied to the free surface of the fluid. To describe the fluid motion in the porous medium, the unsteady Brinkman equation is used, and the motion of the fluid outside the porous medium is described by the Navier–Stokes equation. Examples of numerical calculations of the fluid velocity and filtration velocity profiles are presented. The existence of fluid layers with counter-directed velocities is revealed.     

Lifting of Disperse Particles from a Cavity Behind the Front of an Unsteady Shock Wave with a Triangular Velocity Profile

The gas flow in plane shock waves slipping along an impermeable surface with a rectangular cavity where solid disperse particles are suspended is considered numerically. The motion of the gas and particles (gas suspension) is modeled by equations of mechanics of multiphase media. Some laws of the behavior of the dusty cloud in the cavity are established for the case of wave interaction with the cavity.     

The velocity of the collective motion of sedimentation of sand and clay

In this article.we shall discuss the velocity of the col-lective motion of sedimentation when the concentration ofclay particles is very high.we believe that the problemof the so-called velocity of the collective motion of sedi-mentation is not a real problem of sedimentation in physi-cal nature.but it is a problem of filtration in the porousmedium which is composed of colloidal clusters.That is tosay,those clusters consist of water and clay particles.Itis a problem of the water passing through the porous medium.Based upon this consideration.we put down the mathematicalexpression of the velocity of the collective motion of se-dimentation.In comparing with the theoretical curves withthe experimental data of the Institute of Hydraulic Research,of the Yellow River Conservancy Commission,we get congruousresults.     

A multiparameter family of phase portraits in the dynamics of a rigid body interacting with a medium

The problem of plane-parallel motion of a uniform symmetric rigid body interacting with a medium only through a flat region of its outer surface is studied. The force field is constructed on the basis of information on the properties of jet flow under quasistationarity conditions. The motion of the medium is not studied. The problem of rigid body dynamics is considered for the case when the characteristic time of motion of the body relative to its center of mass is comparable with the characteristic time of motion of this mass center.     

Inertial migration of sedimenting particles in a suspension flow through a Hele-Shaw cell

Within the framework of the model of two interpenetrating continua, a horizontal laminar dilute-suspension flow in a vertical Hele-Shaw cell is investigated. Using the method of matched asymptotic expansions, an asymptotic model of the transverse migration of sedimenting particles is constructed. The particle migration in the horizontal section of the cell is caused by an inertial lateral force induced by the particle sedimentation and the shear flow of the carrier phase. A characteristic longitudinal length scale is determined, on which the particles migrate across the slot through a distance of the order of the slot half-width. The evolution of the particle number concentration and velocity fields along the channel is studied using the full Lagrangian method. Depending on the particle inertia parameter, different particle migration regimes (with and without crossing of the channel central plane by the particles) are detected. A critical value of the particle inertia parameter corresponding to the change in migration regime is found analytically. The possibility of intersection of the particle trajectories and the formation of singularities in the particle number concentration is demonstrated.     

Dynamics of two elliptical particles sedimentation in a vertical channel: chaotic state

The LB-DF/FD method derived from the Lattice Boltzmann Method and direct forcing/fictitious domain method is used to numerically investigate the dynamics and interaction of two elliptical particles settling in an infinitely long channel. One particle (EP0) is initially kept horizontal (major axis perpendicular to sedimentation) for all simulations while the other's (EP1) orientation is varied. It is found that if EP1strays away from horizontality, the particles undergo transitions from a steady state to reach a chaotic state. Furthermore, there are two distinct chaotic states for the particle motion when EP1 orientation is varied, in which a turning point is observed to distinguish the two states.     

Structure and stability of the interface between magnetic and conventional fluids. Model of a three-component medium

The structure of a flat interphase boundary between a magnetic suspension and a conventional immiscible fluid is investigated within the framework of the model of a three-component medium taking the dependence of the free energy of the system on the concentration gradients into account. It is shown that for certain values of the constitutive parameters the bulk magnetic particle concentration increases significantly inside the interfacial layer, i.e., the particles are significantly adsorbed on the interface. The dependence of the surface tension on the magnetic field strength is determined. It is shown that for certain problem parameters this dependence qualitatively corresponds to that obtained experimentally and described in the phenomenological theory developed by Golubyatnikov and Subkhankulov in 1986. In the case of strong particle adsorption the dependence of the surface tension on the magnetic particle concentration on the phase interface is significantly nonlinear. A refined model of the interface as a two-dimensional continuum with surface magnetization is constructed. Constitutive equations, conditions on the interface, and necessary stability conditions are obtained.     

Numerical simulation of rarefied suspension sedimentation in a container

The evolution of the large-scale velocity perturbations in a homogeneous suspension sedimenting in a rectangular container with rigid horizontal walls and periodic conditions on the vertical boundaries is considered. Numerical simulation of the point-particle motion showed that the density and velocity fluctuations decrease with time. The perturbations are damped due to reshaping of the sedimentation front and the nonlinear interaction of the different modes.     

Effect of convection on the stability of a liquid with a nonuniformly distributed heavy admixture

The effect of the transverse temperature gradient on the stability of steady motion of a viscous incompressible liquid in a plane vertical layer bounded by two infinite solid surfaces is studied. The motion of the liquid is caused by sedimentation of heavy solid spherical particles distributed nonuniformly across the layer and by the horizontal temperature gradient. Spectra of decrements of small normal perturbations are calculated for different particle sizes and different degrees of nonuniformity of the distribution of admixture particles. The stability of a steady flow of the liquid with an admixture decreases with increasing temperature gradient and increasing particle radius and increases with a tendency of the particles to a uniform distribution. Chelyabinsk State University, Chelyabinsk 454021. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 5, pp. 180–187, September–October, 2000.     

Model of a gas of solid particles with allowance for inelastic collisions

A study is made of the structure of the interaction region of a flow of solid particles with a surface (or another flow) in the case when the width of this region is much greater than the mean free path of the particles between collisions. The disperse phase is described by means of a model of an ideal non-heat-conducting gas of imperfectly elastic spherical particles. For simplicity, the influence of the carrier medium on the motion of the particles is ignored.     

Viscosity of the liquid phase in a dispersion

The motion of a dispersion (continuous medium and particles) may be described [1] via the equations ot conservation of matter and momentum for the two phases separately. Here it is necessary to know how the viscosity, pressure in the solid, and other quantities vary with the parameters of the motion. This difficulty occurs even for the very simple model where the internal stresses in the dispersed phase are taken as zero, as there is then an uncertainty as to the viscosity of the medium, which is not a material constant and is dependent on the concentration. There is also uncertainty as to the forces of interaction between the phases. There are numerous empirical relationships for these forces, and also a theoretical one [2]. Here an analogous method is applied to derive an expression for the viscosity of the liquid. This viscosity applies to a liquid filtering through a porous medium in the particular case where the concentration is such as to produce close packing of the solid particles. The result corresponds to standard formulas in the case of low concentrations.     

Discrete particle simulation of mixed sand transport

An Eulerian/Lagrangian numerical simulation is performed on mixed sand transport. Volume averaged Navier–Stokes equations are solved to calculate gas motion, and particle motion is calculated using Newton's equation, involving a hard sphere model to describe particle-to-particle and particle-to-wall collisions. The influence of wall characteristics, size distribution of sand particles and boundary layer depth on vertical distribution of sand mass flux and particle mean horizontal velocity is analyzed, suggesting that all these three factors affect sand transport at different levels. In all cases, for small size groups, sand mass flux first increases with height and then decreases while for large size groups, it decreases exponentially with height and for middle size groups the behavior is in-between. The mean horizontal velocity for all size groups well fits experimental data, that is, increasing logarithmically with height in the middle height region. Wall characteristics greatly affects particle to wall collision and makes the flat bed similar to a Gobi surface and the rough bed similar to a sandy surface. Particle size distribution largely affects the sand mass flux and the highest heights they can reach especially for larger particles.     

Nonlinear Dynamics of Turbulent Thermals in Shear Flow

The nonlinear integral model of a turbulent thermal is extended to the case of the horizontal component of its motion relative to the medium (e.g., thermal floating-up in shear flow). In contrast to traditional models, the possibility of a heat source in the thermal is taken into account. For a piecewise constant vertical profile of the horizontal velocity of the medium and a constant vertical velocity shear, analytical solutions are obtained which describe different modes of dynamics of thermals. The nonlinear interaction between the horizontal and vertical components of thermal motion is studied because each of the components influences the rate of entrainment of the surrounding medium, i.e., the growth rate of the thermal size and, hence, its mobility. It is shown that the enhancement of the entrainment of the medium due to the interaction between the thermal and the cross flow can lead to a significant decrease in the mobility of the thermal.     

Interaction of local inhomogeneities of a fluidized bed

In connection with an analysis of transport processes in fluidized beds a study is made of the steady motion of a system of local inhomogeneities of the bed porosity, which are modeled by packets of particles. The interaction of the inhomogeneities is taken into account on the basis of a cell model. The velocity fields and the pressure distributions of the solid and gas phases together with the rising and sinking velocities of the system of packets are determined in the approximation of a double continuum. The flow regimes of the fluidizing agent are investigated. It is shown that with increasing concentration of packets in the bed the velocity of their motion decreases, and the circulation region of the dispersion medium surrounding a packet or trapped within it contracts. The dependence of the rate of flow of the fluidizing agent through the transverse section of the reactor on the concentration of packets in the bed is found.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 42–49, January–February, 1991.     

Structure of the interface between magnetic and conventional fluids: Model of immiscible phases

The structure of the flat interface between a two-component magnetic suspension and a conventional nonmagnetizable fluid immiscible with it is investigated with account for the dependence of the free energy of the system on the magnetization gradients, the concentration of magnetic particles, and the bearing phase density. It is shown that at certain values of the problem parameters the volume concentration of magnetic particles strongly increases near the interface, that is, the particles are substantially adsorbed at this surface. The dependence of the surface tension tensor components on the magnetic field stress is determined.     

Mass transfer to particle clusters and bubbles in a fluidized bed at low Reynolds numbers

The process of mass transfer to a particle cluster or bubble rising in a developed fluidized bed rapidly enough for a region of closed circulation of the fluidizing agent (cloud) to be formed is investigated in the Stokes approximation on the basis of a model of the steady-state motion of the fluid and solid phases near the cluster or bubble [1]. Within the cloud surroundinga local inhomogeneity of the fluidized bed intense mixing of the fluid phase takes place and the mass transfer between the cloud and the surrounding medium is determined by diffusion. The method of matched asymptotic expansions is used to obtain an analytic solution of the problem of the concentration field and the diffusion mass flux to the surface of the cloud at small and large values of the Péclet number. The latter is determined from the relative velocity of the cluster, the radius of the cloud, and the effective diffusion coefficient. In the limiting case of zero concentration of the solid phase within the cluster the solution obtained describes the mass transfer to a bubble in the fluidized bed. A comparison is made with the corresponding results previously obtained within the framework of a model of the solid phase as an inviscid fluid [2]. It is shown that the effect of viscosity on the mass transfer to the bubble is most important at large Péclet numbers, and that the correction to the total diffusion flux to the surface of the closed circulation zone due to viscosity effects may reach 40%.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 60–67, July–August, 1986.     

激波驱动下固体颗粒抛撒的实验研究

本文使用阴影照相技术、高速摄像技术及压力测试手段,实验记录和研究了激波与固体颗粒群的作用及激波作用后固体颗粒群的抛撒和云团的形成过程.结果表明:在激波与固体颗粒群作用过程中,存在着清晰的激波透射、反射及绕射现象,同时激波强度在作用后有明显的下降趋势;在固体颗粒抛撒及云团形成过程中,实验发现对同一粒径的颗粒抛撒来说,抛撒的颗粒群质量越大,云团形成的均匀性及稳定性越好,而对不同粒径的颗粒群来说,粒径越大,形成的云.团集中性越强.     

Experimental and numerical study of the rotation and the erosion of fillers suspended in viscoelastic fluids under simple shear flow

When a porous agglomerate immersed in a fluid is submitted to a shear flow, hydrodynamic stresses acting on its surface may cause a size reduction if they exceed the cohesive stress of the agglomerate. The aggregates forming the agglomerate are slowly removed from the agglomerate surface. Such a behaviour is known when the suspending fluid is Newtonian but unknown if the fluid is viscoelastic. By using rheo-optical tools, model fluids, carbon black agglomerates and particles of various shapes, we found that the particles had a rotational motion around the vorticity axis with a period which is independent on shape (flat particles not considered), but which is exponentially increasing with the elasticity of the medium expressed by the Weissenberg number (We). Spherical particles are always rotating for We up to 2.6 (largest investigated We in this study) but elongated particles stop rotating for We>0.9 while orienting along the flow direction. Erosion is strongly reduced by elasticity. Since finite element numerical simulation shows that elasticity increases the local stress around a particle, the origin of the erosion reduction is interpreted as an increase of cohesiveness of the porous agglomerate due to the infiltration of a viscoelastic fluid.     

Interaction forces of particles dispersed in an electrolyte

Expressions are derived for the forces acting in a disperse medium in the presence of interaction of the double layers surrounding particles or drops of the dispersed phase when the potential of the dispersed particles is small. It is found that the force produced by the presence of double layers is proportional to the concentration gradient of the dispersed particles. It is shown that this force is comparable with the force produced by Brownian motion of the particles and may even exceed it. The equations of motion for the dispersed phase are derived with allowance for the convective terms, the pressure gradient, and the forces caused by Brownian motion and the presence of the double layers. A generalized Fick's law is obtained with effective diffusion coefficient. The equilibrium distribution of the particle concentration in a uniformly rotating cylinder is found with allowance for the interaction of the double layers.Translated from Izvestiya Akademii Nauk SSSR, Meklianika Zhidkosti i Gaza, No. 5, pp. 98–102, September–October, 1984.