Numerical study of concentrated fluid–particle suspension flow in a wavy channel

The particle migration effects and fluid–particle interactions occurring in the flow of highly concentrated fluid–particle suspension in a spatially modulated channel have been investigated numerically using a finite volume method. The mathematical model is based on the momentum and continuity equations for the suspension flow and a constitutive equation accounting for the effects of shear‐induced particle migration in concentrated suspensions. The model couples a Newtonian stress/shear rate relationship with a shear‐induced migration model of the suspended particles in which the local effective viscosity is dependent on the local volume fraction of solids. The numerical procedure employs finite volume method and the formulation is based on diffuse‐flux model. Semi‐implicit method for pressure linked equations has been used to solve the resulting governing equations along with appropriate boundary conditions. The numerical results are validated with the analytical expressions for concentrated suspension flow in a plane channel. The results demonstrate strong particle migration towards the centre of the channel and an increasing blunting of velocity profiles with increase in initial particle concentration. In the case of a stenosed channel, the particle concentration is lowest at the site of maximum constriction, whereas a strong accumulation of particles is observed in the recirculation zone downstream of the stenosis. The numerical procedure applied to investigate the effects of concentrated suspension flow in a wavy passage shows that the solid particles migrate from regions of high shear rate to low shear rate with low velocities and this phenomenon is strongly influenced by Reynolds numbers and initial particle concentration. Copyright © 2008 John Wiley & Sons, Ltd.     

CFD simulation of particle suspension in a stirred tank

Particle suspension characteristics are predicted computationally in a stirred tank driven by a Smith turbine. In order to verify the hydrodynamic model and numerical method, the predicted power number and flow pattern are compared with designed values and simulated results from the literature, respectively. The effects of particle density, particle diameter, liquid viscosity and initial solid loading on particle suspension behavior are investigated by using the Eulerian–Eulerian two-fluid model and the standard k–? turbulence model. The results indicate that solid concentration distribution depends on the flow field in the stirred tank. Higher particle density or larger particle size results in less homogenous distribution of solid particles in the tank. Increasing initial solid loading has an adverse impact on the homogeneous suspension of solid particles in a low-viscosity liquid, whilst more uniform particle distribution is found in a high-viscosity liquid.     

Multi-fluid model of suspension filtration in a porous medium

In the framework of a three-fluid approach, a new model of suspension filtration in a porous medium is constructed with account for the formation of a dense packing of trapped particles with finite permeability and porosity. The following three continua are considered: the carrier fluid, the suspended particles, and the deposited particles. For a one-dimensional transient flow of suspension, a system of equations for the concentrations of the suspended and deposited particles, the suspension velocity, and the pressure is constructed. Two cases of the flow in a porous medium are considered: plane and radial. Numerical solution is found using a finite-difference method. Numerical calculations are shown to be in agreement with an analytical solution for the simplest case of filtration with a constant velocity and constant porosity and permeability. A comparison is performed with the classic filtration models for five sets of experimental data on the contamination of a porous sample. It is shown that near the inlet boundary, where an intense deposition of particles takes place, the new model describes the concentration profile of the deposited particles more accurately than the classical model.     

Effective Solution through the Streamline Technique and HT-Splitting for the 3D Dynamic Analysis of the Compositional Flows in Oil Reservoirs

In this article we suggest a new phenomenological mathematical model for the groundwater transport of colloid particles through porous media which is able to describe some significant effects experimentally observed but not captured within the framework of the classic approach. Our basic idea is to consider both the pure water and the colloid suspension as two thermodynamic phases. Using the network models of porous media, we simulated numerically the transport process at the pore-scale. By averaging the result derived, we have obtained the relative permeabilities for both phases, the percolation threshold for suspension flow, and the effective suspension viscosity. Due to specific laws of colloid particles repartition between various classes of pores, the relative permeability of suspension happens to be a highly nonlinear function of saturation, very far from the diagonal straight line. This determines a difference between the macroscale phase velocities. The suspension velocity is shown to be higher than that of water in major cases, only if the colloid particles are not too large. The suggested model predicts and describes in a closed form the effect of colloid transport facilitation observed experimentally.     

Brownian dynamics simulation of multiphase suspension of disc-like particles and polymers

A computational model is proposed for simulating the flow of polymer nanocomposites. This model is based on a multiphase suspension of disc-like particles and polymers. The particles are represented by oblate spheroid particles that interact with each other via the Gay-Berne (GB) potential, and the polymers are modeled by finitely extensible nonlinear elastic (FENE) chains that interact with each other via the repulsive Lennard-Jones potential. The interaction between an oblate spheroid particle and a FENE chain is also considered using a modified GB potential. A Brownian dynamics simulation of the shear flows of this system was conducted to investigate the orientation behavior of disc-like particles and the rheological properties of this system. The orientation of disc-like particles was affected by polymers, and the particles in a suspension were well aligned in flows because of the flow orientation property of polymers. The predicted shear viscosity exhibited shear thinning, and the normal stress differences agree qualitatively with experimental measurements of polymer/clay nanocomposites. The simulation results suggest that the present model has the potential to be used as a computational model for polymer nanocomposites.     

Rheology of a suspension of particles in viscoelastic fluids

In this paper we study the bulk stress of a suspension of rigid particles in viscoelastic fluids. We first apply the theoretical framework provided by Batchelor [J. Fluid Mech. 41 (1970) 545] to derive an analytical expression for the bulk stress of a suspension of rigid particles in a second-order fluid under the limit of dilute and creeping flow conditions. The application of the suspension balance model using this analytical expression leads to the prediction of the migration of particles towards the centerline of the channel in pressure-driven flows. This is in agreement with experimental observations. We next examine the effects of inertia (or flow Reynolds number) on the rheology of dilute suspensions in Oldroyd-B fluids by two-dimensional direct numerical simulations. Simulation results are verified by comparing them with the analytical expression in the creeping flow limit. It is seen that the particle contribution to the first normal stress difference is positive and increases with the elasticity of the fluid and the Reynolds number. The ratio of the first normal stress coefficient of the suspension and the suspending fluid decreases as the Reynolds number is increased. The effective viscosity of the suspension shows a shear-thinning behavior (in spite of a non-shear-thinning suspending fluid) which becomes more pronounced as the fluid elasticity increases.     

Filtration in a Porous Granular Medium: 2. Application of Bubble Model to 1-D Column Experiments

This paper describes the formulation of a quasi-1-D network model, referred to as the ‘bubble model’, and its application for simulating particle transport and filtration through a granular filter bed. The model comprises a series of homogeneous sites linked through bundles of cylindrical bonds that represent flow pathways through distributions of pores and pore throats. This model incorporates pore scale processes of particle sieving and infiltration are based on numerical simulations described in a companion paper. The modeling of infiltration is further refined based on detailed experimental observations and measurements of the filtration of a dilute suspension of acrylic particles through a column of glass beads reported by Yoon et al. (2005 Water Resour. Res., to appear). Their data distinguish (a) between the collection of particles on grain surfaces and at grain-to-grain contact points, and (b) between particles that are fully entrapped and those that are hindered (temporarily collected) and can later become detached. These effects are represented by two parameters that characterize the probability of attachment and are linked to the surface roughness of the grains; one that describes the minimum particle size that can be fully entrapped, and one that describes the detachment rate. These parameters can be readily calibrated from conventional measurements of effluent concentration and effluent particle size distribution. Detailed comparisons with the data reported by Yoon et al. show that the proposed bubble model is able to achieve reliable predictions of the spatial distribution of particles within the filter bed following phases of particle injection and washing.     

Stability of the plane Couette flow of a disperse medium with a finite volume fraction of the particles

The linear hydrodynamic stability of the plane Couette flow of a suspension with a finite volume fraction of the particles is considered. The two-phase medium flow is described within the framework of the model of mutually penetrating continua which allows for the finiteness of the volume occupied by the particles. In the main flow the phase velocities are the same, while gravity is not taken into account. The stability of disperse flows with both uniform and nonuniform particle distributions is studied. The linearized system of the equations of suspension motion with the no-slip boundary conditions imposed on solid walls is reduced to the eigenvalue problem for an ordinary differential fourth-order equation in the stream function. The eigenvalues are sought using the orthogonolization method. The parametric investigation of the stability characteristics of the disperse flow is performed. It is shown that in the case of the uniform spatial distribution of the particles in the main flow, the presence of an admixture in the flow leads to a slight variation in the wave decay rates, while the flow remains stable for any permissible combinations of the dimensionless governing parameters. In the case of nonuniform distribution of inclusions the flow loses stability already for low Reynolds numbers on a wide range of the dimensionless governing parameters.     

Instability of a horizontal plane-channel flow of a dilute suspension

The stability of a horizontal plane-channel flow of a dilute suspension is studied theoretically. It is shown that the mechanism of action of the sedimenting particles on the flow stability parameters is equivalent to the effect of a distributed flow stratification and is attributable to the vertical nonuniformity of the body force induced by the excess weight of the sedimenting particles. A strong dependence of the disturbance growth rate on the location of the interface between the suspension and the pure liquid is detected.     

Capillary flow of a suspension of non-magnetic particles in a ferrofluid under highly non-uniform magnetic field

When a ferromagnetic suspension flows through a capillary placed between two small strong permanent magnets, the magnetic force acts upon the non-magnetic (silica) particles dispersed in a ferrofluid and they tend to be extruded from the zone of high magnetic field. Particles get concentrated at the entrance section between magnets and form a plug. The increase of hydraulic resistance is due to the relative motion between particulate and ferrofluid phases in the presence of a field. If we keep the pressure difference constant, the flow rate will decrease when the field is applied and can eventually completely stop. In order to restart the flow a pressure difference, high enough to push the silica plug out of the capillary, is needed. The critical pressure of the flow blockage is nearly two times less than the pressure of the flow onset, both pressures being independent of the particle concentration in the suspension (except for near zero concentration). Such hysteresis of the flow onset/blockage has also been predicted in the frame of the proposed two-phase flow model, which has been used to calculate steady-state concentration profiles and discharge characteristics.     

The settling of suspensions promoted by rigid buoyant particles

When substantial numbers of buoyant particles are mixed with a sedimenting suspension of heavy particles, the two types of solids undergo rapid lateral segregation from each other. Streams 3–5 mm in diameter containing the less populous species form and flow through a concentrated continuum suspension of the more populous species. This buoyancy driven convection results in greatly enhanced sedimentation rates. A theory is developed on the premise that streams move in plug flow with all resistance to motion confined to a thin lubricating layer at the strean-continuum interface. The model is compared with experimental data and is shown to account correctly for the effect of each system parameter on the observed settling rates.     

Particle suspension in a simple shear flow

A hydrodynamic model describing the particle distribution over the cross-section of a finely dispersed flow is proposed. The model is constructed on the basis of notions concerning the diffusion of particles induced by their random displacements in the process of relative motion of neighboring layers at constant shear velocity. It is shown that the suspension capacity of the flow is large for small particles due to thermal fluctuations and for relatively large particles due to shear-induced particle pulsations. There are critical particle sizes for which the particles are suspended and transported by the flow less effectively than larger or smaller particles.Ekaterinburg. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 112–121, January–February, 1995.     

Formation of waves on a horizontal erodible bed of particles

The mechanisms responsible for the initial growth of sand waves on the surface of a settled layer of particles are studied experimentally and theoretically. Experiments employ water-glycerin solutions of 1–14 cP and glass spheres (_s = 2.4 g/cm³) that are either 100 or 300 μm in diameter. The particle Reynolds number and Shields parameter are of order one and the flow Reynolds number is of order 1000 to 10,000. Experimentally obtained regime maps of sand wave behavior and data on the wavelengths of the sand waves that first appear on the surface of the settled bed are presented. Turbulence in the clear liquid is not necessary for formation of waves and there is no dramatic change in behavior as the flowrate is increased across the turbulent transition. The initial wavelength varies as the Fronde number to the first power. Because a flowing suspension phase is observed before waves form, linear stability analysis of the clear-layer—suspension-layer cocurrent two-phase flow is presented. The suspension phase is modeled as a continuum that has an either constant or exponentially increasing viscosity. Neither of the models correctly predicts the wavelength for the first observed waves, their growth rate or their speed. However, the initial wavelength is found to agree well with the trajectory length for a saltating particle obtained from a model for forces on individual particles.     

Transient phenomena in thixotropic systems

A nonlinear rheological model which accounts for the time-dependent elastic, viscous and yielding phenomena is developed in order to describe the flow behavior of thixotropic materials which exhibit yield stress. A key feature of the formulation is a smooth transition from an ‘elastically’ dominated response to a ‘viscous’ response without a discontinuity in the stress–strain curve. The model is phenomenological and is based on the kinetic processes responsible for structural changes within the thixotropic material. As such, it can predict thixotropic effects, such as stress overshoot during start-up of a steady shear flow and stress relaxation after cessation of flow. Thus this model extends a previously proposed viscoplastic model [J. Rheol. 34 (1991) 647] to include thixotropy.An analysis and comparison to experimental data involving oscillatory shear flow are provided to evaluate the accuracy of the model and to estimate the model parameters in a prototype concentrated suspension. The experiments were conducted using a series of concentrated suspensions of silicon particles and silicon carbide whiskers in polyethylene. The data obtained with this experimental system indicated much better agreement between the theory and experiments that obtained in earlier work.     

Verification of a model to predict the influence of particle inertia and gravity on a decaying turbulent particle-laden flow

In an earlier publication some of the authors presented a theoretical model for the calculation of the influence of particle inertia and gravity on the turbulence in a stationary particle-laden flow. In the present publication the model is extended for application to a decaying suspension. Also a comparison is given between predictions made with the model and experimental data (own data and data reported in the literature) on a decaying turbulent flow with particles in a water tunnel or in a wind tunnel. For most of the experiments a prediction with reasonable accuracy and an interpretation is possible by means of the model.     

Single-Fluid Model of a Mixture for Laminar Flows of Highly Concentrated Suspensions

A model of laminar flow of a highly concentrated suspension is proposed. The model includes the equation of motion for the mixture as a whole and the transport equation for the particle concentration, taking into account a phase slip velocity. The suspension is treated as a Newtonian fluid with an effective viscosity depending on the local particle concentration. The pressure of the solid phase induced by particle-particle interactions and the hydrodynamic drag force with account of the hindering effect are described using empirical formulas. The partial-slip boundary condition for the mixture velocity on the wall models the formation of a slip layer near the wall. The model is validated against experimental data for rotational Couette flow, a plane-channel flow with neutrally buoyant particles, and a fully developed flow with heavy particles in a horizontal pipe. Based on the comparison with the experimental data, it is shown that the model predicts well the dependence of the pressure difference on the mixture velocity and satisfactorily describes the dependence of the delivered particle concentration on the flow velocity.     

Flow of a suspension in a flat channel with porous walls

In the flow of a suspension in a channel with porous walls, when the size of particles of a suspended phase is much less than the width of the channel but greatly exceeds the size of the pores (in particular, in the flow of blood in the plasma separator used in an artificial kidney system [1, 2]), phenomena are observed which apparently cannot be satisfactorily explained by means of the well-known solutions of problems on the motion of a Newtonian fluid [3]. For example, the flow rate of the liquid phase of the suspension through the walls depends on the velocity of the main flow and does not depend on the pressure drop on the wall at fairly high values of it [1, 2]. The present study gives below the formulation and an approximate solution, which explains this effect, of the problem of an incompressible two-phase suspension in a long slit with porous walls which are impermeable in relation to the suspended phase and through which the liquid phase is pumped. Certain effects are taken into account which are caused by the high volume concentration of the suspended phase.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 37–43, November–December, 1987.     

Semi-active hydro-gas suspension system for a tracked vehicle

A semi-active hydro-gas suspension is proposed for a tracked vehicle to improve ride comfort performance, without compromising the road holding and load carrying capabilities of the passive suspension. This is achieved through an active damper used in parallel with a gas spring. The suspension damper parameters are varied by a control mechanism based on sky-hook damping theory, which alters the flow characteristics. A damper prototype has been developed, tested for its flow characteristics, after which it has been integrated into an existing hydro-gas suspension system. An analytical model has been proposed from first principles rather than developing a phenomenological model based on experimental characteristics. This model is validated with experiments carried out on a suspension test rig. In order to compare the performance with the original passive system, an in-plane vehicle model is developed and the simulations clearly show that the semi-active system performance is superior to the passive system.     

Experimental study on the pressure drop and the entry length of the gas-solid suspension flow in a circular tube

This paper presents an experimental study on the flow characteristics of a dilute gas-solid suspension medium in a circular tube, wherein the inlet entry length and the frictional pressure drop are measured systematically. In particular, the effects of particle size on the frictional pressure-drop are examined in some detail. The spherical copper particles, whose average diameter ranges from about 45 to 170 μm, are used for the dispersed medium and the ranges of gas Reynolds number and solid loading ratio are up to 50,000 and 5 respectively. The experimental results show that the entry lengths are feasible to be correlated briefly to the apparent Reynolds number of suspension flow and also that the reduction of the frictional pressure drop is observed at the lower loading ratio for the smallest size particles.     

Internal pulsations in flows of finely dispersed suspensions

The results are given of calculations for the flow of a finely dispersed suspension when momentum and energy are exchanged between individual particles exclusively through the ambient fluid, i.e., the role of direct collisions between particles is negligible. Using the results, one can not only calculate the rms characteristics of the pseudoturbulence and, in particular, find the stresses in the dispersed phase but also to give a natural explanation of the phenomena observed in experiments on sedimentation and hydrodynamic self-diffusion of particles in monodisperse suspensions.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 91–100, May–June, 1993.