Sphere Settling Research Articles

The motion of a single heavy sphere in ambient fluid: A benchmark for interface-resolved particulate flow simulations with significant relative velocities

Detailed data describing the motion of a rigid sphere settling in unperturbed fluid is generated by means of highly-accurate spectral/spectral-element simulations with the purpose of serving as a future benchmark case. A single solid-to-fluid density ratio of 1.5 is chosen, while the value of the Galileo number is varied from 144 to 250 such as to cover the four basic regimes of particle motion (steady vertical, steady oblique, oscillating oblique, chaotic). This corresponds to a range of the particle Reynolds number from 185 to 365. In addition to the particle velocity data, extracts of the fluid velocity field are provided, as well as the pressure distribution on the sphere’s surface. Furthermore, the same solid–fluid system is simulated with a particular non-boundary-conforming approach, i.e. the immersed boundary method proposed by Uhlmann (2005a), using various spatial resolutions. It is shown that the current benchmark case allows to adjust the resolution requirements for a given error tolerance in each flow regime.

Wake Field Flow of a Sphere Falling in a Finite Duct

We report on a simple study involving a single non-Brownian sphere settling under the influence of gravity in a quiescent viscous fluid housed in a finite square duct. Spheres are shown to achieve terminal velocity in a fraction of the time predicted by infinite fluid dynamics. Terminal velocities agree well with right cylinder equations for spheres with diame-ter-to-width ratios less than 0.45. The finite chamber length results a two-phase flow and interesting wake field dynamics for spheres with diameter-to-width ratios greater than 0.255.

Near-wall hydrodynamic interactions between a settling sphere and a wall

The near-wall dynamics of a sphere is experimentally resolved at microscale using a specially designed interferometry-related device. The millimeter size sphere, settling in a viscous oil towards a horizontal wall, is used as a reflector. The accuracy on the sphere displacement is smaller than 0.2 μm. Since its first implementation, this technique has been used to investigate various issues related to near-wall hydrodynamics. This presentation focuses on two recent developments of this device. In the first case of a sphere entering the near-wall region with a Reynolds number of order unity, it is shown how the dynamics of the sphere is modified by small inertia effects. In the second case, if a textured wall with microscopic periodic grooves is used, the velocity of the sphere increases, as compared to the velocity near a "smooth" wall. This velocity enhancement is measured accurately and correlated to the texture geometry. The results are useful for specifying effective boundary conditions at microscale, with possible applications in the field of microfluidics.

Settling of spherical particles in unbounded and confined surfactant-based shear thinning viscoelastic fluids: An experimental study

An experimental study is performed to understand and quantify settling velocity of spherical particles in unbounded and confined surfactant-based shear thinning viscoelastic fluids. Experimental data is presented to show that elastic effects can increase or decrease the settling velocity of particles, even in the creeping flow regime. Experimental data shows that a significant drag reduction occurs with increase in Weissenberg number. This is followed by a transition to increasing drag at higher Weissenberg numbers. A new correlation is presented for the sphere settling velocity in unbounded viscoelastic fluids as a function of the fluid rheology and the proppant properties.The wall factors for sphere settling velocities in viscoelastic fluids confined between solid parallel plates are calculated from experimental measurements made on these fluids over a range of Weissenberg numbers. Results indicate that elasticity reduces the effect of the confining walls and this reduction is more pronounced at higher ratios of the particle diameter to spacing between the walls. Shear thinning behavior of fluids is observed to reduce the retardation effect of the confining walls. A new empirical correlation for wall factors for spheres settling in a viscoelastic fluid confined between two parallel walls is presented.

Sedimentation of a sphere in a viscoelastic fluid: a multiscale simulation approach

AbstractA long-standing problem in non-Newtonian fluid mechanics, namely the relationship between drag experienced by a sphere settling in a tube filled with a dilute polymeric solution and the sphere sedimentation velocity, is investigated via self-consistent multiscale flow simulations. Comparison with experimental measurements by Arigo et al. (J. Non-Newtonian Fluid Mech., vol. 60, 1995, pp. 225–257) have revealed that the evolution of the drag coefficient as a function of fluid elasticity can be accurately predicted when the macromolecular dynamics is described by realistic micromechanical models that closely capture the transient extensional viscosity of the experimental fluid at high extension rates. Specifically, for the first time we have computed the drag coefficient on the sphere at high Weissenberg number $\mathit{Wi}$ utilizing multi-segment bead–spring chain models with appropriate molecular parameters and have demonstrated that a hi-fidelity multiscale simulation is not only capable of accurately describing the drag on the sphere as a function of $\mathit{Wi}$ at various sphere-to-tube diameter ratios but also it can closely reproduce the experimentally observed velocity and stresses in the wake of the sphere.

Sedimentation of a charged colloidal sphere in a charged cavity

An analytical study is presented for the quasisteady sedimentation of a charged spherical particle located at the center of a charged spherical cavity. The overlap of the electric double layers is allowed, and the polarization (relaxation) effect in the double layers is considered. The electrokinetic equations that govern the ionic concentration distributions, electric potential profile, and fluid flow field in the electrolyte solution are linearized assuming that the system is only slightly distorted from equilibrium. Using a perturbation method, these linearized equations are solved for a symmetric electrolyte with the surface charge densities of the particle and cavity as the small perturbation parameters. An analytical expression for the settling velocity of the charged sphere is obtained from a balance among the gravitational, electrostatic, and hydrodynamic forces acting on it. Our results indicate that the presence of the particle charge reduces the magnitude of the sedimentation velocity of the particle in an uncharged cavity and the presence of the fixed charge at the cavity surface increases the magnitude of the sedimentation velocity of an uncharged particle in a charged cavity. For the case of a charged sphere settling in a charged cavity with equivalent surface charge densities, the net effect of the fixed charges will increase the sedimentation velocity of the particle. For the case of a charged sphere settling in a charged cavity with their surface charge densities in opposite signs, the net effect of the fixed charges in general reduces/increases the sedimentation velocity of the particle if the surface charge density of the particle has a greater/smaller magnitude than that of the cavity. The effect of the surface charge at the cavity wall on the sedimentation of a colloidal particle is found to increase with a decrease in the particle-to-cavity size ratio and can be significant in appropriate situations.

Nonmonotonic settling of a sphere in a cornstarch suspension

Cornstarch suspensions exhibit remarkable behavior. Here, we present two unexpected observations for a sphere settling in such a suspension: In the bulk of the liquid the velocity of the sphere oscillates around a terminal value, without damping. Near the bottom the sphere comes to a full stop, but then accelerates again toward a second stop. This stop-go cycle is repeated several times before the object reaches the bottom. We show that common shear thickening or linear viscoelastic models cannot account for the observed phenomena, and propose a minimal jamming model to describe the behavior at the bottom.

A contact model for normal immersed collisions between a particle and a wall

AbstractThe incompressible Navier–Stokes equations are solved numerically to predict the coupled motion of a falling particle and the surrounding fluid as the particle impacts and rebounds from a planar wall. The method is validated by comparing the numerical simulations of a settling sphere with experimental measurements of the sphere trajectory and the accompanying flow field. The normal collision process is then studied for a range of impact Stokes numbers. A contact model of the liquid–solid interaction and elastic effect is developed that incorporates the elasticity of the solids to permit the rebound trajectory to be simulated accurately. The contact model is applied when the particle is sufficiently close to the wall that it becomes difficult to resolve the thin lubrication layer. The model is calibrated with new measurements of the particle trajectories and reproduces the observed coefficient of restitution over a range of impact Stokes numbers from 1 to 1000.

Simulations of natural and forced convection flows with moving embedded object using immersed boundary method

In the present study, an immersed-boundary method is adopted to simulate natural and forced convection within a domain with complex geometry at low Reynolds numbers. The method is based on the direct momentum and energy forcing on a Cartesian grid, and issues involving the influence of the solid-body-forcing and the implementation of the isothermal and isoflux boundary conditions are addressed. It was shown that the second-order accuracy of global error norms of the method was ascertained only when the solid-body-forcing in velocity is implemented. On the other hand, the solid-body-forcing of the thermal field is not essential. Also, the solid-body-forcing in velocity has negligible influence on the thermal and dynamic field with stationary embedded object. Further test problems are simulated to examine the validity of the present technique: 2-D flows induced by natural convection in the annulus between two horizontal concentric cylinders, moving heated cylinder within a stationary fluid, transversely oscillating cylinder with different excitation frequencies, and 3-D simulation of a heated sphere settling under gravity in a static fluid. All computed results are in general good agreement with previous experimental measurements and numerical simulations.

Settling of particles in liquids: Effects of material properties

AbstractThis article presents a numerical study on the settling of uniform spheres in liquids by means of the discrete element method. The effects of particle and liquid properties, such as particle size, Hamaker constant, liquid density, and viscosity, on the formation of packed beds or cakes were studied in terms of packing fraction, radial distribution function (RDF), and coordination number (CN). The results showed that the packing fraction of a cake increases with increasing particle size but decreases with increasing the Hamaker constant, liquid density, and viscosity. RDF and CN also change correspondingly: packings with lower packing fraction generally have RDFs with fewer peaks and smaller mean CNs. A good correlation between packing fraction and other structural properties was identified. The analysis of the particle‐particle and liquid‐particle interactions showed that the packing properties are mainly affected by the ratio of the interparticle cohesion to the effective gravity of particles. The previously proposed equation linking packing fraction with the interparticle forces has been extended to incorporate the impact‐induced pressure force in a settling process. Based on the modified equation, the effects of key variables on the relationship between packing fraction and particle size were re‐examined for general application. © 2011 American Institute of Chemical Engineers AIChE J, 2012.

Trans-Mesh Method and Its Application to Simulations of Incompressible Fluid-Rigid Bodies Interaction

In this paper, a Trans-mesh method is presented for simulating three-dimensional incompressible fluid-rigid body interaction with collisions. In the Trans-mesh method, the bodies can move freely in a main mesh that covers the entire flow field. The method is constructed based on the four-dimensional control volume in space-time unified domain such that the method assures to be divergence-free in the space-time unified domain and thus satisfies both the physical and geometrical conservation laws simultaneously. First of all, it is confirmed that the present method satisfies the geometric conservation law. Next, we did calculations for a single sphere settling under gravity in the stationary fluid to evaluate the present method. The method was applied to a flow around bodies driven by a flow in a square duct and the unsteady behavior of the flow is shown. The results indicate that this method is promising in such simulations.

Wall effects on the velocities of a single sphere settling in a stagnant and counter-current fluid and rising in a co-current fluid

Experimental results were obtained on the steady settling of spheres in quiescent media in a range of cylindrical tubes to ascertain the wall effects over a relatively wide range of Reynolds number values. For practical considerations, the retardation effect is important when the ratio of the particle diameter to the tube diameter ( λ) is higher than about 0.05. A new empirical correlation is presented which covers a Reynolds number range Re = 53–15,100 and a particle to tube diameter ratio λ < 0.88. The absolute mean deviation between the experimental data and the presented correlation was 1.9%. The well-known correlations of Newton, Munroe and Di Felice agree with the presented data reasonably well. For steady settling of spheres in a counter-current water flow, the slip velocity remains practically the same as in quiescent media. However, for rising spheres in a co-current water flow, the slip velocity decreases with increasing co-current water velocity, i.e., the wall factor decreases with increasing co-current water velocity. Consequently, the drag coefficient for rising particles in co-current water flow increases with increasing water velocity.

Direct numerical simulations of a freely falling sphere using fictitious domain method: Breaking of axisymmetric wake

In the present paper, numerical simulations of the wake generated by a freely falling sphere, under the action of gravity, are performed. Simulations have been carried out in the range of Reynolds numbers from 1 to 210 for understanding the formation, growth and breakup of the axisymmetric wake. The in-house code used is based on a non-Lagrange multiplier fictitious-domain method, which has been developed and validated by Veeramani et al. (2007). The onset of instability in the wake and its growth along with the dynamic behavior of a settling sphere is examined at Reynolds number ( Re) of 210. It is found that at the onset of instability the sphere starts to rotate and gives rise to a lift force due to the break of the axisymmetry in the wake which in turns triggers a lateral migration of the sphere. The lift coefficient of a freely falling sphere is 1.8 times that of a fixed sphere at a given sphere density of 4000 kg m −3 and sphere to fluid density ratio of 4. This is attributed to the Robin's force which arises due to the rotation of the sphere. At this Reynolds number ( Re=210) a double threaded wake is observed, which resembles the experimental observations of Magarvey and MacLatchy (1965).

Simulating flows with moving rigid boundary using immersed-boundary method

The present study is to apply the immersed-boundary method to simulate 2- and 3-D viscous incompressible flows interacting with moving solid boundaries. Previous studies indicated that for stationary-boundary problems, different treatments inside the solid body did not affect the external flow. However, the relationship between internal treatment of the solid body and external flow for moving-boundary problems was not studied extensively and is investigated here. This is achieved via direct-momentum forcing on a Cartesian grid by combining “solid-body forcing” at solid nodes and interpolation on neighboring fluid nodes. The influence of the solid body forcing within the solid nodes is first examined by computing flow induced by an oscillating cylinder in a stationary square domain, where significantly lower amplitude oscillations in computed lift and drag coefficients are obtained compared with those without solid-body-forcing strategy. Grid-function convergence tests also indicate second-order accuracy of this implementation with respect to the L 1 norm in time and the L 2 norm in space. Further test problems are simulated to examine the validity of the present technique: 2-D flows over an asymmetrically-placed cylinder in a channel, in-line oscillating cylinder in a fluid at rest, in-line oscillating cylinder in a free stream, two cylinders moving with respect to one another, and 3-D simulation of a sphere settling under gravity in a static fluid. All computed results are in generally good agreement with various experimental measurements and with previous numerical simulations. This indicates the capability of the present simple implementation in solving complex-geometry flow problems and the importance of solid body forcing in computing flows with moving solid objects.

Enhanced drag of a sphere settling in a stratified fluid at small Reynolds numbers

We present a combined experimental and numerical investigation of a sphere settling in a linearly stratified fluid at small Reynolds numbers. Using time-lapse photography and numerical modelling, we observed and quantified an increase in drag due to stratification. For a salt stratification, the normalized added drag coefficient scales as Ri0.51, where Ri = a3N2/(νU) is the viscous Richardson number, a the particle radius, U its speed, ν the kinematic fluid viscosity and N the buoyancy frequency. Microscale synthetic schlieren revealed that a settling sphere draws lighter fluid downwards, resulting in a density wake extending tens of particle radii. Analysis of the flow and density fields shows that the added drag results from the buoyancy of the fluid in a region of size (ν/N)1/2 surrounding the sphere, while the bulk of the wake does not influence drag. A scaling argument is provided to rationalize the observations. The enhanced drag can increase settling times in natural aquatic environments, affecting retention of particles at density interfaces and vertical fluxes of organic matter.

Sphere settling in an aging yield stress fluid: link between the induced flows and the rheological behavior

The flow pattern induced by the settling of a non-Brownian sphere in a fluid depends on the rheological properties of that fluid. For instance, at small Reynolds numbers, the pattern presents a fore–aft symmetry in a Newtonian fluid, whereas, in some viscoelastic polymer solutions, it can exhibit a negative wake, i.e., an upward flow in the sphere’s wake. This study is an experimental work on the settling of a sphere in a suspension of a synthetic colloidal clay, laponite. The fluid is a yield stress fluid that ages, i.e., whose rheological properties evolve over time. We show that the settling velocity of a given sphere, as well as the induced flow pattern, are strongly modified as the fluid ages. In particular, the flow pattern asymmetry increases with the age of the fluid, and a negative wake eventually forms. We relate those modifications to rheological measurements and suggest, in line with works dealing with polymer solutions, that it is the increase in the fluid viscoelasticity that is responsible for the formation of a negative wake. The flow field measurements are also compared with flow-induced birefringence measurements, and we show that very slow relaxation processes are involved in the sphere settling.

Wall Effects on a Sphere Falling in Quiescent Power Law Fluids in Cylindrical Tubes

The effect of finite boundaries on the drag experienced by a sphere settling in quiescent power law fluids in cylindrical vessels has been investigated numerically. In particular, the momentum equations have been solved numerically over the following ranges of conditions: sphere Reynolds number, 1−100; power law index, 0.2−1; and sphere-to-tube diameter ratio, 0−0.5. Due to the backflow of the fluid caused by a falling sphere and the corresponding changes in the velocity field close to the sphere, the presence of finite boundaries leads to an increase in the drag force acting on a falling sphere thereby slowing its descent. The effect, however, is more significant at low Reynolds numbers than at high Reynolds numbers. Similarly, the additional drag due to the walls increases with the increasing degree of confinement, i.e., the sphere-to-tube diameter ratio. Overall, all else being equal, the wall effect is less severe in power law fluids than in Newtonian fluids. Furthermore, the confining walls also influence the onset of flow separation and subsequently the size of the recirculation region. The present numerical predictions are consistent with the experimental results available in the literature.

Simulating deformable particle suspensions using a coupled lattice-Boltzmann and finite-element method

A novel method is developed to simulate suspensions of deformable particles by coupling the lattice-Boltzmann method (LBM) for the fluid phase to a linear finite-element analysis (FEA) describing particle deformation. The methodology addresses the need for an efficient method to simulate large numbers of three-dimensional and deformable particles at high volume fraction in order to capture suspension rheology, microstructure, and self-diffusion in a variety of applications. The robustness and accuracy of the LBM–FEA method is demonstrated by simulating an inflating thin-walled sphere, a deformable spherical capsule in shear flow, a settling sphere in a confined channel, two approaching spheres, spheres in shear flow, and red blood cell deformation in flow chambers. Additionally, simulations of suspensions of hundreds of biconcave red blood cells at 40% volume fraction produce continuum-scale physics and accurately predict suspension viscosity and the shear-thinning behaviour of blood. Simulations of fluid-filled spherical capsules which have red-blood-cell membrane properties also display deformation-induced shear-thinning behaviour at 40% volume fraction, although the suspension viscosity is significantly lower than blood.

Effect of underfill filler settling on thermo-mechanical fatigue analysis of flip-chip eutectic solders

Underfills containing filler particles exhibit filler settling during the (capillary-based) wicking and curing processes, thus causing the reliability estimation to deviate from that of the presumed base of no filler settling. This paper examines the thermo-mechanical responses of the solder joints in flip-chip packaging to various conditions of filler settling. We built five y-dependent profiles for describing the uniform, bilayered, and gradual settling of filler spheres in the through-depth direction of the underfill and used the Mori–Tanaka method to calculate the effective material properties of the filler–resin underfill compound by considering a linearly elastic, temperature-dependent resin with a glass transition temperature range of 70–130 °C. For each settling profile we analyzed the fatigue indicators, referred to as the inelastic shear strain ranges and the inelastic shear strain energy densities of the solder joints, and compared their magnitudes against the extent of filler settling. The results show that the fatigue indicators depend on the extent of filler settling. A greater extent of bilayered filler settling produced larger (in magnitude) fatigue indicators. The fatigue indicators associated with gradual filler settling, however, were almost always smaller, on average, than those associated with no filler settling, indicating that some types of filler settling might favor a longer solder fatigue life. This preliminary but intriguing finding may be partially explained by considering the asymmetric thermal mismatch in the through-depth direction of the underfill; a comparatively good thermal match near the bottom side of the solder joints may compensate for the thermal mismatch at the top side, thus contributing to an overall better thermal match in the solder joint.

21216 界面活性剤水溶液での連続沈降球の速度変化(マイクロバブル・界面現象,一般講演,学術講演)

In the present study, surfactant solution systems of CTAB/NaSal were used to examine the motion of a solid sphere falling through the fluid At relatively low salt concentrations, the terminal settling velocity increases with the number of successive sphere drops, and the oscillations in the velocity of a falling sphere are observed. On the other hand, at relatively high salt concentrations, no discernible change in the settling velocity is observed, and the spheres reach a terminal velocity without oscillations. The sudden acceleration of the sphere during sedimentation is related to the formation and breaking of shear-induced-state at the trail of a settling sphere.