Sedimentation Of Sphere Research Articles

Influence of concentration on thermophoresis of spherical aerosol particles within a Brinkman medium

Abstract We are examining the thermophoretic movement of a uniform mixture of spherical aerosol particles, all with the same properties, as they are situated within a porous material. These particles can have various thermal conductivity and surface characteristics. This analysis focuses on situations where the Péclet and Reynolds numbers are small. The influence of particle interactions is carefully considered by using a unit cell model, a well-established method known for its accurate predictions in the context of sedimentation for monodisperse suspensions of spherical particles. The porous medium is represented as a Brinkman fluid characterized by a Darcy permeability, which can be determined directly from experimental observations. This medium is considered to be uniform and isotropic, and the solid matrix is in thermal equilibrium with the fluid flowing through the voids of the medium. The Knudsen number is assumed to be low, enabling the description of fluid flow through the porous medium using a continuum model that includes temperature jump, thermal creep, frictional slip, and thermal stress slip at the aerosol particle’s surface. The conservation equations for energy and momentum are individually tackled within each cell. In this model, each cell represents a spherical particle enclosed by a concentric shell of surrounding fluid. The thermophoretic particle migration velocity is determined across different cases. We derive analytical expressions for this average particle velocity, expressing it in terms of the particle volume fraction. It is observed that different cell models yield somewhat varied results for particle velocity. Generally, with a fixed permeability parameter characterizing the porous medium, an increase in the thermal stress slip coefficient tends to decrease the normalized thermophoretic velocity across the different cell models. The results are in good agreement with the available data as documented in the existing literature. Additionally, a parallel examination of aerosol sphere sedimentation is provided.

Emergence of lobed wakes during the sedimentation of spheres in viscoelastic fluids

The motion of rigid particles in complex fluids is ubiquitous in natural and industrial processes. The most popular toy model for understanding the physics of such systems is the settling of a solid sphere in a viscoelastic fluid. There is general agreement that an elastic wake develops downstream of the sphere, causing the breakage of fore-and-aft symmetry, while the flow remains axisymmetric, independent of fluid viscoelasticity and flow conditions. Using a continuum mechanics model, we reveal that axisymmetry holds only for weak viscoelastic flows. Beyond a critical value of the settling velocity, steady, non-axisymmetric disturbances develop peripherally of the rear pole of the sphere, giving rise to a four-lobed fingering instability. The transition from axisymmetric to non-axisymmetric flow fields is characterized by a regular bifurcation and depends solely on the interplay between shear and extensional properties of the viscoelastic fluid under different flow regimes. At higher settling velocities, each lobe tip is split into two new lobes, resembling fractal fingering in interfacial flows. For the first time, we capture an elastic fingering instability under steady-state conditions, and provide the missing information for understanding and predicting such instabilities in the response of viscoelastic fluids and soft media.

Sedimentation of a Charged Soft Sphere within a Charged Spherical Cavity.

The sedimentation of a soft particle composed of an uncharged hard sphere core and a charged porous surface layer inside a concentric charged spherical cavity full of a symmetric electrolyte solution is analyzed in a quasi-steady state. By using a regular perturbation method with small fixed charge densities of the soft sphere and cavity wall, a set of linearized electrokinetic equations relevant to the fluid velocity field, electrical potential profile, and ionic electrochemical potential energy distributions are solved. A closed-form formula for the sedimentation velocity of the soft sphere is obtained as a function of the ratios of core-to-particle radii, particle-to-cavity radii, particle radius-to-Debye screening length, and particle radius-to-porous layer permeation length. The existence of the surface charge on the cavity wall increases the settling velocity of the charged soft sphere, principally because of the electroosmotic enhancement of fluid recirculation within the cavity induced by the sedimentation potential gradient. When the porous layer space charge and cavity wall surface charge have the same sign, the particle velocity is generally enhanced by the presence of the cavity. When these fixed charges have opposite signs, the particle velocity will be enhanced/reduced by the presence of the cavity if the wall surface charge density is sufficiently large/small relative to the porous layer space charge density in magnitude. The effect of the wall surface charge on the sedimentation of the soft sphere increases with decreases in the ratios of core-to-particle radii, particle-to-cavity radii, and particle radius-to-porous layer permeation length but is not a monotonic function of the ratio of particle radius-to-Debye length.

Dynamics of the surface growth resulted from sedimentation of spheres in a Hele–Shaw cell containing a low-viscosity fluid

In this paper, we investigate the dynamics of surface growth resulting from sedimentation of spherical granular particles in a fluid environment, using experiments and simulations. In the experimental part, spherical polystyrene particles are poured down from the top of a vertical Hele–Shaw cell and form a 1 + 1-dimensional growing surface. The surface roughness is obtained from the images, and the growth and roughness exponents are measured. In the numerical simulation part, the surface growth process is simulated using the discrete element method, considering the interactions between the grains, and the exponents are calculated. In this method, unlike conventional simulation models, instead of a discrete deposition law, the dynamics of the individual particles throughout the process are obtained, considering different forces acting on the particles. Finally, the simulation results are compared with the experiment, and we see a very good agreement between them. We find different values for the exponents using different methods, indicating that the system is multi-affine and does not obey the scaling laws of affine models.

An eigenvalue-free implementation of the log-conformation formulation

The log-conformation formulation, although highly successful, was from the beginning formulated as a partial differential equation that contains an, for PDEs unusual, eigenvalue decomposition of the unknown field. To this day, most numerical implementations have been based on this or a similar eigenvalue decomposition, with Knechtges et al. (2014) being the only notable exception for two-dimensional flows.In this paper, we present an eigenvalue-free algorithm to compute the constitutive equation of the log-conformation formulation that works for two- and three-dimensional flows. Therefore, we first prove that the challenging terms in the constitutive equations are representable as a matrix function of a slightly modified matrix of the log-conformation field. We give a proof of equivalence of this term to the more common log-conformation formulations. Based on this formulation, we develop an eigenvalue-free algorithm to evaluate this matrix function. The resulting full formulation is first discretized using a finite volume method, and then tested on the confined cylinder and sedimenting sphere benchmarks.

Holographic analysis of colloidal spheres sedimenting in horizontal slit pores.

The mobility of a colloidal particle in a slit pore is modified by the particle's hydrodynamic coupling to the bounding surfaces and therefore depends on the particle's position within the pore and its direction of motion. We report holographic particle tracking measurements of colloidal particles' diffusion and sedimentation between parallel horizontal walls that yield the mobility for motions perpendicular to the walls, including its dependence on height within the channel. These measurements complement previous studies that probed colloidal mobility parallel to confining surfaces. When interpreted with effective-medium theory, holographic characterization measurements yield estimates for the sedimenting spheres' densities that can be compared with kinematic values to draw insights into the spheres' compositions. This comparison suggests, for example, that the silica spheres used in this study are slightly porous, but that their pores are too small for water to penetrate.

LBM-DEM simulation of particle deposition and resuspension of pre-deposited dynamic membrane

Pre-deposited dynamic membrane (DM) has been widely applied in filtration processes, yet the microscale quantitative analysis of DM particles deposition and resuspension has not been studied. In this study, the fundamental physics of DM particles deposition and resuspension on the support filter surface in a channel is numerically revealed by a user-defined solver via combining the Discrete Element Method with Lattice Boltzmann Method (LBM-DEM). After comprehensive model validations in terms of a single sphere sedimentation and two spheres sedimentation in water, the effect of particle mass proportion, particle size, and fluid inlet velocity on DM particles deposition on support filter surface is studied. A critical and a threshold particle mass proportion and fluid inlet velocity are identified under the given conditions. Furthermore, the impact of dynamic membrane thickness (L) and backwash velocity (Vf,b) on DM particles resuspension from support filter surface is explored. The time required for DM complete resuspension is correlated as a function of L and Vf,b. The proposed correlation is then verified against the empirical correlations in the open literature, demonstrating its reasonability in predicting DM particles behaviours during the backwashing. The microscale findings guide the design and optimization of dynamic membranes.

Interface-resolved simulations of the confinement effect on the sedimentation of a sphere in yield-stress fluids

We perform three-dimensional numerical simulations to investigate the confinement effect on the sedimentation of a single sphere in an otherwise quiescent yield stress fluid, in the presence of finite elasticity and weak inertia. The carrier fluid is modeled using the elastoviscoplastic constitutive laws proposed by Saramito (2009). The additional elastic stress tensor is fully coupled with the flow equation, while the rigid particle is represented by an immersed boundary method. The simulations show the faster relaxation of the fluid velocity and the progressive translation of the location of the negative wake downstream of the sphere as the bounding walls are brought closer to the particle. Moreover, the sphere drag decreases by increasing the particle–wall distance. We show that the confinement ratio (ratio of the gap between rigid confining walls and the sphere radius) reaches a critical value beyond which the wall-effect on the particle and flow dynamics becomes negligible. The key finding here is that the critical confinement ratio and the maximum variation of the Stokes drag with confinement ratio are weakly dependent on the level of material elasticity and plasticity for a certain range of material parameters. Finally, we propose an expression for the Stokes drag coefficient, as a function of material plasticity and confinement ratio.

Dynamics of elastic, nonheavy spheres sedimenting in a rectangular duct.

Understanding of sedimentation dynamics of particles in bounded fluids is of crucial importance for a wide variety of processes. While there is a profound knowledge base regarding the sedimentation of rigid solid particles, the fundamental principles of sedimentation dynamics of elastic spheres in bounded fluids are not well understood. This especially applies to nonheavy spheres, whose density is close to that of the surrounding medium and which therefore show extended inertial effects upon acceleration. Here, we present model experiments of the sedimentation dynamics of deformable, nonheavy spheres in the presence of walls. Despite the deformations of the particles being small, the particle dynamics of elastic spheres differed fundamentally from that of rigid spheres. Initially, the sedimentation of elastic spheres is comparable with the sedimentation of rigid spheres. From a characteristic onset position of about 10·R, deformability effects kick in and a second acceleration appears. Finally, the deformable spheres reach a terminal sedimentation velocity. The softer the spheres are (in terms of Young's elastic modulus), the higher the terminal velocity is. In the present setup, a terminal velocity up to 9% higher than the velocity for comparable rigid spheres was reached. By analyzing the obtained data, insights into the dynamics are given that could serve as basic approaches for modelling the dynamics of elastic spheres in bounded fluids.

Unsteady motion past a sphere translating steadily in wormlike micellar solutions: a numerical analysis

Abstract

Sedimentation of two spheres in a square tube

In this study, we simulated the sedimentation of two identical spheres having the same density in a square tube. Compared with the center-line and the diagonal planes (including the reverse diagonal plane), the sedimentation of spheres on other planes is more complicated. Results show that at relatively low and high Reynolds number, the spheres will deflect and eventually move to the diagonal plane of the square tube. At the medium Reynolds number, the spheres settle near the initial plane. The possible mechanisms underlying these behaviors are examined. Finally, it is shown that the distance between the spheres increases with an increase in the Reynolds number, which is applicable to all the initial settlement planes studied.

Waves, Algebraic Growth, and Clumping in Sedimenting Disk Arrays

An array of spheres descending slowly through a viscous fluid always clumps [J.M. Crowley, J. Fluid Mech. {\bf 45}, 151 (1971)]. We show that anisotropic particle shape qualitatively transforms this iconic instability of collective sedimentation. In experiment and theory on disks, aligned facing their neighbours in a horizontal one-dimensional lattice and settling at Reynolds number $\sim 10^{-4}$ in a quasi-two-dimensional slab geometry, we find that for large enough lattice spacing the coupling of disk orientation and translation rescues the array from the clumping instability. Despite the absence of inertia the resulting dynamics displays the wavelike excitations of a mass-and-spring array, with a conserved "momentum" in the form of the collective tilt of the disks and an emergent spring stiffness from the viscous hydrodynamic interaction. However, the non-normal character of the dynamical matrix leads to algebraic growth of perturbations even in the linearly stable regime. Stability analysis demarcates a phase boundary in the plane of wavenumber and lattice spacing, separating the regimes of algebraically growing waves and clumping, in quantitative agreement with our experiments. Anisotropic shape thus suppresses the classic linear instability of sedimenting sphere arrays, introduces a new conserved variable, and opens a window to the physics of transient growth of linearly stable modes.

An improved immersed moving boundary for hydrodynamic force calculation in lattice Boltzmann method

SummaryParticles suspension is considerably prevalent in petroleum industry and chemical engineering. The efficient and accurate simulation of such a process is always a challenge for both the traditional computational fluid dynamics and lattice Boltzmann method. Immersed moving boundary (IMB) method is promising to resolve this issue by introducing a particle‐fluid interaction term in the standard lattice Boltzmann equation, which allows for the smooth hydrodynamic force calculation even for a large grid size relative to the solid particle. Although the IMB method was proved good for stationary particles, the deviation of hydrodynamic force on moving particles exists. In this work, we reveal the physical origin of this problem first and figure out that the internal fluid effect on the hydrodynamic force calculation is not counted in the previous IMB. An improved immersed moving boundary method is therefore proposed by considering the internal fluid correction, which is easy to implement with the little extra computation cost. A 2D single elliptical particle and a 3D sphere sedimentation in Newtonian fluid is simulated directly for the validation of the corrected model by excellent agreements with the standard data.

Lubricated contact model for numerical simulations of suspensions

Discrete granular models are a natural choice when simulating dense suspensions, when the small distances between the particles lead to dominant contributions by lubrication and contact forces. In such case one can get rid of the costly resolution of Navier-Stokes equations, using closed form expressions for lubrication terms. However, those terms diverge when two hard spheres approach contact, and there are issues when integrating them directly with the finite precision of floating point calculations. In this paper, we introduce a visco-elasto-plastic interaction model for suspended spheres, which combines lubrication and elastic-frictional contact behaviour depending on surface roughness. An integration scheme is proposed for that model. Unlike earlier methods, the scheme enables an unconditionally stable time-integration of the interactions. The case of perfectly smooth spheres (null roughness), namely, is integrated correctly. The theoretical results are well reproduced in benchmark tests on two-sphere systems: one sphere sedimenting on one other and two spheres in a shear flow. From these benchmark tests, we propose phase diagrams showing the interplay between viscosity, roughness and stiffness. The second test case highlights the origin of non-reversibility particle trajectories. It is controlled by the particle roughness for rigid particles, and by the particle deformation when the capillary number is higher than the relative roughness.

Numerical simulations of a sphere settling in simple shear flows of yield stress fluids

We perform $3$D numerical simulations to investigate the sedimentation of a single sphere in the absence and presence of a simple cross shear flow in a yield stress fluid with weak inertia. In our simulations, the settling flow is considered to be the primary flow, whereas the linear cross shear flow is a secondary flow with amplitude $10\%$ of the primary flow. To study the effects of elasticity and plasticity of the carrying fluid on the sphere drag as well as the flow dynamics, the fluid is modeled using the elastovisco-plastic (EVP) constitutive laws proposed by \cite{saramito2009new}. The extra non-Newtonian stress tensor is fully coupled with the flow equation and the solid particle is represented by an immersed boundary (IB) method. Our results show that the fore-aft asymmetry in the velocity is less pronounced and the negative wake disappears when a linear cross shear flow is applied. We find that the drag on a sphere settling in a sheared yield stress fluid is reduced significantly as compared to an otherwise quiescent fluid. More importantly, the sphere drag in the presence of a secondary cross shear flow cannot be derived from the pure sedimentation drag law owing to the non-linear coupling between the simple shear flow and the uniform flow. Finally, we show that the drag on the sphere settling in a sheared yield-stress fluid is reduced at higher material elasticity mainly due to the form and viscous drag reduction.

Simulation of sedimentation of two spheres with different densities in a square tube

We simulated the sedimentations of two unequal spheres with different densities in a square tube for Galileo numbers (Ga) from 5 to 25, resulting in a Reynolds number range of based on the terminal settling velocity. The sedimentation of spheres with different densities is dynamically more complex than that of identical spheres. At high Ga the spheres oscillate in the centreline plane of the tube, where they are initially released from rest. By contrast, the spheres move to the diagonal or reverse–diagonal plane of the tube at low Ga, reaching a steady or periodic state depending on the density difference between them. A phase diagram illustrates the transitions between different sedimentation behaviours depending on Ga and the density difference. A possible mechanism for these behaviours is also presented. Furthermore, we compare two-dimensional (2-D) and three-dimensional computations for our system to attain a better understanding of the hydrodynamic interactions between two unequal spheres at low but finite Reynolds number. Comparing relative trajectories, periods of oscillation and flow features shows that 2-D circular cylinders oscillate much more strongly and frequently than spheres under the same flow conditions. In particular, spheres do not have the discontinuity in period that arises in the 2-D case from the change in rotation sign of a heavy particle.

Microsphere Adhesion on Rubber Films Accompanied by Sphere Sedimentation.

Understanding the adhesion process between a rigid material (filler) and a viscoelastic material is important for designing an enhanced industrial material. However, the adhesion process is not simple because the properties of the adhesive, adherend, and interface are intricately influenced by this process. Here we investigate the adhesion of microspheres onto rubber films to clarify the dominant factor in the adhesion process. A rubber meniscus first forms on the sphere surface, followed by sedimentation of the sphere into the rubber film. This sedimentation is even observed when the surface free energy of the sphere is lower than that of the rubber film, which indicates that the driving force of meniscus formation obeys Young's equation on a tangential line of the sphere. The dominant factor of the sedimentation behavior is investigated by using atomic force microscopy force-sample deformation curve measurements and creep tests on the rubber films. These experimental results demonstrate that the adhesion process is strongly dominated by the viscoelastic property of the bulk rubber as opposed to the sphere and interface properties.

Numerical simulation of jet mode in electrospraying of Newtonian and viscoelastic fluids

This study aims to model the role of the viscoelastic properties of polymeric solutions in the jet mode of the electrospray process numerically. For this purpose, first, the computational challenges in front of a numerical investigation of such complex physics are explained, and the recent advances in rectifying these problems, especially the high Weissenberg instability, is described. Then, a numerical procedure based on the Log-conformation method (LCM) is introduced, and several numerical simulations are performed to model the behavior of electrified Newtonian and viscoelastic jets. In the first step, the discrete model for the LCM equations is validated by simulating the benchmark problem of the sedimenting sphere. The obtained results show that the proposed algorithm can improve the previous Weissenberg threshold reported for a stable solution. Then, an electrified deionized (DI) water jet is simulated, and the obtained jet profile is compared with the experimental data. Finally, the proposed algorithm is used to simulate viscoelastic electrified jets, where the effect of the Weissenberg number (Wi) on the jet profile is examined. In agreement with the previously obtained experimental results, the numerical results show that by increasing the solution concentration, the asymptotic jet profile is reached at a smaller length from the nozzle, while the final thickness of the jet is slightly reduced.

Flow behaviour of an agitated tubular reactor using a novel dynamic mesh based CFD model

Agitated tubular reactor (ATR) is an intensified continuous reactor in which radial mixing is decoupled from axial flow. This study proposed a novel CFD model to study fluid dynamics and mixing mechanisms in the ATR system. It combines a soft-sphere collision model with a dynamic meshing approach to tackle structure-structure and fluid-structure interactions (FSI) simultaneously. Its ability to handle FSI is validated on the sedimentation of a large sphere in a quiescent fluid. A periodic motion of the agitator was predicted, which is mainly driven by collision with the external oscillating tube. The fluid motion is complex, with vortex structures forming behind the agitator at its highest point; and the highest fluid velocities seen as fluid is squeezed under the agitator. Three Reynolds stress models were evaluated and the pressure-strain term shows little influence on the agitator’s motion. Viscous dissipation was found to be the main mechanism for energy dissipation.

Sedimentation of a small sphere in stratified fluid

An extensive theoretical and numerical study of gravitational settling of small particles in a stratified fluid was performed. Adopting the integral equation on surface traction, we derived the drag enhancement due to stratification at low Reynolds and Peclet numbers, and confirmed with simulations.