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Numerical Simulations of Particulate Suspensions via a Discretized Boltzmann Equation Part II. Numerical Results
Abstract
A new and very general technique for simulating solid-fluid suspensions has been described in a previous paper (Part I); the most important feature of the new method is that the computational cost scales with the number of particles. In this paper (Part II), extensive numerical tests of the method are described; for creeping flows, both with and without Brownian motion, and at finite Reynolds numbers. Hydrodynamic interactions, transport coefficients, and the short-time dynamics of random dispersions of up to 1024 colloidal particles have been simulated. Comment: Text and figures in uuencode-tar-compressed postcript Email tony_ladd@llnl.gov
... In this paper, the isothermal incompressible fluid flows are primarily focused on, and the no-penetration and no-slip boundary conditions should be satisfied on the wall for the macroscopic fluid velocity. Hence, the bounce-back (BB) rule [59] is used to determine the desired f + and f − on both sides of the IB as follows: ...
... For all cases in this work, the local Ma number is always less than 0.3, so the flows could be considered as incompressible. For the boundary conditions, the bounce-back rule [59] is used for the straight wall boundary, while the nonequilibrium extrapolation method [62] is used for the open boundary. The number of IBM corrections N IBM , unless otherwise stated, is set as N IBM = 20. ...
In this paper, a general immersed boundary force density is introduced for the Boltzmann equation and finally expressed as the desired particle distribution function discontinuity across the immersed boundary. Because of its independence of any specific boundary conditions and any specific solvers for the Boltzmann equation, it actually establishes a unified framework to incorporate various types of boundary conditions and several different kinds of solvers for the Boltzmann equation. Hence, a particle distribution function discontinuity-based kinetic immersed boundary method (KIBM) for the Boltzmann equation is proposed based on this general immersed boundary force density. Subsequently, this paper primarily focuses on the isothermal incompressible fluid-solid flows, and uses the discrete unified gas kinetic scheme to solve the Boltzmann Bhatnagar-Gross-Krook model equation. Meanwhile, the regularized delta function and the bounce-back rule combined with an iterative IBM correction procedure are employed in obtaining the general immersed boundary force density to enforce the no-penetration and no-slip boundary conditions on the solid wall. Finally, some numerical experiments for typical incompressible fluid-solid flows show that the present KIBM could provide good agreement with other numerical and experimental results.
... The colloids are modelled as spherical particles, with a no-slip boundary condition. The no-slip boundary condition at the fluid/solid interface is realized by a standard method of bounce-back on links (BBL) [20,21] and it can be modified to take into account the moving particle surface [22]. ...
We study the dynamics of a driven spherical colloidal particle moving in a fluid with a broken rotational symmetry. Using a nematic liquid crystal as a model, we demonstrate that when the applied force is not aligned along or perpendicular to the orientational order, the colloidal velocity does not align with the force, but forms an angle with respect to the pulling direction. This leads to blue an anisotropic hydrodynamic drag tensor which depends on the material parameters. In the case of nematic liquid crystal, we give an analytical expression and discuss the resulting implications for active microrheology experiments on fluids with broken rotational symmetry.
... We used a lattice Boltzmann method (LBM) to solve the Navier-Stokes equations that have become an established method to perform DNS of forced homogeneous isotropic turbulence 32,37,38 and complex multiphase flows. 12,13,[39][40][41][42] We performed simulations in the fully periodic cubic simulation domain of 128 3 and 256 3 lattice unit (lu) size. Once the HIT was generated, we evaluated the turbulence statistics and energy-related quantities and balanced to compare and study the effects of different forcing schemes on the HIT flow. ...
Studies of forced homogeneous isotropic turbulence (HIT) of multiphase systems rely on a comprehensive understanding of the single-phase HIT flow to quantify any turbulence modifications due to injection of the dispersed phase. Here, we compare external forcing schemes to generate and sustain single-phase HIT. The considered forcing schemes, Lundgren, Arnold–Beltrami–Childress, and Mallouppas, are based on the application of the body force in physical space to inject energy into the flow at large length scales. Direct numerical simulations are performed in cubic periodic domains of 1283 and 2563 size using a lattice Boltzmann method. The range of the Taylor's Reynolds number achieved is ReλT=24.4–75.4. The Lundgren force takes the longest time to generate turbulence and produces significant fluctuations in the turbulence properties in the statistically stationary state. Additionally, this force interacts with the velocity field in the entire range of wavenumbers, which is not the case for the other two forces. However, the scale-by-scale analysis shows that for the considered forces, the behavior of the non-linear energy transfer, dissipation, and energy injection terms differs only within the initial 16% of the wavenumbers that represent large length scales. After that, all terms behave consistently among each other for different forcing schemes. We conclude that the three considered large-scale forcing schemes do not affect the generated turbulent flow fields at small scales and can be used to study turbulence modification by the dispersed phase.
... To account for the no-slip boundary condition on a moving particle surface, the 'bounce back on links' algorithm is applied 38 . Specifically, we consider a density-matched suspension (ρ ¼ ρ fluid ¼ ρ particle ¼ 1) and set the LBM lattice spacing, Δx; to be 1 and time unit, Δt, to be 1. ...
Swarms of soft microrobots controlled by minimally invasive magnetic fields show promise as biomedical agents. The collective behaviour of such swarms, governed by magnetic and hydrodynamic interactions, emerges from the properties of their individual constituents. The introduction of both magnetic and structural anisotropy into microrobots expands the possibilities for tailoring and predetermining interactions and collective behaviours that result. Unfortunately, current methods for large-scale production of soft microrobots, typically result in isotropic properties. Herein, by combining simulation-guided design and droplet-based microfluidics, we present a versatile, high-throughput technique for fabricating soft microrobots with programmable structural and magnetic anisotropy. Such microrobots consist of iron oxide nanoparticles organized into supra-domain structures and entrapped in a hydrogel matrix that can be elongated independently of its magnetic properties. By applying rotating magnetic fields to resulting swarms, distinct collective behaviours are produced, including gas-like formations, variable crystals, and heterogeneous motions.
... Here the original LB method [42,49] is considered as the one with the flux m = ρu, and the corrected LB method denotes the one with the flux m = ρu and the additional interface force [40,47]. In the following simulations, the MRT collision operator with D2Q9 lattice (the transformation matrix M and the moments of distribution functions are shown in Appendix D) is adopted for all LB methods, and the half-way bounce-back scheme [50,51] is applied for the no-flux and no-slip velocity boundary conditions. The relaxation parameters corresponding to mobility and viscosity are given by s 3 ...
In this work, we consider a general consistent and conservative phase-field model for the incompressible two-phase flows. In this model, not only the Cahn-Hilliard or Allen-Cahn equation can be adopted, but also the mass and the momentum fluxes in the Navier-Stokes equations are reformulated such that the consistency of reduction, consistency of mass and momentum transport, and the consistency of mass conservation are satisfied. We further develop a lattice Boltzmann (LB) method, and show that through the direct Taylor expansion, the present LB method can correctly recover the consistent and conservative phase-field model. Additionally, if the divergence of the extra momentum flux is seen as a force term, the extra force in the present LB method would include another term which has not been considered in the previous LB methods. To quantitatively evaluate the incompressibility and the consistency of the mass conservation, two statistical variables are introduced in the study of the deformation of a square droplet, and the results show that the present LB method is more accurate. The layered Poiseuille flow and a droplet spreading on an ideal wall are further investigated, and the numerical results are in good agreement with the analytical solutions. Finally, the problems of the Rayleigh-Taylor instability, a single rising bubble, and the dam break with the high Reynolds numbers and/or large density ratios are studied, and it is found that the present consistent and conservative LB method is robust for such complex two-phase flows.
... The hydrodynamic force acted on the square cylinder is calculated by the momentum-exchange methods. 13,65,66 Figure 17 exhibits the time histories of the drag and lift coefficients under various K. When K ¼ 9 2 % 1=1000 and 1/500, a single oscillation frequency of both drag and lift coefficients can be observed in Figs A power spectrum analysis for the time histories of the drag and lift coefficients is performed. ...
Bounce-back schemes represent the most popular boundary treatments in the lattice Boltzmann method (LBM) when reproducing the no-slip condition at a solid boundary. While the lattice Boltzmann equation used in LBM for interior nodes is known to reproduce the Navier-Stokes (N-S) equations under the Chapman-Enskog (CE) approximation, the unknown distribution functions reconstructed from a bounce-back scheme at boundary nodes may not be consistent with the CE approximation. This problem could lead to undesirable effects such as non-physical slip velocity, grid-scale velocity and pressure noises, the local inconsistency with the N-S equations, and sometimes even a reduction of the overall numerical-accuracy order of LBM. Here we provide a systematic study of these undesirable effects. We first derive the explicit structure of the mesoscopic distribution function for interior nodes. Then the bounce-back distribution function is examined to identify the hidden errors. It is shown that the relaxation parameters in the collision models play a key role in determining the magnitude of the hidden error terms, and there exists an optimal setting which can suppress or eliminate most of these undesirable effects. While the existence of this optimal setting is derived previously for unidirectional flows, here we show that this optimal setting can be extended to non-uniform flows under certain conditions. Finally, a systematic numerical benchmark study is carried out, including non-uniform and unsteady flows. It is shown that, in all these flows, our theoretical analyses of the hidden errors can guide us to significantly improve the quality of the simulation results.
... In [4], Stokes obtained an analytical solution for sedimentation of a single spherical particle in an unbounded medium. A lot of numerical approaches in recent years are established to simulate settling particles including the finite element [5-7], Lattice-Boltzmann [8,9], Immersed Boundary method(IBM) [10] and boundary element methods [11,12], etc. In [10], authors have extensively studied numerical simulations of circular particle in two dimensions, and the authors found the results got were quite close to the existence analytical results. ...
In this research, we studied the settling of an impermeable and permeable planktonic shaped structure in a 2D rectangular fluid domain filled with viscous, Newtonian, incompressible fluid. We conducted numerical studies of the problem using Immersed boundary method (IBM). Parametric studies were conducted to study the variations in parameters, namely structure’s density, permeability of the immersed structure and fluid’s dynamic viscosity on settling velocity(terminal velocity) of the particle. In this research, the authors examined the impact of structure’s initial orientation on terminal velocity. In this work, we also established analytically, as well as numerically, that terminal velocity of permeable structure is higher than as compared to an impermeable structure and with the increasing permeability, the terminal velocity also increases.
... In order to deal with the issue, the particle reflection approach, named 211 as simple bounce-back method of first-order accuracy is utilized. 57 However, two error sources were 212 found relating to staircase discretization and artificial slip velocity due to the usage of relaxation rate. ...
The flow around three elliptic cylinders with equal spacing and aspect ratio in tandem arrangements was numerically investigated through direct numerical simulation. The spacing ratio (L/D, where D and L are the major axis and the center-to-center distance of two adjacent elliptic cylinders, respectively) ranging from 1.5 to 10 and the Reynolds numbers of 𝑅𝑒=65−160 (based on D) are examined. The analysis aims at the effects of L/D and Re on wake structures, hydrodynamic forces, and Strouhal numbers and correlates them with the underlying flow physics. The flow is highly changeable to Re and L/D, classifying into five distinct regimes, namely, meandering, overshoot, reattachment, quasi-coshedding, and coshedding. Two vortex shedding frequencies for middle and downstream cylinders are observed in the latter two regimes, indicating the significant wake interference, where three vortex shedding modes are spatially observed including primary, two-layered, and secondary. The transition between two adjacent modes forms two boundaries. At the first boundary, vortices divert from the cylinder centerline and follow two layers, while vortices converge the cylinder centerline at the second boundary. The first boundary location is not stationary at 𝑅𝑒=65–100, while it is stationary at Re = 160. Otherwise, the second boundary location moves upstream with an increase in L/D, while the range of movement decreases with an increase in Re. The increase in Re advances the disturbance level and urges the transition between vortex shedding modes. The time-mean lift and drag coefficients for three cylinders are highly sensitive with an increase in L/D.
... The top wall (y=H) is at constant temperature while remaining walls are assumed 235 as adiabatic. At solid boundaries, we consider no-slip boundary condition while the bounce-back 236 boundary condition[57] is used for distribution functions, for the left boundary as example:where n being the lattice on the boundary. The bounce-back boundary scheme is applied to the 238 adiabatic walls while the top wall temperature is set constant temperature, θh=1, where the ...
The aim of this paper is to study the potential of two heat transfer enhancement techniques, namely adding fins and hybrid nanoparticle (Cu and Al2O3), for improvement of the melting process of PCM (water) in an inclined rectangular enclosure as model of cold energy storage systems. Phase change and heat transfer of PCM are modelled using an in-house built code based on Lattice Boltzmann Method, which is validated with experimental and numerical results from literature. Enhancement effect is investigated through the main parameters including number of fins, fin length ratio (W/H) and nanoparticle volume fraction (ϕ). Heat transfer and fluid flow, liquid fraction, transient evolution of melting front, required time for full melting, and thermal energy storage are analysed in detail. The results show that the melting rate is about two times higher for the fin length ratio of W/H = 0.75, compared to W/H = 0.25. Besides, nanoparticle loading further enhances the melting rate. The highest melting rate is attained in the case of 3-fins and nanoparticle concentration of ϕ = 6 vol% with a 33.5% reduction in the complete melting time. Incrementing the fin length ratio increases the thermal energy stored, and the melting time can be shortened by up to 64%.
... Grad [31] 等 的矩方法在一维空间至少到五阶矩是吻合 的，从第六阶矩开始存在偏差。累计量碰撞模型确实 取得了良好的数值稳定性，并被应用到湍流边界层模 拟 [32] 、气动声学模拟 [33] ，以及自由面两相流模拟 [34] 。 I. V. Karlin 等 [35][36][37][38] Ladd [39] 于1994年提出，并成功应用到粒子沉降流动 中 [40][41] 殊版本。几乎同时，H. Chen等 [43] 也讨论了与J. ...
Numerical stability issue has always been the bottleneck that plagues the wide application of lattice Boltzmann methods in the flow scenarios of high Reynolds number and high Mach number. To improve the numerical stability and accuracy of the collision models in the field of lattice Boltzmann methods has been a research hotspot and difficulty in the past three decades. As an important branch of many collision models, regularized collision model has made massive theoretical progress in recent years, and has been widely applied in the fields of high Reynolds number turbulent flow, high Mach number compressible flow and aeroacoustics flow. The present paper systematically reviews the development history of the regularized collision models, and establishes a systematic theoretical framework to illustrate the underlying theoretical connections among different regularized collision models. Meanwhile, the present work reveals the theoretical essence of the regularized multi-relaxation collision model, which is to rapidly relax the higher-order kinetic moments that cannot be expressed by the discrete lattice model in the coordinate-independent and self-similar moment space, so as to avoid the fake modes affecting the hydrodynamic equations.
Comprehensive coherent structures around surface-mounted low aspect ratio square cylinder in uniform flow of oblique angle of $45^o$ were investigated for cylinder-width-based Reynolds numbers of 3000 and 10000 by direct numerical simulation based on topology-confined mesh refinement framework. High-resolution simulations and critical-point concept were scrutinized to reveal for the first time the reasonable and compatible topologies of flow separation and complete near-wall structures, due to their extensive impact on various engineering applications. Large-scale horseshoe vortices are observed at two notable foci in viscous sublayer. Within this layer, wall-parallel jet is formed by downflow intruding into bottom surface at half-saddle point, then deflecting in upstream direction, and finally penetrating bottom surface until the half-saddle point. Pair of conical vortex on cylinder's top surface switches themselves on two sides of square cylinder, where switching frequency is identical with that of the sway of side shear layer. The undulation of Kelvin Helmholtz instability is identified in the instantaneous development of conical vortex and side shear layer, where scaling of ratio of Kelvin Helmholtz and von Karman frequency follows the power-law relation obtained by Lander \textit{et al.} (\textit{J. Fluid Mech.}, vol. 849, 2018, p. 1096–1119). Large-scale arch-shaped vortex is often detected in intermediate wake region of square cylinder, involving two interconnected portions, such as leg portion separated from leeward surfaces and head portion rolled up from top surface. The leg portion of arch-shaped vortex was rooted by two foci near the bottom-surface plane.
The hydrodynamic stresses created by active particles can destabilize orientational order present in the system. This is manifested, for example, by the appearance of a bend instability in active nematics or in quasi-two-dimensional living liquid crystals consisting of swimming bacteria in thin nematic films. Using large-scale hydrodynamics simulations, we study a system consisting of spherical microswimmers within a three-dimensional nematic liquid crystal. We observe a spontaneous chiral symmetry breaking, where the uniform nematic state is kneaded into a continuously twisting state, corresponding to a helical director configuration akin to a cholesteric liquid crystal. The transition arises from the hydrodynamic coupling between the liquid crystalline elasticity and the swimmer flow fields, leading to a twist-bend instability of the nematic order. It is observed for both pusher (extensile) and puller (contractile) swimmers. Further, we show that the liquid crystal director and particle trajectories are connected: in the cholesteric state the particle trajectories become helicoidal.
The lymphatic system plays a vital role in maintaining fluid balance in living tissue and serves as a pathway for the transport of antigen, immune cells and metastatic cancer cells. Here, we investigate how the movement of cells through a contracting lymphatic vessel differs from steady flow using a lattice Boltzmann-based computational model. Our model consists of cells carried by flow in a 2D vessel with regularly spaced, bi-leaflet valves that ensure net downstream flow as the vessel walls contract autonomously in response to calcium and nitric oxide levels regulated by stretch and shear stress levels. The orientation of the vessel with respect to gravity, which may oppose or assist fluid flow, significantly modulates cellular motion due to its effect on the contraction dynamics of the vessel, even when the cells themselves are neutrally buoyant. Additionally, our model shows that cells are carried along with the flow, but when the vessel is actively contracting, they move faster than the average fluid velocity. We also find that the fluid forces cause significant deformation of the compliant cells, especially in the vicinity of the valves. Our study highlights the importance of considering the complex, transient flows near the valves in understanding cellular motion in lymphatic vessels.
With the rapid development of studies involving droplet microfluidics, drug delivery, cell detection, and microparticle synthesis, among others, many scientists have invested significant efforts to model the flow of these fluid-filled bodies. Motivated by the intricate coupling between hydrodynamics and the interactions of fluid-filled bodies, several methods have been developed. The objective of this review is to present a compact foundation of the methods used in the literature in the context of lattice Boltzmann methods. For hydrodynamics, we focus on the lattice Boltzmann method due to its specific ability to treat time- and spatial-dependent boundary conditions and to incorporate new physical models in a computationally efficient way. We split the existing methods into two groups with regard to the interfacial boundary: fluid-structure and fluid-fluid methods. The fluid-structure methods are characterised by the coupling between fluid dynamics and mechanics of the flowing body, often used in applications involving membranes and similar flexible solid boundaries. We further divide fluid-structure-based methods into two subcategories, those which treat the fluid-structure boundary as a continuum medium and those that treat it as a discrete collection of individual springs and particles. Next, we discuss the fluid-fluid methods, particularly useful for the simulations of fluid-fluid interfaces. We focus on models for immiscible droplets and their interaction in a suspending fluid and describe benchmark tests to validate the models for fluid-filled bodies.
Brownian motion (BM) is pivotal in natural science for the stochastic motion of microscopic droplets. In this study, we investigate BM driven by thermal composition noise at submicro scales, where intermolecular diffusion and surface tension both are significant. To address BM of microscopic droplets, we develop two stochastic multiphase-field models coupled with the full Navier-Stokes equation, namely, Allen-Cahn-Navier-Stokes and Cahn-Hilliard-Navier-Stokes. Both models are validated against capillary-wave theory; the Einstein's relation for the Brownian coefficient D*∼kBT/r at thermodynamic equilibrium is recovered. Moreover, by adjusting the co-action of the diffusion, Marangoni effect, and viscous friction, two nonequilibrium phenomena are observed. (I) The droplet motion transits from the Brownian to Ballistic with increasing Marangoni effect which is emanated from the energy dissipation mechanism distinct from the conventional fluctuation-dissipation theorem. (II) The deterministic droplet motion is triggered by the noise induced nonuniform velocity field which leads to a novel droplet coalescence mechanism associated with the thermal noise.
Using a hybrid simulation approach that combines a lattice-Boltzmann method for fluid flow and a molecular dynamics model for polymers, we investigate the inertial migration of star-like and crew-cut polymer micelles in a square microchannel. It is found that they exhibit two types of equilibrium positions, which shift further away from the center of the microchannel when the Reynolds number (Re) increases, as can be observed for soft particles. What differs from the behaviors of soft particles is that here, the blockage ratio is no longer the decisive factor. When the sizes are the same, the star-like micelles are always relatively closer to the microchannel wall as they gradually transition from spherical to disc-like with the increase of Re. In comparison, the crew-cut micelles are only transformed into an ellipsoid. Conversely, when the hydrophobic core sizes are the same, the equilibrium position of the star-like micelles becomes closer to that of the crew-cut micelles. Our results demonstrate that for polymer micelles with a core-shell structure, the equilibrium position is no longer solely determined by their overall dimensions but depends on the core and shell's specific dimensions, especially the hydrophobic core size. This finding opens up a new approach for achieving the separation of micelles in inertial migration.
We study the morphology of the Saturn ring defect and director structure around a colloidal particle with normal anchoring conditions and within the flow of the nematic host phase through...
In this paper, we propose a numerical model to simulate gas–liquid–solid interaction problems, coupling the lattice Boltzmann method and discrete element method (LBM–DEM). A cascaded LBM is used to simulate the liquid–gas flow field using a pseudopotential interaction model for describing the liquid–gas multiphase behaviour. A classical DEM resorting to fictitious overlaps between the particles is used to simulate the multiple-solid-particle system. A multiphase fluid–solid two-way coupling algorithm between LBM and DEM is constructed. The model is validated by four benchmarks: (i) single disc sedimentation, (ii) single floating particle on a liquid–gas interface, (iii) sinking of a horizontal cylinder and (iv) self-assembly of three particles on a liquid–gas interface. Our simulations agree well with the numerical results reported in the literature. Our proposed model is further applied to simulate droplet impact on deformable granular porous media at pore scale. The dynamic droplet spreading process, the deformation of the porous media (composed of up to 1277 solid particles) as well as the invasion of the liquid into the pores are well captured, within a wide range of impact Weber number. The droplet spreading dynamics on particles is analysed based on the energy budget, which reveals mechanisms at play, showing the evolution of particle energy, surface energy and viscous dissipation energy. A scaling relation based on the impact Weber number is proposed to describe the maximum spreading ratio.
We revisit force evaluation methodologies on rigid solid particles suspended in a viscous fluid that is simulated via the lattice Boltzmann method (LBM). We point out the noncommutativity of streaming and collision operators in the force evaluation procedure due to the presence of a solid boundary, and provide a theoretical explanation for this observation. Based on this analysis, we propose a discrete force calculation scheme with enhanced accuracy. The proposed scheme is essentially a discrete version of the Reynolds transport theorem (RTT) in the context of a lattice Boltzmann formulation. Besides maintaining satisfactory levels of reliability and accuracy, the method also handles force evaluation on complex geometries in a simple and transparent way. We run benchmark simulations for flow past cylinder and NACA0012 airfoil (for Reynolds numbers ranging from 102 to 0.5×106) and show that the current approach significantly reduces the grid size requirement for accurate force evaluation.
This mathematical model studies the dynamics of tumor growth, one of the most complex dynamics problems that relates several
interrelated processes over multiple ranges of spatial and temporal scales. In order to construct a tumor growth model, an
angiogenesis model is used with focus on controlling the tumor volume, preventing new establishment, dissemination, and
growth. The lattice Boltzmann method (LBM) is effectively applied to Navier-Stokes’ equation for obtaining the numerical
simulation of blood flow through vasculature. It is observed that the flow features are extremely sensitive to stenosis severity,
even at small strains and stresses, and that a severe effect on flow patterns and wall shear stresses is noticed in the tumor blood
vessels. It is noted that based on the nonlinear deformation of the blood vessel’s wall, the flow rate conditions became unstable
or distorted and affect the complex blood vessel’s geometry and it changes the blood flow pattern. When the blood flows inside
the stenotic artery, depending on the presence of moderate or severe stenosis, it can lead to insufficient blood supply to the
tissues in the downstream. Consequently, the highly disturbed flow occurs in the downstream of the stenosed artery, or even
plaque ruptures happen when the flow pattern becomes very irregular and complex as it transits to turbulent which cannot be
described without assumptions on the geometry. The results predicted by LBM-based code surpassed the expectations, and
thus, the numerical results are found to be in great accord with the relevant established results of others.
We carried out 3D simulations of monodisperse particle suspensions subjected to a constant shear rate with the view to investigate the effect of electrical double layers around the particles on apparent suspension viscosities. To this end, expressions for Debye length, zeta potential, and ionic strength (pH) of the liquid were incorporated into our in‐house lattice Boltzmann code that uses the immersed boundary method and includes subgrid lubrication models. We varied the solids concentration and particle radius, keeping the particle Reynolds number equal to 0.1. We report on results with respect to the effect of pH in the range 9 through 12 and of Debye length on apparent viscosity and spatial suspension structures, particularly at higher solids volume fractions, and on the effect of flow reversals.
The elastic response of dense suspensions under an impact is studied using coupled lattice Boltzmann method and discrete element method (LBM-DEM) and its reduced model. We succeed to extract the elastic force acting on the impactor in dense suspensions, which can exist even in the absence of percolating clusters of suspended particles. We then propose a reduced model to describe the motion of the impactor and demonstrate its relevancy through the comparison of the solution of the reduced model and that of LBM-DEM. Furthermore, we illustrate that the perturbation analysis of the reduced model captures the short-time behavior of the impactor motion quantitatively. We apply this reduced model to the impact of a foot-spring-body system on a dense suspension, which is the minimal model to realize walking on the suspension. Due to the spring force of the system and the stiffness of the suspension, the foot undergoes multiple bounces. We also study the parameter dependencies of the hopping motion and find that multiple bounces are suppressed as the spring stiffness increases.
Hydrodynamic interactions can give rise to a collective motion of rotating particles. This, in turn, can lead to coherent fluid flows. Using large scale hydrodynamic simulations, we study the coupling between these two in spinner monolayers at weakly inertial regime. We observe an instability, where the initially uniform particle layer separates into particle void and particle rich areas. The particle void region corresponds to a fluid vortex, and it is driven by a surrounding spinner edge current. We show that the instability originates from a hydrodynamic lift force between the particle and fluid flows. The cavitation can be tuned by the strength of the collective flows. It is suppressed when the spinners are confined by a no-slip surface, and multiple cavity and oscillating cavity states are observed when the particle concentration is reduced.
The motivation of this paper is to examine the evaluation of local surface stresses and hydrodynamic forces acting on a stationary or moving body using a diffuse interface immersed boundary method (IBM). This task is not trivial for the diffuse IBM because it uses a smoothed regularized delta function in the transfer steps between Lagrangian and Eulerian locations. In our earlier work [D. Xu et al., Phys. Rev. E 105, 035306 (2022)], a particle distribution function (PDF) discontinuity-based kinetic immersed boundary method (KIBM) was proposed based on the Boltzmann equation. This paper is a continuation of our work on the improvement of the KIBM in the framework of the diffuse interface IBM. In the present study, the concept of the immersed boundary layer (IBL) is brought forward, and the dynamic effects of particle advection and collision in the IBL are coupled and evaluated within a numerical time step scale in a kinetic manner. Consequently, the PDFs on both sides of the IBL are reconstructed, and the general immersed boundary force density can be obtained accurately and efficiently. Meantime, the local surface stress distribution acting on the body wall from the actual fluid can be conveniently and accurately calculated by the moment of the PDFs. Finally, some commonly used problems involving incompressible fluid flows in the continuum flow regime with stationary and moving boundaries are simulated by the present KIBM, and the results show that the present KIBM can significantly accelerate the rate of convergence and has a good agreement with other numerical and experimental results.
A numerical method for describing droplet-driven particle motion at low Weber numbers in fluids is proposed. A coupling strategy combining the Shan–Chen (SC) model and the smoothed profile method (SPM) is employed. The proposed scheme correctly resolves the momentum transfer between solid particles and fluid phases, while effectively controlling the wetting condition. An interaction zone where three phases coexist is introduced to solve the contradiction between the SC model and SPM (i.e. whether fluid particles are allowed at the solid nodes). In the interaction zone, partial solid particles are entrapped in the fluids owing to the coexistence of solid and fluid particles, causing solid particles to exchange momentum with fluid particles. Furthermore, the interaction forces between fluid and solid particles near the solid surface are considered at nodes near pure solid particles within this region. Based on the proposed scheme, the interactions between freely moving particles and freely moving droplets were investigated, along with the effects of wettability (Gads) and volume forces (Fp*) on particle–droplet interactions. When a liquid droplet comes into contact with a solid particle, the adhesion force between the droplet and the solid surface promotes the movement of solid particles toward the droplet. When a double emulsion comes into contact with solid particles, the adhesion force between the emulsion film and solid surface causes the double emulsion to break at the connecting point, and the volume force of the emulsion film Fd* causes the solid particles to penetrate the emulsion.
The accuracy of the free‐surface lattice Boltzmann method (FSLBM) depends significantly on the boundary condition employed at the free interface. Ideally, the chosen boundary condition balances the forces exerted by the liquid and gas pressure. Different variants of the same boundary condition are possible, depending on the number and choice of the particle distribution functions (PDFs) to which it is applied. This study analyzes and compares four variants, in which (i) the boundary condition is applied to all PDFs oriented in the opposite direction of the free interface's normal vector, including or (ii) excluding the central PDF. While these variants overwrite existing information, the boundary condition can also be applied (iii) to only missing PDFs without dropping available data or (iv) to only missing PDFs but at least three PDFs as suggested in the literature. It is shown that neither variant generally balances the forces exerted by the liquid and gas pressure at the free surface. The four variants' accuracy was compared in five different numerical experiments covering various applications. These include a standing gravity wave, a rectangular and cylindrical dam break, a rising Taylor bubble, and a droplet impacting a thin pool of liquid. Overall, variant (iii) was substantially more accurate than the other variants in the numerical experiments performed in this study.
We study synchronization in bulk suspensions of spherical microswimmers with chiral trajectories using large scale numerics. The model is generic. It corresponds to the lowest order solution of a general model for self-propulsion at low Reynolds numbers, consisting of a nonaxisymmetric rotating source dipole. We show that both purely circular and helical swimmers can spontaneously synchronize their rotation. The synchronized state corresponds to velocity alignment with high orientational order in both the polar and azimuthal directions. Finally, we consider a racemic mixture of helical swimmers where intraspecies synchronization is observed while the system remains as a spatially uniform fluid. Our results demonstrate hydrodynamic synchronization as a natural collective phenomenon for microswimmers with chiral trajectories.
In this work, a multiple‐distribution‐function lattice Boltzmann method (MDF‐LBM) for the incompressible two‐phase flows with large density ratios is proposed by considering the governing equations as a coupled convection‐diffusion system. In this method, the phase‐field and convection‐diffusion based Navier‐Stokes equations can be correctly recovered through the direct Taylor expansion. In addition, the pressure is obtained from the first‐order moment of the distribution function, and the incompressibility is satisfied automatically. Furthermore, we also develop a local computing scheme for the velocity gradient, which can be used to calculate some physical quantities, e.g., the strain rate tensor, velocity divergence and vorticity. In our simulations, two simple phase‐capturing benchmark tests without including flow field are first conducted to validate the present MDF‐LBM. When coupling the flow field, we consider the deformation of a square droplet, and find that compared to the commonly used single LBM, the present MDF‐LBM can preserve the incompressibility and mass conservation much better. Then the layered Poiseuille flow is also used to test the present MDF‐LBM, and the numerical results are in good agreement with the analytical solutions and some available results. Finally, the problems of the droplet impact on a thin liquid film as well as the two‐dimensional and three‐dimensional Rayleigh‐Taylor instability with the large deformation are further investigated to show the capability of present MDF‐LBM in the study of the complex two‐phase flows.
Rising bubble systems in porous media exist in a variety of industrial processes. However, the flow characteristics of the issue are not well understood. In this work, the rising of bubble/bubbles through two types of porous structures, namely, in-line structured pore and staggered structured pore, are studied using a large density ratio lattice Boltzmann model. The effects of Eötvös number, pore shape, viscosity ratio, initial bubble number, and arrangement manner of the initial bubbles on the bubble deformation, bubble rising velocity, residual bubble mass, bubble perimeter, and the number of bubble breakups are investigated. It is found that as the Eötvös number increases, the bubbles are more easily broken during the process of passing through the porous media, the shapes of the sub-bubbles deviate from the original ones more and more, the bubble perimeter increases, and the difference between the bubble dynamics obtained by the in-line and staggered porous media decreases. Compared to the results of circular and rectangular pores, the bubble rising through the diamondoid pore has a more considerable deformation, which causes a slower rising speed. Furthermore, in the case that two bubbles are originally placed under the porous medium, the bubble deformation is greater and the bubble fracture times increase if the initial bubbles are aligned vertically. The findings of this work can contribute to the understanding of gas–liquid two-phase flow in porous media.
Non-spherical shape is a general appearance feature for bioparticles. Therefore, a mechanical mechanism study of non-spherical particle migration in a microfluidic chip is essential for more precise isolation of target particles. With the manipulation of non-spherical particles, refined disease detection or medical intervention for human beings will be achievable in the future. In this review, fabrication and manipulation of non-spherical particles are discussed. Firstly, various fabrication methods for non-spherical microparticle are introduced. Then, the active and passive manipulation techniques for non-spherical particles are briefly reviewed, including straight inertial microchannels, secondary flow inertial microchannels and deterministic lateral displacement microchannels with extremely high resolution. Finally, applications of viscoelastic flow are presented which obviously increase the precision of non-spherical particle separation. Although various techniques have been employed to improve the performance of non-spherical particle manipulation, the universal mechanism behind this has not been fully discussed. The aim of this review is to provide a reference for non-spherical particle manipulation study researchers in every detail and inspire thoughts for non-spherical particle focused device design.
We provide a Boltzmann-type kinetic description for dilute polymer solutions based on two-fluid theory. This Boltzmann-type description uses a quasiequilibrium based relaxation mechanism to model collisions between a polymer dumbbell and a solvent molecule. The model reproduces the desired macroscopic equations for the polymer-solvent mixture. The proposed kinetic scheme leads to a numerical algorithm which is along the lines of the lattice Boltzmann method. Finally, the algorithm is applied to describe the evolution of a perturbed Kolmogorov flow profile, whereby we recover the major elastic effect exhibited by a polymer solution, specifically, the suppression of the original inertial instability.
We present an immersed boundary/multi-relaxation time lattice Boltzmann method for the particle-resolved simulation of particle-laden flows. The no-slip boundary condition is handled by an explicit feedback immersed boundary method for fixed and moving particles with arbitrary shape. Two special treatments are applied to improve the simulation stability and accuracy: (i) a smoothed discrete delta function (Yang et al., 2009) is adopted to suppress the spurious force oscillations and (ii) a multi-relaxation time collision operator is utilized to fix the viscosity-dependent numerical slip error of the immersed boundaries. To further reduce the computational effort, we extend the method from fixed and uniform Cartesian grids to adaptive quadtree/octree grids by implementing it in the open-source software Basilisk. A Lax-Wendroff streaming operator is adopted in our method to retain second-order accuracy in both space and time on non-uniform grids, as well as to maintain a unified time scale over the entire computational domain. Consequently, one collision-streaming operation only is required per time step at all grid levels. We test our proposed method on a set of validation cases corresponding to assorted incompressible flows with fixed and moving rigid particles in both 2D and 3D. The accuracy and robustness of our method are demonstrated via thorough comparison with the analytical, experimental and numerical data provided in the literature.
In this paper, we present an improved lattice Boltzmann (LB) model for fluid-fluid-solid (FFS) flows with a high viscosity ratio. The bounce-back particle model is combined with the Shan-Chen (SC) multicomponent model. We extend the bounce-back scheme based on velocity interpolation and a fresh-node initialization approach with second-order accuracy to moving particles within the framework of the multicomponent model. An improved virtual solid density model for wetting boundary conditions is employed to implement contact angles on curved boundaries. We examine the factors that lead to the violation of mass conservation, and an easy redistributing method is developed to fix the mass leakage issue. The combined multiphase particle model is able to simulate FFS flows with a high viscosity ratio of up to 1000 while preserving the total mass of the two fluids. The performance of the approach is tested by a variety of numerical experiments. The dynamic behaviors of moving contact lines on the curved boundary are validated by a droplet wetting on a solid particle. The model is then applied to simulate dynamic FFS problems, such as particle wetting at the fluid interface and particle motion through a fluid-fluid interface. According to the simulation results, the present model is capable of capturing the total force exerted on a particle by the fluid and the interface. However, the SC-type fluid-solid interaction force does not equal the capillary force in the present model. Finally, the self-assembly process of two floating particles on a liquid-liquid interface is investigated.
Bubbles are existent everywhere and of great importance for the daily life and industry process, such as heat exchange rate influenced by bubbles in the tube, battery life partially decided by bubbles of chemical reaction in it, etc. With the further requirement for miniaturization, physical mechanisms behind bubble behaviors in microchannels become crucial. In the present work, the lattice Boltzmann method is used to investigate the behavior of bubbles as they rise in complex microchannels under the action of buoyancy. The channel is placed with two asymmetric obstacles on its left and right side. Initially, the lattice Boltzmann model is tested for its reliability and accuracy by Laplace law. Then a few parameters of flow field, i.e. the Eötvös number, the viscosity ratio, the vertical distance between the obstacles, the horizontal distance between the obstacles, are employed to study the characteristics of the bubble during the movement, including the deformation, the rising speed, the residual mass, and the time of bubble passing through the channel. The results are shown below. First, the trend of the bubble's velocity changing with time in the process of passing through the channel corresponds to the change process of the dynamic behavior of the interface, i.e. the bubble velocity decreases when the bubble shape changes significantly under the same channel width. Second, with the increase of \begin{document}$ Eo $\end{document} number, the bubble deformation as well as the bubble velocity increases and the bubble residual mass decreases. Besides, the gas-to-liquid viscosity ratio has a significant effect on the bubble velocity. Under the condition of high viscosity ratio, the bubble shape is difficult to maintain a round shape, while the bubble rise velocity increases and the residual mass of the bubble decreases with the viscosity ratio. What is more, when the obstacle setting is changed, the longer the vertical distance between the two asymmetric obstacles, the shorter the bubble passing time is, and the faster it will return to the original shape after passing through the obstacle, while the residual mass of the bubble shows a change trend of approximately unchanged-increase-decrease-increase with the augment of the vertical distance between the obstacles. In the study of changing the horizontal spacing, two cases: the two obstacles are changed at the same time (Case A) and only the one-sided obstacle is changed (Case B), are considered. The results show that under the same small horizontal interval, the obstruction effect caused by changing only the length of one side obstacle is stronger. Finally, the study shows that when the width of the right obstacle is long enough, although the width of the obstacle continues to increase, the passing time of the bubble increases slowly, and the position of the bubble leaving from the obstacle is always approximately the same.
We introduce a scheme to simulate the spatial and temporal evolution of the densities of charged species, taking into account diffusion, thermal fluctuations, coupling to a carrier fluid, and chemical reactions. To this end, the diffusive fluxes in the electrokinetic model by Capuani et al. (2004) are supplemented with thermal fluctuations. Chemical reactions are included via an additional source term in the mass balance equation. The diffusion-reaction model is then coupled to a solver for fluctuating hydrodynamics based on the lattice Boltzmann method. This combination is particularly useful for soft matter simulations, due to the ability to couple particles to the lattice-Boltzmann fluid. These could, e.g., be charged colloids or polymers, which then interact with an ion distribution. We describe one implementations based on the automatic code generation tools pystencils and lbmpy, and another one that is contained in the molecular dynamics package ESPResSo and that allows for an easy coupling of particles to the density fields. We validate our implementations by comparing to several known analytic results. Our method can be applied to coarse-grained catalysis problems as well as to many other multi-scale problems that require the coupling of explicit-particle simulations to flow fields, diffusion, and reaction problems in arbitrary geometries.
We introduce a novel mesoscopic computational model based on a multiphase-multicomponent lattice Boltzmann method for the simulation of self-phoretic particles in the presence of liquid–liquid interfaces. Our model features fully resolved solvent hydrodynamics, and, thanks to its versatility, it can handle important aspects of the multiphysics of the problem, including particle wettability and differential solubility of the product in the two liquid phases. The method is extensively validated in simple numerical experiments, whose outcome is theoretically predictable, and then applied to the study of the behavior of active particles next to and trapped at interfaces. We show that their motion can be variously steered by tuning relevant control parameters, such as the phoretic mobilities, the contact angle, and the product solubility.
The steady state of motion of two particles in Poiseuille flow of power-law fluid is numerically studied using the lattice Boltzmann method in the range of Reynolds number 20 ≤ Re ≤ 60, diameter ratio of two particles 0.125 ≤ β ≤ 2.4, and power-law index of the fluid 0.4 ≤ n ≤ 1.2. Some results are validated by comparing with other available results. The effects of Re, β, and n on the steady state of motion of two particles are discussed. The results show that, for two particles of the same diameter, the particle spacing l in the steady state is independent of n. In shear-thinning fluid, l increases rapidly at first and then slowly, finally approaching a constant for different Re. In shear-thickening fluid, although l tends to be stable in the end, the values of l after stabilization are different. For two particles of different sizes, l does not always reach a stable state, and whether it reaches a stable state depends on n. When the small particle is downstream, l increases rapidly at first and then slowly in shear-thickening fluid, but increases rapidly at first and then decreases slowly, finally approaching a constant in a shear-thinning fluid. In shear-thinning fluid, the larger n is, the smaller l is. In shear-thickening fluid, β has no effect on l in steady-state. When the large particle is downstream, l increases rapidly at first and then slowly in shear-thinning fluid but increases rapidly at first and then decreases in a shear-thickening fluid. The effect of n on l in the steady state is obvious. In shear-thinning fluid, l increases rapidly at first and then slowly, the larger Re is, the smaller l is. In shear- thickening fluid, l will reach a stable state.
We present LBcuda, a GPU accelerated version of LBsoft, our open-source MPI-based software for the simulation of multi-component colloidal flows. We describe the design principles, the optimization and the resulting performance as compared to the CPU version, using both an average cost GPU and high-end NVidia GPU cards (V100 and the latest A100). The results show a substantial acceleration for the fluid solver reaching up to 200 GLUPS (Giga Lattice Updates Per Second) on a cluster made of 512 A100 NVIDIA cards simulating a grid of eight billion lattice points. These results open attractive prospects for the computational design of new materials based on colloidal particles.
Program summary
Program Title: LBcuda
CPC Library link to program files: https://doi.org/10.17632/v6fvmzpcrn.1
Developer's repository link: https://github.com/copmat/LBcuda
Licensing provisions: 3-Clause BSD License
Programming language: CUDA Fortran
Nature of problem: Hydro-dynamics of colloidal multi-component systems and Pickering emulsions.
Solution method: Lattice-Boltzmann method solving the Navier-Stokes equations for the fluid dynamics within an Eulerian description. Particle solver describing colloidal particles within a Lagrangian representation coupled to the fluid solver. The numerical solution of the coupling algorithm includes the back reaction effects for each force terms according to a fluid-particle multi-scale paradigm.
Non-isothermal particles suspended in a fluid lead to complex interactions – the particles respond to changes in the fluid flow, which in turn is modified by their temperature anomaly.
Here, we perform a novel proof-of-concept numerical study based on tracer particles that are thermally coupled to the fluid. We imagine that particles can adjust their internal temperature reacting to some local fluid properties and follow simple, hard-wired active control protocols. We study the case where instabilities are induced by switching the particle temperature from hot to cold depending on whether it is ascending or descending in the flow. A macroscopic transition from a stable to unstable convective flow is achieved, depending on the number of active particles and their excess negative/positive temperature. The stable state is characterized by a flow with low turbulent kinetic energy, strongly stable temperature gradient, and no large-scale features. The convective state is characterized by higher turbulent kinetic energy, self-sustaining large-scale convection, and weakly stable temperature gradients. The particles individually promote the formation of stable temperature gradients, while their aggregated effect induces large-scale convection.When the Lagrangian temperature scale is small, a weakly convective laminar system forms. The Lagrangian approach is also compared to a uniform Eulerian bulk heating with the same mean injection profile and no such transition is observed. Our empirical approach shows that thermal convection can be controlled by pure Lagrangian forcing and opens the way for other data driven particle-based protocols to enhance or deplete large-scale motion in thermal flows.
The sedimentation process in an active suspension is the result of the competition between gravity and the autonomous motion of particles. We carry out simulations of run-and-tumble squirmers that move in a fluid medium, focusing on the dependence of the non-equilibrium steady state on the swimming properties. We find that for large enough activity, the density profiles are no longer simple exponentials; we recover the numerical results through the introduction of a local effective temperature, suggesting that the breakdown of the Perrin-like exponential form is a collective effect due to fluid-mediated dynamic correlations among particles. We show that analogous concepts can also fit the case of active non-motile particles, for which we report the first study of this kind. Moreover, we provide evidence of scenarios where the solvent hydrodynamics induces non-local effects which require the full three-dimensional dynamics to be taken into account in order to understand sedimentation in active suspensions. Finally, analyzing the statistics of the orientations of microswimmers, the emergence of a height-dependent polar order in the system is discussed.
This chapter proposes an original alternative to Lagrange extrapolation in the hydrodynamic force and heat transfer computation in order to avoid contaminated solution of the Navier‐Stokes equations near the solid‐fluid interface. It is devoted to setting up Aslam extension numerical parameters and estimating drag and Nusselt coefficients for uniform flows around an isolated sphere, and comparing them to those given by Lagrange extrapolation in order to assess the improvements made by Aslam extension. The chapter is a presentation of the numerical method (i.e. viscous penalty method) to simulate various particulate flows and the method used to extract from these simulations the momentum and heat transfers between the two phases. A viscous penalty method is used to simulate the interaction between spherical particles and the surrounding carrier fluid using particle‐resolved direct numerical simulations. The chapter compares the drag coefficient values computed with Lagrange extrapolation with those computed with Aslam extension.
We generalize the hydrodynamic lattice gas model to include arbitrary numbers of particles moving in each lattice direction. For this generalization we derive the equilibrium distribution function and the hydrodynamic equations, including the equation of state and the prefactor of the inertial term that arises from the breaking of galilean invariance in these models. We show that this prefactor can be set to unity in the generalized model, therby effectively restoring galilean invariance. Moreover, we derive an expression for the kinematic viscosity, and show that it tends to decrease with the maximum number of particles allowed in each direction, so that higher Reynolds numbers may be achieved. Finally, we derive expressions for the statistical noise and the Boltzmann entropy of these models. I. LATTICE GASES Lattice gas automata (LGA) are a class of dynamical systems in which particles move on a lattice in discrete time steps. If the collisions between the particles conserve mass and momentum, the coarse-grained behavior of the system can be shown to be that of a viscous fluid in the appropriate scaling limit 1-4. Used as an algorithm for simulating hydrodynamics, the method has the virtues of exact conservation laws, and of unconditional numerical stability. In a typical LGA, there is an association between the lattice vectors and the particles at each site. If there are n lattice vectors, then the state of the site is represented by n bits. Each bit represents the presence or absence of a particle in the corresponding direction. At each time step, a particle propagates along its corresponding lattice vector and then collides with other arriving particles at the new site 1. The collisions 1 Note that rest particles can be subsumed into this scheme by associating them with null lattice vectors.
Artificial microswimmers, nano- and microrobots, are essential in many applications from engineering to biology and medicine. We present a Stokesian dynamics study of the dynamical properties and efficiency of one of the simplest artificial swimmers, the three linked spheres swimmer (TLS), extensively shown to be an excellent and model example of a deformable micromachine. Results for two different swimming strokes are compared with an approximate solution based on point force interactions. While this approximation accurately reproduces the solutions for swimmers with long arms and strokes of small amplitude, it fails when the amplitude of the stroke is such that the spheres come close together, a condition where indeed the largest efficiencies are obtained. We find that swimmers with a “square stroke cycle” result more efficient than those with “circular stroke cycle” when the swimmer arms are long compared with the sphere radius, but the differences between the two strokes are smaller when the arms of the swimmers are short. This extended theoretical research of TLS incorporates a much precise description of the swimmer hydrodynamics, demonstrating the relevance of considering the finite size of the constitutive microswimmers spheres. This work expects to trigger future innovative steps contributing to the design of micro- and nanomachines and its applications.
In this paper, we present an improved phase-field-based lattice Boltzmann (LB) method for thermocapillary flows with large density, viscosity, and thermal conductivity ratios. The present method uses three LB models to solve the conservative Allen-Cahn equation, the incompressible Navier-Stokes equations, and the temperature equation. To overcome the difficulty caused by the convection term in solving the convection-diffusion equation for the temperature field, we first rewrite the temperature equation as a diffuse equation where the convection term is regarded as the source term and then construct an improved LB model for the diffusion equation. The macroscopic governing equations can be recovered correctly from the present LB method; moreover, the present LB method is much simpler and more efficient. In order to test the accuracy of this LB method, several numerical examples are considered, including the planar thermal Poiseuille flow of two immiscible fluids, the two-phase thermocapillary flow in a nonuniformly heated channel, and the thermocapillary Marangoni flow of a deformable bubble. It is found that the numerical results obtained from the present LB method are consistent with the theoretical prediction and available numerical data, which indicates that the present LB method is an effective approach for the thermocapillary flows.
Despite many particle shapes and motion patterns that can be present in viscoplastic settling, the scientific studies are mostly limited to spherical particles and cases in which the solid body motion is purely vertical or rotational. Here we tackle the problem of an ellipsoidal particle settling in Bingham fluid through fully-resolved simulations. Oblate and prolate spheroids of varied aspect ratios and initial orientations are analyzed. We use the Lattice Boltzmann Method (LBM), in which the Bingham constitutive equation is solved exactly. We employ the Immersed Boundary Method (IBM) to delineate the particle boundaries and enforce the no-slip condition over them. For the case of creeping flow around a settling ellipsoid, we benchmark results of the critical yield number for oblate and prolate spheroids against augmented Lagrangian method (ALM) data and provide values for different aspect ratios and initial orientations. Then, we analyze the dynamics of prolate and oblate spheroids settling in Bingham fluid at an inertial flow. We present the evolution of trajectory, orientation, angular momentum, Reynolds number, and drift angle for various aspect ratios and initial orientations.
The hydrodynamic interactions (HIs) of two colloidal spheres characterized by the translation–translation (T–T) couplings have been studied under various confinements, but little is known regarding the HIs of anisotropic particles and rotational motions, which are common in nature and industry. Here, we study the T–T, rotation–rotation (R–R) and translation–rotation (T–R) hydrodynamic couplings of two colloidal ellipsoids sediment on the bottoms of channels in experiment, theory and simulation. We find that the degree of confinement and the particle shape anisotropy are critical tuning factors resulting in anomalous hydrodynamic and diffusive behaviours. The negative R–R coupling reflects the tendency of opposite rotations of two neighbouring ellipsoids. The positive T–R coupling reflects that an ellipsoid rotates away from the channel axis as another ellipsoid approaches. As the channel width increases, the positive T–T coupling changes to an abnormal negative coupling, indicating that the single-file diffusion can exist even in wide channels. By contrast, only positive T–T couplings were observed for spheres in channels. The T–T coupling increases with the aspect ratio p . The R–R coupling is the maximum at a moderate p ~ 2.8. The T–R coupling is the maximum at a moderate degree of confinement. The spatial range of HIs is longer than that of spheres and increases with p . We propose a simple model which reproduces some coupling phenomena between two ellipsoids, and it is further confirmed by low-Reynolds-number hydrodynamic simulation. These findings shed new light on anisotropic particle diffusion in porous media, transport through membranes, microfluidics and microrheology.
Hydrodynamical phenomena can be simulated by discrete lattice gas models obeing cellular automata rules (U. Frisch, B. Hasslacher, and Y. Pomeau, Phys. Rev. Lett. 56, 1505, (1986); D. d'Humieres, P. Lallemand, and U. Frisch, Europhys. Lett. 2, 291, (1986)). It is here shown for a class of D-dimensional lattice gas models how the macrodynamical (large-scale) equations for the densities of microscopically conserved quantities can be systematically derived from the underlying exact ''microdynamical'' Boolean equations. With suitable restrictions on the crystallographic symmetries of the lattice and after proper limits are taken, various standard fluid dynamical equations are obtained, including the incompressible Navier-Stokes equations in two and three dimensions. The transport coefficients appearing in the macrodynamical equations are obtained using variants of fluctuation-dissipation and Boltzmann formalisms adapted to fully discrete situations.
The standard theory of fluctuations in thermodynamic variables in various ensembles is generalized to nonthermodynamic variables: e.g., the mean-square fluctuations of the kinetic energy K in a classical microcanonical ensemble at fixed energy E is given, for large systems, by 〈(δK)2〉/〈K〉=T[1-3/2C), where T is the temperature (corresponding to the energy E) and C is the specific heat per particle (in units of Boltzmann's constant). The general results may be expressed in terms of the asymptotic behavior of the Ursell functions in various ensembles. Applications are made to molecular dynamic computations where time averages correspond (via ergodicity) to phase averages in an ensemble with fixed energy and momentum. The results are also useful for time-dependent correlations.
A new Hamiltonian method for deformation simulations is related to the Green-Kubo fluctuation theory through perturbation theory and linear-response theory. Numerical results for the bulk and shear viscosity coefficients are compared to corresponding Green-Kubo calculations. Both viscosity coefficients depend similarly on frequency, in a way consistent with enhanced ''long-time tails.''
Spatially periodic fundamental solutions of the Stokes equations of motion for a viscous fluid past a periodic array of obstacles are obtained by use of Fourier series. It is made clear that the divergence of the lattice sums pointed out by Burgers may be rescued by taking into account the presence of the mean pressure gradient. As an application of these solutions the force acting on any one of the small spheres forming a periodic array is considered. Cases for three special types of cubic lattice are investigated in detail. It is found that the ratios of the values of this force to that given by the Stokes formula for an isolated sphere are larger than 1 and do not differ so much among these three types provided that the volume concentration of the spheres is the same and small. The method is also applied to the two-dimensional flow past a square array of circular cylinders, and the drag on one of the cylinders is found to agree with that calculated by the use of elliptic functions.
The Langevin equation describing Brownian motion is considered as a contraction from the more fundamental, but still phenomenological, description of an incompressible fluid governed by fluctuating hydrodynamics in which a Brownian particle with stick boundary condition is immersed. First, the derivation of fluctuating hydrodynamics is reconsidered to clarify certain ambiguities as to the treatment of boundaries. Subsequently the contraction is carried out. Since Brownian particles of arbitrary shape are considered, rotations and translations are in general coupled. The symmetry of the 66 friction tensor
ij
(t) is proved for arbitrary shape without appeal to microscopic arguments. This symmetry is then used to prove that the fluctuation-dissipation theorem on the contracted level (nonwhite noise in general) follows from the corresponding statement on the level of fluctuating hydrodynamics (white noise). The condition under which the contracted description reduces to the classical Langevin equation is given, and the connection between our theory and related work is discussed.
On a short time scale, Brownian particles undergo a transtion from the initial ballistic trajectories to diffusive motion. Hydrodynamic interactions with the surrounding fluid lead to a complex time dependence of this transition. We directly probe this transition for colloidal particles by measuring the autocorrelation function of multiply scattered, transmitted light. We show that a quantitative interpretation is possible because the transport of the light is diffusive, resolving a conflict in previous measurements.
The mean-square displacement 〈Δr2(τ)〉 of particles in concentrated suspensions is measured at times sufficiently short to observe the transient nature of hydrodynamic interactions. For all volume fractions φ, the velocity autocorrelation function decays as a power law R(τ)∼τ-3/2. A remarkable scaling with φ is observed for the time-dependent self-diffusion coefficient Ds(τ)=〈Δr2(τ)〉/6τ: If Ds(τ) is scaled by its asymptotic value and if time is scaled by a viscous time inversely proportional to the shear viscosity of the suspension, all the data fall onto a single master curve.
Low Reynolds number flow theory finds wide application in such diverse fields as sedimentation, fluidization, particle-size classification, dust and mist collection, filtration, centrifugation, polymer and suspension rheology, flow through porous media, colloid science, aerosol and hydrosal technology, lubrication theory, blood flow, Brownian motion, geophysics, meteorology, and a host of other disciplines. This text provides a comprehensive and detailed account of the physical and mathematical principles underlying such phenomena, heretofore available only in the original literature.
On a short time scale, Brownian particles undergo a transition from initially ballistic trajectories to diffusive motion. Hydrodynamic interactions with the surrounding fluid lead to a complex time dependence of this transition. We directly probe this transition for colloidal particles by measuring the autocorrelation function of multiply scattered light and observe the effects of the slow power-law decay of the velocity autocorrelation function.
Liquid metals differ from the other classes of liquids considered so far primarily through the presence of the conduction electrons. Although the theory of liquid metals has much in common with that of the other ionic liquids discussed in Chapter 10, the problem is complicated by the fact that the electronic component requires a quantum-mechanical treatment. Many elements show metallic behaviour in the liquid state, but their electronic band structures can differ widely. We shall restrict ourselves to the so-called “simple” metals. The class of simple metals comprises those in which the electronic valence states are well separated in energy from the tightly-bound core states; their properties are reasonably well described by the nearly-free-electron model. Metals that are classified as simple in this sense include the alkali metals, magnesium, zinc, mercury, gallium and aluminium. Other liquid metals (noble and transition metals, alkaline earths, lanthanides and actinides) have more complicated electronic structures, and the theory of such systems is correspondingly less well advanced. More complete accounts of the properties of liquid metals than we are able to give here can be found in a number of excellent monographs (Faber, 1972; Shimoji, 1977) and review articles (Ashcroft and Stroud, 1978; Evans, 1978), and in the proceedings of regular conferences on the subject (Takeuchi, 1973; Evans and Greenwood, 1977; Cyrot-Lackmann and Desre, 1980).
A long standing problem in numerical statistical mechanics has been the incorporation of the long range, many-body hydrodynamic forces between particles in suspension. In this paper I describe a general computational method for calculating the forces and torques exerted by slowly moving spheres suspended in an incompressible fluid. In particular, the method correctly incorporates the effect of periodic boundary conditions on the hydrodynamic flow field. Results are presented for the friction and mobility matrices of small clusters of spheres as a function of the size of the periodic unit cell. An expression for the viscosity of a suspension of freely moving spheres is derived for a system with periodic boundary conditions, and numerical results are obtained for a suspension of spheres arranged in a simple-cubic lattice.
The sedimentation of monodisperse suspensions of rigid spheres has been studied by dynamical simulation; computational techniques are described and numerical results are reported. It has been found that there is a slow relaxation of the suspension microstructure during sedimentation, so that compared with the initial equilibrium distribution, there is an increased number of pairs of particles near contact; this leads to a 5%–10% increase in the average sedimentation velocity. Individual particle velocities fluctuate about the mean fall speed; these fluctuations are large and persist for long times. The resulting hydrodynamically induced dispersion of the particles can be characterized by strongly anisotropic diffusion coefficients; however, the dispersion process is non-Fickian at high solids concentrations.
Equations of motion are given that are suitable for setting up non-equilibrium momentum fluxes in molecular fluids. In a recent article Marechal and Ryckaert have pointed out that equilibrium time correlation functions depend on whether the stress is measured in a centre of mass or atomic representation. In non-equilibrium simulations of molecular systems, a corresponding ambiguity arises between perturbations applied to the individual atoms and perturbations applied to the centres-of-mass of the molecules. I use linear response theory to demonstrate the equivalence of the non-equilibrium equations of motion with the equilibrium correlation functions, for small strain rates. I also show that the two sets of equations of motion produce identical trajectories in the low-frequency limit. In this limit the equations of motion are equivalent to the Lees-Edward's method, but at finite frequencies the moving boundary method is incorrect.
Solutions for the slow flow past a square and a hexagonal array of cylinders are determined using a somewhat non-conventional numerical method. The calculated values of the drag on a cylinder as a function of c, the volume fraction of the cylinders, are shown to be in excellent agreement with the corresponding asymptotic expressions for c ⪡ 1 and for , the maximum volume fraction. These solutions are then used to calculate the average temperature difference between the bulk and the cylinders which are heated uniformly under conditions of small Reynolds and Péclet numbers.
Molecular-dynamic studies of the behavior of the diffusion coefficient after a long time s have shown that the velocity autocorrelation function decays as s-1 for hard disks and as s-3/2 for hard spheres, at least at intermediate fluid densities. A hydrodynamic similarity solution of the decay in velocity of an initially moving volume element in an otherwise stationary compressible viscous fluid agrees with a decay of (ηs)-d/2, where η is the viscosity and d is the dimensionality of the system. The slow decay, which would lead to a divergent diffusion coefficient in two dimensions, is caused by a vortex flow pattern which has been quantitatively compared for the hydrodynamic and molecular-dynamic calculations.
Accurate values for the hydrodynamic transport properties of random dispersions of hard spheres have been determined by numerical simulation. The many‐body hydrodynamic interactions are calculated from a multipole‐moment expansion of the force density on the surface of the solid particles; the singular lubrication forces are included exactly for pairs of particles near contact. It has been possible to calculate the transport properties of small periodic systems, at all packing fractions, with uncertainties of less than 1%; but for larger systems we are limited computationally to lower order, and therefore less accurate, moment approximations to the induced force density. Nevertheless, since the higher‐order moment contributions are short range they are essentially independent of system size and we can use small system data to correct our results for larger systems. Numerical calculations show that this is a reliable and accurate procedure. The ensemble‐averaged mobility tensors are strongly dependent on system size, with deviations from the thermodynamic limit varying as N−1/3. We show that these deviations can be accounted for by a straightforward calculation based on the length of the unit cell and the suspension viscosity and mobility. The remaining number dependencies are small. Problems involved in implementing this method for larger numbers of particles are considered, and alternative methods are discussed.
Brownian motion of particles suspended in a fluid is studied, and expressions derived for the particle diffusivity and velocity autocorrelation function. The theory of thermal noise in a general linear system is applied to both the particles and the fluid in a novel formulation. This enables the recent modification of the Langevin equation to include the effect of fluid inertia to be seen as just a necessary but simple reinterpretation of the original analysis, without introducing the theory of non-Markovian processes.
We treat the problem of slow flow through a periodic array of spheres. Our interest is in the drag force exerted on the array, and hence the permeability of such arrays. It is shown to be convenient to formulate the problem as a set of two-dimensional integral equations for the unknown surface stress vector, thus lowering the dimension of the problem. This set is solved numerically to obtain the drag as a function of particle concentration and packing characteristics. Results are given over the full concentration range for simple cubic, body-centred cubic and face-centred cubic arrays and these agree well with previous limited experimental, asymptotic and numerical results.
The dispersion considered consists of a large number of identical small rigid spheres with random positions which are falling through Newtonian fluid under gravity. The volume fraction of the spheres ( c ) is small compared with unity. The dispersion is statistically homogeneous, and the axes of reference are chosen so that the mean volume flux across any stationary surface is zero. The problem is to determine the mean value of the velocity of a sphere ( U ). In §3 there is described a systematic and rigorous procedure which overcomes the familiar difficulty presented by the occurrence of divergent integrals, essentially by the choice of a quantity V whose mean value can be found exactly and which has the same long-range dependence on the position of a second sphere as U so that the mean of U – V can be expressed in terms of an absolutely convergent integral. The result is that, correct to order c , the mean value of U is U 0 (1 – 6.55 c ), where U 0 , is the velocity of a single sphere in unbounded fluid. The only assumption made in the calculation is that the centres of spheres in the dispersion take with equal probability all positions such that no two spheres overlap; arguments are given in support of this assumption, which is expected to be valid only when the spheres are identical. Calculations which assume a simple regular arrangement of the spheres or which adopt a cell model of the hydrodynamic interactions give the quite different result that the change in the mean speed of fall is proportional to $c^{\frac{1}{3}}$ , for reasons which are made clear.
The general procedure described here is expected to be applicable to other problems concerned with the effect of particle interactions on the average properties of dispersions with small volume fraction of the particles.
Numerical solutions are presented for steady two-dimensional incompressible flow past an infinite row of cylinders (of unit radii, with distances W between their centres). The calculations cover R [less-than-or-eq, slant] 700 for 5 [less-than-or-eq, slant] W [less-than-or-eq, slant] [infty infinity] and also R = 800 for 5 [less-than-or-eq, slant] W [less-than-or-eq, slant] 100 (where R denotes the Reynolds number based on the cylinder diameters). The recirculation regions (wake bubbles) are found to grow in length approximately linearly with R in all cases. For high values of R, a major change occurs in their character when W is increased past Wcrit [approximate] 16. While they have remained slender up to this point (essentially only stretching in length in proportion to R), their centres of circulation have moved towards their ends. As W is further increased, the wake bubbles widen rapidly, beginning from the rear of the wakes. In the limit of W[rightward arrow][infty infinity], the present results agree with the previous ones for a single cylinder as reported by Fornberg (1985).
At present, human society is facing a health care crisis that is affecting patients worldwide. In the United States, it is generally believed that the major problem is lack of affordable access to health care (i.e. health insurance). This book takes an unprecedented approach to address this issue by proposing that the major problem is not lack of affordable access to health care per se, but lack of access to better, safer, and more affordable medicines. The latter problem is present not only in the United States and the developing world but also in countries with socialized health care systems, such as Europe and the rest of the industrialized world. This book provides a comparative analysis of the health care systems throughout the world and also examines the biotechnology and pharmaceutical industries. Examines the health care structure of the United States, Europe, and the third world, both separately and comparatively Offers primary source insight through in-depth interviews with pharmaceutical and health care industry leaders from around the world Carefully explains, in clear terms, the intricacies of the health care and pharmaceutical system and how these intricacies have led to the current crisis Offers concrete, comprehensive solutions to the health care crisis.
A review of recent work and new developments are presented for the penalty-function/finite element formulation of incompressible viscous flows. Basic features of the penalty method are described in the context of the steady and unsteady Navier-Stokes equations. Galerkin and “upwind” treatments of convection terms are discussed. Numerical results indicate the versatility and effectiveness of the new methods.
A five-mode truncation of the Navier—Stokes equation for incompressible flow through a cubic array of spheres is analyzed for its stationary and time-dependent solutions. Under the imposed symmetry requirements, the stationary solutions accurately depict the appearance of a vortex pair, and the volume-averaged velocity is in excellent agreement with the Ergun equation up to Re = 250. At the point the deviation from empirical data begins, the unique stationary solution destabilizes via a Hopf bifurcation point. A numerical analysis of the resulting periodic solution shows that for a certain range of imposed pressure gradient the system exhibits chaotic behavior, approached through a period-doubling subharmonic bifurcation sequence.
Computer simulation is an essential tool in studying the chemistry and physics of liquids. Simulations allow us to develop models and to test them against experimental data. They can be used to evaluate approximate theories of liquids, and to provide detailed information on the structure and dynamics of model liquids at the molecular level. This book is an introduction and practical guide to the molecular dynamics and Monte Carlo methods. The first four chapters describe these methods in detail, and provide the essential background in intermolecular forces and statistical mechanics. Chapters 5 and 6 emphasize the practical aspects of writing efficient programs and analysing the simulation results. The remaining chapters cover advanced techniques, non-equilibrium methods, Brownian dynamics, quantum simulations, and some important applications. FORTRAN code is presented in the text.
The hydrodynamic force acting on a rigid spherical particle translating with arbitrary time-dependent motion in a time-dependent flowing fluid is calculated to O(Re) for small but finite values of the Reynolds number, Re, based on the particle's slip velocity relative to the uniform flow. The corresponding expression for an arbitrarily shaped rigid particle is evaluated for the case when the timescale of variation of the particle's slip velocity is much greater than the diffusive scale, a^2/v, where a is the characteristic particle dimension and v is the kinematic viscosity of the fluid. It is found that the expression for the hydrodynamic force is not simply an additive combination of the results from unsteady Stokes flow and steady Oseen flow and that the temporal decay to steady state for small but finite Re is always faster than the t^-½ behaviour of unsteady Stokes flow. For example, when the particle accelerates from rest the temporal approach to steady state scales as t^-2.
The self-diffusivity of Brownian hard spheres in a simple shear flow is studied by numerical simulation. Particle trajectories are calculated by Stokesian dynamics, with an accurate representation of the suspension hydrodynamics that includes both many-body interactions and lubrication. The simulations are of a monolayer of identical spheres as a function of the Péclet number: Pe =gamma-dot a^2/D0, which measures the relative importance of shear and Brownian forces. Here gamma-dot is the shear rate, a the particle radius, and D0 the diffusion coefficient of a single sphere at infinite dilution. In the absence of shear, using only hydrodynamic interactions, the simulations reproduce the pair-distribution function of the equivalent hard-disk system. Both short- and long-time self-diffusivities are determined as a function of the Péclet number. The results show a clear transition from a Brownian motion dominated regime (Pe<1) to a hydrodynamically dominated regime (Pe>10) with a dramatic change in the behavior of the long-time self-diffusivity.
The hydrodynamic transport properties of hard-sphere dispersions are calculated for volume fractions (φ) spanning the dilute limit up to the fluid–solid transition at φ=0.49. Particle distributions are generated by a Monte Carlo technique and the hydrodynamic interactions are calculated by Stokesian dynamics simulation. The effects of changing the number of particles in the simulation cell are investigated, and the scaling laws for the finite-size effects are determined. The effects of using various levels of approximation in computing both the far- and near-field hydrodynamic interactions are also examined. The transport properties associated with porous media—permeabilities and hindered diffusion coefficients—are determined here. The corresponding properties of freely mobile suspensions are determined in a companion paper [Phys. Fluids 31, 3462 (1988)].
A new and general technique for simulating solid-fluid suspensions, which combines molecular dynamics for the solid particles with a lattice-Boltzmann model for the fluid, is described. The many-body hydrodynamic interactions are fully accounted for, both for small particle velocities and at higher Reynolds numbers. Brownian motion of the solid particles is included by adding a fluctuating component to the fluid stress tensor. Simulations of the dynamics of colloidal particles at short times compare favorably with recent diffusing-wave spectroscopy experiments.
We resolve the mean square displacement of particles in concentrated suspensions during the first 20 ns of their motion, before hydrodynamic interactions between particles have had time to fully develop. Nevertheless, we find that the concentrated systems exhibit clear deviations from the isolated-particle theory. These deviations are well described by several scaled versions of the isolated particle theory. The measurements also demonstrate a new multiple-scattering photon correlation technique with ultrashort time resolution.
A new and very general technique for simulating solid-fluid suspensions is described; its most important feature is that the computational cost scales linearly with the number of particles. The method combines Newtonian dynamics of the solid particles with a discretized Boltzmann equation for the fluid phase; the many-body hydrodynamic interactions are fully accounted for, both in the creeping-flow regime and at higher Reynolds numbers. Brownian motion of the solid particles arises spontaneously from stochastic fluctuations in the fluid stress tensor, rather than from random forces or displacements applied directly to the particles. In this paper, the theoretical foundations of the technique are laid out, illustrated by simple analytical and numerical examples; in the companion paper, extensive numerical tests of the method, for stationary flows, time-dependent flows, and finite Reynolds number flows, are reported. Comment: Text and figures in uuencode-tar-compressed postcript Email tony_ladd@llnl.gov
Fluctuating hydrodynamics and Brownian motion Application of the Langevin equation to fluid suspensions
- H Hasimoto
- E H Hauge
- A Martin-Lof
- E J Hinch
- W G Hoover
- D J Evans
- R B Hickman
- A J C Ladd
- W T Ashurst
- €3 8~ Moran
HASIMOTO, H. 1959 On the periodic fundamental solutions of the Stokes equations and their HAUGE, E. H. & MARTIN-LOF, A. 1973 Fluctuating hydrodynamics and Brownian motion. J. Statist. HINCH, E. J. 1975 Application of the Langevin equation to fluid suspensions. J. Fluid Mech. 72,499. HOOVER, W. G., EVANS, D. J., HICKMAN, R. B., LADD, A. J. C., ASHURST, W. T. 8~ MORAN, €3. 1980
Green-Kubo theory, Hamiltonian mechanics, and nonequilibrium molecular dynamics Finite-element analysis of incompressible viscous flows by the penalty-function formulation
- T J R Hughes
- W K Lin
- A Brookes
Lennard-Jones triple-point bulk and shear viscosities. Green-Kubo theory, Hamiltonian mechanics, and nonequilibrium molecular dynamics. Phys. Rev. A 22, 1690. HUGHES, T. J. R., LIN, W. K. & BROOKES, A. 1979 Finite-element analysis of incompressible viscous flows by the penalty-function formulation. J. Comput. Phys. 30, 1.
LAMFLOW: Three dimensional, laminar, incompressible flow
- A F B Tompson
Tompson, A. F. B. (1983). LAMFLOW: Three dimensional, laminar, incompressible ow.
Technical Report WR-83-3, Department of Civil Engineering, Princeton University.
Finite-element analysis of incompressible viscous ows by the penalty-function formulation
- T J R Hughes
- W K Lin
- A Brookes
Hughes, T. J. R., Lin, W. K., and Brookes, A. (1979). Finite-element analysis of incompressible viscous ows by the penalty-function formulation. J. Comp. Phys., 30:1.
Lattice gas hydrodynamics in two and three dimensions
- U Frisch
- D D'humières
- B Hasslacher
- P Lallemand
- Y Pomeau
- J.-P Rivet
Numerical simulations of particulate suspensions via a discretized Boltzmann equation. Part 1. Theoretical foundation
- A J C Ladd