Force-coupling Method Research Articles

Simulating Brownian suspensions with fluctuating hydrodynamics.

Fluctuating hydrodynamics has been successfully combined with several computational methods to rapidly compute the correlated random velocities of Brownian particles. In the overdamped limit where both particle and fluid inertia are ignored, one must also account for a Brownian drift term in order to successfully update the particle positions. In this paper, we present an efficient computational method for the dynamic simulation of Brownian suspensions with fluctuating hydrodynamics that handles both computations and provides a similar approximation as Stokesian Dynamics for dilute and semidilute suspensions. This advancement relies on combining the fluctuating force-coupling method (FCM) with a new midpoint time-integration scheme we refer to as the drifter-corrector (DC). The DC resolves the drift term for fluctuating hydrodynamics-based methods at a minimal computational cost when constraints are imposed on the fluid flow to obtain the stresslet corrections to the particle hydrodynamic interactions. With the DC, this constraint needs only to be imposed once per time step, reducing the simulation cost to nearly that of a completely deterministic simulation. By performing a series of simulations, we show that the DC with fluctuating FCM is an effective and versatile approach as it reproduces both the equilibrium distribution and the evolution of particulate suspensions in periodic as well as bounded domains. In addition, we demonstrate that fluctuating FCM coupled with the DC provides an efficient and accurate method for large-scale dynamic simulation of colloidal dispersions and the study of processes such as colloidal gelation.

Large-scale simulation of steady and time-dependent active suspensions with the force-coupling method

We present a new development of the force-coupling method (FCM) to address the accurate simulation of a large number of interacting micro-swimmers. Our approach is based on the squirmer model, which we adapt to the FCM framework, resulting in a method that is suitable for simulating semi-dilute squirmer suspensions. Other effects, such as steric interactions, are considered with our model. We test our method by comparing the velocity field around a single squirmer and the pairwise interactions between two squirmers with exact solutions to the Stokes equations and results given by other numerical methods. We also illustrate our method's ability to describe spheroidal swimmer shapes and biologically-relevant time-dependent swimming gaits. We detail the numerical algorithm used to compute the hydrodynamic coupling between a large collection (104–105) of micro-swimmers. Using this methodology, we investigate the emergence of polar order in a suspension of squirmers and show that for large domains, both the steady-state polar order parameter and the growth rate of instability are independent of system size. These results demonstrate the effectiveness of our approach to achieve near continuum-level results, allowing for better comparison with experimental measurements while complementing and informing continuum models.

Migration of finite sized particles in a laminar square channel flow from low to high Reynolds numbers

The migration of neutrally buoyant finite sized particles in a Newtonian square channel flow is investigated in the limit of very low solid volumetric concentration, within a wide range of channel Reynolds numbers Re = [0.07-120]. In situ microscope measurements of particle distributions, taken far from the channel inlet (at a distance several thousand times the channel height), revealed that particles are preferentially located near the channel walls at Re > 10 and near the channel center at Re < 1. Whereas the cross-streamline particle motion is governed by inertia-induced lift forces at high inertia, it seems to be controlled by shear-induced particle interactions at low (but finite) Reynolds numbers, despite the low solid volume fraction (<1%). The transition between both regimes is observed in the range Re = [1-10]. In order to exclude the effect of multi-body interactions, the trajectories of single freely moving particles are calculated thanks to numerical simulations based on the force coupling method. With the deployed numerical tool, the complete particle trajectories are accessible within a reasonable computational time only in the inertial regime (Re > 10). In this regime, we show that (i) the particle undergoes cross-streamline migration followed by a cross-lateral migration (parallel to the wall) in agreement with previous observations, and (ii) the stable equilibrium positions are located at the midline of the channel faces while the diagonal equilibrium positions are unstable. At low flow inertia, the first instants of the numerical simulations (carried at Re = O(1)) reveal that the cross-streamline migration of a single particle is oriented towards the channel wall, suggesting that the particle preferential positions around the channel center, observed in the experiments, are rather due to multi-body interactions.

Fluctuating force-coupling method for simulations of colloidal suspensions

The resolution of Brownian motion in simulations of micro-particle suspensions can be crucial to reproducing the correct dynamics of individual particles, as well as providing an accurate characterisation of suspension properties. Including these effects in simulations, however, can be computationally intensive due to the configuration dependent random displacements that would need to be determined at every time step. In this paper, we introduce the fluctuating force-coupling method (FCM) to overcome this difficulty, providing a fast approach to simulate colloidal suspensions at large-scale. We show explicitly that by forcing the surrounding fluid with a fluctuating stress and employing the FCM framework to obtain the motion of the particles, one obtains the random particle velocities and angular velocities that satisfy the fluctuation–dissipation theorem. This result holds even when higher-order multipoles, such as stresslets, are included in the FCM approximation. Through several numerical experiments, we confirm our analytical results and demonstrate the effectiveness of fluctuating FCM, showing also how Brownian drift can be resolved by employing the appropriate time integration scheme and conjugate gradient method.

Numerical investigation of channel blockage by flowing microparticles

The dynamic formation of 3D structures of microparticle aggregates blocking the flow through straight microchannels is investigated by direct numerical simulation of the coupled motion of particles and fluid. We use the Force Coupling Method to handle simultaneously multibody hydrodynamic interactions of confined flowing suspension together with particle–particle and particle–wall surface interactions leading to adhesion and aggregation of particles. The basic idea of the Force Coupling Method relies on a multipole expansion of forcing terms (added to the Navier–Stokes equations) accounting for the velocity perturbation induced by the presence of particles in the fluid flow. When a particle reaches the wall or an attached particle, we consider that the adhesion is irreversible and this particle remains fixed. We investigate the kinetics of the microchannel blockage for several solid volumetric concentrations and different surface interaction forces. Many physical quantities such as the temporal evolution of the bulk permeability, capture efficiency, modification of the fluid flow and forces acting on attached particles are analyzed. We show that physical–chemical interactions, modeled by DLVO forces, are essential features which control the blockage dynamics and aggregate structure.

The effect of neutrally buoyant finite-size particles on channel flows in the laminar-turbulent transition regime

The presence of finite-size particles in a channel flow close to the laminar-turbulent transition is simulated with the Force Coupling Method which allows two-way coupling with the flow dynamics. Spherical particles with channel height-to-particle diameter ratio of 16 are initially randomly seeded in a fluctuating flow above the critical Reynolds number corresponding to single phase flow relaminarization. When steady-state is reached, the particle volume fraction is homogeneously distributed in the channel cross-section (ϕ ≅ 5%) except in the near-wall region where it is larger due to inertia-driven migration. Turbulence statistics (intensity of velocity fluctuations, small-scale vortical structures, wall shear stress) calculated in the fully coupled two-phase flow simulations are compared to single-phase flow data in the transition regime. It is observed that particles increase the transverse r.m.s. flow velocity fluctuations and they break down the flow coherent structures into smaller, more numerous and sustained eddies, preventing the flow to relaminarize at the single-phase critical Reynolds number. When the Reynolds number is further decreased and the suspension flow becomes laminar, the wall friction coefficient recovers the evolution of the laminar single-phase law provided that the suspension viscosity is used in the Reynolds number definition. The residual velocity fluctuations in the suspension correspond to a regime of particulate shear-induced agitation.

A meshless boundary method for Stokes flows with particles: Application to canalithiasis

We propose to couple the method of fundamental solutions (MFS) to the force coupling method (FCM). The resulting method is an efficient, easy to program, meshless method for flows at low Reynolds numbers with finite-size particles. In such an approach, the flow domain is extended across the solid particle phase, and the flow is approximated by a superposition of singular Stokeslets positioned outside the flow domain and finite-size multipoles collocated with the particle. To improve the efficiency of the coupling, we propose new MFS quadratures for the computation of the volume integrals required for the FCM. These are exact and do not require the expensive evaluation of Stokeslets. The proposed method has been developed in the context of investigations of the fluid dynamics of canalithiasis, that is, a pathological condition of the semicircular canals of the inner ear. Numerical examples are presented to illustrate the applicability of the method.

Hydrodynamic interactions among large populations of swimming micro-organisms

Bio-convection, biofilm forming or mechanics of reproduction are connected to the motility and collective behaviour of micro-organisms. For instance, spermatozoa suspensions exhibit coherent motion whose frequency and lifespan are strongly correlated to semen fertility (Moore et al. 2002). As interplays in many-bodied systems result in intricate patterns, the understanding of these requires an in-depth knowledge of the suspension microstructure and statistics. Representative and reliable statistics require a large number Np of interactive swimmers that many simulation methods can hardly afford [e.g. Np ¼ 40 in Mehandia and Nott (2008) or Np # 216 in Ishikawa et al. (2008)]. In the following, a spherical swimmer model is derived from the classical low Reynolds number framework and implemented in the force-coupling method (FCM) for large populations. Resulting statistics reveals non-trivial spatial arrangements of swimmers depending on their swimming gait.

Dynamics and rheology of concentrated, finite-Reynolds-number suspensions in a homogeneous shear flow

We present the lubrication-corrected force-coupling method for the simulation of concentrated suspensions under finite inertia. Suspension dynamics are investigated as a function of the particle-scale Reynolds number \documentclass[12pt]{minimal}\begin{document}$Re_{\dot{\gamma }}$\end{document}Reγ̇ and the bulk volume fraction ϕ in a homogeneous linear shear flow, in which \documentclass[12pt]{minimal}\begin{document}$Re_{\dot{\gamma }}$\end{document}Reγ̇ is defined from the density ρf and dynamic viscosity μ of the fluid, particle radius a, and the shear rate \documentclass[12pt]{minimal}\begin{document}$\dot{\gamma }$\end{document}γ̇ as \documentclass[12pt]{minimal}\begin{document}$Re_{\dot{\gamma }}= \rho _f \dot{\gamma } a^2 / \mu$\end{document}Reγ̇=ρfγ̇a2/μ. It is shown that the velocity fluctuations in the velocity-gradient and vorticity directions decrease at larger \documentclass[12pt]{minimal}\begin{document}$Re_{\dot{\gamma }}$\end{document}Reγ̇. However, the particle self-diffusivity is found to be an increasing function of \documentclass[12pt]{minimal}\begin{document}$Re_{\dot{\gamma }}$\end{document}Reγ̇ as the motion of the suspended particles develops a longer auto-correlation under finite fluid inertia. It is shown that finite-inertia suspension flows are shear-thickening and the particle stresses become highly intermittent as \documentclass[12pt]{minimal}\begin{document}$Re_{\dot{\gamma }}$\end{document}Reγ̇ increases. To study the detailed changes in the suspension microstructure and rheology, we introduce a particle-stress-weighted pair-distribution function. The stress-weighted pair-distribution function clearly shows that the increase of the effective viscosity at high \documentclass[12pt]{minimal}\begin{document}$Re_{\dot{\gamma }}$\end{document}Reγ̇ is mostly related to the strong normal lubrication interaction in the compressive principal axis of the shear flow.

A Meshless Boundary Method for Computation of Stokes Flow with Particles in the Semicircular Canals of the Inner Ear

AbstractBenign paroxismal positional vertigo (BPPV) is modeled by introducing free‐floating particles (canaliths) which settle inside the semicircular canals (SCC). The Stokes flow induced by a canalith is evaluated by coupling the force coupling method (FCM) to the method of fundamental solutions (MFS). The proposed methodology results in a straightforward meshless boundary method for the simulation of bounded Stokes flow with finite‐size particles. (© 2010 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Dynamics of concentrated suspensions of non-colloidal particles in Couette flow

Fully three-dimensional numerical simulations of concentrated suspensions of O(1000) particles in a Couette flow at zero Reynolds number are performed with the goal of determining the wall effects on concentrated suspensions of non-colloidal particles. The simulations, based on the force-coupling method, are performed for 0.2 ≤ φ ≤ 0.4 and 10 < Ly/a < 30, where φ denotes the volume fraction and Ly and a are, respectively, the channel height and the particle radius. It is shown that the suspensions can be divided into three regions depending on the microstructures; the wall region where a structured particle layering is dominant, the core region in which the suspension field is quasi-homogeneous, and the buffer region which shows the characteristics of both the particle layer and the shear structure. The width of the inhomogeneous region (wall and buffer) is a function of φ and not sensitive to Ly/a, once Ly/a is larger than a threshold. Rheological properties in the inhomogeneous and quasi-homogeneous regions are investigated. The particle stresses are compared with previous rheological models.

Simulation of concentrated suspensions using the force-coupling method

Abstract The force-coupling method (FCM) represents the dynamics of low Reynolds number suspension flows through a distributed, low-order, finite force-multipole expansion and provides an efficient, matrix-free method to solve the mobility problem for the particle motion. In concentrated suspensions, strong short-range lubrication forces are generated between particles in close proximity as fluid in the intervening gap is displaced by the relative motion of the particles. These forces, together with near-surface contact forces, play an important role in the suspension rheology and self-diffusion of particles. However these forces lead to ill-conditioned problems for determining the particle stresses and particle motion in large systems of particles at higher volume fractions. A robust and effective iteration scheme for determining the particle stresslets is described together with a new scheme for including lubrication forces as near-field corrections to the FCM resistance problem. Both the lubrication and far-field interactions are solved as fully coupled systems in O ( N p log ( N p ) ) operations, for N p particles, using preconditioned conjugate gradient solvers. Numerical results for particles settling under gravity, particle pairs in linearly varying flows and in concentrated suspensions are compared with previous theoretical results and simulations. Numerical simulations with more than 4000 non-Brownian, spherical particles in a homogeneous shear flow provide results on the pair-distribution function and Lagrangian velocity correlations. The extension of the methods to simulate bidisperse systems or wall-bounded suspensions are discussed.

Modulation of homogeneous turbulence seeded with finite size bubbles or particles

The dynamics of homogeneous, isotropic turbulence seeded with finite sized particles or bubbles is investigated in a series of numerical simulations, using the force-coupling method for the particle phase and low wavenumber forcing of the flow to sustain the turbulence. Results are given on the modulation of the turbulence due to massless bubbles, neutrally buoyant particles and inertial particles of specific density 1.4 at volumetric concentrations of 6%. Buoyancy forces due to gravity are excluded to emphasize finite size and inertial effects for the bubbles or particles and their interactions with the turbulence. Besides observing the classical entrapment of bubbles and the expulsion of inertial particles by vortex structures, we analyze the Lagrangian statistics for the velocity and acceleration of the dispersed phase. The turbulent fluctuations are damped at mid-range wavenumbers by the bubbles or particles while the smallscale kinetic energy is significantly enhanced. Unexpectedly, the modulation of turbulence depends only slightly on the dispersion characteristics (bubble entrapment in vortices or inertial sweeping of the solid particles) but is closely related to the stresslet component (finite size effect) of the flow disturbances. The pivoting wavenumber characterizing the transition from damped to enhanced energy content is shown to vary with the size of the bubbles or particles. The spectrum for the energy transfer by the particle phase is examined and the possibility of representing this, at large scales, through an additional effective viscosity is discussed.

Force-coupling method for flows with ellipsoidal particles

The force-coupling method, previously developed for spherical particles suspended in a liquid flow, is extended to ellipsoidal particles. In the limit of Stokes flow, there is an exact correspondence with known analytical results for isolated particles. More generally, the method is shown to provide good approximate results for the particle motion and the flow field both in viscous Stokes flow and at finite Reynolds number. This is demonstrated through comparison between fully resolved direct numerical simulations and results from the numerical implementation of the force-coupling method with a spectral/hp element scheme. The motion of settling ellipsoidal particles and neutrally buoyant particles in a Poiseuille flow are discussed.

Smoothed profile method for particulate flows: Error analysis and simulations

We re-formulate and analyze a new method for particulate flows, the so-called “smoothed profile” method (SPM) first proposed in [Y. Nakayama, R. Yamamoto, Simulation method to resolve hydrodynamic interactions in colloidal dispersions, Phys. Rev. E 71 (2005) 036707], which uses a fixed computational mesh without conformation to the geometry of the particles. The method represents the particles by certain smoothed profiles to construct a body force term added into the Navier–Stokes equations. SPM imposes accurately and efficiently the proper conditions at the particle–fluid interface. In particular, while the original method employs a fully-explicit time-integration scheme we develop a high-order semi-implicit splitting scheme, which we implement in the context of spectral/ hp element discretization. First, we show that the modeling error of SPM has a non-monotonic dependence on the time step size Δ t ; it is a function of ν Δ t / ξ , where ν is the kinematic viscosity of fluid and ξ is the interface thickness of the smoothed profile. Subsequently, we present several steady and unsteady simulations, including flow past 3D complex-shaped particles, and compare against direct numerical simulations and the force coupling method (FCM).

Numerical Modeling of a Moving Particle in a Continuous Flow Subjected to Microwave Heating

Microwave heating of a food particle and carrier liquid as they flow continuously in a circular pipe is investigated numerically. The three-dimensional transient fluid flow as well as electromagnetic and temperature fields are described by a model that includes coupled Maxwell's, continuity, Navier-Stokes, and energy equations. The electromagnetic power and temperature distributions in both the liquid and the particle are taken into account. The hydrodynamic interaction between the solid particle and the carrier fluid is simulated by the force-coupling method (FCM). This article explores the effects of dielectric properties and the inlet position of the particle on microwave energy and temperature distributions inside the particle. The effect of the particle on power absorption in the carrier liquid is studied as well. The results show that electromagnetic power absorption by the particle is greatly influenced by the ratio of dielectric properties of the particle and the liquid as well as the distance between the particle and the location in the applicator where the electromagnetic power takes on its maximum value.

Investigation of a particulate flow containing spherical particles subjected to microwave heating

Microwave heating of a liquid and large spherical particles that it carries while continuously flowing in a circular applicator pipe is investigated. A three-dimensional model that includes coupled Maxwell, continuity, Navier–Stokes, and energy equations is developed to describe transient temperature, electromagnetic, and fluid velocity fields. The hydrodynamic interaction between the solid particles and the carrier liquid is simulated by the force-coupling method (FCM). Computational results are presented for the microwave power absorption, temperature distribution inside the liquid and the particles, as well as the velocity distribution in the applicator pipe and trajectories of particles. The effect of the time interval between consecutive injections of two groups of particles on power absorption in particles is studied. The influence of the position of the applicator pipe in the microwave cavity on the power absorption and temperature distribution inside the liquid and the particles is investigated as well.

Mathematical modelling of two-phase non-Newtonian flow in a helical pipe

Governing equations for a two-phase 3D helical pipe flow of a non-Newtonian fluid with large particles are derived in an orthogonal helical coordinate system. The Lagrangian approach is utilized to model solid particle trajectories. The interaction between solid particles and the fluid that carries them is accounted for by a source term in the momentum equation for the fluid. The force-coupling method (FCM), developed by M.R. Maxey and his group, is adopted; in this method the momentum source term is no longer a Dirac delta function but is spread on a numerical mesh by using a finite-sized envelop with a spherical Gaussian distribution. The influence of inter-particle and particle–wall collisions is also taken into account. Copyright © 2005 John Wiley & Sons, Ltd.

Modeling and optimization of colloidal micro-pumps

Manipulating micro-particles is very important in microfluidic applications, such as biomedical flows and self-assembled structures. Here, flows generated by the forced motion of colloidal micro-particles in a microchannel are investigated. The force coupling method combined with the spectral/hp element method is used to numerically simulate the dynamics of the flow, while a penalty method is used to determine the required forces on the particles. The pumping motion is investigated for two specific systems: a peristaltic micro-pump and a gear micro-pump. We verify the accuracy of the simulations and then for each system, we investigate the net flow rate as a function of pump frequency and channel dimension, and present optimization results. The results for the net flow rate are comparable to and within the range of the experimental data.

Spectral distributed Lagrange multiplier method: algorithm and benchmark tests

We extend the formulation of the distributed Lagrange multiplier (DLM) approach for particulate flows to high-order methods within the spectral/ hp element framework. We implement the rigid-body motion constraint inside the particle via a penalty method. The high-order DLM method demonstrates spectral convergence rate, i.e. discretization errors decrease exponentially as the order of spectral polynomials increases. We provide detailed comparisons between the spectral DLM method, direct numerical simulations, and the force coupling method for a number of 2D and 3D benchmark flow problems. We also validate the spectral DLM method with available experimental data for a transient problem. The new DLM method can potentially be very effective in many-moving body problems, where a smaller number of grid points is required in comparison with low-order methods.