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Diffuse interface model to simulate the rise of a fluid droplet across a cloud of particles
Abstract
A large variety of industrial and natural systems involve the adsorption of solid particles to the fluidic interface of droplets in motion. A diffuse interface model is here suggested to directly simulate the three-dimensional dynamics of a fluid droplet rising across a cloud of large particles. In this three-phase model the two solid-fluid boundaries and the fluidic boundary are replaced with smoothly spreading interfaces. The capillary effects and the three-phase flow hydrodynamics are fully resolved. A special treatment is adopted for the interparticle collisions. The effect of the particle concentration on the terminal velocity of a rising fluid droplet is then investigated. It is found that, at low Reynolds number, the terminal velocity of a rising fluid droplet decreases exponentially with the particle concentration. This exponential decay is confirmed by a simple rheological model.
... The accuracy of the SP method has been extensively studied in previous works [20,[22][23][24][25]. For example, the friction and mobility tensors of nonspherical particle assemblies obtained from simulations at low Reynolds number (Re) are within 5% of experimental values and high-precision solutions of the Stokes equation [24]. ...
... A similar degree of accuracy is found for the angular velocities of spherical particles under shear flow, for Re 10 [26], and the ζ potential of charged colloidal dispersions [27]. Likewise, simulation results for the terminal velocity of a rising droplet in a binary fluid (density ratio ρ A /ρ B 1) were within 8% of the theoretical predictions [25]. Finally, we note that these benchmark simulations were performed using relatively coarse resolutions for particles of size a = 4-5 with interfacial thickness ξ = 2 ( is the grid spacing). ...
Few simulations exist for microswimmers near deformable interfaces. Here, we present numerical simulations of the hydrodynamic flows associated with a single microswimmer embedded in a binary fluid mixture. The two fluids demix, separated by a penetrable and deformable interface that we assume to be initially prepared in its planar ground state. We find that the microswimmer can either penetrate the interface, move parallel to it, or bounce back off it. We analyze how the trajectory depends on the swimmer type (pusher/puller) and the angle of incidence with respect to the interface. Our simulations are performed in a system with periodic boundary conditions, corresponding to an infinite array of fluid interfaces. A puller reaches a steady state in which it either swims parallel to the interface or selects a perpendicular orientation, repeatedly penetrating through the interface. In contrast, a pusher follows a bouncing trajectory between two interfaces. We discuss several examples in biology in which swimmers penetrate soft interfaces. Our paper can be seen as a highly simplified model of such processes.
... Hence, a modified Allen-Cahn model [31], which relies on the minimization of a free energy E(ϕ, ∇ϕ) following the Ginzburg-Landau theory [40], is here employed. Dropping the subscript associated with the liquid for better readability, that is ϕ = ϕ ℓ , the spatio-temporal evolution of the liquid concentration is obtained by solving: where u is an Eulerian velocity field associated with the folding of the structural net, τ d a response time associated with the mobility of the drop, Ẽ = E/E 0 the free energy in its non-dimensional form, E 0 = k B T 0 /L 0 3 the reference energy, k B the Boltzmann constant, T 0 the temperature and L 0 a reference length [41]. The advection term u ⋅ ∇ϕ on the left-hand side of equation (2.12) contributes to a change in the drop shape caused by the displacement of each side panel, while the right-hand side term strives for an energetically stable drop shape. ...
We present a numerical framework bringing together a structural and a surface-energy minimization model to compute nano- and microorigami self-folding processes, that are three-dimensional in nature. A liquid drop, initially at rest on the template, triggers the spontaneous folding and deforms dynamically with the template. As application, the self-folding of thin two-dimensional templates into pyramidal microstructures is simulated. Each template is composed of a fixed regular base connected to rigid triangular side panels by two elastic hinges. We presently determine the condition, at which the transition from partial to full drop encapsulation occurs. The present model makes use of phase fields to numerically represent the drop and the template. This allows to: (i) develop an efficient computational method, which incorporates time-derivative terms and (ii) study three-dimensional complex self-folding processes.
... Broadly, one-fluid formulations can be categorised into interface-capturing or front-tracking (interface-tracking) schemes. Although both formulations have seen extensive application to phase-change problems [13,14,15,16,17,18,19,20,21,22], when one considers FSI in tandem with phase-change, the extension of interface-capturing methods to accommodate FSI is non-trivial, although we note some recent works which have attempted this [23,24]. The key difficulty in applying interface-capturing schemes to FSI are attributed to the fact that the interface location is not explicitly tracked, but instead captured, typically through an Eulerian phase-indicator field. ...
In so-called Lagrangian melting problems, a solid immersed in a fluid medium is free to rotate and translate in tandem with its phase-change from solid to liquid. Such configurations may be classified as a fluid-solid interaction (FSI) problem coupled to phase-change. Our present work proposes a numerical method capable of simulating these Lagrangian melting problems and adopts a front-tracking immersed-boundary (IB) method. We use the moving least squares IB framework, a well-established method for simulating a diverse range of FSI problems and extend this framework to accommodate melting by additionally imposing the Stefan condition at the interface. In the spirit of canonical front-tracking methods, the immersed solid is represented by a discrete triangulated mesh which is separate from the Eulerian mesh in which the governing flow equations are solved. A known requirement for these methods is the need for comparable Eulerian and Lagrangian grid spacings to stabilise interpolation and spreading operations between the two grids. For a melting object, this requirement is inevitably violated unless interventional remeshing is introduced. Our work therefore presents a novel dynamic remeshing procedure to overcome this. The remeshing is based on a gradual coarsening of the triangulated Lagrangian mesh and amounts to a negligible computational burden per timestep owing to the incremental and local nature of its operations, making it a scalable approach. Moreover, the coarsening is coupled to a volume-conserving smoothing procedure detailed by Kuprat et al. (2001), ensuring a zero net volume change in the remeshing step to machine precision. This added feature makes our present method highly specialised to the study of melting problems, where precise measurements of the melting solid's volume is often the primary predictive quantity of interest.
... Recently, ad hoc formulations have been proposed for the study of phase change problems, from solid-liquid [38,39] to evaporation/boiling phenomena [86][87][88]. Finally, phase-field methods have been also used to describe the dynamics of solid finite size particles [89] or particle-interface interactions [90,91]. ...
... Recently, ad hoc formulations have been proposed for the study of phase change problems, from solid-liquid [38,39] to evaporation/boiling phenomena [86][87][88]. Finally, phase-field methods have been also used to describe the dynamics of solid finite size particles [89] or particle-interface interactions [90,91]. ...
Turbulent flows laden with large, deformable drops are ubiquitous in nature and in a wide range of industrial processes. Prediction of the interactions between drops, which deform under the action of turbulence, exchange momentum via surface tension, and that can also exchange heat or mass, are complicated due to the wide range of scales involved: from the largest scales of the flow, down to the Kolmogorov scales of turbulence, and further down to the molecular scale of the interface. Due to this wide range of scales, the numerical description of these flows is challenging and requires robust and accurate numerical schemes that are able to capture both the turbulence characteristics and the dynamics of ever-moving and deforming interfaces including their topological changes (i.e., coalescence and breakage). In the past decades, various numerical methods have been proposed for simulating two-phase flows, from interface-tracking methods, where the interface is explicitly tracked with the use of marker points to interface-capturing methods, where the interface is identified as the isovalue of a color/marker function. Phase-field methods belong to the category of interface-capturing methods, and have emerged as promising approaches to simulate complex two-phase flows. In phase-field methods, the transport equation to describe the drop motion is obtained from first thermodynamics principles, and phenomena acting at the interface scale can be conveniently modeled. Although in realistic case scenarios, the physical thickness of the interface cannot be directly simulated, this family of methods offers desirable properties that have attracted the interest of researchers in recent years. In this work, we describe the fundamentals of the phase-field modeling associated with the direct numerical simulation of turbulence in the context of drop-laden flows. We discuss the potentials of the phase-field method with reference to breakage and coalescence phenomena, and to the corresponding drop size distribution; we examine how to model surface tension changes due to surfactant distribution, and we outline the framework to model heat and mass transfer fluxes. Finally, we present our perspectives for future developments of phase-field modeling of drop-laden turbulent flows in the context of the current available literature.
... Since previously, the coupled level set VOF method has shown great advantageous over the VOF method, so the present work aims to apply the same methodology for multiphase fluid flow systems. This kind of solver can then be used in many application areas like emulsions [4][5]. ...
Dynamics of compound droplets consisting of more than two liquids is of engineering importance, in particular to bio-medical applications. This work reports a numerical study on the deformation and breakup of such compound droplets. An OpenFOAM based Coupled Level VOF finite volume solver was used in this study. A 2D axisymmetric isolated droplet consisting of two fluids was taken. Two set of simulations were performed. In the first case, the inner fluid was assumed to be Newtonian and the outer fluid was non-Newtonian. In the second set, the arrangement was reversed where the inner fluid was non-Newtonian and the outer fluid was Newtonian. In both cases, the non-Newtonian fluid was assumed to be a power-law fluid with a rheology of the form τ = Ký^{n}. The density ratio between the liquids is unity and the liquid to gas density ratio is 10^{3}. The compound droplet was subjected to an ambient air flow stream. Effect of outer fluid film thickness and interfacial shear rate on droplet deformation and breakup for different Reynolds number is studied. Both shear thinning and shear thickening fluids were considered where the Power fluid index, n is varied in the range of 0.5 to 1.5. Detailed analysis of the flow field inside the droplet was also analyzed.
... Nonetheless, this first work is an alternative and promising method for future foam simulations. Extension of this foam model to account for better flow conditions is currently on-going (Lecrivain et al. 2018), yet it requires additional efforts for gas fractions approaching unity, such as a dynamic mesh refinement algorithm to compute the liquid velocity in the interstitial spaces. ...
A modified phase-field model is presented to numerically study the dynamics of flowing foam in an obstructed channel. The bubbles are described as smooth deformable fields interacting with one another through a repulsive potential. A strength of the model lies in its ability to simulate foams with wide range of gas fraction. The foam motion, composed of about hundred two-dimensional gas elements, was analyzed for gas fractions ranging from 0.4 to 0.99, that is below and beyond the jamming transition. Simulations are preformed near the quasi-static limit, indicating that the bubble rearrangement in the obstructed channel is primarily driven by the soft collisions and not by the hydrodynamics. Foam compression and relaxation upstream and downstream of the obstacle are reproduced and qualitatively match previous experimental and numerical observations. Striking dynamics, such as bubbles being squeezed by their neighbors in negative flow direction, are also revealed at intermediate gas fractions.
A general method is presented for computing the motions of hydrodynamically interacting particles in various kinds of host fluids for arbitrary Reynolds numbers. The method follows the standard procedure for performing direct numerical simulations (DNS) of particulate systems, where the Navier-Stokes equation must be solved consistently with the motion of the rigid particles, which defines the temporal boundary conditions to be satisfied by the Navier-Stokes equation. The smoothed profile (SP) method provides an efficient numerical scheme for coupling the continuum fluid mechanics with the dispersed moving particles, which are allowed to have arbitrary shapes. In this method, the sharp boundaries between solid particles and the host fluid are replaced with a smeared out thin shell (interfacial) region, which can be accurately resolved on a fixed Cartesian grid utilizing a SP function with a finite thickness. The accuracy of the SP method is illustrated by comparison with known exact results. In the present paper, the high degree of versatility of the SP method is demonstrated by considering several types of active and passive particle suspensions.
The capillary-induced bending of flexible fibres, a process also known as elasto-capillary deformation, is central to a variety of industrial and non-industrial applications, among which stand out textile flotation, stabilization of emulsions, micro-folding of elastic structures, and clogging of feather fibres by oil droplets. A consistent formulation for the direct numerical simulation of a flexible fibre interacting with a fluidic interface is presently suggested. The fibre is geometrically decomposed into a chain of spherical beads, which undergo stretching, bending, and twisting. interactions. The capillary force, acting at the three-phase contact line, is calculated using a ternary diffuse-interface model. In a first stage, the fibre deformation model and the ternary diffuse-interface model are validated against theoretical solutions. In a second stage, the two- and three-dimensional elasto-capillary bending of a fibre by an immersed droplet are numerically investigated. Partial wrapping and complete encapsulation could be simulated. The results show that the fibre curvature increases linearly with the square of the elasto-capillary length, for both low and large structural deformation.
The attachment of solid particles to the surface of immersed gas bubbles plays a fundamental role in surface science, and hence plays key roles in various engineering fields ranging from industrial separation processes to the fabrication of functional materials. However, detailed investigation from a microscopic view on how a single particle attaches to a bubble surface and how the particle properties affect the attachment behavior has been so far scarcely addressed. Here, we observed the attachment of a single particle to a bubble surface using a high-speed camera and systematically investigated the effects of the wettability and shape of particles. We found that hydrophobic particles abruptly “jumped into” the bubble while sliding down the bubble surface to eventually satisfy their static contact angles, the behavior of which induced a much stronger attachment to the bubble surface. Interestingly, the determinant factor for the attachment efficiency of spherical particles was not the wettability of the spherical particles but the location of the initial collision with the bubble surface. In contrast, the attachment efficiency of anisotropically-shaped particles was found to increase with the hydrophobicity caused by a larger contact area to the bubble surface. Last but not least, a simple formulation is suggested to recover the contact angle based on the jump-in.
One of the biggest problems when computational approaches are applied for novel engineering targets is that the software often does not support necessary inter-particle force models or boundary conditions needed to reproduce proper situations. Many previous simulation studies are based mostly on scientific and academic interests or focusing only on whether already known experimental results are correctly reproduced or whether the reliability of the model is acceptable. This prevents computational approaches to be used as an effective tool to develop new materials and processes. In this project, we aimed to enhance the applicability and functionality of a colloid simulator KAPSEL, which has been developed by ourselves. Specifically, we have achieved the following points.
1) DLVO model was implemented as an interparticle force model for efficiently simulating charged particle dispersions with thinner electric double layer compared to the particle diameter.
2) The simulation code is parallelized so that more realistic larger-scale systems can be simulated.
3) The external-electric-field induced anisotropic interaction between like-charged colloidal particles is calculated.
Direct numerical simulation of liquid-gas-solid flows is uncommon due to the considerable computational cost. As the grid spacing is determined by the smallest involved length scale, large grid sizes become necessary -- in particular if the bubble-particle aspect ratio is on the order of 10 or larger. Hence, it arises the question of both feasibility and reasonability. In this paper, we present a fully parallel, scalable method for direct numerical simulation of bubble-particle interaction at a size ratio of 1-2 orders of magnitude that makes simulations feasible on currently available super-computing resources. With the presented approach, simulations of bubbles in suspension columns consisting of more than fully resolved particles become possible. Furthermore, we demonstrate the significance of particle-resolved simulations by comparison to previous unresolved solutions. The results indicate that fully-resolved direct numerical simulation is indeed necessary to predict the flow structure of bubble-particle interaction problems correctly.
A numerical extension of the “smooth profile method” is presently suggested to simulate the attachment of a colloidal particle to the surface of an immersed bubble. In this approach, the two fluid-particle boundaries and the fluidic boundary are replaced with diffuse interfaces. The method is tested under various capillary numbers. Upon attachment to a stable bubble, it is found that the method is capable of reproducing the three microprocesses associated with the particle attachment. The change in the trajectory as the particle approaches the fluidic interface, the collision process, and the sliding down the bubble surface are all captured. Potential application of the present method shows great promise in the field of froth flotation, where the capture of hydrophobic particles by rising bubbles is of primary importance.
Bubbles are dynamic objects that grow and rise or shrink and disappear, often on the scale of seconds. This conflicts with their uses in foams where they serve to modify the properties of the material in which they are embedded. Coating the bubble surface with solid particles has been demonstrated to strongly enhance the foam stability, although the mechanisms for such stabilization remain mysterious. In this paper, we reduce the problem of foam stability to the study of the behavior of a single spherical bubble coated with a monolayer of solid particles. The behavior of this armored bubble is monitored while the ambient pressure around it is varied, in order to simulate the dissolution stress resulting from the surrounding foam. We find that above a critical stress, localized dislocations appear on the armor and lead to a global loss of the mechanical stability. Once these dislocations appear, the armor is unable to prevent the dissolution of the gas into the surrounding liquid, which translates into a continued reduction of the bubble volume, even for a fixed overpressure. The observed route to the armor failure therefore begins from localized dislocations that lead to large-scale deformations of the shell until the bubble completely dissolves. The critical value of the ambient pressure that leads to the failure depends on the bubble radius, with a scaling of ΔPcollapse∝R−1, but does not depend on the particle diameter. These results disagree with the generally used elastic models to describe particle-covered interfaces. Instead, the experimental measurements are accounted for by an original theoretical description that equilibrates the energy gained from the gas dissolution with the capillary energy cost of displacing the individual particles. The model recovers the short-wavelength instability, the scaling of the collapse pressure with bubble radius, and the insensitivity to particle diameter. Finally, we use this new microscopic understanding to predict the aging of particle-stabilized foams, by applying classical Ostwald ripening models. We find that the smallest armored bubbles should fail, as the dissolution stress on these bubbles increases more rapidly than the armor strength. Both the experimental and theoretical results can readily be generalized to more complex particle interactions and shell structures.
Particles added to a fluid interface can be used as a surface stabilizer in
the food, oil and cosmetic industries. As an alternative to rigid particles, it
is promising to consider highly deformable particles that can adapt their
conformation at the interface. In this study we compute the shapes of soft
elastic particles using molecular dynamics simulations of a cross-linked
polymer gel, complemented by continuum calculations based on linear elasticity.
It is shown that the particle shape is not only affected by the Young's modulus
of the particle, but also strongly depends on whether the gel is partially or
completely wetting the fluid interface. We find that the molecular simulations
for the partially wetting case are very accurately described by the continuum
theory. By contrast, when the gel is completely wetting the fluid interface the
linear theory breaks down and we reveal that molecular details have a strong
influence on the equilibrium shape.
The beauty and complexity of the shapes and dynamics of bubbles rising in liquid have fascinated scientists for centuries. Here we perform simulations on an initially spherical bubble starting from rest. We report that the dynamics is fully three-dimensional, and provide a broad canvas of behaviour patterns. Our phase plot in the Galilei-Eötvös plane shows five distinct regimes with sharply defined boundaries. Two symmetry-loss regimes are found: one with minor asymmetry restricted to a flapping skirt; and another with marked shape evolution. A perfect correlation between large shape asymmetry and path instability is established. In regimes corresponding to peripheral breakup and toroid formation, the dynamics is unsteady. A new kind of breakup, into a bulb-shaped bubble and a few satellite drops is found at low Morton numbers. The findings are of fundamental and practical relevance. It is hoped that experimenters will be motivated to check our predictions.
This paper presents a new direct simulation method for a drying colloidal suspension on a substrate. A key issue of the present method is the immersed free surface model proposed by the authors, which enables us to estimate accurately and efficiently capillary forces exerted on particles on a free surface. Using the immersed free surface model along with immersed boundary method and level set method, the present method leads to a three-way coupling of the fluid flow, the free surface motion and the particle motion. In addition, the present method includes a way of curvature estimation using virtual grid differencing to calculate accurately a surface tension. The way of curvature estimation is quantitatively validated through the simulation of a still droplet. The immersed free surface model is quantitatively validated through the simulation of a sphere moving across a free surface and the simulation of two spheres moving along a free surface. Finally, simulations of drying colloidal suspension containing 130 particles are performed to demonstrate the applicability of the present method to actual systems.
We present a review and critique of several methods for the simulation of the dynamics of colloidal suspensions at the mesoscale. We focus particularly on simulation techniques for hydrodynamic interactions, including implicit solvents (Fast Lubrication Dynamics, an approximation to Stokesian Dynamics) and explicit/particle-based solvents (Multi-Particle Collision Dynamics and Dissipative Particle Dynamics). Several variants of each method are compared quantitatively for the canonical system of monodisperse hard spheres, with a particular focus on diffusion characteristics, as well as shear rheology and microstructure. In all cases, we attempt to match the relevant properties of a well-characterized solvent, which turns out to be challenging for the explicit solvent models. Reasonable quantitative agreement is observed among all methods, but overall the Fast Lubrication Dynamics technique shows the best accuracy and performance. We also devote significant discussion to the extension of these methods to more complex situations of interest in industrial applications, including models for non-Newtonian solvent rheology, non-spherical particles, drying and curing of solvent and flows in complex geometries. This work identifies research challenges and motivates future efforts to develop techniques for quantitative, predictive simulations of industrially relevant colloidal suspension processes.
The energy required to detach a single particle from a fluid-fluid interface
is an important parameter for designing certain soft materials, for example,
emulsions stabilised by colloidal particles, colloidosomes designed for
targeted drug delivery, and bio-sensors composed of magnetic particles adsorbed
at interfaces. For a fixed particle volume, prolate and oblate spheroids attach
more strongly to interfaces because they have larger particle-interface areas.
Calculating the detachment energy of spheroids necessitates the difficult
measurement of particle-liquid surface tensions, in contrast with spheres,
where the contact-angle suffices. We develop a simplified detachment energy
model for spheroids which depends only the particle aspect-ratio and the height
of the particle centre-of-mass above the fluid-fluid interface. We use lattice
Boltzmann simulations to validate the model and provide quantitative evidence
that the approach can be applied to simulate particle-stabilized emulsions, and
highlight the experimental implications of this validation.
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.
Interfaces between two fluids are ubiquitous and of special importance for
industrial applications, e.g., stabilisation of emulsions. The dynamics of
fluid-fluid interfaces is difficult to study because these interfaces are
usually deformable and their shapes are not known a priori. Since experiments
do not provide access to all observables of interest, computer simulations pose
attractive alternatives to gain insight into the physics of interfaces. In the
present article, we restrict ourselves to systems with dimensions comparable to
the lateral interface extensions. We provide a critical discussion of three
numerical schemes coupled to the lattice Boltzmann method as a solver for the
hydrodynamics of the problem: (a) the immersed boundary method for the
simulation of vesicles and capsules, the Shan-Chen pseudopotential approach for
multi-component fluids in combination with (b) an additional
advection-diffusion component for surfactant modelling and (c) a molecular
dynamics algorithm for the simulation of nanoparticles acting as emulsifiers.
Subject collections (18 articles) geology (97 articles) fluid mechanics (87 articles) materials science Articles on similar topics can be found in the following collections We present data for the rheology of suspensions of monodisperse particles of varying aspect ratio, from oblate to prolate, and covering particle volume fractions φ from dilute to highly concentrated. Rheology is characterized by fitting the experimental data to the model of Herschel & Bulkley (Herschel & Bulkley 1926 Kolloid Z. 39, 291–300 (doi:10.1007/BF01432034)) yielding three rheometric parameters: consistency K (cognate with viscosity); flow index n (a measure of shear-thinning); yield stress τ 0 . The consistency K of suspensions of particles of arbitrary aspect ratio can be accurately predicted by the model of Maron & Pierce (Maron & Pierce 1956 J. Colloid Sci. 11, 80–95 (doi:10.1016/0095-8522(56)90023-X)) with the maximum packing fraction φ m as the only fitted parameter. We derive empirical relationships for φ m and n as a function of average particle aspect ratio r p and for τ 0 as a function of φ m and a fitting parameter τ * . These relationships can be used to predict the rheology of suspensions of prolate particles from measured φ and r p . By recasting our data in terms of the Einstein coefficient, we relate our rheological observations to the underlying particle motions via Jeffery's (Jeffery 1922 Proc. R. Soc. Lond. A 102, 161–179 (doi:10.1098/rspa.1922.0078)) theory. We extend Jeffery's work to calculate, numerically, the Einstein coefficient for a suspension of many, initially randomly oriented particles. This provides a physical, microstructural explanation of our observations, including transient oscillations seen during run start-up and changes of rheological regime as φ increases.
The authors review the development of diffuse-interface models of hydrodynamics and their application to a wide variety of interfacial phenomena. The authors discuss the issues involved in formulating diffuse-interface models for single-component and binary fluids. Recent applications and computations using these models are discussed in each case. Further, the authors address issues including sharp-interface analyses that relate these models to the classical free-boundary problem, related computational approaches to describe interfacial phenomena, and related approaches describing fully-miscible fluids.
A computational method is presented to resolve hydrodynamic interactions acting on solid particles immersed in incompressible host fluids. In this method, boundaries between solid particles and host fluids are replaced with a continuous interface by assuming a smoothed profile. This enabled us to calculate hydrodynamic interactions both efficiently and accurately, without neglecting many-body interactions. The validity of the method was tested by calculating the drag force acting on a single cylindrical rod moving in an incompressible Newtonian fluid. This method was then applied in order to simulate sedimentation process of colloidal dispersions.
A new description of the binary fluid problem via the lattice Boltzmann method is presented which highlights the use of the moments in constructing two equilibrium distribution functions. This offers a number of benefits, including better isotropy, and a more natural route to the inclusion of multiple relaxation times for the binary fluid problem. In addition, the implementation of solid colloidal particles suspended in the binary mixture is addressed, which extends the solid-fluid boundary conditions for mass and momentum to include a single conserved compositional order parameter. A number of simple benchmark problems involving a single particle at or near a fluid-fluid interface are undertaken and show good agreement with available theoretical or numerical results. Comment: 10 pages, 4 figures, ICMMES 2004
A consistent formulation is presented for the direct numerical simulation of an arbitrarily shaped colloidal particle at a deformable fluidic interface. The rigid colloidal particle is decomposed into a collection of solid spherical beads and the three-phase boundaries are replaced with smoothly spreading interfaces. The major merit of the present formulation lies in the ease with which the geometrical decomposition of the colloidal particle is implemented, yet allows the dynamic simulation of intricate three-dimensional colloidal shapes in a binary fluid. The dynamics of a rodlike, a platelike, and a ringlike particle are presently tested. It is found that platelike particles attach more rapidly to a fluidic interface and are subsequently harder to dislodge when subject to an external force. Using the Bond number, i.e., the ratio of the gravitational force to the reference capillary force, a spherical particle with equal affinity for the two fluids breaks away from a fluidic interface at the critical value Bo=0.75. This value is in line with our numerical experiments. It is here shown that a plate and a ring of equivalent masses detach at greater critical Bond numbers approximately equal to Bo=1.3. Results of this study will find applications in the stabilization of emulsions by colloids and in the recovery of colloidal particles by rising bubbles.
We study the process of coating the interface of a long gas bubble, which is translating in a horizontal circular capillary tube filled with a colloidal suspension. A typical elongated confined bubble is comprised of three distinct regions: a spherical front cap, a central body that is separated from the tube wall by a thin liquid film, and a spherical cap at the back. These three regions are connected by transitional sections. Particles gradually coat the bubble from the back to the front. We investigate the mechanisms that govern the initial accumulation of the particles and the growth of the particle-coated area on the interface of the bubble. We show that the initial accumulation of particles starts at the stable stagnation ring on the rear cap of the bubble, and the particles will completely coat the spherical cap at the back of the bubble before accumulating on the central body. Armoring the central interface of the bubble with particles thickens the liquid film around the bubble relative to that around the particle-free interface. This effect creates a rather sharp step on the interface of the bubble in the central region, which separates the armored region from the particle-free region. After the bubble is completely coated, the liquid film around the body of the bubble will adjust again to an intermediate thickness. We show that the three distinct thicknesses that the liquid film acquires during the armoring process can be well described analytically.
In this study, a numerical method is developed to perform the direct numerical simulation (DNS) of gas-solid-liquid flows involving capillary effects. The volume-of-fluid method employed to track the free surface and the immersed boundary method is adopted for the fluid-particle coupling in three-phase flows. This numerical method is able to fully resolve the hydrodynamic force and capillary force as well as the particle motions arising from complicated gas-solid-liquid interactions. We present its application to liquid bridges among spherical particles in this paper. By using the DNS method, we obtain the static bridge force as a function of the liquid volume, contact angle, and separation distance. The results from the DNS are compared with theoretical equations and other solutions to examine its validity and suitability for modeling capillary bridges. Particularly, the nontrivial liquid bridges formed in triangular and tetrahedral particle clusters are calculated and some preliminary results are reported. We also perform dynamic simulations of liquid bridge ruptures subject to axial stretching and particle motions driven by liquid bridge action, for which accurate predictions are obtained with respect to the critical rupture distance and the equilibrium particle position, respectively. As shown through the simulations, the strength of the present method is the ability to predict the liquid bridge problem under general conditions, from which models of liquid bridge actions may be constructed without limitations. Therefore, it is believed that this DNS method can be a useful tool to improve the understanding and modeling of liquid bridges formed in complex gas-solid-liquid flows.
Gas–liquid–solid three phase flows are commonly found in (bio-)chemical processes, e.g. in bubble slurry columns in the Fischer–Tropsch process. In order to facilitate the rational scale-up and the design of such columns, a detailed understanding of the complex phase interactions is required. In this work, we focus on the effective drag acting on particles and bubbles, using Direct Numerical Simulations. We combined the Front Tracking method of Roghair et al. (2013b) and the second order implicit Immersed Boundary method of Deen et al. (2012). The resulting hybrid Front Tracking Immersed Boundary method is able to simulate dense three phase flows and is used to study swarm effects in terms of drag. A correlation has been obtained for the drag coefficient for a system consisting of 2 mm bubbles and 1 mm particles at several phase volume fractions. In future research, the developed methodology can be applied to study the effect of the bubble and particle size and their ratio.
In gas-cooled high temperature reactors, the diffusion of the fission products into the graphite matrix causes a radioactive contamination of the carbonaceous dust. The contaminated graphite aerosol particles often exhibit large aspect ratios and deposit in complex geometries, which hinders a detailed experimental investigation. The use of quasi Direct Numerical Simulation (quasi-DNS) to simulate the turbulent flow in nuclear reactors has seen an increased interest over the last few years. The capabilities of a quasi-DNS to simulate the transport and the deposition of elongated particles in a wavy channel flow are presently tested. It is shown that quasi-DNS effectively predicts deposition and that, unlike the deposition in a plane channel flow, the particle aspect ratio has no significant effect on the overall deposition rate in a wavy channel. It is suggested that in numerical studies of particle deposition on a significantly roughened channel, the particle can be assumed to be spherical without affecting the results of the deposition study.
Froth flotation is a separation process which plays a major role in the mining industry. It is essentially employed to recover a vast array of different valuable commodities such as rare earth minerals essential to the manufacture of high-tech products. Owing to its simplicity, the process is also widely used for de-inking recycled paper fibres and for waste water treatment. The flotation process essentially relies on the attachment of solid particles on the surface gas bubbles immersed in water. The present study seeks to investigate the effect of the particle shape on the attachment mechanism. Using an in-house optical micro-bubble sensor the approach, the sliding and the adhesion of micron milled glass fibres on the surface of a stationary air bubble immersed in stagnant water is thoroughly investigated. The translational and rotational velocities were measured for fibres of various aspect ratios. The results are compared with a theoretical model and with experimental data obtained with spherical glass beads. It is found that the fibre orientation during the sliding motion largely depends on the collision area. Upon collision near the upstream pole of the gas bubble the major axis of the fibre aligns with the local bubble surface (tangential fibre alignment). If collision occurs at least 30° further downstream only head of the fibre is in contact with the gas-liquid interface (radial fibre alignment).
The equilibrium behavior of ellipsoidal Janus nanoparticles adsorbed at spherical oil/water interfaces was investigated using dissipative particle dynamics simulations. Several phenomena were documented that were not observed on similar simulations for planar oil/water interfaces. The nanoparticles were found to yield isotropic, radial nematic phases, and axial nematic domains, depending on the nanoparticle characteristics (aspect ratio and surface chemistry), particle density at the interface, and droplet properties (curvature of the interface, and, surprisingly, liquid type). When adsorbed on water droplets, the nanoparticles with high aspect ratio and few nonpolar beads on their surface can show two preferred orientation angles. Only one equilibrium orientation was found for such nanoparticles adsorbed on oil droplets. These observations might help explain a discrepancy previously reported between experimental and simulation results concerning the preferential orientation of particles at liquid-liquid interfaces. Different driving forces are responsible for the phenomena just summarized, including nanoparticle-nanoparticle and nanoparticle-solvent interactions, nanoparticle density at the interface, and droplet curvature via the Laplace pressure. The simulation results we present could be useful for engineering Pickering emulsions towards practical applications, and perhaps also for guiding new technologies for material synthesis.
An improved formulation of the "Smoothed Profile" method is introduced to perform direct numerical simulations of arbitrary rigid body dispersions in a Newtonian host solvent. Previous implementations of the method were restricted to spherical particles, severely limiting the types of systems that could be studied. The validity of the method is carefully examined by computing the friction/mobility tensors for a wide variety of geometries and comparing them to reference values obtained from accurate solutions to the Stokes-Equation.
We experimentally investigate the effect of particles on the dynamics of a gas bubble rising in a liquid-solid suspension while the particles are equally sized and neutrally buoyant. Using the Stokes number as a universal scale, we show that when a bubble rises through a suspension characterized by a low Stokes number (in our case, small particles), it will hardly collide with the particles and will experience the suspension as a pseudoclear liquid. On the other hand, when the Stokes number is high (large particles), the high particle inertia leads to direct collisions with the bubble. In that case, Newton’s collision rule applies, and direct exchange of momentum and energy between the bubble and the particles occurs. We present a simple theory that describes the underlying mechanism determining the terminal bubble velocity.
Buoyancy-driven motion of bubbles is examined by direct numerical simulations. Two cases, with 48 monodispersed bubbles at two different gas∕liquid combinations, of deformable and nearly spherical bubbles, are simulated. For the nearly spherical bubbles Eo = 0.5, and for the deformable ones Eo = 4, and for both cases N = 8000 and α = 5.8%. This results in = 91.5 and = 0.53 for the nearly spherical system and = 77.6 and = 3 for the deformable one. The simulations show path oscillations of the bubbles in both cases and shape oscillations of the deformable bubbles. At quasi-steady-state, the distribution of the deformable bubbles is relatively uniform but the spherical bubbles are distributed nonuniformly as a result of the formation of horizontal “rafts.” For both cases, however, the probability density functions of the fluctuation velocities of the bubbles are found to be approximately Gaussian. The temporal autocorrelation functions of the fluctuation velocities show that the horizontal components become uncorrelated faster than the vertical component and the correlation time for the vertical autocorrelation for deformable bubbles is at least twice larger than that for nearly spherical ones.
The present paper gives an overview of two methodologies for the computer simulation of wetting and capillary forces as well as particle-stabilized emulsions, which usually appear in a two-phase fluid system including solid bodies. One method is the molecular simulation with a coarse-grained model. The other is the computational fluid dynamics with a phase-field model. The main emphasis of this review is concerning our idea how to model the solid surfaces with wettability for the different simulation methods in the same spirit of “a simple modeling with focusing on hydrophilicity and hydrophobicity”.
The interactions between two individual particle-stabilized bubbles were investigated, in the absence of surfactant, using a combination of coalescence rig and high speed video camera. This combination allows the visualization of bubble coalescence dynamics which provide information on bubble stability. Experimental data suggested that bubble stability is enhanced by both the adsorption of particles at the interface as indicated by the long induction time, and the increase in damping coefficient at high surface coverage. The interaction between an armoured bubble and a bare bubble (asymmetric interaction) can be destabilized through the addition of a small amount of salt which suggested that electrostatic interactions play a significant role in bubble stability. Interestingly, the DLVO theory cannot be used to describe the bubble stability in the case of a symmetric interaction as coalescence was inhibited at 0.1 M KCl both in the absence and presence of particles at the interfaces. Furthermore, bubbles can also be destabilized by increasing the particle hydrophobicity. This behaviour is due to thinner liquid films between particles and an increase in film drainage rate. The fraction of particles detached from the bubble surface after film rupture was found to be very similar within the range of solution ionic strength, surface coverage and particle hydrophobicity studied. This lack of dependence implies that the kinetic energy generated by the coalescing bubbles is larger than the attachment energy of the particles and dominates the detachment process. This study illuminates the stability behaviour of individual particle-stabilized bubbles and has potential impact on processes which involve their interaction.
The self-assembly behavior of shape-anisotropic particles at curved fluid interfaces is computationally investigated by diffuse interface field approach (DIFA). A Gibbs-Duhem-type thermodynamic formalism is introduced to treat heterogeneous pressure within the phenomenological model, in agreement with Young-Laplace equation. Computer simulations are performed to study the effects of capillary forces (interfacial tension and Laplace pressure) on particle self-assembly at fluid interfaces in various two-dimensional cases. For isolated particles, it is found that the equilibrium liquid interface remains circular and particles of different shapes do not disturb the homogeneous curvature of liquid interface, while the equilibrium position, orientation and stability of a particle at the liquid interface depend on its shape and initial location with respect to the liquid interface. For interacting particles, the curvature of local liquid interfaces is different from the apparent curvature of the particle shell; nevertheless, irrespective of the particle shapes, a particle-coated droplet always tends to deform into a circular morphology under positive Laplace pressure, loses mechanical stability and collapses under negative Laplace pressure, while adapts to any morphology and stays in neutral equilibrium under zero Laplace pressure. Finally, the collective behaviors of particles and Laplace pressure evolution in bicontinuous interfacially jammed emulsion gels (bijels) are investigated.
Benchmark configurations for quantitative validation and comparison of incompressible interfacial flow codes, which model two-dimensional bubbles rising in liquid columns, are proposed. The benchmark quantities: circularity, center of mass, and mean rise velocity are defined and measured to monitor convergence toward a reference solution. Comprehensive studies are undertaken by three independent research groups, two representing Eulerian level set finite-element codes and one representing an arbitrary Lagrangian–Eulerian moving grid approach.
The first benchmark test case considers a bubble with small density and viscosity ratios, which undergoes moderate shape deformation. The results from all codes agree very well allowing for target reference values to be established. For the second test case, a bubble with a very low density compared to that of the surrounding fluid, the results for all groups are in good agreement up to the point of break up, after which all three codes predict different bubble shapes. This highlights the need for the research community to invest more effort in obtaining reference solutions to problems involving break up and coalescence.
Other research groups are encouraged to participate in these benchmarks by contacting the authors and submitting their own data. The reference data for the computed benchmark quantities can also be supplied for validation purposes. Copyright
Slow flow through a periodic array of spheres is studied theoretically, and the drag force by the fluid on a sphere forming the periodic array is calculated using a modification of the method developed by Hashimoto (1959). Results for the complete range of volume fraction c of spheres are given for simple cubic, body-centered cubic, and face-centered cubic arrays and these agree well with the corresponding values reported by previous investigators. Also, series expansions for the drag force to 0(c10) are derived for each of these cubic arrays. The method is also applied to determine the drag force to 0(c3) on infinitely long cylinders in square and hexagonal arrays.
This paper presents an extensive comparison between numerical and experimental results for moderate deformed bubbles concerning their drag coefficient and deformation in the range (Re,We)=[1,100]×[0,5]. Experiments were made in tap water/glycerol mixtures and the numerical model is derived under the surfactant-free interface assumption. The experimental procedure and numerical model are presented in details; the corresponding results are compared with other analytical, experimental and numerical studies. Finally, experimental and numerical data are compared in order to assess the ability of the numerical model to predict the bubble drag coefficient and aspect ratio in a practical situation. It is found that the values predicted by the numerical model are generally in good agreement with the experimental results.
Phase field models offer a systematic physical approach for investigating complex multiphase systems behaviors such as near-critical interfacial phenomena, phase separation under shear, and microstructure evolution during solidification. However, because interfaces are replaced by thin transition regions (diffuse interfaces), phase field simulations require resolution of very thin layers to capture the physics of the problems studied. This demands robust numerical methods that can efficiently achieve high resolution and accuracy, especially in three dimensions. We present here an accurate and efficient numerical method to solve the coupled Cahn–Hilliard/Navier–Stokes system, known as Model H, that constitutes a phase field model for density-matched binary fluids with variable mobility and viscosity. The numerical method is a time-split scheme that combines a novel semi-implicit discretization for the convective Cahn–Hilliard equation with an innovative application of high-resolution schemes employed for direct numerical simulations of turbulence. This new semi-implicit discretization is simple but effective since it removes the stability constraint due to the nonlinearity of the Cahn–Hilliard equation at the same cost as that of an explicit scheme. It is derived from a discretization used for diffusive problems that we further enhance to efficiently solve flow problems with variable mobility and viscosity. Moreover, we solve the Navier–Stokes equations with a robust time-discretization of the projection method that guarantees better stability properties than those for Crank–Nicolson-based projection methods. For channel geometries, the method uses a spectral discretization in the streamwise and spanwise directions and a combination of spectral and high order compact finite difference discretizations in the wall normal direction. The capabilities of the method are demonstrated with several examples including phase separation with, and without, shear in two and three dimensions. The method effectively resolves interfacial layers of as few as three mesh points. The numerical examples show agreement with analytical solutions and scaling laws, where available, and the 3D simulations, in the presence of shear, reveal rich and complex structures, including strings.
In this paper a hybrid model is presented for the numerical simulation of gas–liquid–solid flows using a combined front tracking (FT) and discrete particle (DP) approach applied for, respectively, dispersed gas bubbles and solid particles present in the continuous liquid phase. The hard sphere DP model, originally developed by Hoomans et al. [1996. Chemical Engineering Science 51, 99–108] for dense gas–solid systems, has been extended to account for all additional forces acting on particles suspended in a viscous liquid and has been combined with the FT model presented recently by Deen et al. [2004a. CD-ROM Proceedings of Fifth International Conference on Multiphase Flow, ICMF’04, Yokohama, Japan, May 30–June 4, 2004; 2004b. Chemical Engineering Science 59, 1853–1861] for complex free surface flows. In this paper, the physical foundation of the combined FT-DP model will be presented together with illustrative computational results highlighting the capabilities of this hybrid model. The effect of bubble-induced particle mixing has been studied focusing on the effect of the volumetric particle concentration. In addition the retarding effect (drag modification) on the bubble rise velocity due to the presence of the suspended solid particles has been quantified.
We present a novel mesoscale simulation approach to modeling the evolution of solid particles segregated at fluid-fluid interfaces. The approach involves a diffuse-interface field description of each fluid phase in addition to the set of solid particles. The unique strength of the model is its generality to include particles of arbitrary shapes and orientations, as well as the ability to incorporate electrostatic particle interactions and external forces via a previous work [P.C. Millett, Y.U. Wang, Acta Mater. 57 (2009) 3101]. In this work, we verify that the model produces the correct capillary forces and contact angles by comparing with a well-defined analytical solution. In addition, simulation results of rotations of various-shaped particles at fluid-fluid interfaces, external force-induced capillary attraction/repulsion between particles, and spinodal decomposition arrest due to colloidal particle jamming at the interfaces are presented.
Interface stretching during mixing of a two phase fluid in shear flow is investigated numerically by introducing a mesoscopic description of the fluid. The classical infinitely thin boundary of separation between the two phases is replaced by a transition region of small but finite width, across which the order parameter of the two-phase fluid changes continuously. We consider the case of a conserved scalar order parameter, and a fluid velocity that satisfies a modified Navier-Stokes equation that includes an explicit coupling term to the order parameter. In the macroscopic limit of a very thin interface, this coupling term gives rise to capillary forces. We focus on the limit of low Reynolds number flow and compute the interface stretching as a function of time for a range of parameters of the fluid. At early times and small coupling, our calculation agrees with the classical case of a material line passively advected by the flow. At later times, the interface stretching i...
Inclusion of solid particles drastically affects the pattern evolution of phase separation of a binary fluid mixture, via preferential wetting of one of the phases to the particles. Here we study this problem by numerical simulation, which incorporates interparticle hydrodynamic interactions properly. When particles favor one of the components of a mixture, wetting layers are quickly formed on the particle surfaces and all particles are eventually included into the more wettable phase. For immobile particles, domains of the more wettable phase are pinned to the particles and the domain growth is thus suppressed. For this case, the domain size at a certain phase-separation time decreases monotonically with increasing the particle concentration. For mobile particles, on the other hand, the reentrant morphological transformation is observed as a function of the particle concentration: With an increase in the particle concentration, the domain morphology of the more wettable phase sequentially changes from network, droplet to network. We found that the final morphological transition is induced by wetting-induced depletion interaction: strong attractive interactions act among particles when the total volume of the more wettable phase is not enough to cover all the particles by wetting layers.
Bubbles of less than 1 micrometer and as large as 13.5 micrometers in diameter, stabilized by an apparent compression of substances sorbed onto their surfaces, were examined to determine their physical and temporal stability. Their ease of formation is related to the qualities of the water in which they are formed. Their presence in the water column must now be considered when interpreting acoustic data gathered to determine marine bubble populations.