17 results
Dynamics and active mixing of a droplet in a Stokes trap
- Gesse Roure, Alexander Z. Zinchenko, Robert H. Davis
-
- Journal:
- Journal of Fluid Mechanics / Volume 985 / 25 April 2024
- Published online by Cambridge University Press:
- 22 April 2024, A15
-
- Article
-
- You have access Access
- Open access
- HTML
- Export citation
-
Particle trapping and manipulation have a wide range of applications in biotechnology and engineering. Recently, a flow-based, particle-trapping device called the Stokes trap was developed for trapping and control of small particles in the intersection of multiple branches in a microfluidic channel. This device can also be used to perform rheological experiments to determine the viscoelastic response of an emulsion or suspension. We show that besides these applications, the various flow modes produced by the Stokes trap are able to manipulate drop shapes and induce active mixing inside droplets. To this end, we analyse the dynamics of a droplet in a Stokes trap through boundary-integral simulations. We also explore the dynamic response of drop shape with respect to distinct external flow modes, which allows us to perform numerical experiments such as step strain and oscillatory extension. A linear controller is used to manipulate drop position, and the drop deformation is characterized by a spherical-harmonic decomposition. For small drop deformations, we observe a linear superposition of harmonics, which, surprisingly, seems to hold even for moderate deformations. This result indicates that such a device can be used for shape control of droplets. We also investigate how the different flow modes may be combined to induce mixing inside the droplets. The transient combination of modes produces an effective chaotic mixing, which is characterized by a mixing number. The mixing inside the droplet can be further enhanced for lower viscosity ratios and low, but non-zero capillary number and flow frequencies.
Flow of a concentrated emulsion with surfactant through a periodic porous medium
- Alexander Z. Zinchenko, Jacob R. Gissinger, Robert H. Davis
-
- Journal:
- Journal of Fluid Mechanics / Volume 953 / 25 December 2022
- Published online by Cambridge University Press:
- 07 December 2022, A21
-
- Article
- Export citation
-
High-resolution, long-time three-dimensional simulations are presented for slow, pressure-driven flow of a periodic emulsion of deformable drops through a dense, simple cubic array of solid spheres (one drop and one particle per periodic cell). The drops, covered with insoluble, non-diffusive surfactant, are large compared with pores, and they squeeze with high resistance, very closely coating the solids to overcome surface tension and lubrication effects. The solid volume fraction is 50 %, the emulsion concentration $c_{em}$ in the pore space is 36 % or 50 %, the drop-to-medium viscosity ratio $\lambda$ is 0.25 to 4. The contamination measure $\beta \leq 0.1$ keeps the linear surfactant model (assumed in most of the work) physically relevant. The boundary-integral solution requires extreme resolutions (tens of thousands of boundary elements per surface) achieved by multipole acceleration with special desingularizations, combined with flow-biased surfactant transport algorithms for numerical stability. The time-periodic regime is typically attained after a few squeezing cycles; the motion period is used in the extrapolation scheme to evaluate critical capillary numbers $Ca_{crit}$ demarcating squeezing from trapping. Due to Marangoni stresses, even light ($\beta =0.05$) to moderate ($\beta =0.1$) contaminations significantly reduce the average drop-phase migration velocity (up to 2.8 times, compared with clean drops), especially at small $\lambda =0.25$. In contrast, $Ca_{crit}$ is weakly sensitive to contamination and levels off completely at $\beta =0.05$. At $\lambda =0.25$ and $c_{em}=0.36$, the average drop-phase velocities are much different for lightly and moderately contaminated emulsions, except for near-critical squeezing when they become the same. Nonlinear surfactant models (Langmuir, Frumkin) are used to validate the linear model.
A pairwise hydrodynamic theory for flow-induced particle transport in shear and pressure-driven flows
- Rodrigo B. Reboucas, Alexander Z. Zinchenko, Michael Loewenberg
-
- Journal:
- Journal of Fluid Mechanics / Volume 952 / 10 December 2022
- Published online by Cambridge University Press:
- 18 November 2022, A2
-
- Article
-
- You have access Access
- Open access
- HTML
- Export citation
-
An exact pairwise hydrodynamic theory is developed for the flow-induced spatial distribution of particles in dilute polydisperse suspensions undergoing two-dimensional unidirectional flows, including shear and planar Poiseuille flows. Coupled diffusive fluxes and a drift velocity are extracted from a Boltzmann-like master equation. A boundary layer is predicted in regions where the shear rate vanishes with thickness set by the radii of the upstream collision cross-sections for pair interactions. An analysis of this region yields linearly vanishing drift velocities and non-vanishing diffusivities where the shear rate vanishes, thus circumventing the source of the singular particle distribution predicted by the usual models. Outside of the boundary layer, a power-law particle distribution is predicted with exponent equal to minus half the exponent of the local shear rate. Trajectories for particles with symmetry-breaking contact interactions (e.g. rough particles, permeable particles, emulsion drops) are analytically integrated to yield particle displacements given by quadratures of hard-sphere (or spherical drop) mobility functions. Using this analysis, stationary particle distributions are obtained for suspensions in Poiseuille flow. The scale for the particle distribution in monodisperse suspensions is set by the collision cross-section of the particles but its shape is almost universal. Results for polydisperse suspensions show size segregation in the central boundary layer with enrichment of smaller particles. Particle densities at the centreline scale approximately with the inverse square root of particle size. A superposition approximation reliably predicts the exact results over a broad range of parameters. The predictions agree with experiments in suspensions up to approximately 20 % volume fraction without fitting parameters.
Drop squeezing between arbitrary smooth obstacles
- Jacob R. Gissinger, Alexander Z. Zinchenko, Robert H. Davis
-
- Journal:
- Journal of Fluid Mechanics / Volume 908 / 10 February 2021
- Published online by Cambridge University Press:
- 10 December 2020, A33
-
- Article
- Export citation
-
A fully three-dimensional boundary-integral method (BIM) is developed for the interaction of drops, suspended in a uniform far-field flow at small Reynolds number, with arbitrary Lyapunov surfaces. The close approach of fluid interfaces to solid surfaces poses significant challenges for numerical BIM implementations, due to the highly singular behaviour of single- and double-layer boundary integrals. Two new methods are described that generalize the accurate calculation of the highly singular surface integrals used by high-order desingularization techniques. The first method is semi-analytical, and applies to axisymmetric solid obstacles (in an arbitrary three-dimensional configuration). An axisymmetric particle can be divided into a series of characteristic disks along its axis, for which closed-form expressions for single and double layers are derived in terms of elliptic integrals. To accommodate arbitrary smooth surfaces, a multimesh desingularization method is introduced that calculates surface integrals utilizing a hierarchy of embedded mesh resolutions, together with distance-activated mesh interactions. Several particle shapes, including spherocylinders (capsules) and flat plates, are used to represent major classes characteristic of porous media. A droplet approaching a capsule will break up after forming two lobes, connected by a thin filament, on either side of the capsule. The cross-sectional shape of the filament affects lubrication behaviour. A constriction made of two parallel capsules, even of low aspect ratio, significantly retards drop passage compared to two spheres. Trends in drop squeezing between two capsules are summarized over a range of capillary number, viscosity ratio, drop size and capsule length. A constriction of two coplanar plates results in notably different lubrication and squeezing behaviour. Flow rectification is demonstrated for constrictions that are non-symmetrical with respect to flow reversal, for several non-axisymmetric particles.
Drops with insoluble surfactant squeezing through interparticle constrictions
- Jacob R. Gissinger, Alexander Z. Zinchenko, Robert H. Davis
-
- Journal:
- Journal of Fluid Mechanics / Volume 878 / 10 November 2019
- Published online by Cambridge University Press:
- 10 September 2019, pp. 324-355
-
- Article
- Export citation
-
The interfacial behaviour of surfactant-laden drops squeezing through tight constrictions in a uniform far-field flow is modelled with respect to capillary number, drop-to-medium viscosity ratio and surfactant contamination. The surfactant is treated as insoluble and non-diffusive, and drop surface tension is related to surfactant concentration by a linear equation of state. The constriction is formed by three solid spheres held rigidly in space. A characteristic aspect of this confined and contaminated multiphase system is the rapid development of steep surfactant-concentration gradients during the onset of drop squeezing. The interplay between two physical effects of surfactant, namely the greater interface deformability due to decreased surface tension and interface immobilization due to Marangoni stresses, results in particularly rich drop-squeezing dynamics. A three-dimensional boundary-integral algorithm is used to describe drop hydrodynamics, and accurate treatment of close squeezing and trapped states is enabled by advanced singularity subtraction techniques. Surfactant transport and hydrodynamics are coupled via the surface convection equation (or convection–diffusion equation, if artificial diffusion is included), the interfacial stress balance and a solid-particle contribution based on the Hebeker representation. For extreme conditions, such as drop-to-medium viscosity ratios significantly less than unity, it is found that upwind-biased methods are the only stable approaches for modelling surfactant transport. Two distinct schemes, upwind finite volume and flow-biased least squares, are found to provide results in close agreement, indicating negligible numerical diffusion. Surfactant transport is enhanced by low drop-to-medium viscosity ratios, at which extremely sharp concentration gradients form during various stages of the squeezing process. The presence of surfactant, even at low degrees of contamination, significantly decreases the critical capillary number for droplet trapping, due to the accumulation of surfactant at the downwind pole of the drop and its subsequent elongation. Increasing the degree of contamination significantly affects surface mobility and further decreases the critical capillary number as well as drop squeezing times, up to a threshold above which the addition of surfactant negligibly affects squeezing dynamics.
General rheology of highly concentrated emulsions with insoluble surfactant
- Alexander Z. Zinchenko, Robert H. Davis
-
- Journal:
- Journal of Fluid Mechanics / Volume 816 / 10 April 2017
- Published online by Cambridge University Press:
- 08 March 2017, pp. 661-704
-
- Article
- Export citation
-
A general constitutive model is constructed and validated for highly concentrated monodisperse emulsions of deformable drops with insoluble surfactant through long-time, large-scale and high-resolution multidrop simulations. There is the same amount of surfactant on each drop, and the linear model is assumed for the surface tension versus the surfactant concentration. The surfactant surface transport is coupled to multidrop hydrodynamics through the convective–diffusive equation and the interfacial stress balance. Only the limit of small surfactant diffusivities is addressed, when this parameter does not affect the rheology. An Oldroyd constitutive equation is postulated, with five variable coefficients depending on one instantaneous flow invariant (chosen as the drop-phase contribution to the dissipation rate). These coefficients are found by fitting the model to five precise rheological functions from two steady base flows at arbitrary deformation rates. One base flow is planar extension (PE) ($\dot{\unicode[STIX]{x1D6E4}}x_{1},-\dot{\unicode[STIX]{x1D6E4}}x_{2},0$), the other one is planar mixed flow (PM) ($\dot{\unicode[STIX]{x1D6FE}}x_{2}$, $\dot{\unicode[STIX]{x1D6FE}}\unicode[STIX]{x1D712}x_{1}$, 0) with $\unicode[STIX]{x1D712}=0.16$. A small but finite $\unicode[STIX]{x1D712}$ (a precise choice in the range $\unicode[STIX]{x1D712}\sim 0.1$ is unimportant) provides a necessarily perturbation to exclude severe ergodic difficulties and abnormal, kinked behaviour inherent in simple shear for high drop volume fractions $c$, especially at small capillary numbers $Ca$ and small drop-to-medium viscosity ratios $\unicode[STIX]{x1D706}$. The database rheological functions are obtained for $c=0.45{-}0.6$, $\unicode[STIX]{x1D706}=0.25{-}3$ and surfactant elasticities $\unicode[STIX]{x1D6FD}=0.05{-}0.2$ (based on the equilibrium surfactant concentration) from long-time simulations by a multipole-accelerated boundary-integral code with $N=100{-}200$ drops in a periodic cell and 2000–4000 boundary elements per drop. The code is an extension from Zinchenko & Davis (J. Fluid Mech., vol. 779, 2015, pp. 197–244) to account for surfactant transport and Marangoni stresses. Massive drop cusping or (sometimes) drop break-up limit the range of $Ca$ from above in the base flows, but there is no substantial lower limitation owing to the absence of phase transition difficulties. At small $\unicode[STIX]{x1D706}$, even minimal surface contamination may have a strong effect on the rheology. The simulations remain accurate for quite strong drop interactions, when the PE emulsion viscosity is nine times that for the carrier fluid. The model validation against a steady PM flow with a different $\unicode[STIX]{x1D712}=0.5$ shows a very good agreement for various $Ca$, $c$ and $\unicode[STIX]{x1D706}$. In the three PE and PM time-dependent flow tests, the quasi-steady approximation is found to predict stresses poorly. In contrast, the combination of the steady-state results for PE and PM used in the present method to generate the Oldroyd parameters gives a model with much better predictions for these time-dependent flows.
Extensional and shear flows, and general rheology of concentrated emulsions of deformable drops
- Alexander Z. Zinchenko, Robert H. Davis
-
- Journal:
- Journal of Fluid Mechanics / Volume 779 / 25 September 2015
- Published online by Cambridge University Press:
- 14 August 2015, pp. 197-244
-
- Article
- Export citation
-
The rheology of highly concentrated monodisperse emulsions is studied by rigorous multidrop numerical simulations for three types of steady macroscopic flow, (i) simple shear ($\dot{{\it\gamma}}x_{2}$, 0 0), (ii) planar extension (PE) ($\dot{{\it\Gamma}}x_{1},-\dot{{\it\Gamma}}x_{2},0$) and (iii) mixed ($\dot{{\it\gamma}}x_{2}$, $\dot{{\it\gamma}}{\it\chi}x_{1}$, 0), where $\dot{{\it\gamma}}$ and $\dot{{\it\Gamma}}$ are the deformation rates, and ${\it\chi}\in (-1,1)$ is the flow parameter, in order to construct and validate a general constitutive model for emulsion flows with arbitrary kinematics. The algorithm is a development of the multipole-accelerated boundary-integral (BI) code of Zinchenko & Davis (J. Fluid Mech., vol. 455, 2002, pp. 21–62). It additionally incorporates periodic boundary conditions for (ii) and (iii) (based on the reproducible lattice dynamics of Kraynik–Reinelt for PE), control of surface overlapping, much more robust controllable surface triangulations for long-time simulations, and more efficient acceleration. The emulsion steady-state viscometric functions (shear viscosity and normal stress differences) for (i) and extensiometric functions (extensional viscosity and stress cross-difference) for (ii) are studied in the range of drop volume fractions $c=0.45{-}0.55$, drop-to-medium viscosity ratios ${\it\lambda}=0.25{-}10$ and various capillary numbers $\mathit{Ca}$, with 100–400 drops in a periodic cell and 2000–4000 boundary elements per drop. High surface resolution is important for all three flows at small $\mathit{Ca}$. Large system size and strains $\dot{{\it\gamma}}t$ of up to several thousand are imperative in some shear-flow simulations to identify the onset of phase transition to a partially ordered state, and evaluate (although still not precisely) the viscometric functions in this state. Below the phase transition point, the shear viscosity versus $\mathit{Ca}$ shows a kinked behaviour, with the local minimum most pronounced at ${\it\lambda}=1$ and $c=0.55$. The ${\it\lambda}=0.25$ emulsions flow in a partially ordered manner in a wide range of $\mathit{Ca}$ even when $c=0.45$. Increase of ${\it\lambda}$ to 3–10 shifts the onset of ordering to much smaller $\mathit{Ca}$, often outside the simulation range. In contrast to simple shear, phase transition is never observed in PE or mixed flow. A generalized five-parameter Oldroyd model with variable coefficients is fitted to our extensiometric and viscometric functions at arbitrary flow intensities (but outside the phase transition range). The model predictions compare very well with precise simulation results for strong mixed flows, ${\it\chi}=0.25$. Time-dependent PE flow is also considered. Ways to overcome the phase transition and drop breakup limitations on constitutive modelling are discussed.
Growth of multiparticle aggregates in sedimenting suspensions
- Alexander Z. Zinchenko, Robert H. Davis
-
- Journal:
- Journal of Fluid Mechanics / Volume 742 / 10 March 2014
- Published online by Cambridge University Press:
- 24 February 2014, pp. 577-617
-
- Article
- Export citation
-
The process of multiparticle aggregation in a dilute sedimenting suspension is rigorously simulated, with precise hydrodynamical interactions. The primary particles are monodisperse non-Brownian spheres at zero Reynolds number, with short-range molecular attractions. The rigid aggregates grow, as they settle downwards, by sequential particle addition – a valid assumption for dilute suspensions during the initial stages. The growth starts from doublet–particle interaction, but the indeterminate initial doublet concentration does not affect the results for cluster geometry and settling velocity. A new particle is generated far below a cluster with uniform probability density, and many trial particle–cluster relative trajectories are computed with high accuracy until a collision is found. The new cluster is then assumed to be rigid and allowed to reach a steady sedimentation regime (which is a spiral motion around the axis of steady rotation, ASR) before another particle is added, and so on. The ASR is typically far away from the cluster centre of mass. The Stokes flow solution algorithm for particle–cluster interaction works very efficiently with high-order multipoles (to order 100) and is extended to arbitrarily small particle–cluster separations by a geometry perturbation adapted from the conductivity simulations of Zinchenko (Phil. Trans. R. Soc. Lond. A, 1998, vol. 356, pp. 2953–2998). Clusters are generated to $N=100$ spheres, with extensive averaging over many growth realizations. The fractal scaling $\sim N^{0.48}$ for the cluster settling speed is quickly attained once $N\geq 25$, and the exponent 0.48 is practically independent of the strength of molecular forces. The cluster fractal dimension is predicted to be $d_f=1.91\pm 0.02$ (in contrast to the existing views that sequential addition can only produce high-$d_f$ clusters). Several average characteristics of the cluster size are also computed. The theoretical settling speed has no adjustable parameters and agrees reasonably well with prior experiments for a moderately polydisperse system in a broad range of cluster sizes.
Emulsion flow through a packed bed with multiple drop breakup
- Alexander Z. Zinchenko, Robert H. Davis
-
- Journal:
- Journal of Fluid Mechanics / Volume 725 / 25 June 2013
- Published online by Cambridge University Press:
- 22 May 2013, pp. 611-663
-
- Article
- Export citation
-
Pressure-driven squeezing of a concentrated emulsion of deformable drops through a randomly packed granular material is studied by rigorous three-dimensional multidrop–multiparticle simulations at low Reynolds numbers. The drops are comparable in size with granular particles, so the drop phase and the carrier fluid have different permeabilities, and the emulsion cannot be treated as single phase. Squeezing requires significant drop deformation and can meet much resistance, depending on the capillary number $\boldsymbol{Ca}$. The granular material is modelled as a random loose packing (RLP) of many highly-frictional rigid monodisperse spheres in a periodic cell in mechanical equilibrium. Flow simulations for many drops squeezing through the network of solid spheres are performed by an extension of the multipole-accelerated boundary-integral (BI) algorithm of Zinchenko & Davis (J. Comput. Phys., vol. 227, 2008, pp. 7841–7888). A major improvement is robust mesh control on drop surfaces combined with a novel fragmentation algorithm, now allowing for long-time simulations with intricate drop shapes and multiple breakups. A major challenge is that up to $O(1{0}^{5} )$ time steps are required in a simulation for time averaging, and $O(1{0}^{4} )$ boundary elements per surface to sufficiently resolve lubrication and breakups. Such simulations are feasible due to multipole acceleration, with two orders-of-magnitude gain over the standard BI coding. For initial drop-to-particle size ratio 0.51–0.52, emulsion concentration 41–42 % in the available space, and matching viscosities, time- and ensemble-averaged permeabilities of the drop phase and the continuous phase are studied versus $\boldsymbol{Ca}$ for systems of different size (up to 36 particles and 100 drops in a periodic cell). An avalanche of drop breakups observed at sufficiently large $\boldsymbol{Ca}$ does not preclude the permeabilities from reaching a statistical steady state in a feasible simulation time. The critical, system-size-independent $\boldsymbol{Ca}$, when the drop-phase flow effectively stops due to blockage in the pores by capillary forces, is estimated from simulations. For a sample RLP configuration, deep distinctions are found between the flow of concentrated emulsions and single-drop motion.
Gravity-induced collisions of spherical drops covered with compressible surfactant
- ALEXANDER Z. ZINCHENKO, MICHAEL A. ROTHER, ROBERT H. DAVIS
-
- Journal:
- Journal of Fluid Mechanics / Volume 667 / 25 January 2011
- Published online by Cambridge University Press:
- 14 January 2011, pp. 369-402
-
- Article
- Export citation
-
Gravity-induced collisions of two spherical drops covered with an insoluble surfactant at low Reynolds numbers are considered. Unlike in previous collision studies, the present work accounts for nonlinear coupling between the surfactant distribution and drop hydrodynamics by solving the full unsteady convective–diffusion equation for the surfactant transport. Our method includes high-order three-dimensional multipole expansions for hydrodynamics and a Galerkin-type approach for the surfactant transport with implicit marching. The efficiency of the algorithm allows for calculating thousands of trajectories to very close contact and determining the collision efficiency (related to the critical initial horizontal offset) by trial and error. The solution is valid for arbitrary surface Péclet (Pes) and Marangoni (Ma) numbers and sets limitations on approximations used in prior work for collision-efficiency calculations. Two limiting cases are observed: at small Pes or large Ma, the variation in surfactant coverage is small, and the results for the incompressible surfactant model are recovered, while for large Pes and small Ma, the collision efficiency approaches the clean-interface value. For moderate drop-size ratios (radius ratio k ≤ 0.5), the results generally fall between these limits. At larger size ratios, however, the collision efficiency may even exceed the geometrical Smoluchowski limit for both drops and bubbles. Moreover, with even moderate redistribution of the surfactant, equal-sized drops can move relative to one another and collide. These novel effects do not exist for clean drops or drops covered with an incompressible surfactant, and they are due to the nonlinear coupling between surfactant dynamics and flow. This surfactant-enhanced coalescence takes place, for example, in a physical system of air bubbles in water if the surfactant surface concentration is dilute (Γ ≈ 1×10−9 mol m−2, much smaller than the typical maximum-packing value of 10−5−10−6 mol m−2).
A boundary-integral study of a drop squeezing through interparticle constrictions
- ALEXANDER Z. ZINCHENKO, ROBERT H. DAVIS
-
- Journal:
- Journal of Fluid Mechanics / Volume 564 / 10 October 2006
- Published online by Cambridge University Press:
- 15 September 2006, pp. 227-266
-
- Article
- Export citation
-
A three-dimensional boundary-integral algorithm is developed to study the squeezing of a deformable drop through a tight constriction formed by several solid particles rigidly held in space. The drop is freely suspended and driven by a flow that is uniform away from the solid obstacles. Particular emphasis is on the trapping mechanism and flow conditions close to critical, when the drop squeezes with high resistance. The problem is a close prototype of drop–solid interactions for emulsion flow through a granular material; such interactions are much more lubrication-sensitive than drop–drop interactions and require advanced numerical tools to succeed. The algorithm is based on the Hebeker representation for the solid–particle contribution, leading to a well-behaved system of second-kind integral equations, combined with novel regularization techniques for singular and near-singular boundary integrals; high-order near-singularity subtraction for the solid-to-drop double-layer contribution is the most crucial element. Simulations are performed for drop squeezing between (i) two close spheres, (ii) two parallel spheroidal disks, and (iii) three close spheres forming an equilateral triangle (including the case of close solid–solid contact). The drop non-deformed diameter is from two to several times larger than the inner constriction diameter and, in some simulations, the drop decelerates $10^3$–$10^4$ times in the throat before being able to pass through. The effects of the constriction type, capillary number, and viscosity ratio on the drop velocity in the throat, exit time, and drop–solid spacing (of the order of 1% of the particle size) are explored in detail; critical capillary numbers (below which trapping occurs) are accurately determined. Even for a substantially supercritical capillary number, the drop has to nearly coat solid particles to be able to pass through a tight constriction. The ability of the algorithm to simulate both supercritical and subcritical conditions (when the drop is trapped, with a small but non-zero drop–solid spacing) is vital for future applications to large-scale simulations of emulsion flow through granular media.
The collision rate of small drops in linear flow fields
- Hua Wang, Alexander Z. Zinchenko, Robert H. Davis
-
- Journal:
- Journal of Fluid Mechanics / Volume 265 / 25 April 1994
- Published online by Cambridge University Press:
- 26 April 2006, pp. 161-188
-
- Article
- Export citation
-
A dilute dispersion containing small, force-free drops of one fluid dispersed in a second, immiscible in a linear flow field is considered for small Reynolds numbers and large Péclet numbers under isothermal conditions. The emphasis of our analysis is on the effects of pairwise drop interactions on their collision rate, as described by the collision efficiency, using a trajectory analysis. Simple shear flow and uniaxial extensional or compressional flow are considered. For both flows, the collision efficiency decreases with increasing drop viscosity due to the effects of hydrodynamic interactions. It also decreases as the ratio of the smaller drop radius to the larger radius decreases. For uniaxial flow, finite collision rates are predicted in the absence of interdroplet forces for all finite values of the drop size ratio and the ratio of the viscosities of the drop and suspending medium. In contrast, several kinds of relative trajectories exist for a pair of drops in simple shear flow, including open trajectories, collision trajectories, and closed and semi-closed trajectories, in the absence of interdroplet forces. When the ratio of small to large drop diameters is smaller than a critical value, which increases with increasing drop viscosity, all of the relative trajectories that start with the two drops far apart remain open (no collisions), unless in the presence of attractive forces. Attractive van der Walls forces are shown to increase the collision rates.
Gravity-induced coalescence of drops at arbitrary Péclet numbers
- Alexander Z. Zinchenko, Robert H. Davis
-
- Journal:
- Journal of Fluid Mechanics / Volume 280 / 10 December 1994
- Published online by Cambridge University Press:
- 26 April 2006, pp. 119-148
-
- Article
- Export citation
-
The collision efficiency in a dilute suspension of sedimenting drops is considered, with allowance for particle Brownian motion and van der Waals attractive force. The drops are assumed to be of the same density, but they differ in size. Drop deformation and fluid inertia are neglected. Owing to small particle volume fraction, the analysis is restricted to binary interactions and includes the solution of the full quasi-steady Fokker—Planck equation for the pair-distribution function. Unlike previous studies on drop or solid particle collisions, a numerical solution is presented for arbitrary Péclet numbers, Pe, thus covering the whole range of particle size in typical hydrosols. Our technique is mainly based on an analytical continuation into the plane of complex Péclet number and a special conformal mapping, to represent the solution as a convergent power series for all real Péclet numbers. This efficient algorithm is shown to apply to a variety of convection—diffusion problems. The pair-distribution function is expanded into Legendre polynomials, and a finite-difference scheme with respect to particle separation is used. Two-drop mobility functions for hydrodynamic interactions are provided from exact bispherical coordinate solutions and near-field asymptotics. The collision efficiency is calculated for wide ranges of the size ratio, the drop-to-medium viscosity ratio, and the Péclet number, both with and without interdroplet forces. Solid spheres are considered as a limiting case; attractive van der Waals forces are required for non-zero collision rates in this case. For Pe [Gt ] 1, the correction to the asymptotic limit Pe → ∞ is O(Pe−1/2). For Pe [Lt ] 1, the first two terms in an asymptotic expansion for the collision efficiency are C/Pe + ½C2, where the constant C is determined from the Brownian solution in the limit Pe → 0. The numerical results are in excellent agreement with these limits. For intermediate Pe, the numerical results show that Brownian motion is important for Pe ≤ O(102). For Pe = 10, the trajectory analysis for Pe → ∞ may underestimate the collision rate by a factor of two. A simpler, approximate solution based on neglecting the transversal diffusion is also considered and compared to the exact solution. The agreement is within 2–3% for all conditions investigated. The effect of van der Waals attractions on the collision efficiency is studied for a wide range of droplet sizes. Except for very high drop-to-medium viscosity ratios, the effect is relatively small, especially when electromagnetic retardation is accounted for.
Dynamic simulation of spheroid motion between two parallel plane walls in low-Reynolds-number Poiseuille flow
- MICHELLE E. STABEN, ALEXANDER Z. ZINCHENKO, ROBERT H. DAVIS
-
- Journal:
- Journal of Fluid Mechanics / Volume 553 / 25 April 2006
- Published online by Cambridge University Press:
- 06 April 2006, pp. 187-226
-
- Article
- Export citation
-
A novel boundary-integral algorithm is used to study the general, three-dimensional motion of neutrally buoyant prolate and oblate spheroids in a low-Reynolds-number Poiseuille flow between parallel plates. Adaptive meshing of the spheroid surface assists in obtaining accurate numerical results for particle–wall gaps as small as 1.3% of the spheroid's major axis. The resistance formulation and lubrication asymptotic forms are then used to obtain results for arbitrarily small particle–wall separations. Spheroids with their major axes shorter than the channel spacing experience oscillating motion when the spheroid's centre is initially located in or near the midplane of the channel. For both two-dimensional and three-dimensional oscillations, the period length decreases with an increase in the initial inclination of the spheroid's major axis with respect to the lower wall. These spheroids experience tumbling motions for centre locations further from the midplane of the channel, with a period length that decreases as the spheroid is located closer to a wall. The transition from two-dimensional oscillating motion to two-dimensional tumbling motion occurs for an initial centre location closer to a wall as the initial inclination of the major axis is increased. For these spheroids, the average translational velocity along the channel length for two-dimensional oscillating motion decreases for an increase in the initial inclination of the major axis, and the average translational velocity for two-dimensional tumbling motion decreases for a decrease in the initial centre location. A prolate spheroid with its major axis 50% longer than the channel spacing and confined to the ($x_2$, $x_3$)-plane (where $x_2$ is the primary flow direction and $x_3$ is normal to the walls) cannot experience two-dimensional tumbling; instead, the spheroid becomes wedged between the walls for initial centre locations near the midplane of the channel when the initial inclination of the large spheroid's major axis is steep, and experiences two-dimensional oscillations for initial centre locations near a wall. When this spheroid's major axis is not confined to the ($x_2$, $x_3$)-plane, it experiences three-dimensional oscillations for initial centre locations in or near the midplane of the channel, and three-dimensional tumbling for initial centre locations near a wall.
Shear flow of highly concentrated emulsions of deformable drops by numerical simulations
- ALEXANDER Z. ZINCHENKO, ROBERT H. DAVIS
-
- Journal:
- Journal of Fluid Mechanics / Volume 455 / 25 March 2002
- Published online by Cambridge University Press:
- 18 April 2002, pp. 21-61
-
- Article
- Export citation
-
An efficient algorithm for hydrodynamical interaction of many deformable drops subject to shear flow at small Reynolds numbers with triply periodic boundaries is developed. The algorithm, at each time step, is a hybrid of boundary-integral and economical multipole techniques, and scales practically linearly with the number of drops N in the range N < 1000, for NΔ ∼ 103 boundary elements per drop. A new near-singularity subtraction in the double layer overcomes the divergence of velocity iterations at high drop volume fractions c and substantial viscosity ratio γ. Extensive long-time simulations for N = 100–200 and NΔ = 1000–2000 are performed up to c = 0.55 and drop-to-medium viscosity ratios up to λ = 5, to calculate the non-dimensional emulsion viscosity μ* = Σ12/(μeγ˙), and the first N1 = (Σ11−Σ22)/(μe[mid ]γ˙[mid ]) and second N2 = (Σ22−Σ33)/(μe[mid ]γ˙[mid ]) normal stress differences, where γ˙ is the shear rate, μe is the matrix viscosity, and Σij is the average stress tensor. For c = 0.45 and 0.5, μ* is a strong function of the capillary number Ca = μe[mid ]γ˙[mid ]a/σ (where a is the non-deformed drop radius, and σ is the interfacial tension) for Ca [Lt ] 1, so that most of the shear thinning occurs for nearly non-deformed drops. For c = 0.55 and λ = 1, however, the results suggest phase transition to a partially ordered state at Ca [les ] 0.05, and μ* becomes a weaker function of c and Ca; using λ = 3 delays phase transition to smaller Ca. A positive first normal stress difference, N1, is a strong function of Ca; the second normal stress difference, N2, is always negative and is a relatively weak function of Ca. It is found at c = 0.5 that small systems (N ∼ 10) fail to predict the correct behaviour of the viscosity and can give particularly large errors for N1, while larger systems N [ges ] O(102)show very good convergence. For N ∼ 102 and NΔ ∼ 103, the present algorithm is two orders of magnitude faster than a standard boundary-integral code, which has made the calculations feasible.
Cusping, capture, and breakup of interacting drops by a curvatureless boundary-integral algorithm
- ALEXANDER Z. ZINCHENKO, MICHAEL A. ROTHER, ROBERT H. DAVIS
-
- Journal:
- Journal of Fluid Mechanics / Volume 391 / 25 July 1999
- Published online by Cambridge University Press:
- 25 July 1999, pp. 249-292
-
- Article
- Export citation
-
A three-dimensional boundary-integral algorithm for interacting deformable drops in Stokes flow is developed. The algorithm is applicable to very large deformations and extreme cases, including cusped interfaces and drops closely approaching breakup. A new, curvatureless boundary-integral formulation is used, containing only the normal vectors, which are usually much less sensitive than is the curvature to discretization errors. A proper regularization makes the method applicable to small surface separations and arbitrary λ, where λ is the ratio of the viscosities of the drop and medium. The curvatureless form eliminates the difficulty with the concentrated capillary force inherent in two-dimensional cusps and allows simulation of three-dimensional drop/bubble motions with point and line singularities, while the conventional form can only handle point singularities. A combination of the curvatureless form and a special, passive technique for adaptive mesh stabilization allows three-dimensional simulations for high aspect ratio drops closely approaching breakup, using highly stretched triangulations with fixed topology. The code is applied to study relative motion of two bubbles or drops under gravity for moderately high Bond numbers [Bscr ], when cusping and breakup are typical. The deformation-induced capture efficiency of bubbles and low-viscosity drops is calculated and found to be in reasonable agreement with available experiments of Manga & Stone (1993, 1995b). Three-dimensional breakup of the smaller drop due to the interaction with a larger one for λ=O(1) is also considered, and the algorithm is shown to accurately simulate both the primary breakup moment and the volume partition by extrapolation for moderately supercritical conditions. Calculations of the breakup efficiency suggest that breakup due to interactions is significant in a sedimenting emulsion with narrow size distribution at λ=O(1) and [Bscr ][ges ]5–10. A combined capture and breakup phenomenon, when the smaller drop starts breaking without being released from the dimple formed on the larger one, is also observed in the simulations. A general classification of possible modes of two-drop interactions for λ=O(1) is made.
Buoyancy-driven coalescence of slightly deformable drops
- MICHAEL A. ROTHER, ALEXANDER Z. ZINCHENKO, ROBERT H. DAVIS
-
- Journal:
- Journal of Fluid Mechanics / Volume 346 / 10 September 1997
- Published online by Cambridge University Press:
- 10 September 1997, pp. 117-148
-
- Article
- Export citation
-
The simultaneous effect of small deformation and short-range van der Waals attraction on the coalescence efficiency of two different-sized slowly sedimenting drops is considered. For spherical drops, it has been shown previously that the tangential mobility of drop surfaces makes collision possible even without van der Waals attraction; on the other hand, even a small amount of deformation precludes drops from coming into contact unless van der Waals attraction is accounted for. In the present work, the conditions are delineated when these two small-scale factors, acting in opposite directions, have a considerable combined effect on the coalescence efficiency. The problem is solved by matched asymptotic expansions valid for small capillary numbers (Ca). The outer solution, for two spherical drops moving in apparent contact without van der Waals attraction, determines the contact force as a function of time. This force is used as the driving force for the inner solution of the relevant integro-differential thin-film equations (coupling the flow in the small-gap region to that inside the drops) to determine whether coalescence occurs during the apparent contact motion. The initial gap profile for the inner solution is provided by matching with the outer trajectory for spherical drops approaching contact.
The analysis shows that, for Ca[Lt ]1, the near-contact deformation is mainly axisymmetric, greatly simplifying the inner solution; nevertheless, determination of the critical horizontal offsets leading to coalescence and the parametric analysis are computationally very intensive. To facilitate these tasks, a substantially new, highly efficient, and absolutely stable numerical method for solving stiff thin-film equations is developed. Unlike for spherical drops, when the upstream intersection area is a circle, the existence of a second coalescence zone for deformable drops is found over much of the parameter space. Results are mapped out for a range of four dimensionless parameters (capillary number, size and drop-to-medium viscosity ratios, dimensionless Hamaker parameter). As a physical application, predicted coalescence efficiencies are shown for a system of ethyl salicylate drops in diethylene glycol.
The present solution extends the range of drop sizes where the coalescence efficiencies are known theoretically and can be used in drop population dynamics. Comparison with full three-dimensional boundary-integral calculations for deformable drops without van der Waals attraction is also made to demonstrate that, when the drop-to-medium viscosity ratio is of the order of unity, the present asymptotic approach is valid in a wide range of small and moderately small capillary numbers.