Orest Shardt

Simulations of Droplet Coalescence in Simple Shear Flow

Simulating droplet coalescence is challenging because small-scale (tens of nanometers) phenomena determine the behavior of much larger (micrometer- to millimeter-scale) droplets. In general, liquid droplets colliding in a liquid medium coalesce when the capillary number is less than a critical value. We present simulations of droplet collisions and coalescence in simple shear flow using the free-energy binary-liquid lattice Boltzmann method. In previous simulations of low-speed collisions, droplets coalesced at unrealistically high capillary numbers. Simulations of noncoalescing droplets have not been reported, and therefore, the critical capillary number for simulated collisions was unknown. By simulating droplets with radii up to 100 lattice nodes, we determine the critical capillary number for coalescence and quantify the effects of several numerical and geometric parameters. The simulations were performed with a well-resolved interface, a Reynolds number of one, and capillary numbers from 0.01 to 0.2. The ratio of the droplet radius and interface thickness has the greatest effect on the critical capillary number. As in experiments, the critical capillary number decreases with increasing droplet size. A second numerical parameter, the interface diffusivity (Péclet number) also influences the conditions for coalescence: coalescence occurs at higher capillary numbers with lower Péclet numbers (higher diffusivity). The effects of the vertical offset between the droplets and the confinement of the droplets were also studied. Physically reasonable results were obtained and provide insight into the conditions for coalescence. Simulations that match the conditions of experiments reported in the literature remain computationally impractical. However, the scale of the simulations is now sufficiently large that a comparison with experiments involving smaller droplets (≈10 μm) and lower viscosities (≈10(-6) m(2)/s, the viscosity of water) may be possible. Experiments at these conditions are therefore needed to determine the interface thickness and Péclet number that should be used for predictive simulations of coalescence phenomena.

Simulations of Janus droplets at equilibrium and in shear

Janus droplets are compound droplets that consist of two adhering drops of different fluids that are suspended in a third fluid. We use the Shan-Chen lattice Boltzmann method for multicomponent mixtures to simulate Janus droplets at rest and in shear. In this simulation model, interfacial tensions are not known a priori from the model parameters and must be determined using numerical experiments. We show that interfacial tensions obtained with the Young-Laplace law are consistent with those measured from the equilibrium geometry. The regimes of adhering, separated, and engulfing droplets were explored. Two different adhesion geometries were considered for two-dimensional simulations of Janus droplets in shear. The first geometry resembles two adhering circles with small overlap. In the second geometry, the two halves are semicircular. For both geometries, the rotation rate of the droplet depends on its orientation. The width of the periodic simulation domain also affects the rotation rate of both droplet types up to an aspect ratio of 6:1 (width:height). While the droplets with the first geometry oscillated about the middle of the domain, the droplets of the second geometry did not translate while rotating. A four-pole vortex structure inside droplets of the second geometry was found. These simulations of single Janus droplets reveal complex behaviour that implies a rich range of possibilities for the rheology of Janus emulsions.

Membraneless water filtration using CO2

Water purification technologies such as microfiltration/ultrafiltration and reverse osmosis utilize porous membranes to remove suspended particles and solutes. These membranes, however, cause many drawbacks such as a high pumping cost and a need for periodic replacement due to fouling. Here we show an alternative membraneless method for separating suspended particles by exposing the colloidal suspension to CO2. Dissolution of CO2 into the suspension creates solute gradients that drive phoretic motion of particles. Due to the large diffusion potential generated by the dissociation of carbonic acid, colloidal particles move either away from or towards the gas–liquid interface depending on their surface charge. Using the directed motion of particles induced by exposure to CO2, we demonstrate a scalable, continuous flow, membraneless particle filtration process that exhibits low energy consumption, three orders of magnitude lower than conventional microfiltration/ultrafiltration processes, and is essentially free from fouling.

The critical conditions for coalescence in phase field simulations of colliding droplets in shear.

Simulations have been performed using the free-energy binary-liquid lattice Boltzmann method with sufficient resolution that the critical capillary number for coalescence was determined for collisions between droplets in simple shear with a small initial offset in the shear gradient direction. The simulations were used to study the behavior of the interacting interfaces and the film between them during collisions over a wide range of capillary numbers with emphasis on near-critical conditions. From these three-dimensional simulations with deforming interfaces, several features of the evolution of the film between the drops were observed. The critical film thickness was determined to be similar to the interface thickness, a power law described the dependence of the minimum film thickness on the capillary number in collisions without coalescece, and an inflection point was found in the dynamics of the minimum distance between drops that eventually coalesce. The rotation of the film and the flow in it were also studied, and a reversal in the flow was found to occur before coalescence. The mobility of the phase field was therefore important in the continued thinning of the film at the points of minimum thickness after the flow reversal. A comparison of the critical capillary number and critical film thickness in the simulations with the values for experiments in confined simple shear indicated that the effective physical radius of the simulated droplets was on the order of several micrometers. The results are significant for simulations of droplet interactions and emulsion flows in complex geometries and turbulence because they demonstrate the necessary scale of the computations and how parameters, such as the interface thickness and phase field mobility, should be selected for accurate results.

Surface parameterization of nonsimply connected planar Bézier regions

A technique is described for constructing three-dimensional vector graphics representations of planar regions bounded by cubic Bezier curves, such as smooth glyphs. It relies on a novel algorithm for compactly partitioning planar Bezier regions into nondegenerate Coons patches. New optimizations are also described for Bezier inside-outside tests and the computation of global bounds of directionally monotonic functions over a Bezier surface (such as its axis-aligned bounding box or optimal field-of-view angle). These algorithms underlie the three-dimensional illustration and typography features of the TeX-aware vector graphics language Asymptote.

Direct Simulation of Dense Suspensions of Non-Spherical Particles

We describe a method for the direct simulation of high-solids-volume-fraction (up to 45%) suspensions of non-spherical rigid particles that are non-colloidal and slightly denser than the interstitial fluid. The lattice-Boltzmann method is used to solve for the flow of the interstitial Newtonian fluid, and the immersed boundary method is used to enforce a no-slip boundary condition at the surface of each particle. The surface points for the immersed boundary method are also employed for collision handling by applying repulsive forces between the surface points of nearby particles. We also discuss methods for integrating the equations of particle motion at low density ratios and propose a method with improved accuracy. The methods are used to simulate rigid particles shaped like red blood cells. We report on the effect of the solids volume fraction on the sedimentation rate using a Richardson–Zaki model, and we describe the orientation of the particles during sedimentation. The particles settle in a preferentially vertical orientation at terminal particle Reynolds numbers near one. We compare a simulation at a 35% solids volume fraction with typical erythrocyte sedimentation rates, a common blood test. We find an order of magnitude lower sedimentation rate than the value for healthy adults. The discrepancy is attributed to the omission of agglomeration-inducing inter-cellular forces and the treatment of the red blood cells as rigid particles in the simulations.

Implementation of LB Simulations

After reading this chapter, you will understand the fundamentals of high-performance computing and how to write efficient code for lattice Boltzmann method simulations. You will know how to optimise sequential codes and develop parallel codes for multi-core CPUs, computing clusters, and graphics processing units. The code listings in this chapter allow you to quickly get started with an efficient code and show you how to optimise your existing code.

Lattice Boltzmann for Advection-Diffusion Problems

After reading this chapter, you will understand how the lattice Boltzmann equation can be adapted from flow problems to advection-diffusion problems with only small changes. These problems include thermal flows, and you will know how to simulate these as two interlinked lattice Boltzmann simulations, one for the flow and one for the thermal advection-diffusion. You will understand how advection-diffusion problems require different boundary conditions from flow problems, and how these boundary conditions may be implemented.

The Lattice Boltzmann Equation

After reading this chapter, you will know the basics of the lattice Boltzmann method, how it can be used to simulate fluids, and how to implement it in code. You will have insight into the derivation of the lattice Boltzmann equation, having seen how the continuous Boltzmann equation is discretised in velocity space through Hermite series expansion, before being discretised in physical space and time through the method of characteristics. In particular, you will be familiar with the various simple sets of velocity vectors that are available, and how the discrete BGK collision model is applied.

Numerical Methods for Fluids

After reading this chapter, you will have insight into a number of other fluid simulation methods and their advantages and disadvantages. These methods are divided into two categories. First, conventional numerical methods based on discretising the equations of fluid mechanics, such as finite difference, finite volume, and finite element methods. Second, methods that are based on microscopic, mesoscopic, or macroscopic particles, such as molecular dynamics, lattice gas models, and multi-particle collision dynamics. You will know where the particle-based lattice Boltzmann method fits in the landscape of fluid simulation methods, and you will have an understanding of the advantages and disadvantages of the lattice Boltzmann method compared to other methods.