Schedule for: 17w5155 - Complex Creeping Fluids: Numerical Methods and Theory

Soft particles such as drops, vesicles, and red blood cells, display rich dynamics in externally applied flow and electric fields. For example, vesicles have multiple dynamical states in shear flow (tank-treading, tumbling); drops in uniform electric field undergo various symmetry-breaking instabilities. I will overview the complex flow behavior of these inertialess systems and discuss how it arises from the nonlinear interaction of a deformable microstructure and external forcing.

Rigid or flexible particles suspended in viscoelastic fluids are ubiquitous in the food industry (e.g. pastes), industrial molding applications (all composites and 3-D printed parts), the energy industry (e.g. fracking fluids), and biological fluids (i.e. swimming of bacteria in mucous). The mathematics of the description of these suspensions is in its infancy. However, while the mathematics of this subject is subtle a major breakthrough in this area has been the development of computational simulations of such viscoelastic suspensions, with particle level resolution, such that predictions can be made and tested at all volume fraction loadings. I describe the use of an Immersed Boundary methodology that allows the simulation of hundreds of particles in elastic fluids, with particle level flow and stress field resolution. This simulation capability is unique and overcomes the major hurdle in understanding the physics of these suspensions – which in many cases are simply qualitatively different than that of Newtonian suspensions. The simplest flows of such suspensions are not understood at a fundamental level, primarily because the collective behavior of particles in an elastic liquid has no foundation – this will change dramatically in the next few years. I will describe three foundational problems that have now been analyzed using this new computational method – including fracking fluid design and swimming in mucous.

Boundary integral methods are among the most popular methods for computing interfacial fluid flow, and have the advantage that they can be made high-order accurate. Thus, they are useful for investigating phenomena that require high accuracy to resolve features, such as "pinching" or topological singularities that can occur on the interface. However, standard BI methods lose accuracy when two parts of an interface are near touching. In this talk, we present a new algorithm based on the QBX method of Klockner et al. for the accurate computation of boundary integrals with singular or nearly singular kernels in 3D. The QBX method is typically based on a spherical harmonics expansion which when truncated at $\mathcal{O}(p)$ has $\mathcal{O}(p^2)$ terms. This expansion can equivalently be written with $\mathcal{O}(p)$ terms, but paying the price that the expansion coefficients will depend on the target point. Based on this observation, we develop a target specific QBX method. We give error estimates for our method, and illustrate its performance in several examples. This work is joint with Anna-Karin Tornberg.

The level of complexity in blood flow simulation depends on the application. The focus here is on cellular blood where major cells and proteins must be included in the analysis. The progress in cellular blood flow simulation based on lattice-Boltzmann method (LBM) will be discussed. Advantages of this method in terms of scalability, accuracy and flexibility for multiscale analysis critical to many applications will be presented.

A central part of integral equation methods are the quadrature methods used to evaluate boundary integrals that are singular or have nearby singularities. A relatively new such method is quadrature by expansion (QBX), which evaluates the solution using local expansions formed at points inside the domain. I will in this talk discuss how QBX can be used for Stokes flow simulations in both two and three dimensions. In particular, I will discuss techniques for accelerating the method such that it can be combined with existing fast methods, and also how parameter selection can be simplified through accurate estimation of quadrature errors.

Fluid-mechanical erosion of solid material occurs across many scales, from massive geological structures down to tiny granular constituents. Here we examine the erosion of a granular medium in Stokes flow - the typical flow regime of groundwater - using numerical simulations. We combine a highly-accurate boundary-integral formulation (for the Stokes flow) with stable interface-evolution methods (to treat the eroding bodies). A single eroding body tends toward a slender, for-aft-symmetric morphology which can be described analytical. Supplementing the Stokes solver with the Fast Multiple Method allows us to simulate 10-100 bodies. We find that the erosion of many bodies naturally leads to the formation of channels, as well as anisotropy in the medium conductivity. This latter feature we connect to the single-body limiting morphology.

We will describe minimal models and fast computations to study the individual and collective motion of micro-swimmers in a variety of confinements such as drops, channels, around obstacles and on surfaces. The dynamics results from a complex interplay of direct collisions, hydrodynamics, noise, the swimmer body geometry. We validate our results to experiments and show that to correctly capture the dynamics, minimal models need to resolve species particulars such as the body shape asymmetry, cell or flagella spinning.

The electrohydrodynamics (EHD) of vesicle suspensions is characterized by studying their pairwise interactions in applied DC electric fields in two dimensions. A boundary integral equation (BIE) based formulation for vesicle EHD is introduced, followed by a solution scheme based on Stokes and Laplace potential theory. In the dilute limit, the rheology of the suspension is shown to vary nonlinearly with the electric conductivity ratio of the interior and exterior fluids. We demonstrate our capability of simulating EHD phenomena including one vesicle deformation, pairwise interaction, and multiple vesicle interactions.

Pronuclear centering and spindle positioning is a fundamental dynamics problem in organismal development, and constitutes a very complex fluid-structure interaction problem involving bodies being moved by immersed biopolymers and motor-proteins. I will discuss specialized computational methods, based on singularity and boundary integral methods, we have developed for efficiently studying such problems, as well as coarse-graining methods for evolving suspensions of microtubules. I'll end by discussing open areas and problems.

We will discuss a parallel boundary integral method for simulating highly concentrated vesicle suspensions in a Stokesian fluid. The simulation of high volume fraction vesicle suspensions which are representative of real biological systems (such as blood with 35% - 50% volume fraction for RBCs) presents several challenges. It requires computing accurate vesicle-vesicle interactions at length scales where standard quadratures are too expensive. The inter-vesicle separation can become arbitrarily small leading to vesicle collisions. Numerical errors can accumulate over time, making long-timescale simulations inaccurate. We tackle these challenges by developing state-of-the-art parallel algorithms for efficient computation of boundary integrals and an adaptive time-stepping scheme. We will also discuss algorithms for handling vesicle collisions and re-meshing of vesicle surfaces.

We performed combined experimental and computational study of human mammary epithelial cells developing a new computational model that is suitable for modeling of cells with wide range of viscoelastic properties and, at the same time, computationally efficient to be employed to large and complex flow domains. To the best of our knowledge, it is the first mesoscale particle-based computational model for simulations of flow-induced deformations of cells which explicitly takes account cell membrane, nucleus, and cytoskeleton. Using micropipette aspiration experiment, we first probed cells' elastic properties and used these data to set up parameters of the computational model. We then validated the model using data from microfluidic experiments, where cells were flown through microfluidic devices with different flow properties and different degree of cell deformations. This work was done in collaboration with the group of CT Lim from NUS.

In droplet based micro-fluidics, tiny water droplets are immersed in oil, stabilized by surfactants. Numerical methods based on integral equations for Stokes flow are attractive to use for this and similar applications as jumps in pressure and velocity gradients are naturally taken care of, viscosity ratios enter only in coefficients of the equations, and only the drop surfaces must be discretized and not the volume inside nor in between. We present numerical methods for drops with insoluble surfactants, both in two and three dimensions. We discretize the integral equations using Nyström methods, and special care is taken in the evaluation of singular and also nearly singular integrals that is needed in the case of close drop interactions. A spectral method is used to solve the advection-diffusion equation on each drop surface that describes the evolution of surfactant concentration. The drop velocity and surfactant concentration couple together through an equation of state for the surface tension coefficient. An adaptive time-stepping strategy is developed for the coupled problem. For high quality discretizations of the drops throughout the simulations, a hybrid method is used in two dimensions, offering an arc-length parameterization of the interface. In three dimensions, a reparameterization procedure is developed to optimize the spherical harmonics representation of the drop, while conserving the drop volume and amount of surfactant. The next step in the surfactant treatment is to consider surfactants that are soluble in the oil phase. This imples that an advection-diffusion equation must be solved in the bulk with boundary conditions coupling to the concentration of surfactants on the drop surfaces. A key ingredient to extend the integral equation methodology to this problem is the ability to efficiently compute a high-regularity extension of a function outside the domain where it is given. We introduce a novel method called partition of unity extension (PUX) for this purpose.

Suspensions of rigid particles can be found in a wide range of natural and industrial settings. Detailed models of the movement of individual particles can be a useful tool to predict and understand macroscopic properties of these suspensions. Boundary integral equations have been shown to be an efficient method to do this, however for dense suspensions avoiding particle overlaps requires excessively small time steps. By applying a constraint requiring no contact at each time step to the variational Stokes equations we are able to maintain separation of all the particles without requiring a small time step. This enables long time horizon simulations of dense suspensions of rigid particles. After presenting the method and demonstrating its robustness we demonstrate its applicability to modeling physical phenomena such as effective viscosity and particle alignment angles.

The cellular cytoplasm consists a viscous fluid filled with fibrous networks that also have their own dynamics. Such fluid-structure interactions have been modeled as a soft porous material immersed in a viscous fluid. In this talk we focus on the hydrodynamics of a viscous drop filled with soft porous material inside. Suspended in a Stokes flow, such a porous viscous drop is allowed to deform, both the drop interface and the porous structures inside. Special focus is on the deformation dynamics of both the porosity and the shape of the drop under simple flows such as a uniform streaming flow and linear flows. We examine the effects of flow boundary conditions at interface between the porous drop and the surrounding viscous fluid. We also examine the dynamics of a porous drop with active stress from the porous network.

Recent experiments show intriguing surface patterns when a uniform electric field is applied to a droplet covered with colloidal particles. Depending on the particle properties and the electric field intensity, particles organize into an equatorial belt, pole-to-pole chains, or dynamic vortices. Here we present simulations of the collective particle dynamics, which account for electrohydrodynamic and dielectrophoresis of particles. In addition, in order to better understand the collective particle dynamics, we will also discuss the forces on a single dielectric particle on an interface in an applied electric field.

Dynamics and patterns of RBCs under flow will be described, such as "blood crystals" (spontaneous order of RBCs under pure hydrodynamic interactions), similarities with traffic flow, or the existence of an optimal hematocrit. The second part of the talk will be devoted to amoeboid swimming, both for single swimmers and collective effects. This type of swimming is common to a wide variety of cells (e.g. the immune system cells, many cancer cells....)

Stiff particles flowing in a suspension of highly deformable red blood cells are pushed to the channel walls, an effect called margination. Here we first use Lattice Boltzmann simulations to study the margination of particles near constrictions. We show that the density of marginated particles directly in front of a constriction increases strongly, while the concentration of red blood cells does not. We then propose a novel mechanism dubbed ultrasound-triggered margination by which oscillating microbubbles can be deliberately pushed to the walls of the blood vessel. This property makes microbubbles the ideal drug delivery agent: flowing far away from the biochemically active walls during transport and directly interacting with the endothelium at the target organ. To investigate this phenomenon we develop a novel volume-changing-object boundary-integral-method (VCO-BIM).

We present an efficient, accurate, and robust method for simulation of dense suspensions of deformable and rigid particles immersed in Stokesian fluid in two dimensions. We use a well-established boundary integral formulation for the problem as the foundation of our approach. This type of formulation, with a high-order spatial discretization and an implicit and adaptive time discretization, have been shown to be able to handle complex interactions between particles with high accuracy. Yet, for dense suspensions, very small time-steps or expensive implicit solves as well as a large number of discretization points are required to avoid non-physical contact and intersections between particles, leading to infinite forces and numerical instability. Our method maintains the accuracy of previous methods at a significantly lower cost for dense suspensions. The key idea is to ensure interference-free configuration by introducing explicit contact constraints into the system. While such constraints are unnecessary in the formulation, in the discrete form of the problem, they make it possible to eliminate catastrophic loss of accuracy by preventing contact explicitly. Introducing contact constraints results in a significant increase in stable time-step size for explicit time-stepping, and a reduction in the number of points adequate for stability. This is joint work with Libin Lu and Abtin Rahimian

Blood is a suspension of objects of various shapes, sizes and mechanical properties, whose distribution during flow is important in many contexts. Red blood cells tend to migrate toward the center of a blood vessel, leaving a cell-free layer at the vessel wall, while white blood cells and platelets are preferentially found near the walls, a phenomenon called margination that is critical for the physiological responses of inflammation and hemostasis. Additionally, drug delivery particles in the bloodstream also undergo margination – the influence of these phenomena on the efficacy of such particles is unknown. In this talk a mechanistic theory is developed to describe segregation in blood and other confined multicomponent suspensions. It incorporates the two key phenomena arising in these systems at low Reynolds number: hydrodynamic pair collisions and wall-induced migration. The theory predicts that the cell-free layer thickness follows a master curve relating it in a specific way to confinement ratio and volume fraction. Results from experiments and detailed simulations with different parameters (flexibility of different components in the suspension, viscosity ratio, confinement, among others) collapse onto the same curve. In simple shear flow, several regimes of segregation arise, depending on the value of a "margination parameter" M. Most importantly, there is a critical value of M below which a sharp "drainage transition" occurs: one component is completely depleted from the bulk flow to the vicinity of the walls. Direct simulations also exhibit this transition as the size or flexibility ratio of the components changes. Results are presented for both Couette and plane Poiseuille flow. Experiments performed in the laboratory of Wilbur Lam indicate the physiological and clinical importance of these observations.

The three-dimensional spatiotemporal organization of genetic material inside the cell nucleus remains an open question in cellular biology. During the time between two cell divisions, the functional form of DNA in cells, known as chromatin, fills the cell nucleus in its uncondensed polymeric form, which allows the transcriptional machinery to access DNA. Recent in vivo imaging experiments have cast light on the existence of coherent chromatin motions inside the nucleus, in the form of large-scale correlated displacements on the scale of microns and lasting for seconds. To elucidate the mechanisms for such motions, we have developed a coarse-grained active polymer model where chromatin is represented as a confined flexible chain acted upon by active molecular motors, which perform work and thus exert dipolar forces on the system. Numerical simulations of this model that account for steric and hydrodynamic interactions as well as internal chain mechanics demonstrate the emergence of coherent motions in systems involving extensile dipoles, which are accompanied by large-scale chain reconfigurations and local nematic ordering. Comparisons with experiments show good qualitative agreement and support the hypothesis that long-ranged hydrodynamic couplings between chromatin-associated active motors are responsible for the observed coherent dynamics.

When discretizing the Navier-Stokes equations with an IMEX scheme in time, the modified Stokes equation must be solved to advance the solution from one time step to the next. The Green's function for the modified Stokes equation is given in terms of the difference of the Green's functions for the Laplace and modified Helmholtz equations. Unfortunately, it is unstable, particularly on fine grids or at low Reynolds number, to evaluate the Green's function by first evaluating the Laplace and modified Helmholtz Green's functions and then taking the difference. As an alternative, we present a fast multipole method for the modified Stokes kernel itself, which leverages new special functions to stably evaluate the interactions.

Microfluidic cell separation techniques are of great interest since they help rapid medical diagnoses and tests. Deterministic lateral displacement (DLD) is one of them. A DLD device consists of arrays of pillars. Main flow and alignment of the pillars define two different directions. Size-based separation of rigid spherical particles is possible as they follow one of these directions depending on their sizes. However, the separation of non- spherical deformable particles such as red blood cells (RBCs) is more complicated than that due to their intricate dynamics. We study the separation of RBCs in DLD using an in-house integral equation solver. We systematically investigate the effects of the interior fluid viscosity and the membrane elasticity of an RBC on its behavior. These mechanical properties of a cell determine its deformability, which can be altered by several diseases. We particularly consider deep devices in which an RBC can show rich dynamics such as tank-treading and tumbling. It turns out that strong hydrodynamic lift force moves the tank-treading cells along the pillars and weak or negative lift force leads the tumbling ones to move with the flow. Thereby, deformability-based separation of RBCs is possible. We also assess the efficiency of the technique for dense suspensions.

Recent work has demonstrated the interesting dynamics possible when considering multicomponent vesicles. Up to now, the dynamics of two dimensional vesicles have been studied. In this work, the dynamics of fully three dimensional, multicomponent vesicles will be investigated. Building upon a volume and surface area conserving Navier-Stokes projection method, the appropriate forcing terms are derived from the energy of a multicomponent membrane. The evolution of the surface components is done via a conserving, surface Cahn-Hilliard model. Time permitting, further extensions to the vesicle model will also be discussed.

We present a fluctuating boundary integral method (FBIM) for Brownian Dynamics (BD) of suspensions of rigid particles of complex shape immersed in a Stokes fluid. Our approach relies on a first-kind boundary integral formulation of a Stochastic Stokes Boundary Value Problem (SSBVP) in which a random surface velocity is prescribed on the particle surface. This random surface velocity has zero mean and covariance proportional to the Green’s function for the Stokes flow (Stokeslet). Furthermore, we demonstrate that discretizing the first-kind formulation using standard boundary integral techniques leads to an efficient numerical method that strictly preserves discrete fluctuation-dissipation balance (DFDB). We develop fast linear-scaling algorithms for performing matrix-vector products and for rapidly generating the random surface velocity by employing the Hasimoto splitting of the Stokeslet, which guarantees that the near-field (short-ranged) and far-field (long-ranged) contributions of the Stokeslet are independently symmetric and positive-definite. This ensures that the computaitonal cost of Brownian simulation is only marginally larger than the cost of deterministic simulations, in stark contrast to traditional BD approaches. To handle the inherent ill-conditioning of the linear system due to the first-kind formulation, we employ an effective block-diagonal preconditioner that ignores all the hydrodynamic interactions between distinct particles. FBIM provides the key ingredient for time integration of the overdamped Langevin equations for Brownian suspensions of rigid particles. We demonstrate that FBIM obeys DFDB by performing equilibrium BD simulations of suspensions of starfish-shaped bodies using a random finite difference temporal integrator. We numerically demonstrate the linear scaling of FBIM and discuss the computational cost of the various components of the algorithm.

In this talk, we will discuss efficient boundary integral equation (BIE) formulations for different classes of problems involving rigid objects in Stokes flow. For spherical particles, the boundary integral operators diagonalize on a specific set of vector spherical harmonics; we derive their spectra and formulas for evaluating particle-particle close interactions. Then we will discuss optimal complexity algorithms for solving BIEs on rotationally-symmetric geometries.

A common challenge in simulating dense suspension of rigid particles in Stokes flow is the numerical inaccuracies and instabilities that arises due to particle collisions. To overcome this problem, a strong repulsive potential between particles is often prescribed. This in turn leads to numerical stiffness and dramatic reduction in stable time-step sizes. In this work, we eliminate such stiffness by introducing contact constraints explicitly and solving the hydrodynamic equations in tandem with a linear complementarity problem with inequality constraints. Satisfaction of Newtons third law for the collision force is explicitly guaranteed, allowing the consistent calculation of collision stresses. Efficient parallelization for shared-memory and distributed-memory architectures is also implemented. This method can be coupled to any Stokes hydrodynamics solver for particles with various shapes and allows us to simulate $10^4-10^7$ spheres on a laptop, depending on the cost of the Stokes hydrodynamics solver. We demonstrate its performance on a range of applications of rigid suspensions.

While the field of fast direct solvers solvers for boundary integral equations has been growing, the development of these methods for fluid applications is limited. Since most fluid application involve evolving geometries, it is not possible to amortize the high computational cost of building a direct solver over many solves out of the box. In this talk, we will present extensions of the standard fast direct solver framework to fluid applications.