A boundary condition-enhanced direct-forcing immersed boundary method for simulations of three-dimensional phoretic particles in incompressible flows

In this paper we propose an improved three-dimensional immersed boundary method coupled with a finite-difference code to simulate self-propelled phoretic particles in viscous incompressible flows. We focus on the phenomenon of diffusiophoresis which, using the driving of a concentration gradient, can generate a slip velocity on a surface. In such a system, both the Dirichlet and Neumann boundary conditions are involved. In order to enforce the boundary conditions, we propose two improvements to the basic direct-forcing immersed boundary method. The main idea is that the immersed boundary terms are corrected by adding the force of the previous time step, in contrast to the traditional method which relies only on the instantaneous forces in each time step. For the Neumann boundary condition, we add two auxiliary layers inside the body to precisely implement the desired concentration gradient. To verify the accuracy of the improved method, we present problems of different complexity: The first is the pure diffusion around a sphere with Dirichlet and Neumann boundary conditions. Then we show the flow past a fixed sphere. In addition, the motion of a self-propelled Janus particle in the bulk and the spontaneously symmetry breaking of an isotropic phoretic particle are reported. The results are in very good agreements with the data that are reported in previously published literature.</p

Fast, High-Order Accurate Integral Equation Methods and Application to PDE-Constrained Optimization

Over the last several decades, the development of fast, high-order accurate, and robust integral equation methods for computational physics has gained increasing attention. Using integral equation formulation as a global statement in contrast to a local partial differential equation (PDE) formulation offers several unique advantages. For homogeneous PDEs, the boundary integral equation (BIE) formulation allows accurate handling of complex and moving geometries, and it only requires a mesh on the boundary, which is much easier to generate as a result of the dimension reduction. With the acceleration of fast algorithms like the Fast Multipole Method (FMM), the computational complexity can be reduced to O(N), where N is the number of degrees of freedom on the boundary. Using standard potential theory decomposition, inhomogeneous PDEs can be solved by evaluating a volume potential over the inhomogeneous source domain, followed by a solution of the homogeneous part. Despite the advantages of BIE methods in easy meshing, near-optimal efficiency, and well conditioning, the accurate evaluation of nearly singular integrals is a classical problem that needs to be addressed to enable simulations for practical applications. In the first half of this thesis, we develop a series of product integration schemes to solve this close evaluation problem. The use of differential forms provides a dimensional-agnostic way of integrating the nearly singular kernels against polynomial basis functions analytically. So the problem of singular integration gets reduced to a matter of source function approximation. In 2D, this procedure has been traditionally portrayed by building a connection to complex Cauchy integral, then supplemented by a complex monomial approximation. In $3$D, the closed differential form requirement leads to the design of a new function approximation scheme based on harmonic polynomials and quaternion algebra. Under a similar framework, we develop a high-order accurate product integration scheme for evaluating singular and nearly singular volume integral equations (VIE) in complex domains using regular Cartesian grids discretization. A high-order accurate source term approximation scheme matching smooth volume integrals on irregular cut cells is developed, which requires no function extension. BIE methods have been widely used for studying Stokes flows, incompressible flows at low Reynolds' number, in both biological systems and microfluidics. In the second half of this thesis, we employ the BIE methods to simulate and optimize Stokes fluid-structure interactions. In 2D, a hybrid computational method is presented for simulating cilia-generated fluid mixing as well as the cilia-particle hydrodynamics. The method is based on a BIE formulation for confining geometries and rigid particles, and the method of regularized Stokeslets for the cilia. In 3D, we use the time-independent envelop model for arbitrary axisymmetric microswimmers to minimize the power loss while maintaining a target swimming speed. This is a quadratic optimization problem in terms of the slip velocity due to the linearity of Stokes flow. Under specified reduced volume constraint, we find prolate spheroids to be the most efficient micro-swimmer among various families of shapes we considered. We then derive an adjoint-based formulation for computing power loss sensitivities in terms of a time-dependent slip profile by introducing an auxiliary time-periodic function, and find that the optimal swimmer displays one or multiple traveling waves, reminiscent of the typical metachronal waves observed in ciliated microswimmers.PhDApplied and Interdisciplinary MathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/169695/1/hszhu_1.pd

Driven soft matter at the nanoscale

[eng] This thesis will present a body of articles on the research topic of soft matter. Soft matter is a research subfield of condensed matter, where the energy required for deforming the media is comparable to that of thermal fluctuations. Tipical lengthscales of these systems are of the order of micrometer (10^(-6) m) and the nanometer (10^(-6) m). It encompasses a broad range of topics, since in soft matter fluids, life and interacting matter meet. The complexity of the coupling between different interactions in soft matter can result in complex emergent responses and in a rich variety of laws that we need to understand if we want to control matter at the micro and nanoscales. In the last years, thanks to the rise of computational power, soft matter has progresively included more and more simulation methodologies to predict experimental results and pose new challenges for understanding complex behaviours at these scales. Here, the emphasis is put on the computational modelling of driven soft matter. We will present a compendium of publications, where different simulation methodologies are exploited for explaining experiments and for setting experimental challenges to be tested. We classify the presented works in two parts, the scientific approach of which differ notorously. In the first part, experiments were available and the objective was to understand emergent responses reported in the lab. Since the outcome was alreadyknown, we used simple simulation methodologies that delved into the fundamental mechanisms that lead to the response of interest. The focus of the subjects, althought diverse, was centered around dynamics of colloidal suspensions, hence a mixture of a majoritary liquid phase with a minoritary solid phase. In the second part of the thesis, we employed simulations that rigorously solved the hydrodynamics coupled to the physics of the free energy of interest. The goal was to investigate novel experimental setups, the outcome of which was unknown due to the early stage of the subject. With the simulation results, we built theories that explained the observed phenomena, setting the basis for future experimental explorations. This last part focused on two independent topics, namely, capillary driven spontaneous in lubricant infused surfaces and electrolites in charge-patterned confined nanochannels.[spa] En esta tesis se presentarán una serie de artículos en el área de investigación de la materia blanda. La materia blanda es un subcampo de la materia condensada, en el que la energía típica de los sistemas es del orden de magnitud del de las fluctuaciones térmicas. Las escalas en las que se trabaja la materia blanda suelen ser la escala micrométrica (10^(-6) m) y la nanométrica (10^(-9) m), y en estas escalas la física de fluidos convive con la física de la vida y la de la materia interacuante. Esta mezcla de interacciones puede resultar en un alto grado de complejidad y en un sín fin de respuestas emergentes que aún quedan por entender. En estos últimos años, gracias al avanze del poder computacional, se han desarrollado en la materia blanda muchas metodologías de simulación que ahora se pueden utilizar para estudiar muchas de las preguntas que aun quedan sin respuesta en la materia blanda. En esta tesis, haremos énfasis en la modelizacion computacional en este tipo de materia. Presentaremos un compendio de publicaciones, en las que hemos utilizado diferentes métodos de simulación para explicar experimentos y para postular nuevos desafíos experimentales. La tesis se divide en dos partes donde el enfoque científico varía: En la primera parte, mas centrada en coloides y micronadadores, se utilizan modelos computacionales simples para explicar efectos emergentes en experimentos de materia blanda. En la segunda parte, centrada en dinámica de fluidos, capilaridad y electrolitos se utilizan métodos mas sofisticados para intentar predecir, esta vez sin evidencia experimental alguna, los posíbles escenarios a los que podría llevar una realización experimental. Tomada en su totalidad, esta tesis se puede entender como un enfoque práctico a la hora de escoger métodos de simulación en la micro y nanoescala

Microhydrodynamics of Droplets and Particles: Applications in Microfluidics and Agglomeration

Low-Reynolds-number flows are present throughout several fields in applied sciences and engineering, such as in the study of colloidal suspensions, cell motility, and other small-scale phenomena. This project concerns the investigation of two distinct systems involving the motion of particles and droplets in such flows. In the first part, we consider the problem of capturing small particles suspended in a fluid by using an emulsion of saltwater droplets covered by a semi-permeable oil layer. This problem is motivated by a recently proposed mineral-recovery technique. A theoretical investigation of binary interactions between droplets and particles provides us insight on how the physical parameters such as permeability and drop expansion due to osmotic swelling may affect particle capture. We observe that drop expansion considerably increases the capture efficiency of particle capture. Expansion limitation due to the diffusion of salt inside the droplets are also considered. In the second part, we investigate the motion of droplets in microchannels. This problem was motivated by the increasing number of applications of drop-based microfluidic systems, ranging from emulsion generation to medical diagnosis. To this end, we have designed a boundary-integral algorithm to simulate the droplet motion through three dimensional channels with complex geometries. The algorithm also uses a moving frame that follows the droplet throughout its motion in the channel to reduce computational time. Physical parameters such as capillary number, viscosity ratio, and drop size can affect drop motion and breakup conditions. We investigate the effects of channel depth on drop motion. For regular geometries of uniform cross-section, the infinite-depth limit is approached only slowly with increasing depth, though we show much faster convergence by scaling with maximum versus average velocities. For non-regular channel geometries, features such as different branch heights can affect drop partitioning, as the flow rate required to make a droplet go through a smaller branch of a channel is larger than the one required for making the same droplet go through a smaller branch, in contrast to the symmetrical behavior usually found in regular geometries. Moreover, non-regular geometries present further challenges when comparing the results for deep and infinite-depth channels. A simplified approach is also developed to probe inertial effects on drop motion. To this end, the full Navier-Stokes equations are first solved for the entire channel, and the tabulated solution is then used as a boundary condition at the moving-frame surface for the Stokes flow inside the moving frame. We find that, for moderate Reynolds numbers up to Re = 5, inertial effects on the undisturbed flow are small even for a more complex, irregular geometry, meaning that inertial contributions arise only from the transience of drop motion and are likely small. Finally, using our boundary-integral algorithm we also analyze the dynamics of a droplet in a hydrodynamic trap. By changing the fluxes in the different branches, we can manipulate drop shape and position. A linear controller is used to manipulate drop position, and the drop deformation is characterized by a decomposition of the shape into spherical harmonics. For droplets with small deformation (e.g., small radii and/or capillary number), we observe a linear superposition of harmonics that can be used to manipulate drop shape. 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 inside the droplet, which can be further enhanced by changing parameters such as viscosity ratio and flow frequency</p

Flowing matter

This open access book, published in the Soft and Biological Matter series, presents an introduction to selected research topics in the broad field of flowing matter, including the dynamics of fluids with a complex internal structure -from nematic fluids to soft glasses- as well as active matter and turbulent phenomena.Flowing matter is a subject at the crossroads between physics, mathematics, chemistry, engineering, biology and earth sciences, and relies on a multidisciplinary approach to describe the emergence of the macroscopic behaviours in a system from the coordinated dynamics of its microscopic constituents.Depending on the microscopic interactions, an assembly of molecules or of mesoscopic particles can flow like a simple Newtonian fluid, deform elastically like a solid or behave in a complex manner. When the internal constituents are active, as for biological entities, one generally observes complex large-scale collective motions. Phenomenology is further complicated by the invariable tendency of fluids to display chaos at the large scales or when stirred strongly enough. This volume presents several research topics that address these phenomena encompassing the traditional micro-, meso-, and macro-scales descriptions, and contributes to our understanding of the fundamentals of flowing matter.This book is the legacy of the COST Action MP1305 “Flowing Matter”

Numerical modelling of polydispersed flows using an adaptive-mesh finite element method with application to froth flotation

An efficient numerical framework for the macroscale simulation of three-phase polydispersed flows is presented in this thesis. The primary focus of this research is on modelling the polydispersity in multiphase flows ensuring the tractability of the solution framework. Fluidity, an open-source adaptive-mesh finite element code, has been used for solving the coupled equations efficiently. Froth flotation is one of the most widely used mineral processing operations. The multiphase, turbulent and polydispersed nature of flow in the pulp phase in froth flotation makes it all the more challenging to model this process. Considering that two of the three phases in froth flotation are polydispersed, modelling this polydispersity is particularly important for an accurate prediction of the overall process. The direct quadrature method of moments (DQMOM) is implemented in the Fluidity code to solve the population balance equation (PBE) for modelling the polydispersity of the gas bubbles. The PBE is coupled to the Eulerian--Eulerian flow equations for the liquid and gas phases. Polydispersed solids are modelled using separate transport equations for the free and attached mineral particles for each size class. The PBE has been solved using DQMOM in a finite element framework for the first time in this work. The behaviour of various finite element and control volume discretisation schemes in the solution of the PBE is analysed. Rigorous verification and benchmarking is presented along with model validation on turbulent gravity-driven flow in a bubble column. This research also establishes the importance of modelling the polydispersity of solids in flotation columns, which is undertaken for the first time, for an accurate prediction of the flotation rate. The application of fully-unstructured anisotropic mesh adaptivity to the polydispersed framework is also analysed for the first time. Significant improvement in the solution efficiency is reported through its use.Open Acces