Schedule for: 17w5057 - Mathematical Approaches to Interfacial Dynamics in Complex Fluids

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

The wetting of a solid surface by the drop of an emulsion has traditionally been thought to be mediated by the formation of a liquid bridge that connects the drop and the surface. In the current work, we experimentally show that there exists a different mechanism of the spreading of a drop on a surface. Experiments were conducted for simple liquid-liquid systems, wherein drops of higher density were allowed to settle under gravity in a lighter liquid phase under conditions of small Bond numbers. The approach of the drop towards the substrate was visualized using Reflection Interference Contrast Microscopy (RICM), and the details of the film drainage dynamics and the eventual spreading mechanism of the drop on the surface were recorded. Three liquid-liquid systems were used - 1) glycerol-in-silicone oil 1000 cP (SO1000), 2) glycerol in silicone oil 500 cP (SO500), and 3) silicone oil (SO500) in paraffin oil (PO). The substrates were also varied in this study. The film shapes obtained from this were then compared with predictions from scaling analysis. The temporal variation of the minimum film heights matched theoretical expectations, except when the height reached about the order of 10 nm for silicone oil films (system 1 and 2). In this case, cessation of film drainage was observed and was attributed to the formation of an immobilized silicone oil layer due to polymer confinement. While the film appeared to be stable, after an induction period ranging from a few minutes to several hours, deformable islands of glycerol were observed to grow on the substrate. Wetting of the surface then occurs by the formation of a bridge, not between the parent drop and the surface directly, but between the parent drop and the nucleated sites. The fundamental effect discovered here will ultimately guide the tailoring of emulsion- based coatings or paints to have specific spreading times. It also has application in multiphase industrial operations such as froth floatation, where the understanding of the time scale of particle-droplet/bubble attachment is critical.

Abstract: Particles straddling a fluid interface exhibit rich dynamics due to co-existing rigid moving boundaries, deformable fluid interfaces, and moving contact lines. For instance, as a particle falls onto a liquid surface, it may sink, float, or even bounce off depending on a wide range of parameters. To better understand this class of multiphase systems, we develop hybrid methods that tracks solid particles by an arbitrary Eulerian-Lagrangian (ALE) method and captures the fluid interfaces by a phase-field (PF) method or a level set (LS) method. For the ALE-PF method, the governing equations for particles and fluids are solved in a unified variational framework that satisfies an energy law. We first validate our code by computing experiments found in literature: sinking of a sphere through an air-oil interface at small Reynolds numbers and bouncing of a sphere after normal impact onto an air-water interface. Our numerical results show good agreements with experimental data. We then investigate the effect of wetting properties, including static contact angle, slip length, and wall energy relaxation, on particle dynamics at the fluid interface. In the end we will also discuss an ALE-LS method that is still under development. An interface-preserving high-order discontinuous Galerkin algorithm has been developed to reinitialize the LS function; this greatly improves the mass conservation property of the LS component. Compared to ALE-PF, the ALE-LS method is much less demanding in terms of computational mesh.

Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

By methods from nonequilibrium thermodynamics, we derive a diffuse-interface model for two-phase flow of incompressible fluids with dissolved noninteracting polymers. The polymers are modeled by dumbbells subjected to finitely extensible, nonlinear elastic (FENE) spring-force potentials. Their density and orientation are described by a Fokker-Planck-type equation which is coupled to a Cahn-Hilliard and a momentum equation for phase-field and gross velocity/pressure. Henry-type energy functionals are used to describe different solubility properties of the polymers in the different phases or at the liquid-liquid interface. Taking advantage of the underlying energetic/entropic structure of the system, we prove existence of a weak solution globally in time. As a by-product in the case of Hookean spring potentials, we derive a macroscopic diffuse-interface model for two-phase flow of Oldroyd-B-type liquids. We present numerical simulations which highlight the effects exerted by the polymers on the relaxation of nonspherical droplets. These simulations are based on numerical schemes which recently have been proven to be convergent by S. Metzger. Finally, extensions of the model are discussed which take the interaction between polymer and fluid interface orientation into account ("amphiphilic surfactant"). Again, as a proof of concept characteristic numerical simulations are presented.

Abstract. Designing numerical methods with high-order accuracy for problems with interfaces (for example, models for polycrystalline materials or composite fluids, etc.), as well as models in irregular domains is crucial to many applications in fluid dynamics, materials science, biology and physics. In this talk we will present recently developed efficient numerical schemes based on the idea of the Difference Potentials for elliptic and parabolic composite domain/interface problems. Numerical experiments to illustrate high-order accuracy and the robustness of the developed methods will be given. Current and future research will be discussed as well.

Cells with different properties grow, divide, and interact with each other and with Extra-Cellular Matrix (ECM) through mechanical forces and signal transduction, resulting in the formation of complex patterns that are important for processes such as tissue development and wound healing. However, current computational cell models are challenged to account for detailed changes in cellular shapes and physical mechanics when studying thousands of migrating and interacting cells. We discuss recent development of quantitative models and algorithms, with focus on a novel dynamic cellular finite element (dyCelFEM) model. With full account of changes in cellular shapes, cellular topology, and celluar mechanics, and with key intra-cellular signaling networks embedded in individual cells, we can study the full range of cell motion, from motilities of individual cells to collective cell migrations. We discuss our recent findings on effects of micro-patterned geometry on cell elongation and migration. We also described results on examining the effects of biochemical and mechanical cues in regulating cell migration and proliferation, and in controlling tissue patterning in re-epithelialization of skin wound healing (Joint work with Jieling Zhao, Youfang Cao, Wei Tian, Luisa DiPietro, and Margaret Gardel).

A fundamental issue in the modeling of magnetoelastic materials is that elasticity is phrased in Lagrangian coordinates whereas magnetism is phrased in Eulerian coordinates. We discuss a model that is completely phrased in Eulerian coordinates and takes microstructures of the magnetization into account. The model presented is a system of partial differential equations that contains (1) the incompressible Navier-Stokes equations with magnetic and elastic terms in the stress tensor obtained by a variational approach, (2) a regularized transport equation for the deformation gradient and (3) the Landau-Lifshitz-Gilbert equation for the dynamics of the magnetization. We will indicate the derivation of the model and will present results on the analytical properties of the system.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

We present in this talk a new way to construct linear, decoupled, energy stable schemes for gradient flows. The new class of schemes enjoy the following advantages: (i) at each time step, one only has to solve decoupled, linear positive definite systems with constant coefficients; (ii) it is proved to be unconditionally energy stable and appears to be quantitatively more accurate than existing schemes of the same order; (iii) it applies to a wider class of problems.

Abstract: In this talk we discuss a fractional mass-conserving Allen-Cahn phase-field model that describes the mixture of two incompressible fluids. The new fractional model allows controlling the sharpness of the interface, which is typically diffusive in integer-order phase-field models. The model is derived based on an energy variational formulation. An additional constraint is employed to make the Allen-Cahn formulation mass-conserving and comparable to the Cahn-Hilliard formulation but at reduced cost. The spatial discretization is based on a spectral method whereas the temporal discretization is based on a stabilized ADI scheme both for the phase-field equation and for the Navier-Stokes equations. A number of numerical examples are provided to demonstrate the accuracy of the method and the ability to control the interface thickness between two fluids.

Abstract: The Ohta-Kawasaki model is a nonlocal Cahn-Hilliard model for pattern formation driven by competing long- and short-range interactions. The model is commonly used to describe pattern formation in diblock copolymer systems. Recently, we have developed an Otha-Kawasaki type model for ionic liquids and high concentrated electrolyte solutions. This equation involves more general, possibly asymmetric, long- and short-range interactions. In this talk, I will present a systematic study of pattern formation in the classical and the extended Otha-Kawasaki model, which reveals the effect of asymmetric interactions on pattern formation. In particular, we map regions of co-existence of stable solutions (a bi-stable or multi-stable system). Furthermore, we focus on new spatially localized states in 1D and 2D in both infinite and finite domain sizes. We show that such states exist in both the classic and the extended Otha-Kawasaki model, and describe their dependence upon domain size.

Abstract:We describe a Volume-Of-Fluid method for direct numerical simulation of variable surface tension flows. We present an accurate calculation of the surface gradients that is central to the computation of the surface tension which has a gradient on the interface. Numerical validations and convergence of the method are discussed. We also show numerical examples, motivated by experimental observations, of the effect of the concentration-dependent surface tension on the flow field and the interface evolution.

The interplay between diffusion, attraction, and repulsion arises in a variety of contexts in the natural- and life-sciences. Such a system for two (biological) species is introduced, with different inner- and intra-species attraction. Under suitable conditions on this self- and cross-wise attraction, phase separation into neigboring regions can be observed, each of which contains only one of the species. The intersection of the support of the stationary solutions to this continuum model for the two species has zero Lebesgue measure, while the support of the sum of the two densities is simply connected. Spatial sorting of biological/biochemical species plays an important role in developmental processes.

A few study focused on problems involving interaction of fluid with an elastic interface with mass. In this talk, we present an operator splitting scheme for simulating an elastic and massive interface moving in viscous, incompressible flows. We utilize a continuous description of the interface and Helfrich bending elasticity energy for representing its elasticity. An energy approach is employed to derive the fluid-interface coupling condition. To this end, the dynamic evolution of the interface is defined by a geometric partial differential equation, which involve computing curvatures of the surface and surface Laplacian of the mean curvature. We introduce a robust finite difference scheme for solving the geometric partial differential equation. The Lie's scheme then is used to define the operator splitting for solving the fluid-interface interaction problem. Examples will be presented to demonstrate the applicability of the proposed scheme for numerical studying biological problems. (Joint work with Shixin Xu)

The lattice Boltzmann equation (LBE) combined with conservative phase-field equation has become a popular method to simulate various interface phenomena in 2D and 3D. I will demonstrate some recent results obtained with the LBE with adaptive mesh refinement (AMR), including (1) a rising bubble, (2) a falling drop, (3) crown splashing of a droplet on a wet surface; and (d) the partial coalescence mechanism of a liquid drop at a liquid-liquid interface.

Materials science and engineering is the study and manipulation of the hierarchical architectures of lattice, chemical, electric polarization, charge, and magnetic order to achieve the desirable properties for structural and functional applications. This presentation will briefly discuss the applications of the phase-field method to modeling, predicting, and optimizing the materials morphologies and microstructures for materials processes ranging from chemical vapor deposition (CVD) growth of 2D materials such as MoS2 and GaSe and additive manufacturing (AM) of metallic alloys such as Ti-6Al-4V alloys. CVD technique is one of the main techniques for the synthesis of 2D materials that involves complex coupling and interplay of heat transfer, fluid flow, and chemical reactions. Additive manufacturing of metallic alloys leads to non-uniform temperature distributions and rapid thermal cycles that result in microstructures featured with strong anisotropy and critically affect the mechanical properties of the AM builds. We employ a combination of finite-element solutions to heat and mass transport and fluid flow equations as well as finite-difference solutions to the phase-field equations. We demonstrate that the proposed computational framework is able to provide guidance for the experimental control of the growth morphology of 2D materials and microstructures of AM metallic parts.

The stability of two-layer shear flow, one layer yielded and the other unyielded, is studied for a PECN fluid, which combines the partially extending strand convection model (PEC) with a solvent. The ratio of retardation time to relaxation time is a small parameter that expresses the time scales involved in the description of a thixotropic yield stress fluid. A linear stability analysis of shear flow for a single yielded phase shows bulk instabilities. For the shear banded flow,an additional instability enters from the presence of the interface. The interfacial instability is driven by a normal stress difference across the interface. As for the Johnson-Segalman model, which has been studied in earlier literature, the PEC model has a higher normal stress in the high shear rate phase than in the low shear rate phase. To assess the importance of this property, a model which is specifically designed to reverse this is investigated. It is found that instabilities still occur, although the range of unstable wave numbers is reduced. This is joint work with Michael Renardy.

Abstract: In this talk I will introduce the systematic energetic variational approach for dissipative systems applied to multi-component fluid flows. These variational approaches are motivated by the seminal works of Rayleigh and Onsager. The advantage of this approach is that we have to postulate only energy law and some kinematic relations based on fundamental physical principles. The method gives a clear, quick and consistent way to derive the PDE system. I will compare different approaches to three-component flows and introduce the design of efficient numerical simulations for these models. If time permits, I will discuss the importance of boundary effects and its impact on the dynamics.

In this talk, we will discuss some recent progress in the mathematical analysis for the complex fluids, especially the viscoelastic fluid flow. Both the incompressible case and the compressible case are considered. Global existences of either strong solutions or weak solutions are two main subjects.

In this article, momentum conservation was discretely achieved by using the conservative form of incompressible Navier-Stokes equations, instead of the non-conservative form which has been commonly used in most of the literatures about two-phase flow simulations. Without taking the advantage of continuous velocity field resulting from no-slip boundary condition between two phases, the discontinuous/steep momentum flux caused by possible large density ratio was dealt with high-order WENO scheme. Deferred-correction algorithm, which ensures diagonal dominance of all the coefficient matrixes of the discretized system, was applied to deal with the stiffness caused by large density ratio. Numerical tests were conducted to indicate the 5th order of accuracy of the proposed scheme. Coupling with level-set method, the proposed scheme successfully simulated two-phase problems with large density ratio.

We develop a phase-field model for the binary incompressible (quasi-incompressible) fluid with thermocapillary effects, which allows for the different properties (densities, viscosities and heat conductivities) of each fluid component while maintaining thermodynamic consistency. The governing equations of the model including the Navier-Stokes equations with additional stress terms, Cahn-Hilliard equations and energy balance equation are derived within a thermodynamic framework based on entropy generation, which guarantees thermodynamic consistency. A sharp-interface limit analysis is carried out to show that the interfacial conditions of the classical sharp-interface models can be recovered from our phase-field model. Moreover, a few examples including thermocapillary convections in a two-layer fluid system and thermocapillary migration of a drop are computed using a continuous finite element method. The results are compared to the corresponding analytical solutions and the existing numerical results as validations for our model. For the isothermal variable-density case we also show how an energy law preserving continuous finite element scheme can be derived. The talk is based on two recent papers (one joint with ZL Guo, the other joint with ZL Guo and J Lowengrub).

Abstract - Ocular surface disorders are one of major causes of blindness in humans due to lack of proper treatment right at its onset. We use fluid mechanical tools to predict the occurrence of such disorders by analysing the break up times (BUT) of the tear film that acts as a protective cover over the sensitive corneal epithelium. Our model takes into account the disparate properties of the various tear film components, the resulting viscosity stratification together with the Marangoni stresses at the film/air interface and the slip effects at the base of the tear film, in an effort to understand and explain the correlation between morphological changes in the tear film and its rupture time. This is accomplished by carrying out a detailed linear stability analysis along with a non-linear study to determine their BUT, and also to probe the role of various components of the pre-corneal tear film in the dynamics of rupture.

Much attention has been paid recently, both in experiment and simulations to the motion of active nematics accompanied by spontaneous formation and interaction of half-charged defects. We study the defect dynamics analytically by asymptotic matching of solutions in the defect core and the far field. The analysis is facilitated by the correspondence between the two-dimensional nematic and complex scalar field models. Self-propulsion and topological interactions are identified as the primary drivers of the defect motion, surpassing the influence of both passive backflow and active flow induced by other defects. Precision experiments are now carried out to use the theory for identification of rheology of thin active nematic films formed by ATP-driven microtubule bundle suspension.

The ability of matter to self-organize in complex dynamic structures is increasingly used to generate new, active materials. One propulsion mechanism to generate “active” colloids is induced-charge electrophoresis (ICEP). Several aspects of this technique, in which ionic flows interact with dielectrically inhomogeneous particles, continue to be poorly understood. We have recently developed an efficient technique that couples a dielectric solver with particle-based simulations, making possible dynamic simulations that fully incorporate self-consistently calculated polarization charges. I will introduce our approach, which employs a boundary-element method for the dielectrics, and demonstrate how it can be used to clarify the frequency dependence of ionic flow in ICEP and related problems. [1] K. Barros, D. Sinkovits and E. Luijten, J. Chem. Phys. 140, 064903 (2014). [2] K. Barros and E. Luijten, Phys. Rev. Lett. 113, 017801 (2014). [3] Z. Gan, H. Wu, K. Barros, Z. Xu and E. Luijten, J. Comp. Phys. 291, 317 (2015). [4] M. Han, H. Wu and E. Luijten, Eur. Phys. J. Special Topics 225, 685 (2016). [5] J. Yan, M. Han, J. Zhang, C. Xu, E. Luijten and S. Granick, Nature Materials 15, 1095 (2016).

In this talk, we will discuss the Cahn-Hilliard equation subject to a new class of dynamic boundary condition. The system can be derived via an energetic variational approach and it fulfills several important physical constraints, including conservation of mass, dissipation of the total energy, and force balance both in the bulk and on the boundary. We prove global well-posedness of the system and investigate its long-time behavior.

We present a versatile multiphase-multicomponent lattice Boltzmann scheme that allows high density ratio (~1000) between the gas and liquid phases. It combines the free energy approach, which ensures thermodynamic consistency and allows us to define the bulk fluid phases a priori, and the entropic lattice Boltzmann algorithm, which strongly improves the stability for simulations at high density ratios and for high Weber and Reynold numbers. Focusing on a ternary system (one gas and two liquids), we will show several benchmarks: (i) double emulsions and liquid lenses to validate the surface tensions, (ii) ternary fluids in contact with solid walls to compare the contact angles against analytical predictions. Finally, we will highlight how the novel ternary model can be exploited to study the wetting dynamics of droplets on liquid infused surfaces – liquid repellent surfaces made by infusing a lubricant into porous or rough solid surfaces.

Abstract: The Poisson-Nernst-Planck（PNP) system is an important model to describe ion transport in ionic liquids having many applications in biology, chemistry and physics. Using analytical techniques to study the boundary layer solutions of the PNP system, we may derive an explicit formula for the mixture of Na^{+}, Mg^{2+} and Cl^{-} which can be justified by experiments on electric double layers. To describe ion transport through biological channels, we consider the ionic size effect and derive another PNP type model called the PNP_steric system with cross diffusion terms which come from the approximate Lennard-Jones potential. Under specific parameter regimes, the steady state equation of the PNP_steric system may be reduced to the PB type model of D. Andelman (1997) so it would be expected that the PNP_steric system may describe electrolytes with high concentrations of ions. A numerical scheme of the PNP_steric equations is developed to see the flow dynamics of charged particles, and the special feature of the scheme is that the numerical solutions satisfy a discrete energy law mimicking the energy law of the PNP_steric equations. Our numerical results may be comparable with experimental results of ion channels. This is a joint work with Tzyy-Leng Allen Horng, Chiun-Chang Lee and Chun-Hao Teng.

Dynamic permeability of porous media is responsible for the effective drag force in a porous composite. In this talk, we will study the existing results based on the homogenization approach and suggest some open problems that are relevant to the mathematical materials science community.

We present an effective method for simulating wall-bounded multiphase flows consisting of N (N>=2) immiscible incompressible fluids with different densities, viscosities, pairwiase surface tensions, and various contact angles. The N-phase formulation is based on a modified thermodynamically consistent phase field model, and is developed by considering the reduction consistency property that if only a set of M (2<=M<=N-1) fluid components are present in the N-phase system, then the N-phase formulation should reduce to the corresponding M-phase formulation. We explore the implications of this reduction consistency on the N-phase formulation, the governing equations, and the boundary conditions. A reduction consistent N-phase contact-angle boundary condition will be proposed. Several numerical experiments will be presented for problems involving multiple fluid components and solid-wall boundaries to study the wettability effects with a multitude of contact angles. We will demonstrate that the reduction consistency in the formulation and boundary conditions is critical to correctly capturing the multiple contact angles in N-phase systems.

We study the Rolie-Poly model for entangled polymers, using a singular perturbation analysis for the limit of large relaxation time. In this limit, it is shown that the model displays the characteristic features of thixotropic yield stress fluids, including yield stress hysteresis, delayed yielding and long term persistence of a decreased viscosity after cessation of flow. We focus on the startup and cessation of shear flow. We identify dynamic regimes of fast, slow and yielded dynamics, and show how the combination of these regimes can be used to describe the flow.

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.