A diffuse-interface method for two-phase flows with soluble surfactants

Fluid interfaces laden by force dipoles: towards active matter-driven microfluidic flows

In recent years, nonlinear microfluidics in combination with lab-on-a-chip devices has opened a new avenue for chemical and biomedical applications such as droplet formation and cell sorting. In this article, we integrate ideas from active matter into a microfluidic setting, where two fluid layers with identical densities but different viscosities flow through a microfluidic channel. Most importantly, the fluid interface is laden with active particles that act with dipolar forces on the adjacent fluids and thereby generate flows. We perform lattice-Boltzmann simulations and combine them with phase field dynamics of the interface and an advection-diffusion equation for the density of active particles. We show that only contractile force dipoles can destabilize the flat fluid interface. It develops a viscous finger from which droplets break up. For interfaces with non-zero surface tension, a critical value of activity equal to the surface tension is necessary to trigger the instability. Since activity depends on the density of force dipoles, the interface can develop steady deformation. Lastly, we demonstrate how to control droplet formation using switchable activity.

Multiphase Phase-Field Lattice Boltzmann Method for Simulation of Soluble Surfactants

This paper proposes a phase-field model for the lattice Boltzmann method which has discretized symmetrical directions of velocities in a cartesian grid, to simulate the soluble surfactant in a Multicomponent multiphase system. Despite other existing phase-field models following Langmuir relation, the interfacial tension can be calculated analytically in this proposed model. Parameters playing roles in the models and controlling the surfactant’s strength and interaction with other phases are obtained directly from a given initial interfacial tension and bulk surfactant. Consequently, there is no further need for trial-and-error simulations, and a real system, e.g., oil-water-surfactant, can be simulated with given initial parameters. The model is validated with the analytical result for a planar oil–water-surfactant system. Furthermore, the method for reobtaining numerical interfacial tension for five different cases is tested and compared with the given initial values for an oil droplet surrounded by water and surfactant. The results show that the obtained interfacial tension from the method is in good agreement with the given initial interfacial tension. Furthermore, the spurious velocity of the model is calculated and seen that the magnitude of spurious velocities is proportional to interfacial tension.

Electro-Hydrodynamics of Emulsion Droplets: Physical Insights to Applications

The field of droplet electrohydrodynamics (EHD) emerged with a seminal work of G.I. Taylor in 1966, who presented the so-called leaky dielectric model (LDM) to predict the droplet shapes undergoing distortions under an electric field. Since then, the droplet EHD has evolved in many ways over the next 55 years with numerous intriguing phenomena reported, such as tip and equatorial streaming, Quincke rotation, double droplet breakup modes, particle assemblies at the emulsion interface, and many more. These phenomena have a potential of vast applications in different areas of science and technology. This paper presents a review of prominent droplet EHD studies pertaining to the essential physical insight of various EHD phenomena. Here, we discuss the dynamics of a single-phase emulsion droplet under weak and strong electric fields. Moreover, the effect of the presence of particles and surfactants at the emulsion interface is covered in detail. Furthermore, the EHD of multi-phase double emulsion droplet is included. We focus on features such as deformation, instabilities, and breakups under varying electrical and physical properties. At the end of the review, we also discuss the potential applications of droplet EHD and various challenges with their future perspectives.

Decoupled and Energy Stable Time-Marching Scheme for the Interfacial Flow with Soluble Surfactants

In this work, we develop an efficient energy stable scheme for the hydrodynamics coupled phase-field surfactant model with variable densities. The thermodynamically consistent model consists of two Cahn–Hilliard–type equations and incompressible Navier–Stokes equation. We use two scalar auxiliary variables to transform nonlinear parts in the free energy functional into quadratic forms, and then they can be treated efficiently and semi-implicitly. A splitting method based on pressure stabilization is used to solve the Navier–Stokes equation. By some subtle explicit-implicit treatments to nonlinear convection and stress terms, we construct a first-order energy stable scheme for the two-phase system with soluble surfactants. The developed scheme is efficient and easy-to-implement. At each time step, computations of phase-field variables, the velocity and pressure are decoupled. We rigorously prove that the proposed scheme is unconditionally energy stable. Numerical results confirm that our scheme is accurate and energy stable.

Spontaneous shrinkage of droplet on a wetting surface in the phase-field model

Phase-field theory is widely used to model multiphase flow. The fact that a drop can shrink or grow spontaneously due to the redistribution of interface and bulk energies to minimize the system energy may produce ill effects on the simulation. In this Rapid Communication, the spontaneous behavior of a drop on a partially wetting surface is investigated. It is found that there exists a critical radius dependent on the contact angle, the domain size, and the interface width, below which the drop will eventually disappear. In particular, the critical radius can be very large when the surface becomes very hydrophilic. The theoretical prediction of the critical radius is verified numerically by simulating a drop on a surface with various contact angles, the domain sizes, and the interface widths.

The Effect of Oil Viscosity on Droplet Generation Rate and Droplet Size in a T-Junction Microfluidic Droplet Generator

There have been growing interests in droplet-based microfluidics due to its capability to outperform conventional biological assays by providing various advantages, such as precise handling of liquid/cell samples, fast reaction time, and extremely high-throughput analysis/screening. The droplet-based microfluidics utilizes the interaction between the interfacial tension and the fluidic shear force to break continuous fluids into uniform-sized segments within a microchannel. In this paper, the effect of different viscosities of carrier oil on water-in-oil emulsion, particularly how droplet size and droplet generation rate are affected, has been investigated using a commonly used T-junction microfluidic droplet generator design connected to a pressure-controlled pump. We have tested mineral oils with four different viscosities (5, 7, 10, and 15 cSt) to compare the droplet generation under five different flow pressure conditions (i.e., water flow pressure of 30-150 mbar and oil flow pressure of 40-200 mbar). The results showed that regardless of the flow pressure levels, the droplet size decreased as the oil viscosity increased. Average size of the droplets decreased by approximately 32% when the viscosity of the oil changed from 5 to 15 cSt at the flow pressure of 30 mbar for water and 40 mbar for oil. Interestingly, a similar trend was observed in the droplet generation rate. Droplet generation rate and the oil viscosity showed high linear correlation (R2 = 0.9979) at the water flow pressure 30 mbar and oil flow pressure 40 mbar.

Breakups of an encapsulated surfactant-laden aqueous droplet under a DC electric field

We investigate the breakups of an encapsulated conducting aqueous droplet under a direct-current electric field via extensive experiments and theoretical analysis. The encapsulating shell phase and the ambient phase consist of leaky dielectric liquids. We change the surface tension by using an aqueous core with different surfactant (Tween 80) concentrations. Moreover, we vary the core size under different electric-field conditions and observe the core dynamics. We present three different breakup modes of the encapsulated droplet. In the first mode, the encapsulated core forms asymmetric Janus shapes after breakup. In the second and third breakup modes, stable and unstable ternary droplets are formed, respectively. We show that the surfactant molecules significantly alter the dynamics of core stretching. According to the theoretical analysis, we identify the critical conditions of instability leading to breakup. We plot the breakup modes in the form of a phase diagram in the electric capillary number (Ca₂₃ = ε₃r_sE_o²/γ₂₃; ratio of interfacial electric to capillary stresses) vs. radius ratio of the core to the shell (β = r_c/r_s) parametric space at different nondimensional surfactant concentrations (C* = C_Tween 80/C_CMC, where C_CMC represents the critical micellar concentration). The study provides essential physical insight into encapsulated emulsions and is useful for their application in various areas of science and technology.

Conservative phase-field method with a parallel and adaptive-mesh-refinement technique for interface tracking

Based on Fick's second law and Cahn-Hilliard theory, a conservative phase-field model is developed to track interface. The phase-field variable changes in a hyperbolic tangent behavior across the diffuse interface over which the interface curvature can be easily calculated. Different from the frequently used lattice-Boltzmann-based discrete method, the phase-field equation is discretized using a fourth-order Runge-Kutta method. Accordingly, the present numerical scheme alleviates the programming burden, reduces the memory usage, but maintains a high numerical accuracy. To achieve large-scale interface tracking, a parallel and adaptive-mesh-refinement algorithm is developed to reduce the computing overhead. Various cases of the interface evolutions under steady flow fields indicate that the proposed numerical scheme can capture the interface with high accuracy. Furthermore, the robustness of the numerical scheme is validated by simulating the Rayleigh-Taylor instability, and good agreement with previous work is achieved.

Spreading of Micrometer-Sized Droplets under the Influence of Insoluble and Soluble Surfactants: A Numerical Study

Wetting and spreading of surfactant solutions play an important role in many technical applications. In printing processes, the size of individual droplets is typically on the order of a few tens of microns. The purpose of this study is to develop a better understanding of the interaction between spreading and surfactant transport on these small length and related time scales. Therefore, numerical simulations based on the volume-of-fluid method including Marangoni stresses and transport of an insoluble or soluble surfactant are performed. The results for an insoluble surfactant show competing effects of Marangoni flow on the one hand, and a decreasing surfactant concentration as the droplet spreads on the other hand. Even in the case of a soluble surfactant, adsorption and desorption could only partly mitigate these effects, demonstrating the importance of the sorption kinetics for fast, small scale wetting processes.

Numerical Study of Droplet Dynamics on a Solid Surface with Insoluble Surfactants

Surfactants are widely used in many industrial processes, where the presence of surfactants not only reduces the interfacial tension between fluids but also alters the wetting properties of solid surfaces. To understand how the surfactants influence the droplet motion on a solid surface, a hybrid method for interfacial flows with insoluble surfactants and contact-line dynamics is developed. This method solves immiscible two-phase flows through a lattice Boltzmann color-gradient model and simultaneously solves the convection-diffusion equation for surfactant concentration through a finite difference method. In addition, a dynamic contact angle formulation that describes the dependence of the local contact angle on the surfactant concentration is derived, and the resulting contact angle is enforced by a geometrical wetting condition. Our method is first used to simulate static contact angles for a droplet resting on a solid surface, and the results show that the presence of surfactants can significantly modify surface wettability, especially when the surface is more hydrophilic or more hydrophobic. This is then applied to simulate a surfactant-laden droplet moving on a substrate subject to a linear shear flow for varying effective capillary number ( Ca_e), Reynolds number ( Re), and surface wettability, where the results are often compared with those of a clean droplet. For varying Ca_e, the simulations are conducted by considering a neutral surface. At low values of Ca_e, the droplet eventually reaches a steady deformation and moves at a constant velocity. In either a clean or surfactant-laden case, the moving velocity of the droplet linearly increases with the moving wall velocity, but the slope is always higher (i.e., the droplet moves faster) in the surfactant-laden case where the droplet exhibits a bigger deformation. When Ca_e is increased beyond a critical value ( Ca_e,c), the droplet breakup would happen. The presence of surfactants is found to decrease the value of Ca_e,c, but it shows a non-monotonic effect on the droplet breakup. An increase in Re is able to increase not only droplet deformation but also surfactant dilution. The role of surfactants in the droplet behavior is found to greatly depend upon the surface wettability. For a hydrophilic surface, the presence of surfactants can decrease the wetting length and enables the droplet to reach a steady state faster; while for a hydrophobic surface, it increases the wetting length and delays the departure of the droplet from the solid surface.

Tumor growth and calcification in evolving microenvironmental geometries

In this paper, we apply the diffuse domain framework developed in Chen and Lowengrub (Tumor growth in complex, evolving microenvironmental geometries: A diffuse domain approach, J. Theor. Biol. 361 (2014) 14-30) to study the effects of a deformable basement membrane (BM) on the growth of a tumor in a confined, ductal geometry, such as ductal carcinoma in situ (DCIS). We use a continuum model of tumor microcalcification and investigate the tumor extent beyond the microcalcification. In order to solve the governing equations efficiently, we develop a stable nonlinear multigrid finite difference method. Two dimensional simulations are performed where the adhesion between tumor cells and the basement membrane is varied. Additional simulations considering the variation of duct radius and membrane stiffness are also conducted. The results demonstrate that enhanced membrane deformability promotes tumor growth and tumor calcification. When the duct radius is small, the cell-BM adhesion is weak or when the membrane is slightly deformed, the mammographic and pathologic tumor extents are linearly correlated, as predicted by Macklin et al. (J. Theor. Biol. 301 (2012) 122-140) using an agent-based model that does not account for the deformability of the basement membrane and the active forces that the membrane imparts on the tumor cells. Interestingly, we predict that when the duct radius is large, there is strong cell-BM adhesion or the membrane is highly deformed, the extents of the mammographic and pathologic tumors are instead quadratically correlated. The simulations can help surgeons to measure DCIS surgical margins while removing less non-cancerous tissue, and can improve targeting of intra- and post-operative radiotherapy.

Information transmission by Marangoni-driven relaxation oscillations at droplets

Marangoni-driven relaxation oscillations can be observed in many systems where concentration gradients of surface-active substances exist. In the present paper, we describe the experimentally observed coupling between relaxation oscillations at neighboring droplets in a concentration gradient. By a numerical parameter study, we evaluate the oscillation characteristics depending on relevant material parameters and the pairwise droplet distance. Based on these findings, we demonstrate that hydrodynamic interaction in multidroplet configurations can lead to a synchronization of the oscillations over the whole ensemble. This effect has the potential to be used as a novel approach for information transmission in microfluidic applications.

Adaptive Micromixer Based on the Solutocapillary Marangoni Effect in a Continuous-Flow Microreactor

Continuous-flow microreactors are an important development in chemical engineering technology, since pharmaceutical production needs flexibility in reconfiguring the synthesis system rather than large volumes of product yield. Microreactors of this type have a special vessel, in which the convective vortices are organized to mix the reagents to increase the product output. We propose a new type of micromixer based on the intensive relaxation oscillations induced by a fundamental effect discovered recently. The mechanism of these oscillations was found to be a coupling of the solutal Marangoni effect, buoyancy and diffusion. The phenomenon can be observed in the vicinity of an air⁻liquid (or liquid⁻liquid) interface with inhomogeneous concentration of a surface-active solute. Important features of the oscillations are demonstrated experimentally and numerically. The periodicity of the oscillations is a result of the repeated regeneration of the Marangoni driving force. This feature is used in our design of a micromixer with a single air bubble inside the reaction zone. We show that the micromixer does not consume external energy and adapts to the medium state due to feedback. It switches on automatically each time when a concentration inhomogeneity in the reaction zone occurs, and stops mixing when the solution becomes sufficiently uniform.

Numerical Study of Surfactant Dynamics during Emulsification in a T-Junction Microchannel

Microchannel emulsification requires large amounts of surfactant to prevent coalescence and improve emulsions lifetime. However, most numerical studies have considered surfactant-free mixtures as models for droplet formation in microchannels, without taking into account the distribution of surfactant on the droplet surface. In this paper, we investigate the effects of nonuniform surfactant coverage on the microfluidic flow pattern using an extended lattice-Boltzmann model. This numerical study, supported by micro-particle image velocimetry experiments, reveals the likelihood of uneven distribution of surfactant during the droplet formation and the appearance of a stagnant cap. The Marangoni effect affects the droplet breakup by increasing the shear rate. According to our results, surfactant-free and surfactant-rich droplet formation processes are qualitatively different, such that both the capillary number and the Damköhler number should be considered when modeling the droplet generation in microfluidic devices. The limitations of traditional volume and pressure estimation methods for determining the dynamic interfacial tension are also discussed on the basis of the simulation results.

Simulation of Morphogen and Tissue Dynamics

Morphogenesis, the process by which an adult organism emerges from a single cell, has fascinated humans for a long time. Modeling this process can provide novel insights into development and the principles that orchestrate the developmental processes. This chapter focuses on the mathematical description and numerical simulation of developmental processes. In particular, we discuss the mathematical representation of morphogen and tissue dynamics on static and growing domains, as well as the corresponding tissue mechanics. In addition, we give an overview of numerical methods that are routinely used to solve the resulting systems of partial differential equations. These include the finite element method and the Lattice Boltzmann method for the discretization as well as the arbitrary Lagrangian-Eulerian method and the Diffuse-Domain method to numerically treat deforming domains.

Phase-field boundary conditions for the voxel finite cell method: Surface-free stress analysis of CT-based bone structures

The voxel finite cell method uses unfitted finite element meshes and voxel quadrature rules to seamlessly transfer computed tomography data into patient-specific bone discretizations. The method, however, still requires the explicit parametrization of boundary surfaces to impose traction and displacement boundary conditions, which constitutes a potential roadblock to automation. We explore a phase-field-based formulation for imposing traction and displacement constraints in a diffuse sense. Its essential component is a diffuse geometry model generated from metastable phase-field solutions of the Allen-Cahn problem that assumes the imaging data as initial condition. Phase-field approximations of the boundary and its gradient are then used to transfer all boundary terms in the variational formulation into volumetric terms. We show that in the context of the voxel finite cell method, diffuse boundary conditions achieve the same accuracy as boundary conditions defined over explicit sharp surfaces, if the inherent length scales, ie, the interface width of the phase field, the voxel spacing, and the mesh size, are properly related. We demonstrate the flexibility of the new method by analyzing stresses in a human femur and a vertebral body.

Diffusion-Driven Dissolution or Growth of a Liquid Drop Embedded in a Continuous Phase of Another Liquid via Phase-Field Ternary Mixture Model

We simulate the diffusion-driven dissolution or growth of a single-component (resp. two-component) drop embedded in a continuous phase of a binary (resp. single-component) liquid. Our theoretical approach follows a standard diffuse-interface model of partially miscible ternary liquid mixtures, which is based on a regular solution model assumption together with a Flory-Huggins and Cahn-Hilliard representation of the excess and nonlocal components of the Gibbs free energy of mixing. Based on 2D simulation results, we show that for a single-component drop embedded in a continuous phase of a binary liquid (which is highly miscible with either one component of the continuous phase but essentially immiscible with the other) the size of the drop can either shrink to zero or reach a stationary value, depending on whether the global composition of the mixture is within the one-phase region or the unstable range of the phase diagram. On the other hand, for an isolated two-component drop embedded in a continuous phase of a single-component liquid (which is essentially immiscible with either one component of the drop but miscible with the other) the size of the drop can either grow or shrink and, in particular, it will eventually go to zero if the global composition of the mixture is within the one-phase region; otherwise, for system locations in the unstable range the size of the drop tends to a constant value as the composition within the drop reaches its final equilibrium value.

CURVATURE-DRIVEN MOLECULAR FLOW ON MEMBRANE SURFACE

This work presents a mathematical model for the localization of multiple species of diffusion molecules on membrane surfaces. Morphological change of bilayer membrane in vivo is generally modulated by proteins. Most of these modulations are associated with the localization of related proteins in the crowded lipid environments. We start with the energetic description of the distributions of molecules on curved membrane surface, and define the spontaneous curvature of bilayer membrane as a function of the molecule concentrations on membrane surfaces. A drift-diffusion equation governs the gradient flow of the surface molecule concentrations. We recast the energetic formulation and the related governing equations by using an Eulerian phase field description to define membrane morphology. Computational simulations with the proposed mathematical model and related numerical techniques predict (i) the molecular localization on static membrane surfaces at locations with preferred mean curvatures, and (ii) the generation of preferred mean curvature which in turn drives the molecular localization.

Fast Simulation of Lipid Vesicle Deformation Using Spherical Harmonic Approximation

Lipid vesicles appear ubiquitously in biological systems. Understanding how the mechanical and intermolecular interactions deform vesicle membranes is a fundamental question in biophysics. In this article we develop a fast algorithm to compute the surface configurations of lipid vesicles by introducing surface harmonic functions to approximate the membrane surface. This parameterization allows an analytical computation of the membrane curvature energy and its gradient for the efficient minimization of the curvature energy using a nonlinear conjugate gradient method. Our approach drastically reduces the degrees of freedom for approximating the membrane surfaces compared to the previously developed finite element and finite difference methods. Vesicle deformations with a reduced volume larger than 0.65 can be well approximated by using as small as 49 surface harmonic functions. The method thus has a great potential to reduce the computational expense of tracking multiple vesicles which deform for their interaction with external fields.

Numerical simulation of endocytosis: Viscous flow driven by membranes with non-uniformly distributed curvature-inducing molecules

The formation of membrane vesicles from a larger membrane that occurs during endocytosis and other cell processes are typically orchestrated by curvature-inducing molecules attached to the membrane. Recent reports demonstrate that vesicles can form de novo in a few milliseconds. Membrane dynamics at these scales are strongly influenced by hydrodynamic interactions. To study this problem, we develop new diffuse interface models for the dynamics of inextensible vesicles in a viscous fluid with stiff, curvature-inducing molecules. The model couples the Navier-Stokes equations with membrane-induced bending forces that incorporate concentration-dependent bending stiffness coefficients and spontaneous curvatures, with equations for molecule transport and for a Lagrange multiplier to enforce local inextensibility. Two forms of surface transport equations are considered: Fickian surface diffusion and Cahn-Hilliard surface dynamics, with the former being more appropriate for small molecules and the latter being better for large molecules. The system is solved using adaptive finite element methods in 3D axisymmetric geometries. The results demonstrate that hydrodynamics can indeed enable the rapid formation of a small vesicle attached to the membrane by a narrow neck. When the Fickian model is used, this is a transient state with the steady state being a flat membrane with a uniformly distributed molecule concentration due to diffusion. When the Cahn-Hilliard model is used, molecule concentration gradients are sustained, the neck stabilizes and the system evolves to a steady-state with a small, compact vesicle attached to the membrane. By varying the membrane coverage of molecules in the Cahn-Hilliard model, we find that there is a critical (smallest) neck radius and a critical (fastest) budding time. These critical points are associated with changes in the vesicle morphology from spherical to mushroom-like as the molecule coverage on the membrane is increased.

Analysis of Ginzburg-Landau-type models of surfactant-assisted liquid phase separation

In this paper diffuse interface models of surfactant-assisted liquid-liquid phase separation are addressed. We start from the generalized version of the Ginzburg-Landau free-energy-functional-based model of van der Sman and van der Graaf. First, we analyze the model in the constant surfactant approximation and show the presence of a critical point at which the interfacial tension vanishes. Then we determine the adsorption isotherms and investigate the validity range of previous results. As a key point of the work, we propose a new model of the van der Sman/van der Graaf type designed for avoiding both unwanted unphysical effects and numerical difficulties present in previous models. In order to make the model suitable for describing real systems, we determine the interfacial tension analytically more precisely and analyze it over the entire accessible surfactant load range. Emerging formulas are then validated by calculating the interfacial tension from the numerical solution of the Euler-Lagrange equations. Time-dependent simulations are also performed to illustrate the slowdown of the phase separation near the critical point and to prove that the dynamics of the phase separation is driven by the interfacial tension.

Tumor growth in complex, evolving microenvironmental geometries: a diffuse domain approach

We develop a mathematical model of tumor growth in complex, dynamic microenvironments with active, deformable membranes. Using a diffuse domain approach, the complex domain is captured implicitly using an auxiliary function and the governing equations are appropriately modified, extended and solved in a larger, regular domain. The diffuse domain method enables us to develop an efficient numerical implementation that does not depend on the space dimension or the microenvironmental geometry. We model homotypic cell-cell adhesion and heterotypic cell-basement membrane (BM) adhesion with the latter being implemented via a membrane energy that models cell-BM interactions. We incorporate simple models of elastic forces and the degradation of the BM and ECM by tumor-secreted matrix degrading enzymes. We investigate tumor progression and BM response as a function of cell-BM adhesion and the stiffness of the BM. We find tumor sizes tend to be positively correlated with cell-BM adhesion since increasing cell-BM adhesion results in thinner, more elongated tumors. Prior to invasion of the tumor into the stroma, we find a negative correlation between tumor size and BM stiffness as the elastic restoring forces tend to inhibit tumor growth. In order to model tumor invasion of the stroma, we find it necessary to downregulate cell-BM adhesiveness, which is consistent with experimental observations. A stiff BM promotes invasiveness because at early stages the opening in the BM created by MDE degradation from tumor cells tends to be narrower when the BM is stiffer. This requires invading cells to squeeze through the narrow opening and thus promotes fragmentation that then leads to enhanced growth and invasion. In three dimensions, the opening in the BM was found to increase in size even when the BM is stiff because of pressure induced by growing tumor clusters. A larger opening in the BM can increase the potential for further invasiveness by increasing the possibility that additional tumor cells could invade the stroma.

Signaling networks and cell motility: a computational approach using a phase field description

The processes of protrusion and retraction during cell movement are driven by the turnover and reorganization of the actin cytoskeleton. Within a reaction-diffusion model which combines processes along the cell membrane with processes within the cytoplasm a Turing type instability is used to form the necessary polarity to distinguish between cell front and rear and to initiate the formation of different organizational arrays within the cytoplasm leading to protrusion and retraction. A simplified biochemical network model for the activation of GTPase which accounts for the different dimensionality of the cell membrane and the cytoplasm is used for this purpose and combined with a classical Helfrich type model to account for bending and stiffness effects of the cell membrane. In addition streaming within the cytoplasm and the extracellular matrix is taken into account. Combining these phenomena allows to simulate the dynamics of cells and to reproduce the primary phenomenology of cell motility. The coupled model is formulated within a phase field approach and solved using adaptive finite elements.

Particles at fluid-fluid interfaces: A new Navier-Stokes-Cahn-Hilliard surface- phase-field-crystal model

Colloid particles that are partially wetted by two immiscible fluids can become confined to fluid-fluid interfaces. At sufficiently high volume fractions, the colloids may jam and the interface may crystallize. The fluids together with the interfacial colloids form an emulsion with interesting material properties and offer an important route to new soft materials. A promising approach to simulate these emulsions was presented in Aland et al. [Phys. Fluids 23, 062103 (2011)], where a Navier-Stokes-Cahn-Hilliard model for the macroscopic two-phase fluid system was combined with a surface phase-field-crystal model for the microscopic colloidal particles along the interface. Unfortunately this model leads to spurious velocities which require very fine spatial and temporal resolutions to accurately and stably simulate. In this paper we develop an improved Navier-Stokes-Cahn-Hilliard-surface phase-field-crystal model based on the principles of mass conservation and thermodynamic consistency. To validate our approach, we derive a sharp interface model and show agreement with the improved diffuse interface model. Using simple flow configurations, we show that the new model has much better properties and does not lead to spurious velocities. Finally, we demonstrate the solid-like behavior of the crystallized interface by simulating the fall of a solid ball through a colloid-laden multiphase fluid.

Two-phase flow in complex geometries: A diffuse domain approach

We present a new method for simulating two-phase flows in complex geometries, taking into account contact lines separating immiscible incompressible components. We combine the diffuse domain method for solving PDEs in complex geometries with the diffuse-interface (phase-field) method for simulating multiphase flows. In this approach, the complex geometry is described implicitly by introducing a new phase-field variable, which is a smooth approximation of the characteristic function of the complex domain. The fluid and component concentration equations are reformulated and solved in larger regular domain with the boundary conditions being implicitly modeled using source terms. The method is straightforward to implement using standard software packages; we use adaptive finite elements here. We present numerical examples demonstrating the effectiveness of the algorithm. We simulate multiphase flow in a driven cavity on an extended domain and find very good agreement with results obtained by solving the equations and boundary conditions in the original domain. We then consider successively more complex geometries and simulate a droplet sliding down a rippled ramp in 2D and 3D, a droplet flowing through a Y-junction in a microfluidic network and finally chaotic mixing in a droplet flowing through a winding, serpentine channel. The latter example actually incorporates two different diffuse domains: one describes the evolving droplet where mixing occurs while the other describes the channel.