Diffuse interface models of locally inextensible vesicles in a viscous fluid

We present a new diffuse interface model for the dynamics of inextensible vesicles in a viscous fluid. A new feature of this work is the implementation of the local inextensibility condition in the diffuse interface context. Local inextensibility is enforced by using a local Lagrange multiplier, which provides the necessary tension force at the interface. To solve for the local Lagrange multiplier, we introduce a new equation whose solution essentially provides a harmonic extension of the local Lagrange multiplier off the interface while maintaining the local inextensibility constraint near the interface. To make the method more robust, we develop a local relaxation scheme that dynamically corrects local stretching/compression errors thereby preventing their accumulation. Asymptotic analysis is presented that shows that our new system converges to a relaxed version of the inextensible sharp interface model. This is also verified numerically. Although the model does not depend on dimension, we present numerical simulations only in 2D. To solve the 2D equations numerically, we develop an efficient algorithm combining an operator splitting approach with adaptive finite elements where the Navier-Stokes equations are implicitly coupled to the diffuse interface inextensibility equation. Numerical simulations of a single vesicle in a shear flow at different Reynolds numbers demonstrate that errors in enforcing local inextensibility may accumulate and lead to large differences in the dynamics in the tumbling regime and differences in the inclination angle of vesicles in the tank-treading regime. The local relaxation algorithm is shown to effectively prevent this accumulation by driving the system back to its equilibrium state when errors in local inextensibility arise.Comment: 25 page

On the damped oscillations of an elastic quasi-circular membrane in a two-dimensional incompressible fluid

We propose a procedure - partly analytical and partly numerical - to find the frequency and the damping rate of the small-amplitude oscillations of a massless elastic capsule immersed in a two-dimensional viscous incompressible fluid. The unsteady Stokes equations for the stream function are decomposed onto normal modes for the angular and temporal variables, leading to a fourth-order linear ordinary differential equation in the radial variable. The forcing terms are dictated by the properties of the membrane, and result into jump conditions at the interface between the internal and external media. The equation can be solved numerically, and an excellent agreement is found with a fully-computational approach we developed in parallel. Comparisons are also shown with the results available in the scientific literature for drops, and a model based on the concept of embarked fluid is presented, which allows for a good representation of the results and a consistent interpretation of the underlying physics.Comment: in press on JF

An adaptive meshfree method for phase-field models of biomembranes. Part II: A Lagrangian approach for membranes in viscous fluids

We present a Lagrangian phase-field method to study the low Reynolds number dynamics of vesicles embedded in a viscous fluid. In contrast to previous approaches, where the field variables are the phase-field and the fluid velocity, here we exploit the fact that the phasefield tracks a material interface to reformulate the problem in terms of the Lagrangian motion of a background medium, containing both the biomembrane and the fluid. We discretize the equations in space with maximum-entropy approximants, carefully shown to perform well in phase-field models of biomembranes in a companion paper. The proposed formulation is variational, lending itself to implicit time-stepping algorithms based on minimization of a time-incremental energy, which are automatically nonlinearly stable. The proposed method deals with two of the major challenges in the numerical treatment of coupled fluid/phase-field models of biomembranes, namely the adaptivity of the grid to resolve the sharp features of the phase-field, and the stiffness of the equations, leading to very small time-steps. In our method, local refinement follows the features of the phasefield as both are advected by the Lagrangian motion, and large time-steps can be robustly chosen in the variational time-stepping algorithm, which also lends itself to time adaptivity. The method is presented in the axisymmetric setting, but it can be directly extended to 3D. We present a Lagrangian phase-field method to study the low Reynolds number dynamics of vesicles embedded in a viscous fluid. In contrast to previous approaches, where the field variables are the phase-field and the fluid velocity, here we exploit the fact that the phase-field tracks a material interface to reformulate the problem in terms of the Lagrangian motion of a background medium, containing both the biomembrane and the fluid. We discretize the equations in space with maximum-entropy approximants, carefully shown to perform well in phase-field models of biomembranes in a companion paper. The proposed formulation is variational, lending itself to implicit time-stepping algorithms based on minimization of a time-incremental energy, which are automatically nonlinearly stable. The proposed method deals with two of the major challenges in the numerical treatment of coupled fluid/phase-field models of biomembranes, namely the adaptivity of the grid to resolve the sharp features of the phase-field, and the stiffness of the equations, leading to very small time-steps. In our method, local refinement follows the features of the phase-field as both are advected by the Lagrangian motion, and large time-steps can be robustly chosen in the variational time-stepping algorithm, which also lends itself to time adaptivity. The method is presented in the axisymmetric setting, but it can be directly extended to 3D

Fluid Vesicles in Flow

We review the dynamical behavior of giant fluid vesicles in various types of external hydrodynamic flow. The interplay between stresses arising from membrane elasticity, hydrodynamic flows, and the ever present thermal fluctuations leads to a rich phenomenology. In linear flows with both rotational and elongational components, the properties of the tank-treading and tumbling motions are now well described by theoretical and numerical models. At the transition between these two regimes, strong shape deformations and amplification of thermal fluctuations generate a new regime called trembling. In this regime, the vesicle orientation oscillates quasi-periodically around the flow direction while asymmetric deformations occur. For strong enough flows, small-wavelength deformations like wrinkles are observed, similar to what happens in a suddenly reversed elongational flow. In steady elongational flow, vesicles with large excess areas deform into dumbbells at large flow rates and pearling occurs for even stronger flows. In capillary flows with parabolic flow profile, single vesicles migrate towards the center of the channel, where they adopt symmetric shapes, for two reasons. First, walls exert a hydrodynamic lift force which pushes them away. Second, shear stresses are minimal at the tip of the flow. However, symmetry is broken for vesicles with large excess areas, which flow off-center and deform asymmetrically. In suspensions, hydrodynamic interactions between vesicles add up to these two effects, making it challenging to deduce rheological properties from the dynamics of individual vesicles. Further investigations of vesicles and similar objects and their suspensions in steady or time-dependent flow will shed light on phenomena such as blood flow.Comment: 13 pages, 13 figures. Adv. Colloid Interface Sci., 201

Mathematical and Computational Models of Fluctuating Vesicles in Time-Varying Flows

Modeling vesicle dynamics involves a complicated moving boundary problem while the shapes of vesicles are determined dynamically from a balance between interfacial forces and fluid stresses. In this thesis, we investigate the dynamics of a two-dimensional fluctuating vesicle in a viscous fluid.Firstly we develop a two-dimensional stochastic immersed boundary method (SIBM) and analyze thermal fluctuations by matching the numerical results with a theoretical solution. Then we apply the SIBM with fitted thermal fluctuations to study the long term dynamics of an impermeable vesicle in a periodically time-reversed flow. The wrinkling process contains three stages. In the first stage, high-order modes are excited by the negative surface tension and wrinkles appear. In the second stage, low Fourier modes increase, the high-order wrinkles decay, and the shapes of vesicles keep relatively stable. In the last stage, the second Fourier mode grows and dominates. The shapes of the vesicle are ellipse-like with inclination angle θ ≈ 45◦. Then by performing an asymptotic linear analysis of a quasi-circular vesicle, we derive and solve the deterministic and stochastic equations for the motion of membrane interface numerically. The linear theory also indicates this three stage process.Finally, we investigate the nonlinear wrinkling dynamics of a permeable vesicle using an extension of the SIBM. We observe the vesicle shrinkage and the wrinkles on the membrane caused by a large osmosis pressure. We extend the linear theory to account for permeability and find a good agreement between linear and fully nonlinear vesicle dynamics

Hydrodynamic Diffuse Interface Models for Cell Morphology and Motility

In this thesis, we study mathematical models that describe the morphology of a generalized biological cell in equilibrium or under the influence of external forces. Within these models, the cell is considered as a thermodynamic system, where streaming effects in the cell bulk and the surrounding are coupled with a Helfrich-type model for the cell membrane. The governing evolution equations for the cell given in a continuum formulation are derived using an energy variation approach. Such two-phase flow problems that combine streaming effects with a free boundary problem that accounts for bending and surface tension can be described effectively by a diffuse interface approach. An advantage of the diffuse interface approach is that models for e.g. different biophysical processes can easily be combined. That makes this method suitable to describe complex phenomena such as cell motility and multi-cell dynamics. Within the first model for cell motility, we combine a biological network for GTPases with the hydrodynamic Helfrich-type model. This model allows to account for cell motility driven by membrane protrusion as a result of actin polymerization. Within the second model, we moreover extend the Helfrich-type model by an active gel theory to account for the actin filaments in the cell bulk. Caused by contractile stress within the actin-myosin solution, a spontaneous symmetry breaking event occurs that lead to cell motility. In this thesis, we further study the dynamics of multiple cells which is of wide interest since it reveals rich non-linear behavior. To apply the diffuse interface framework, we introduce several phase field variables to account for several cells that are coupled by a local interaction potential. In a first application, we study white blood cell margination, a biological phenomenon that results from the complex relation between collisions, different mechanical properties and lift forces of red blood cells and white blood cells within the vascular system. Here, it is shown that inertial effects, which can become of relevance in various parts of the cardiovascular system, lead to a decreasing tendency for margination with increasing Reynolds number. Finally, we combine the active polar gel theory and the multi-cell approach that is capable of studying collective migration of cells. This hydrodynamic approach predicts that collective migration emerges spontaneously forming coherently-moving clusters as a result of the mutual alignment of the velocity vectors during inelastic collisions. We further observe that hydrodynamics heavily influence those systems. However, a complete suppression of the onset of collective migration cannot be confirmed. Moreover, we give a brief insight how such highly coupled systems can be treated numerically using finite elements and how the numerical costs can be limited using operator splitting approaches and problem parallelization with OPENMP.Diese Dissertation beschäftigt sich mit mathematischen Modellen zur Beschreibung von Gleichgewichts- und dynamischen Zuständen von verallgemeinerten biologischen Zellen. Die Zellen werden dabei als thermodynamisches System aufgefasst, bei dem Strömungseffekte innerhalb und außerhalb der Zelle zusammen mit einem Helfrich-Modell für Zellmembranen kombiniert werden. Schließlich werden durch einen Energie-Variations-Ansatz die Evolutionsgleichungen für die Zelle hergeleitet. Es ergeben sie dabei Mehrphasen-Systeme, die Strömungseffekte mit einem freien Randwertproblem, das zusätzlich physikalischen Einflüssen wie Biegung und Oberflächenspannung unterliegt, vereinen. Um solche Probleme effizient zu lösen, wird in dieser Arbeit die Diffuse-Interface-Methode verwendet. Ein Vorteil dieser Methode ist, dass es sehr einfach möglich ist, Modelle, die verschiedenste Prozesse beschreiben, miteinander zu vereinen. Dies erlaubt es, komplexe biologische Phänomene, wie zum Beispiel Zellmotilität oder auch die kollektive Bewegung von Zellen, zu beschreiben. In den Modellen für Zellmotilität wird ein biologisches Netzwerk-Modell für GTPasen oder auch ein Active-Polar-Gel-Modell, das die Aktinfilamente im Inneren der Zellen als Flüssigkristall auffasst, mit dem Multi-Phasen-Modell kombiniert. Beide Modelle erlauben es, komplexe Vorgänge bei der selbst hervorgerufenen Bewegung von Zellen, wie das Vorantreiben der Zellmembran durch Aktinpolymerisierung oder auch die Kontraktionsbewegung des Zellkörpers durch kontraktile Spannungen innerhalb des Zytoskelets der Zelle, zu verstehen. Weiterhin ist die kollektive Bewegung von vielen Zellen von großem Interesse, da sich hier viele nichtlineare Phänomene zeigen. Um das Diffuse-Interface-Modell für eine Zelle auf die Beschreibung mehrerer Zellen zu übertragen, werden mehrere Phasenfelder eingeführt, die die Zellen jeweils kennzeichnen. Schließlich werden die Zellen durch ein lokales Abstoßungspotential gekoppelt. Das Modell wird angewendet, um White blood cell margination, das die Annäherung von Leukozyten an die Blutgefäßwand bezeichnet, zu verstehen. Dieser Prozess wird dabei bestimmt durch den komplexen Zusammenhang zwischen Kollisionen, den jeweiligen mechanischen Eigenschaften der Zellen, sowie deren Auftriebskraft innerhalb der Adern. Die Simulationen zeigen, dass diese Annäherung sich in bestimmten Gebieten des kardiovaskulären Systems stark vermindert, in denen die Blutströmung das Stokes-Regime verlässt. Schließlich wird das Active-Polar-Gel-Modell mit dem Modell für die kollektive Bewegung vom Zellen kombiniert. Dies macht es möglich, die kollektive Bewegung der Zellen und den Einfluss von Hydrodynamik auf diese Bewegung zu untersuchen. Es zeigt sich dabei, dass der Zustand der kollektiven gerichteten Bewegung sich spontan aus der Neuausrichtung der jeweiligen Zellen durch inelastische Kollisionen ergibt. Obwohl die Hydrodynamik einen großen Einfluss auf solche Systeme hat, deuten die Simulationen nicht daraufhin, dass Hydrodynamik die kollektive Bewegung vollständig unterdrückt. Weiterhin wird in dieser Arbeit gezeigt, wie die stark gekoppelten Systeme numerisch gelöst werden können mit Hilfe der Finiten-Elemente-Methode und wie die Effizienz der Methode gesteigert werden kann durch die Anwendung von Operator-Splitting-Techniken und Problemparallelisierung mittels OPENMP

Approximation of phase-field models with meshfree methods: exploring biomembrane dynamics

Las biomembranas constituyen la estructura de separación fundamental en las celulas animales, y son importantes en el diseño de sistemas bioinspirados. Su simulación presenta desafíos, especialmente cuando ésta implica dinámica y grandes cambios de forma o se estudian sistemas micrométricos, impidiendo el uso de modelos atomísticos y de grano grueso. El objetivo principal de esta tesis es el desarrollo de un marco computacional para entender la dinámica de biomembranas inmersas en fluido viscoso usando modelos de campo de fase. Los modelos de campo de fase introducen un campo escalar contínuo que define una interfase difusa, cuya física está codificada en las ecuaciones en derivadas parciales que la gobiernan. Estos modelos son capaces de soportar cambios dramáticos de forma y topología, y facilitan el acoplamiento de distintos fenómenos físicos. No obstante, presentan desafíos numéricos significativos, como el alto orden de las ecuaciones, la resolución de frentes móviles y abruptos, o una eficiente integración en el tiempo. En esta disertación abordamos estos puntos mediante la combinación de una discretización espacial con métodos sin malla usando las funciones base locales de máxima entropía, y una formulación variacional Lagrangiana para acoplamiento elástico-hidrodinámico. La suavidad del método sin malla genera una aproximación precisa del campo de fase y puede lidiar fácilmente con adaptatividad local, la aproximación Lagrangiana extiende de manera natural esta adaptividad a la dinámica, y la formulación variacional permite una integración variacional temporal no linealmente estable y robusta. La implementación numérica de estos métodos en un entorno de computación de alto rendimiento ha motivado el desarrollo de un nuevo código computacional. Este código integra el estado del arte de las librerías en paralelo e incorpora importantes contribuciones técnicas para solventar cuellos de botella que aparecen con el uso de métodos sin malla en computación a gran escala. El código resultante es flexible y ha sido aplicado a otros problemas científicos en varias colaboraciones que incluyen flexoelectricidad, conformado metálico, fluidos viscosos o fractura en materiales con energía de superficie altamente anisotrópica.Biomembranes are the fundamental separation structure in animal cells, and are also used in engineered bioinspired systems. Their simulation is challenging, particularly when large shape changes and dynamics are involved, or micrometer systems are considered, ruling out atomistic or coarse-grained molecular modeling. The main goal of this thesis is to develop a computational framework to understand the dynamics of biomembranes embedded in a viscous fluid using phase-field models. Phase-field models introduce a scalar continuous field to define a diffuse moving interface, whose physics is encoded in partial differential equations governing it. These models can deal with dramatic shape and topological transformations and are amenable to multiphysics coupling. However, they present significant numerical challenges, such as the high-order character of the equations, the resolution of sharp and moving fronts, or the efficient time-integration. We address all these issues through a combination of meshfree spacial discretization using local maximum-entropy basis functions, and a Lagrangian variational formulation of the coupled elasticity-hydrodynamics. The smooth meshfree approach provides accurate approximations of the phase-field and can easily deal with local adaptivity, the Lagrangian approach naturally extend adaptivity to dynamics, and the variational formulation enables nonlinearly-stable robust variational time integration. The numerical implementation of these methods in a high-performance computing framework has motivated the development of a new computer code, which integrates state-of-the-art parallel libraries and incorporates important technical contributions to overcome bottlenecks that arise in meshfree methods for large-scale problems. The resulting code is flexible and has been applied to other scientific problems in a number of collaborations dealing with flexoelectricity, metal forming, creeping flows, or fracture in materials with strongly anisotropic surface energy