Whilst many vertebrates appear externally left-right symmetric, the arrangement of internal organs is asymmetric. In zebrafish, the breaking of left-right symmetry is organised by Kupffer's Vesicle (KV): an approximately spherical, fluid-filled structure that begins to form in the embryo 10 hours post fertilisation. A crucial component of zebrafish symmetry breaking is the establishment of a cilia-driven fluid flow within KV. However, it is still unclear (a) how dorsal, ventral and equatorial cilia contribute to the global vortical flow, and (b) if this flow breaks left-right symmetry through mechanical transduction or morphogen transport. Fully answering these questions requires knowledge of the three-dimensional flow patterns within KV, which have not been quantified in previous work. In this study, we calculate and analyse the three-dimensional flow in KV. We consider flow from both individual and groups of cilia, and (a) find anticlockwise flow can arise purely from excess of cilia on the dorsal roof over the ventral floor, showing how this vortical flow is stabilised by dorsal tilt of equatorial cilia, and (b) show that anterior clustering of dorsal cilia leads to around 40% faster flow in the anterior over the posterior corner. We argue that these flow features are supportive of symmetry breaking through mechano-sensory cilia, and suggest a novel experiment to test this hypothesis. From our new understanding of the flow, we propose a further experiment to reverse the flow within KV to potentially induce situs inversus.
Fluid mechanics plays a vital role in early vertebrate embryo development, an example being the establishment of left-right asymmetry. Following the dorsal-ventral and anterior-posterior axes, the left-right axis is the last to be established; in several species it has been shown that an important process involved with this is the production of a left-right asymmetric flow driven by 'whirling' cilia. It has previously been established in experimental and mathematical models of the mouse ventral node that the combination of a consistent rotational direction and posterior tilt creates left-right…
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. I…
Lipid vesicles play a key role in fundamental biological processes. Despite recent progress, we lack a complete understanding of the non-equilibrium dynamics of vesicles due to challenges associated with long-time observation of shape fluctuations in strong flows. In this work, we present a flow-phase diagram for vesicle shape and conformational transitions in planar extensional flow using a Stokes trap, which enables control over the center-of-mass position of single or multiple vesicles in precisely defined flows [Shenoy, Rao, Schroeder, \textit{PNAS}, 113(15):3976-3981, 2016]. In this way, …
We present experimental observations and numerical simulations of a wrinkling instability that occurs at sufficiently high strain rates in the trembling regime of vesicle dynamics in steady linear flow. Spectral and statistical analysis of the data shows similarities and differences with the wrinkling instability observed earlier for vesicles in transient elongation flow. The critical relevance of thermal fluctuations for this phenomenon is revealed by a simple model using coupled Langevin equations that reproduces the experimental observations quite well.
Despite the recent upsurge of theoretical reduced models for vesicle shape dynamics, comparisons with experiments have not been accomplished. We review the implications of some of the recently proposed models for vesicle dynamics, especially the Tumbling-Trembling domain regions of the phase plane and show that they all fail to capture the essential behavior of real vesicles for excess areas, Δ, greater than 0.4. We emphasize new observations of shape harmonics and the role of thermal fluctuations.
Red blood cells (RBCs) are the major component of blood and the flow of blood is dictated by that of RBCs. We employ vesicles, which consist of closed bilayer membranes enclosing a fluid, as a model system to study the behavior of RBCs under a confined Poiseuille flow. We extensively explore two main parameters: i) the degree of confinement of vesicles within the channel, and ii) the flow strength. Rich and complex dynamics for vesicles are revealed ranging from steady-state shapes (in the form of parachute and slipper) to chaotic dynamics of shape. Chaos occurs through a cascade of multiple p…
Vesicles are involved in a vast variety of transport processes in living organisms. Additionally, they serve as a model for the dynamics of cell suspensions. Predicting the rheological properties of their suspensions is still an open question, as even the interaction of pairs is yet to be fully understood. Here we analyse the effect of a single vesicle, undergoing tank-treading motion, on its surrounding shear flow by studying the induced disturbance field $δ\vec{V}$, the difference between the velocity field in its presence and absence. The comparison between experiments and numerical simulat…
The viscosity of lipid bilayer membranes plays an important role in determining the diffusion constant of embedded proteins and the dynamics of membrane deformations, yet it has historically proven very difficult to measure. Here we introduce a new method based on quantification of the large-scale circulation patterns induced inside vesicles adhered to a solid surface and subjected to simple shear flow in a microfluidic device. Particle Image Velocimetry based on spinning disk confocal imaging of tracer particles inside and outside of the vesicle, and tracking of phase-separated membrane domai…
Vesicle dynamics in unbounded Poiseuille flow is analyzed using a small-deformation theory. Our analytical results quantitatively describe vesicle migration and provide new physical insights. At low ratio between the inner and outer viscosity $λ$ (i.e. in the tank-treading regime), the vesicle always migrates towards the flow centerline, unlike other soft particles such as drops. Above a critical $λ$, vesicle tumbles and cross-stream migration vanishes. A novel feature is predicted, namely the coexistence of two types of nonequilibrium configurations at the centreline, a bullet-like and a para…
Vesicles are micrometric soft particles whose the membrane is a two-dimensional incompressible fluid governed by bending resistance leading to a zoology of shapes. The dynamics of deflated vesicles in shear flow with a bottom wall, a first minimal configuration to consider confined vesicles is investigated using numerical simulations. Coexistence under flow of oblate (metastable) and prolate (stable) shapes is studied in details. In particular, we discuss the boundaries of the region of coexistence in the (v, Ca) plane where v is the reduced volume of the vesicle and Ca the Capillary number. W…
Soft bodies flowing in a channel often exhibit parachute-like shapes usually attributed to an increase of hydrodynamic constraint (viscous stress and/or confinement). We show that the presence of a fluid membrane leads to the reverse phenomenon and build a phase diagram of shapes --- which are classified as bullet, croissant and parachute --- in channels of varying aspect ratio. Unexpectedly, shapes are relatively wider in the narrowest direction of the channel. We highlight the role of flow patterns on the membrane in this response to the asymmetry of stress distribution.
The hydrodynamic interaction of two deformable vesicles in shear flow induces a net displacement, in most cases an increase of their distance in the transverse direction. The statistical average of these interactions leads to shear-induced diffusion in the suspension, both at the level of individual particles which experience a random walk made of successive interactions, and at the level of suspension where a non-linear down-gradient diffusion takes place, an important ingredient in the structuring of suspension flows. We make an experimental and computational study of the interaction of a pa…
An analytical theory is developed to describe the dynamics of a closed lipid bilayer membrane (vesicle) freely suspended in a general linear flow. Considering a nearly spherical shape, the solution to the creeping-flow equations is obtained as a regular perturbation expansion in the excess area. The analysis takes into account the membrane fluidity, incompressibility and resistance to bending. The constraint for a fixed total area leads to a non-linear shape evolution equation at leading order. As a result two regimes of vesicle behavior, tank-treading and tumbling, are predicted depending on …
The dynamics of an adhesive two-dimensional vesicle doublet under various flow conditions is investigated numerically using a high-order, adaptive-in-time boundary integral method. In a quiescent flow, two nearby vesicles move slowly towards each other under the adhesive potential, pushing out fluid between them to form a vesicle doublet at equilibrium. A lubrication analysis on such draining of a thin film gives the dependencies of draining time on adhesion strength and separation distance that are in good agreement with numerical results. In a planar extensional flow we find a stable vesicle…
We develop a new model, to our knowledge, for the many-body hydrodynamics of amphiphilic Janus particles suspended in a viscous background flow. The Janus particles interact through a hydrophobic attraction potential that leads to self-assembly into bilayer structures. We adopt an efficient integral equation method for solving the screened Laplace equation for hydrophobic attraction and for solving the mobility problem for hydrodynamic interactions. The integral equation formulation accurately captures both interactions for near touched boundaries. Under a linear shear flow, we observe the tan…
We report the experimental studies on interaction of two vesicles trapped in a microfluidic analog of four-roll mill, where a plane linear flow is realized. We found that the dynamics of a single vesicle is significantly altered by the presence of another vesicle at separation distances up to about 3.2 ÷3.7 times of effective radius of the vesicles. This is supported by direct measurements of a single vesicle back-reaction on the velocity field. Thus, the experiment provides the lower bound for the interaction scale of vesicles and so the corresponding upper bound for the volume fraction ϕ=0.0…
We present experimental results on the relaxation dynamics of vesicles subjected to a time-dependent elongation flow. We observed and characterized a new instability, which results in the formation of higher order modes of the vesicle shape (wrinkles), after a switch in the direction of the gradient of the velocity. This surprising generation of membrane wrinkles can be explained by the appearance of a negative surface tension during the vesicle deflation, due to compression in a sign-switching transient. Moreover, the formation of buds in the vesicle membrane has been observed in the vicinity…
Vesicles are soft elastic bodies with distinctive mechanical properties such as bending resistance, membrane fluidity, and their strong ability to deform, mimicking some properties of biological cells. While previous three-dimensional (3D) studies have identified stationary shapes such as slipper and axisymmetric ones, we report a complete phase diagram of 3D vesicle dynamics in a bounded Poiseuille flow with two more oscillatory dynamics, 3D snaking and swirling. 3D snaking is characterized by planar oscillatory motion of the mass center and shape deformations, which is unstable and leads to …
The steady motion and deformation of a lipid-bilayer vesicle translating through a circular tube in low Reynolds number pressure-driven flow are investigated numerically using an axisymmetric boundary element method. This fluid-structure interaction problem is determined by three dimensionless parameters: reduced volume (a measure of the vesicle asphericity), geometric confinement (the ratio of the vesicle effective radius to the tube radius), and capillary number (the ratio of viscous to bending forces). The physical constraints of a vesicle -- fixed surface area and enclosed volume when it i…
Lipid bilayer membranes have a native (albeit small) permeability for water molecules. Under an external load, provided that the bilayer structure stays intact and does not suffer from poration or rupture, a lipid membrane deforms and its water influx/efflux is often assumed negligible in the absence of osmolarity. In this work we use boundary integral simulations to investigate the effects of water permeability on the vesicle hydrodynamics due to a mechanical load, such as the viscous stress from an external flow deforming a vesicle membrane in free space or pushing it through a confinement. …
Dynamics of a single vesicle under shear flow between two parallel plates is studied using two-dimensional lattice-Boltzmann simulations. We first present how we adapted the lattice-Boltzmann method to simulate vesicle dynamics, using an approach known from the immersed boundary method. The fluid flow is computed on an Eulerian regular fixed mesh while the location of the vesicle membrane is tracked by a Lagrangian moving mesh. As benchmarking tests, the known vesicle equilibrium shapes in a fluid at rest are found and the dynamical behavior of a vesicle under simple shear flow is being reprod…
Modern biomedical applications such as targeted drug delivery require a delivery system capable of enhanced transport beyond that of passive Brownian diffusion. In this work an osmotic mechanism for the propulsion of a vesicle immersed in a viscous fluid is proposed. By maintaining a steady-state solute gradient inside the vesicle, a seepage flow of the solvent (e.g., water) across the semipermeable membrane is generated which in turn propels the vesicle. We develop a theoretical model for this vesicle-solute system in which the seepage flow is described by a Darcy flow. Using the reciprocal t…
Recent experiments have shown that when a near-hemispherical lipid vesicle attached to a solid surface is subjected to a simple shear flow it exhibits a pattern of membrane circulation much like a dipole vortex. This is in marked contrast to the toroidal circulation that would occur in the related problem of a drop of immiscible fluid attached to a surface and subjected to shear. This profound difference in flow patterns arises from the lateral incompressibility of the membrane, which restricts the observable flows to those in which the velocity field in the membrane is two-dimensionally diver…
Many vesicles have a spherical resting shape and exposure to fluid flows induces an exchange between sub-optical area and visible (systematic) deformation, while the total area is conserved. The dynamics which controls the exchange between sub-optical and visible area depends on membrane properties such as bending rigidity and initial tension. Conversely, observation of these dynamics can be used to determine the membrane properties. The goal of this work is to create a general numerical model which accounts for the exchange between sub-optical and visible area. Unlike prior modeling efforts, …
We study mixing in Stokesian vesicle suspensions in two dimensions on a cylindrical Couette apparatus using numerical simulations. The vesicle flow simulation is done using a boundary integral method and the advection-diffusion equation for the mixing of the solute is solved using a pseudo-spectral scheme. We study the effect of the area fraction, the viscosity contrast between the inside (the vesicles) and the outside (the bulk) fluid, the initial condition of the solute, and the mixing metric. We compare mixing in the suspension with mixing in the Couette apparatus without vesicles. On the o…
The use of tank-treading as a means of propulsion for microswimmers in viscous shear flows is taken into exam. We discuss the possibility that a vesicle be able to control the drift in an external shear flow, by varying locally the bending rigidity of its own membrane. By analytical calculation in the quasi-spherical limit, the stationary shape and the orientation of the tank-treading vesicle in the external flow, are determined, working to lowest order in the membrane inhomogeneity. The membrane inhomogeneity acts in the shape evolution equation as an additional force term, that can be used t…
A small amplitude perturbation analysis is developed to describe the effect of a uniform electric field on the dynamics of a lipid bilayer vesicle in a simple shear flow. All media are treated as leaky dielectrics and fluid motion is described by the Stokes equations. The instantaneous vesicle shape is obtained by balancing electric, hydrodynamic, bending, and tension stresses exerted on the membrane. We find that in the absence of ambient shear flow, it is possible that an applied step--wise uniform DC electric field could cause the vesicle shape to evolve from oblate to prolate over time if …
Surfactant solutions exhibit multilamellar surfactant vesicles (MLVs) under flow conditions and in concentration ranges which are found in a large number of industrial applications. MLVs are typically formed from a lamellar phase and play an important role in determining the rheological properties of surfactant solutions. Despite the wide literature on the collective dynamics of flowing MLVs, investigations on the flow behavior of single MLVs are scarce. In this work, we investigate a concentrated aqueous solution of linear alkylbenzene sulfonic acid (HLAS), characterized by MLVs dispersed in …
We study the deformation and motion of an erythrocyte in fluid flows via a lattice Boltzmann method. To this purpose, the bending rigidity and the elastic modulus of isotropic dilation are introduced and incorporated with the lattice Boltzmann simulation, and the membrane-flow interactions on both sides of the membrane are carefully examined. We find that the static biconcave shape of an erythrocyte is quite stable and can effectively resist the pathological changes on their membrane. Further, our simulation results show that in shear flow, the erythrocyte will be highly flattened and undergo …
The possibility of microscopic swimming by extraction of energy from an external flow is discussed, focusing on the migration of a simple trimer across a linear shear flow. The geometric properties of swimming, together with the possible generalization to the case of a vesicle, are analyzed.The mechanism of energy extraction from the flow appears to be the generalization to a discrete swimmer of the tank-treading regime of a vesicle. The swimmer takes advantage of the external flow by both extracting energy for swimming and "sailing" through it. The migration velocity is found to scale linearl…
Left-right symmetry breaking is critical to vertebrate embryonic development; in many species this process begins with cilia-driven flow in a structure termed the `node'. Primary `whirling' cilia, tilted towards the posterior, transport morphogen-containing vesicles towards the left, initiating left-right asymmetric development. We review recent theoretical models based on the point-force stokeslet and point-torque rotlet singularities, explaining how rotation and surface-tilt produce directional flow. Analysis of image singularity systems enforcing the no-slip condition shows how tilted rotat…
The mechanism of hydrodynamics-induced pairing of soft particles, namely closed bilayer membranes (vesicles, a model system for red blood cells) and drops, is studied numerically with a special attention paid to the role of the confinement (the particles are within two rigid walls). This study unveils the complexity of the pairing mechanism due to hydrodynamic interactions. We find both for vesicles and for drops that two particles attract each other and form a stable pair at weak confinement if their initial separation is below a certain value. If the initial separation is beyond that distanc…
The concept of internal anatomical asymmetry is familiar; usually in humans the heart is on the left and the liver is on the right, however how does the developing embryo know to produce this consistent laterality? Symmetry breaking initiates with left-right asymmetric cilia-driven fluid mechanics in a small fluid-filled structure called the ventral node in mice. However the question of what converts this flow into left-right asymmetric development remains unanswered. A leading hypotheses is that flow transports morphogen containing vesicles within the node, the absorption of which results in …
Acoustically actuated sessile bubbles can be used as a tool to manipulate microparticles, vesicles and cells. In this work, using acoustically actuated sessile semi-cylindrical microbubbles, we demonstrate experimentally that finite-sized microparticles undergo size-sensitive migration and trapping towards specific spatial positions in three dimensions with high reproducibility. The particle trajectories are successfully reproduced by passive advection of the particles in a steady three-dimensional streaming flow field augmented with volume exclusion from the confining boundaries. For differen…
The dynamics of microcapsules in steady shear flow was studied using a theoretical approach based on three variables: The Taylor deformation parameter $α_{\rm D}$, the inclination angle $θ$, and the phase angle $ϕ$ of the membrane rotation. It is found that the dynamic phase diagram shows a remarkable change with an increase in the ratio of the membrane shear and bending elasticities. A fluid vesicle (no shear elasticity) exhibits three dynamic modes: (i) Tank-treading (TT) at low viscosity $η_{\rm {in}}$ of internal fluid ($α_{\rm D}$ and $θ$ relaxes to constant values), (ii) Tumbling (TB) at…
Deformable bodies in a 3-dimensional viscous flow: Vorticity-Stream vector formulation
Center for Interdisciplinary Research in Biology (CIRB), Collège de France, CNRS, Inserm, Université PSL（跨学科生物研究中心，法兰西学院，法国国家科学研究中心，法国国家健康与医学研究院，巴黎文理研究大学）；Facultat de Física, Universitat de Barcelona（巴塞罗那大学物理学院）；Instituto de Investigación Tecnológica (IIT) and Grupo Interdisciplinar de Sistemas Complejos (GISC), Universidad Pontificia Comillas（技术研究所和复杂系统跨学科小组，科米利亚斯宗座大学）；Institute of Nanoscience and Nanotechnology (IN2UB)（纳米科学与技术研究所）
