Analytical progress in the theory of vesicles under linear flow

Lift at low Reynolds number

Lift forces are widespread in hydrodynamics. These are typically observed for big and fast objects and are often associated with a combination of fluid inertia (i.e. large Reynolds numbers) and specific symmetry-breaking mechanisms. In contrast, the properties of viscosity-dominated (i.e. low Reynolds numbers) flows make it more difficult for such lift forces to emerge. However, the inclusion of boundary effects qualitatively changes this picture. Indeed, in the context of soft and biological matter, recent studies have revealed the emergence of novel lift forces generated by boundary softness, flow gradients and/or surface charges. The aim of the present review is to gather and analyse this corpus of literature, in order to identify and unify the questioning within the associated communities, and pave the way towards future research.

Electrohydrodynamics of Vesicles and Capsules

Giant unilamellar vesicles (GUVs) made up of phospholipid bilayer membranes (liposomes) and elastic capsules with a cross-linked, polymerized membrane, have emerged as biomimetic alternatives to investigating biological cells such as leukocytes and erythrocytes. This feature article looks at the similarities and differences in the electrohydrodynamics (EHD) of vesicles and capsules under electric fields that determines their electromechanical response. The physics of EHD is illustrated through several examples such as the electrodeformation of single and compound, spherical and cylindrical, and charged and uncharged vesicles in uniform and nonuniform electric fields, and the relevance and challenges are discussed. Both small and large deformation results are discussed. The use of EHD in understanding complex interfacial kinetics in capsules and the synthesis of nonspherical capsules using electric fields are also presented. Finally, the review looks at the large electrodeformation of water-in-water capsules and the relevance of constitutive laws in their response.

A theoretical study on the dynamics of a compound vesicle in shear flow

The dynamics of nucleate cells in shear flow is of great relevance in cancer cells and circulatory tumor cells where they determine the flow properties of blood. Buoyed by the success of giant unilamellar vesicles in explaining the dynamics of anucleate cells such as red blood cells, compound vesicles have been suggested as a simple model for nucleate cells. A compound vesicle consists of two concentric unilamellar vesicles with the inner, annular and outer regions filled with aqueous Newtonian solvents. In this work, a theoretical model is presented to study the deformation and dynamics of a compound vesicle in linear shear flow using small deformation theory and spherical harmonics with higher order approximation to the membrane forces. A coupling of viscous and membrane stresses at the membrane interface of the two vesicles results in highly nonlinear shape evolution equations for the inner and the outer vesicles which are solved numerically. The results indicate that the size of the inner vesicle (χ) does not affect the tank-treading dynamics of the outer vesicle. The inner vesicle admits a greater inclination angle than the outer vesicle. However, the transition to trembling/swinging and tumbling is significantly affected. The inner and outer vesicles exhibit identical dynamics in the parameter space defined by the nondimensional rotational (Λ_an) and extensional (S) strength of the general shear flow. At moderate χ, a swinging mode is observed for the inner vesicle while the outer vesicle exhibits tumbling. The inner vesicle also exhibits modification of the TU mode to IUS (intermediate tumbling swinging) mode. Moreover, synchronization of the two vesicles at higher χ and a Capillary number sensitive motion at lower χ is observed in the tumbling regime. These results are in accordance with the few experimental observations reported by Levant and Steinberg. A reduction in the inclination angle is observed with an increase in χ when the inner vesicle is replaced by a solid inclusion. Additionally, a very elaborate phase diagram is presented in the Λ_an-S parameter space, which could be tested in future experiments or numerical simulations.

Effect of ac electric field on the dynamics of a vesicle under shear flow in the small deformation regime

Vesicles or biological cells under simultaneous shear and electric field can be encountered in dielectrophoretic devices or designs used for continuous flow electrofusion or electroporation. In this work, the dynamics of a vesicle subjected to simultaneous shear and uniform alternating current (ac) electric field is investigated in the small deformation limit. The coupled equations for vesicle orientation and shape evolution are derived theoretically, and the resulting nonlinear equations are handled numerically to generate relevant phase diagrams that demonstrate the effect of electrical parameters on the different dynamical regimes such as tank treading (TT), vacillating breathing (VB) [called trembling (TR) in this work], and tumbling (TU). It is found that while the electric Mason number (Mn), which represents the relative strength of the electrical forces to the shear forces, promotes the TT regime, the response itself is found to be sensitive to the applied frequency as well as the conductivity ratio. While higher outer conductivity promotes orientation along the flow axis, orientation along the electric field is favored when the inner conductivity is higher. Similarly a switch of orientation from the direction of the electric field to the direction of flow is possible by a mere change of frequency when the outer conductivity is higher. Interestingly, in some cases, a coupling between electric field-induced deformation and shear can result in the system admitting an intermediate TU regime while attaining the TT regime at high Mn. The results could enable designing better dielectrophoretic devices wherein the residence time as well as the dynamical states of the vesicular suspension can be controlled as per the application.

Prediction of anomalous blood viscosity in confined shear flow

Red blood cells play a major role in body metabolism by supplying oxygen from the microvasculature to different organs and tissues. Understanding blood flow properties in microcirculation is an essential step towards elucidating fundamental and practical issues. Numerical simulations of a blood model under a confined linear shear flow reveal that confinement markedly modifies the properties of blood flow. A nontrivial spatiotemporal organization of blood elements is shown to trigger hitherto unrevealed flow properties regarding the viscosity η, namely ample oscillations of its normalized value [η] = (η-η(0))/(η(0)ϕ) as a function of hematocrit ϕ (η(0) = solvent viscosity). A scaling law for the viscosity as a function of hematocrit and confinement is proposed. This finding can contribute to the conception of new strategies to efficiently detect blood disorders, via in vitro diagnosis based on confined blood rheology. It also constitutes a contribution for a fundamental understanding of rheology of confined complex fluids.

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.

Symmetry breaking and cross-streamline migration of three-dimensional vesicles in an axial Poiseuille flow

We analyze numerically the problem of spontaneous symmetry breaking and migration of a three-dimensional vesicle [a model for red blood cells (RBCs)] in axisymmetric Poiseuille flow. We explore the three relevant dimensionless parameters: (i) capillary number, Ca, measuring the ratio between the flow strength over the membrane bending mode, (ii) the ratio of viscosities of internal and external liquids, λ, and (iii) the reduced volume, ν=[V/(4/3)π]/(A/4π)3/2 (A and V are the area and volume of the vesicle). The overall picture turns out to be quite complex. We find that the parachute shape undergoes spontaneous symmetry-breaking bifurcations into a croissant shape and then into slipper shape. Regarding migration, we find complex scenarios depending on parameters: The vesicles either migrate towards the center, or migrate indefinitely away from it, or stop at some intermediate position. We also find coexisting solutions, in which the migration is inwards or outwards depending on the initial position. The revealed complexity can be exploited in the problem of cell sorting out and can help understanding the evolution of RBCs' in vivo circulation.

Membrane compression in tumbling and vacillating-breathing regimes for quasispherical vesicles

We derive some analytical results of a well-known model for quasispherical vesicles in a linear shear flow at low deformability. Attention is focussed on the oscillatory regimes: the tumbling (TB) mode, vacillating-breathing (VB) mode, and the transition from vacillating-breathing to tumbling, depending on a control parameter Γ. It is shown that, during the VB-to-TB transition (Γ=1), the vesicle momentarily attains its maximal extension in the vorticity direction and transits through a circular profile in the shear plane for which the radius is exactly determined. In addition, we provide an explicit analytical expression for the effective membrane tension for different types of motions. We find a critical bending number below which the membrane undergoes compression at each instant and show that, during the VB-to-TB transition, a fourth-order membrane deformation is possible.

Rheology of a vesicle suspension with finite concentration: a numerical study

Vesicles, closed membranes made of a bilayer of phospholipids, are considered as a biomimetic system for the mechanics of red blood cells. The understanding of their dynamics under flow and their rheology is expected to help the understanding of the behavior of blood flow. We conduct numerical simulations of a suspension of vesicles in two dimensions at a finite concentration in a shear flow imposed by countertranslating rigid bounding walls by using an appropriate Green's function. We study the dynamics of vesicles, their spatial configurations, and their rheology, namely, the effective viscosity η(eff). A key parameter is the viscosity contrast λ (the ratio between the viscosity of the encapsulated fluid over that of the suspending fluid). For small enough λ, vesicles are known to exhibit tank treading (TT), while at higher λ they exhibit tumbling (TB). We find that η(eff) decreases in the TT regime, passes a minimum at a critical λ=λ(c), and increases in the TB regime. This result confirms previous theoretical and numerical works performed in the extremely dilute regime, pointing to the robustness of the picture even in the presence of hydrodynamic interactions. Our results agree also with very recent numerical simulations performed in three dimensions both in the dilute and more concentrated regime. This points to the fact that dimensionality does not alter the qualitative features of η(eff). However, they disagree with recent simulations in two dimensions. We provide arguments about the possible sources of this disagreement.

Noisy nonlinear dynamics of vesicles in flow

We present a model for the dynamics of fluid vesicles in linear flow which consistently includes thermal fluctuations and nonlinear coupling between different modes. At the transition between tank treading and tumbling, we predict a trembling motion which is at odds with the known deterministic motions and for which thermal noise is strongly amplified. In particular, highly asymmetric shapes are observed even though the deterministic flow only allows for axisymmetric ones. Our results explain quantitatively recent experimental observations [Levant and Steinberg, Phys. Rev. Lett. 109, 268103 (2012)].

Analytical and numerical study of three main migration laws for vesicles under flow

Blood flow shows nontrivial spatiotemporal organization of the suspended entities under the action of a complex cross-streamline migration, that renders understanding of blood circulation and blood processing in lab-on-chip technologies a challenging issue. Cross-streamline migration has three main sources: (i) hydrodynamic lift force due to walls, (ii) gradients of the shear rate (as in Poiseuille flow), and (iii) hydrodynamic interactions among cells. We derive analytically these three laws of migration for a vesicle (a model for an erythrocyte) showing good agreement with numerical simulations and experiments. In an unbounded Poiseuille flow, the situation turns out to be quite complex. We predict that a vesicle may migrate either towards the center or away from it, or even show both behaviors for the same parameters, depending on initial position. This finding can both help understanding healthy and pathological erythrocyte behavior in blood circulation and be exploited in biotechnologies for cell sorting out.

Continuum- and particle-based modeling of shapes and dynamics of red blood cells in health and disease

We review recent advances in multiscale modeling of the mechanics of healthy and diseased red blood cells (RBCs), and blood flow in the microcirculation. We cover the traditional continuum-based methods but also particle-based methods used to model both the RBCs and the blood plasma. We highlight examples of successful simulations of blood flow including malaria and sickle cell anemia.

Amplification of thermal noise by vesicle dynamics

A novel noise amplification mechanism resulting from the interaction of thermal fluctuations and nonlinear vesicle dynamics is reported. It is observed in a time-dependent vesicle state called trembling (TR). High spatial resolution and very long time series of TR compared to the vesicle period allow us to quantitatively analyze the generation and amplification of spatial and temporal modes of the vesicle shape perturbations. During a compression part of each TR cycle, a vesicle finds itself on the edge of the wrinkling instability, where thermally excited spatial modes are amplified.

Squaring, parity breaking, and S tumbling of vesicles under shear flow

The numerical study of 3D vesicles with a reduced volume equal to that of human red blood cells leads to the discovery of three types of dynamics: (i) squaring motion, in which the angle between the direction of the longest distance and the flow velocity undergoes discontinuous jumps over time, (ii) spontaneous parity breaking of the shape leading to cross-streamline migration, and (iii) S tumbling where the vesicle tumbles, exhibiting a pronounced S-like shape with a waisted morphology in the center. We report on the phase diagram within a wide range of relevant parameters. Our estimates reveal that healthy and pathological red blood cells are also prone to these types of motion, which may affect blood microcirculation and impact oxygen transport.

Dynamic modes of quasispherical vesicles: exact analytical solutions

In this paper we introduce a simple mathematical analysis to reexamine vesicle dynamics in the quasispherical limit (small deformation) under a shear flow. In this context, a recent paper [Misbah, Phys. Rev. Lett. 96, 028104 (2006)] revealed a dynamic referred to as the vacillating-breathing (VB) mode where the vesicle main axis oscillates about the flow direction and the shape undergoes a breathinglike motion, as well as the tank-treading and tumbling (TB) regimes. Our goal here is to identify these three modes by obtaining explicit analytical expressions of the vesicle inclination angle and the shape deformation. In particular, the VB regime is put in evidence and the transition dynamics is discussed. Not surprisingly, our finding confirms the Keller-Skalak solutions (for rigid particles) and shows that the VB and TB modes coexist, and whether one prevails over the other depends on the initial conditions. An interesting additional element in the discussion is the prediction of the TB and VB modes as functions of a control parameter Γ, which can be identified as a TB-VB parameter.

Effect of thermal noise on vesicles and capsules in shear flow

We add thermal noise consistently to reduced models of undeformable vesicles and capsules in shear flow and derive analytically the corresponding stochastic equations of motion. We calculate the steady-state probability distribution function and construct the corresponding phase diagrams for the different dynamical regimes. For fluid vesicles, we predict that at small shear rates thermal fluctuations induce a tumbling motion for any viscosity contrast. For elastic capsules, due to thermal mixing, an intermittent regime appears in regions where deterministic models predict only pure tank treading or tumbling.

Hydrodynamic interaction between two vesicles in a linear shear flow: asymptotic study

Interactions between two vesicles in an imposed linear shear flow are studied theoretically, in the limit of almost spherical vesicles, with a large intervesicle distance, in a strong flow, with a large inner to outer viscosity ratio. This allows to derive a system of ordinary equations describing the dynamics of the two vesicles. We provide an analytic expression for the interaction law. We find that when the vesicles are in the same shear plane, the hydrodynamic interaction leads to a repulsion. When they are not, the interaction may turn into attraction instead. The interaction law is discussed and analyzed as a function of relevant parameters.

Vesicle dynamics under weak flows: application to large excess area

Dynamics of a vesicle under simple shear flow is studied in the limit of small capillary number. A perturbative approach is used to derive the equation of vesicle dynamics. The expansions are shown to converge for significantly deflated vesicles (with excess area from the sphere as high as 2). In particular, we provide an explicit analytical expression for the tank-treading to tumbling bifurcation point. This expression is valid for excess areas up to 2.5. The results are compared with full 3D numerical simulations. The proposed method can be used for analytical or numerical solution of vesicle dynamics under weak flow of general form.

Three-dimensional numerical simulation of vesicle dynamics using a front-tracking method

Three-dimensional numerical simulation using the front-tracking method is presented on the dynamics of a vesicle in a linear shear flow. The focus here is to elucidate the parametric dependence and the self-similarity of the vesicle dynamics, quantification of vesicle deformation, and the analysis of shape dynamics. A detailed comparison of the numerical results is made with various theoretical models and experiments. It is found that the applicability of the theoretical models is limited despite some general agreement with the simulations and experiments. The deviations between the perturbative results and the simulation results occur even in the absence of thermal noise. Specifically, we find that the vesicle dynamics does not follow a self-similar behavior in a two-parameter phase space, as proposed in a theoretical model. Rather, the dynamics is governed by three controlling parameters, namely, the excess area, viscosity ratio, and dimensionless shear rate. Additionally, we find that a linear scaling of the tank-treading angle, as proposed in the theoretical model, is possible only for nearly spherical vesicles. The breakdown of the scaling occurs at higher values of the excess area even in the absence of thermal noise. We find that the vesicle deformation saturates at large shear rates, and the asymptotic deformation matches well with a theoretical prediction for nearly spherical vesicles. The dependence of the critical viscosity ratio associated with the onset of unsteady dynamics on the vesicle excess area is in excellent agreement with the experimental observation. We show that near the transition between the tank-treading and tumbling dynamics, both the vacillating-breathing-like motion characterized by a smooth ellipsoidal shape and the trembling-like motion characterized by a highly deformed shape are possible. For the trembling-like motion, the shape is highly three-dimensional with concavities and lobes, and the vesicle deforms more in the vorticity direction than in the shear plane. A Fourier spectral analysis of the vesicle shape shows the presence of the odd harmonics and higher order modes beyond fourth order.

Symmetry breaking of vesicle shapes in Poiseuille flow

Vesicle behavior under unbounded axial Poiseuille flow is studied analytically. Our study reveals subtle features of the dynamics. It is established that there exists a stable off-centerline steady-state solution for low enough flow strength. This solution appears as a symmetry-breaking bifurcation upon lowering the flow strength and includes slipper shapes, which are characteristic of red blood cells in the microvasculature. A stable axisymmetric solution exists for any flow strength provided the excess area is small enough. It is shown that the mechanism of the symmetry breaking depends on the geometry of the flow: The bifurcation is subcritical in axial Poiseuille flow and supercritical in planar flow.

Three-dimensional vesicles under shear flow: numerical study of dynamics and phase diagram

The study of vesicles under flow, a model system for red blood cells (RBCs), is an essential step in understanding various intricate dynamics exhibited by RBCs in vivo and in vitro. Quantitative three-dimensional analyses of vesicles under flow are presented. The regions of parameters to produce tumbling (TB), tank-treating, vacillating-breathing (VB), and even kayaking (or spinning) modes are determined. New qualitative features are found: (i) a significant widening of the VB mode region in parameter space upon increasing shear rate γ and (ii) a robustness of normalized period of TB and VB with γ. Analytical support is also provided. We make a comparison with existing experimental results. In particular, we find that the phase diagram of the various dynamics depends on three dimensionless control parameters, while a recent experimental work reported that only two are sufficient.