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.

Soft bodies flowing in a channel often exhibit parachutelike 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.

Many eukaryotic cells undergo frequent shape changes (described as amoeboid motion) that enable them to move forward. We investigate the effect of confinement on a minimal model of amoeboid swimmer. A complex picture emerges: (i) The swimmer's nature (i.e., either pusher or puller) can be modified by confinement, thus suggesting that this is not an intrinsic property of the swimmer. This swimming nature transition stems from intricate internal degrees of freedom of membrane deformation. (ii) The swimming speed might increase with increasing confinement before decreasing again for stronger confinements. (iii) A straight amoeoboid swimmer's trajectory in the channel can become unstable, and ample lateral excursions of the swimmer prevail. This happens for both pusher- and puller-type swimmers. For weak confinement, these excursions are symmetric, while they become asymmetric at stronger confinement, whereby the swimmer is located closer to one of the two walls. In this study, we combine numerical and theoretical analyses.

Microorganisms, such as bacteria, algae, or spermatozoa, are able to propel themselves forward thanks to flagella or cilia activity. By contrast, other organisms employ pronounced changes of the membrane shape to achieve propulsion, a prototypical example being the Eutreptiella gymnastica. Cells of the immune system as well as dictyostelium amoebas, traditionally believed to crawl on a substratum, can also swim in a similar way. We develop a model for these organisms: the swimmer is mimicked by a closed incompressible membrane with force density distribution (with zero total force and torque). It is shown that fast propulsion can be achieved with adequate shape adaptations. This swimming is found to consist of an entangled pusher-puller state. The autopropulsion distance over one cycle is a universal linear function of a simple geometrical dimensionless quantity A/V(2/3) (V and A are the cell volume and its membrane area). This study captures the peculiar motion of Eutreptiella gymnastica with simple force distribution.

Vesicles are locally-inextensible fluid membranes while inextensible capsules are in addition endowed with in-plane shear elasticity mimicking the cytoskeleton of red blood cells (RBCs). Boundary integral (BI) methods based on the Green's function techniques are used to describe their dynamics, that falls into the category of highly nonlinear and nonlocal dynamics. Numerical solutions raise several obstacles and challenges that strongly impact the results. Of particular complexity is (i) the membrane inextensibility, (ii) the mesh stability and (iii) numerical precisions for evaluation of the boundary integral equations. Despite intense research these questions are still a matter of debate.We regularize the single layer integral by subtraction of exact identities for the terms involving the normal and the tangential components of the force. In addition, the regularized kernel remains explicitly self-adjoint. The stability and precision of BI calculation is enhanced by taking advantage of additional quadrature nodes located in vertices of an auxiliary mesh, constructed by a standard refinement procedure from the main mesh. We extend the partition of unity technique to boundary integral calculation on triangular meshes: We split the calculation of the boundary integral between the original and the auxiliary mesh using a smooth weight function, which takes the distance between the source and the target as the argument and falls to zero beyond a certain cut-off distance. We provide an efficient lookup algorithm that allows us to discard most of the vertices of the auxiliary mesh lying beyond the cut-off distance from a given point without actually calculating the distances to them. The proposed algorithm offers the same treatment of near-singular integration regardless if the source and the target points belong to the same surface or not.Additional innovations are used to increase the stability and precision of the method: The bending forces are calculated by differential geometry expressions using local coordinates defined in vicinity of each vertex. The approximation of the surface in vicinity of a vertex is obtained by fitting with a second-degree polynomial of local coordinates.We solve for the Lagrange multiplier associated with membrane incompressibility using two penalization parameters per suspended entity: one for deviation of the global area from prescribed value and another for the sum of squares of local strains defined on each vertex. The proposed advancement is to vary the penalization parameters at each time step in such a way, that the global area of each membrane be conserved and the sum of squares of local strains be at minimum. This optimization is achieved by solving a linear system of rank three times the number of entities involved in the simulation. If no auxiliary mesh is used, the method reduces to steepest descent method thanks to the explicit self-adjointness of the regularized single-layer kernel in the boundary integral equation.Inextensible capsules, a model of RBC, are studied ...

The self-propelled microswimmers have recently attracted considerable attention as model systems for biological cell migration as well as artificial micromachines. A simple and well-studied microswimmer model consists of three identical spherical beads joined by two springs in a linear fashion with active oscillatory forces being applied on the beads to generate self-propulsion. We have extended this linear microswimmer configuration to a triangular geometry where the three beads are connected by three identical springs in an equilateral triangular manner. The active forces acting on each spring can lead to autonomous steering motion; i.e., allowing the swimmer to move along arbitrary paths. We explore the microswimmer dynamics analytically and pinpoint its rich character depending on the nature of the active forces. The microswimmers can translate along a straight trajectory, rotate at a fixed location, as well as perform a simultaneous translation and rotation resulting in complex curved trajectories. The sinusoidal active forces on the three springs of the microswimmer contain naturally four operating parameters which are more than required for the steering motion. We identify the minimal operating parameters which are essential for the motion of the microswimmer along any given arbitrary trajectory. Therefore, along with providing insights into the mechanics of the complex motion of the natural and artificial microswimmers, the triangular three-bead microswimmer can be utilized as a model for targeted drug delivery systems and autonomous underwater vehicles where intricate trajectories are involved.

The dynamics of vesicles under shear flow are carefully analyzed in the regime of a small vesicle excess area relative to a sphere. This regime corresponds to the quasispherical limit, for which several groups have analytically extracted simple nonlinear differential equations. Under shear flow, vesicles are known to exhibit three types of motion: ͑i͒ tank-treading ͑TT͒: the vesicle assumes a steady inclination angle with respect to the flow direction, while its membrane undergoes a tank-treading motion, ͑ii͒ tumbling ͑TB͒, and ͑iii͒ vacillatingbreathing ͑VB͒: the vesicle main axis oscillates about the flow direction, whereas the overall shape undergoes a breathinglike motion. The region of existence for each regime depends on material and control parameters. The whole set of parameters can be cast into three dimensionless control parameters: ͑i͒ the viscosity ratio between the internal and external fluid, , ͑ii͒ the excess area relative to a sphere ͑this parameter measures the degree of the vesicle deflation͒, ⌬, and ͑iii͒ the capillary number ͑the ratio between the vesicle relaxation time toward its equilibrium shape after cessation of the flow and the flow time scale, which is the inverse shear rate͒, Ca. Recent studies ͓Danker et al., Phys. Rev. E 76, 041905 ͑2007͔͒ have focused on the shape of the phase diagram ͑representing the TT, TB, and VB regimes in the Ca-plane͒. In this paper, the physical quantities are analyzed in detail and attention is brought to features that are essential for future experimental studies. It is shown that the boundaries delimiting different dynamical regimes ͑TT, TB, and VB͒ in parameter space depend on the three dimensionless control parameters, in contrast with a recent study ͓V. V. Lebedev et al., Phys. Rev. Lett. 99, 218101 ͑2007͔͒ where it is claimed that only two parameters are relevant. Consideration of the amplitude of oscillation ͑of the vesicle orientation angle and its shape deformation͒ in the VB mode reveals an even more significant dependence on the three parameters. It is also shown that the inclination angle in the TT regime significantly depends on the shear rate ͑Ca͒, which runs contrary to common belief. Finally, we show that the TB and VB periods are quite insensitive to Ca, in marked contrast with a recent study ͓H. Noguchi and G. Gompper, Phys. Rev. Lett. 98, 128103 ͑2007͔͒.

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.