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Three-dimensional (3D) numerical simulations are performed on suspensions composed of puller-like microswimmers that are sensitive to light (phototaxis) mimicking microalgae in a Poiseuille flow. Simulations are based on the numerical resolution of the flow equations at low Reynolds numbers discretized on a 3D grid (finite differences). The model reproduces the formation of a central jet of swimmers by self-focusing [Phys. Rev. Lett. 110, 138106 (2013)] but also predicts an instability of the jet, which leads to its fractionation in clusters. We show that this instability is due to hydrodynamic interactions between microswimmers, which attract each other along the flow direction. This effect was not observed in the experiments conducted on dilute suspensions (i.e., where hydrodynamic interactions are weak). This phenomenon is peculiar for pullers for which collective motions are usually not observed on such a time scale. With this modeling, we hope to pave the way toward a better understanding of concentration techniques of algae (a bottleneck challenge in industrial applications).

Many processes in eukaryotic cells, including cell motility, rely on the growth of branched actin networks from surfaces. Despite its central role the mechanochemical coupling mechanisms that guide the growth process are poorly understood, and a general continuum description combining growth and mechanics is lacking. We develop a theory that bridges the gap between mesoscale and continuum limit and propose a general framework providing the evolution law of actin networks growing under stress. This formulation opens an area for the systematic study of actin dynamics in arbitrary geometries. Our framework predicts a morphological instability of actin growth on a rigid sphere, leading to a spontaneous polarization of the network with a mode selection corresponding to a comet, as reported experimentally. We show that the mechanics of the contact between the network and the surface plays a crucial role, in that it determines directly the existence of the instability. We extract scaling laws relating growth dynamics and network properties offering basic perspectives for new experiments on growing actin networks.

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 shapes) to chaotic dynamics of shape. Chaos occurs through a cascade of multiple periodic oscillations of the vesicle shape. We summarize our results in a phase diagram in the parameter plane (degree of confinement and flow strength). This finding highlights the level of complexity of a flowing vesicle in the small Reynolds number where the flow is laminar in the absence of vesicles and can be rendered turbulent due to elasticity of vesicles.

Universality has been a key concept for the classification of equilibrium critical phenomena, allowing associations among different physical processes and models. When dealing with nonequilibrium problems, however, the distinction in universality classes is not as clear and few are the examples, such as phase separation and kinetic roughening, for which universality has allowed to classify results in a general spirit. Here we focus on an out-of-equilibrium case, unstable crystal growth, lying in between phase ordering and pattern formation. We consider a well-established 2+1-dimensional family of continuum nonlinear equations for the local height h(x,t) of a crystal surface having the general form ∂_{t}h(x,t)=-∇·[j(∇h)+∇(∇^{2}h)]: j(∇h) is an arbitrary function, which is linear for small ∇h, and whose structure expresses instabilities which lead to the formation of pyramidlike structures of planar size L and height H. Our task is the choice and calculation of the quantities that can operate as critical exponents, together with the discussion of what is relevant or not to the definition of our universality class. These aims are achieved by means of a perturbative, multiscale analysis of our model, leading to phase diffusion equations whose diffusion coefficients encapsulate all relevant information on dynamics. We identify two critical exponents: (i) the coarsening exponent, n, controlling the increase in time of the typical size of the pattern, L∼t^{n}; (ii) the exponent β, controlling the increase in time of the typical slope of the pattern, M∼t^{β}, where M≈H/L. Our study reveals that there are only two different universality classes, according to the presence (n=1/3, β=0) or the absence (n=1/4, β>0) of faceting. The symmetry of the pattern, as well as the symmetry of the surface mass current j(∇h) and its precise functional form, is irrelevant. Our analysis seems to support the idea that also space dimensionality is irrelevant.

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.

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.

Many nonlinear partial differential equations (PDEs) display a coarsening dynamics, i.e., an emerging pattern whose typical length scale L increases with time. The so-called coarsening exponent n characterizes the time dependence of the scale of the pattern, L(t)≈t(n), and coarsening dynamics can be described by a diffusion equation for the phase of the pattern. By means of a multiscale analysis we are able to find the analytical expression of such diffusion equations. Here, we propose a recipe to implement numerically the determination of D(λ), the phase diffusion coefficient, as a function of the wavelength λ of the base steady state u(0)(x). D carries all information about coarsening dynamics and, through the relation |D(L)|=/~L(2)/t, it allows us to determine the coarsening exponent. The main conceptual message is that the coarsening exponent is determined without solving a time-dependent equation, but only by inspecting the periodic steady-state solutions. This provides a much faster strategy than a orward time-dependent calculation. We discuss our method for several different PDEs, both conserved and not conserved.

Cross-linked semiflexible polymer networks are omnipresent in living cells. Typical examples are actin networks in the cytoplasm of eukaryotic cells, which play an essential role in cell motility, and the spectrin network, a key element in maintaining the integrity of erythrocytes in the blood circulatory system. We introduce a simple mechanical network model at the length scale of the typical mesh size and derive a continuous constitutive law relating the stress to deformation. The continuous constitutive law is found to be generically nonlinear even if the microscopic law at the scale of the mesh size is linear. The nonlinear bulk mechanical properties are in good agreement with the experimental data for semiflexible polymer networks, i.e., the network stiffens and exhibits a negative normal stress in response to a volume-conserving shear deformation, whereby the normal stress is of the same order as the shear stress. Furthermore, it shows a strain localization behavior in response to an uniaxial compression. Within the same model we find a hierarchy of constitutive laws depending on the degree of nonlinearities retained in the final equation. The presented theory provides a basis for the continuum description of polymer networks such as actin or spectrin in complex geometries and it can be easily coupled to growth problems, as they occur, for example, in modeling actin-driven motility.

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.

Starting from a classical Kröener-Rieder kinematic picture for plasticity, we derive a set of dynamical equations describing plastic flow in a Lagrangian formulation. Our derivation is a natural and straightforward extension of simple fluids, elastic, and viscous solids theories. These equations contain the Maxwell model as a special limit. This paper is inspired by the particularly important work of Langer and coworkers. We shall show that our equations bear some resemblance with the shear-transformation zones model developed by Langer and coworkers. We shall point out some important differences. We discuss some results of plasticity, which can be described by the present model. We exploit the model equations for the simple examples: straining of a slab and a rod. We find that necking manifests always itself (not as a result of instability), except if the very special constant-velocity stretching process is imposed.

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.

Dynamics of a single vesicle under shear flow between two parallel plates is studied in two-dimensions using 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 reproduced. Further, we focus on investigating the effect of the confinement on the dynamics, a question that has received little attention so far. In particular, we study how the vesicle steady inclination angle in the tank-treading regime depends on the degree of confinement. The influence of the confinement on the effective viscosity of the composite fluid is also analyzed. At a given reduced volume (the swelling degree) of a vesicle we find that both the inclination angle, and the membrane tank-treading velocity decrease with increasing confinement. At sufficiently large degree of confinement the tank-treading velocity exhibits a nonmonotonous dependence on the reduced volume and the effective viscosity shows a nonlinear behavior.

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.

Vesicles are becoming a quite popular model for the study of red blood cells. This is a free boundary problem which is rather difficult to handle theoretically. Quantitative computational approaches constitute also a challenge. In addition, with numerical studies, it is not easy to scan within a reasonable time the whole parameter space. Therefore, having quantitative analytical results is an essential advance that provides deeper understanding of observed features and can be used to accompany and possibly guide further numerical development. In this paper, shape evolution equations for a vesicle in a shear flow are derived analytically with precision being cubic (which is quadratic in previous theories) with regard to the deformation of the vesicle relative to a spherical shape. The phase diagram distinguishing regions of parameters where different types of motion (tank treading, tumbling, and vacillating breathing) are manifested is presented. This theory reveals unsuspected features: including higher order terms and harmonics (even if they are not directly excited by the shear flow) is necessary, whatever the shape is close to a sphere. Not only does this theory cure a quite large quantitative discrepancy between previous theories and recent experiments and numerical studies, but also it reveals a phenomenon: the VB mode band in parameter space, which is believed to saturate after a moderate shear rate, exhibits a striking widening beyond a critical shear rate. The widening results from excitation of fourth-order harmonic. The obtained phase diagram is in a remarkably good agreement with recent three-dimensional numerical simulations based on the boundary integral formulation. Comparison of our results with experiments is systematically made.

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: (i) tank-treading (TT): the vesicle assumes a steady inclination angle with respect to the flow direction, while its membrane undergoes a tank-treading motion, (ii) tumbling (TB), and (iii) vacillating-breathing (VB): 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: (i) the viscosity ratio between the internal and external fluid, lambda , (ii) the excess area relative to a sphere (this parameter measures the degree of the vesicle deflation), Delta , and (iii) 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 rate), Ca. Recent studies [Danker, Phys. Rev. E 76, 041905 (2007)] have focused on the shape of the phase diagram (representing the TT, TB, and VB regimes in the Ca-lambda plane). 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 VB) in parameter space depend on the three dimensionless control parameters, in contrast with a recent study [V. V. Lebedev, Phys. Rev. Lett. 99, 218101 (2007)] where it is claimed that only two parameters are relevant. Consideration of the amplitude of oscillation (of the vesicle orientation angle and its shape deformation) 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 (Ca), 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 (2007)].

Instabilities and pattern formation is the rule in nonequilibrium systems. Selection of a persistent length scale or coarsening (increase in the length scale with time) are the two major alternatives. When and under which conditions one dynamics prevails over the other is a long-standing problem, particularly beyond one dimension. It is shown that the challenge can be defied in two dimensions, using the concept of phase diffusion equation. We find that coarsening is related to the lambda dependence of a suitable phase diffusion coefficient, D11(lambda) , depending on lattice symmetry and conservation laws. These results are exemplified analytically on prototypical nonlinear equations.

Vesicles under shear flow exhibit various dynamics: tank treading (TT), tumbling (TB), and vacillating breathing (VB). The VB mode consists in a motion where the long axis of the vesicle oscillates about the flow direction, while the shape undergoes a breathing dynamics. We extend here the original small deformation theory [C. Misbah, Phys. Rev. Lett. 96, 028104 (2006)] to the next order in a consistent manner. The consistent higher order theory reveals a direct bifurcation from TT to TB if Ca identical with taugamma is small enough-typically below 0.5, but this value is sensitive to the available excess area from a sphere (tau=vesicle relaxation time towards equilibrium shape, gamma=shear rate). At larger Ca the TB is preceded by the VB mode. For Ca1 we recover the leading order original calculation, where the VB mode coexists with TB. The consistent calculation reveals several quantitative discrepancies with recent works, and points to new features. We briefly analyze rheology and find that the effective viscosity exhibits a minimum in the vicinity of the TT-TB and TT-VB bifurcation points. At small Ca the minimum corresponds to a cusp singularity and is at the TT-TB threshold, while at high enough Ca the cusp is smeared out, and is located in the vicinity of the VB mode but in the TT regime.

We study the effect of a higher-order nonlinearity in the standard Kuramoto-Sivashinsky equation: partial differentialxG(Hx). We find that the stability of steady states depends on dv/dq , the derivative of the interface velocity on the wave vector q of the steady state. If the standard nonlinearity vanishes, coarsening is possible, in principle, only if G is an odd function of Hx. In this case, the equation falls in the category of the generalized Cahn-Hilliard equation, whose dynamical behavior was recently studied by the same authors. Alternatively, if G is an even function of Hx, we show that steady-state solutions are not permissible.

We develop a general criterion about coarsening for some classes of nonlinear evolution equations describing one-dimensional pattern-forming systems. This criterion allows one to discriminate between the situation where a coarsening process takes place and the one where the wavelength is fixed in the course of time. An intermediate scenario may occur, namely "interrupted coarsening." The power of the criterion on which a brief account has been given [Politi and Misbah, Phys. Rev. Lett. 92, 090601 (2004)], and which we extend here to more general equations, lies in the fact that the statement about the occurrence of coarsening, or selection of a length scale, can be made by only inspecting the behavior of the branch of steady state periodic solutions. The criterion states that coarsening occurs if lambda'(A)>0 while a length scale selection prevails if lambda'(A)<0, where lambda is the wavelength of the pattern and A is the amplitude of the profile (prime refers to differentiation). This criterion is established thanks to the analysis of the phase diffusion equation of the pattern. We connect the phase diffusion coefficient D(lambda) (which carries a kinetic information) to lambda'(A), which refers to a pure steady state property. The relationship between kinetics and the behavior of the branch of steady state solutions is established fully analytically. Another important and new result which emerges here is that the exploitation of the phase diffusion coefficient enables us to determine in a rather straightforward manner the dynamical coarsening exponent. Our calculation, based on the idea that |D(lambda)| approximately lambda2/t, is exemplified on several nonlinear equations, showing that the exact exponent is captured. We are not aware of another method that so systematically provides the coarsening exponent. Contrary to many situations where the one-dimensional character has proven essential for the derivation of the coarsening exponent, this idea can be used, in principle, at any dimension. Some speculations about the extension of the present results are outlined.

We extend our recent phase-field [T. Biben and C. Misbah, Phys. Rev. E 67, 031908 (2003)] approach to 3D vesicle dynamics. Unlike the boundary-integral formulations, based on the use of the Oseen tensor in the small Reynolds number limit, this method offers several important flexibilities. First, there is no need to track the membrane position; rather this is automatically encoded in dynamics of the phase field to which we assign a finite width representing the membrane extent. Secondly, this method allows naturally for any topology change, like vesicle budding, for example. Thirdly, any non-Newtonian constitutive law, that is generically nonlinear, can be naturally accounted for, a fact which is precluded by the boundary integral formulation. The phase-field approach raises, however, a complication due to the local membrane incompressibility, which, unlike usual interfacial problems, imposes a nontrivial constraint on the membrane. This problem is solved by introducing dynamics of a tension field. The first purpose of this paper is to show how to write adequately the advected-field model for 3D vesicles. We shall then perform a singular expansion of the phase field equation to show that they reduce, in the limit of a vanishing membrane extent, to the sharp boundary equations. Then, we present some results obtained by the phase-field model. We consider two examples; (i) kinetics towards equilibrium shapes and (ii) tanktreading and tumbling. We find a very good agreement between the two methods. We also discuss briefly how effects, such as the membrane shear elasticity and stretching elasticity, and the relative sliding of monolayers, can be accounted for in the phase-field approach.

We present a study of the aggregation of interacting particles in one dimension. This situation, for example, applies to atoms trapped along linear defects at the surface of a crystal. Simulations are performed with two lattice models. In the first model, the borders of atoms and islands interact in a vectorial manner via force monopoles. In the second model, each atom carries a dipole. These two models lead to qualitatively similar but quantitatively different behaviors. In both cases, the final average island size S(f) does not depend on the interactions in the limits of very low and very high coverages. For intermediate coverages, S(f) exhibits an asymmetric behavior as a function of the interaction strength: while it saturates for attractive interactions, it decreases for repulsive interactions. A class of mean-field models is designed, which allows one to retrieve the interaction dependence on the coverage dependence of the average island size with a good accuracy.

We derive amplitude equations for interface dynamics in pattern forming systems with long-range interactions. The basic condition for the applicability of the method developed here is that the bulk equations are linear and solvable by integral transforms. We arrive at the interface equation via long-wave asymptotics. As an example, we treat the Grinfeld instability, and we also give a result for the Saffman-Taylor instability. It turns out that the long-range interaction survives the long-wave limit and shows up in the final equation as a nonlocal and nonlinear term, a feature that to our knowledge is not shared by any other known long-wave equation. The form of this particular equation will then allow us to draw conclusions regarding the universal dynamics of systems in which nonlocal effects persist at the level of the amplitude description.