New analytical progress in the theory of vesicles under linear flow Alexander Farutin, 1 Thierry Biben, 2 and Chaouqi Misbah 1 $^{1}$Laboratoire de Spectrométrie Physique, UMR5588, 140 avenue de la physique, Université Joseph Fourier Grenoble, and CNRS, 38402 Saint Martin d’Hères, France 2 Université de Lyon, F-69000, France; Univ. Lyon 1, Laboratoire PMCN; CNRS, UMR 5586; F-69622 Villeurbanne Cedex Vesicles are becoming a quite popular model for the study of red blood cells (RBCs). 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 new 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. PACS numbers: 87.16.D- 83.50.Ha 87.17.Jj 83.80.Lz 87.19.rh strongly depend on its type of motion, it is important to define the phase diagram of a vesicle in a shear flow, predicting which type of motion will be realized for each set of parameters defining the vesicle dynamics. A vesicle is a closed simply connected membrane sepa- rating two liquids. Its dynamics in an ambient flow is an important research target due to various medical appli- cations. For instance, they are employed as biochemical reactors [1, 2], as vectors for targeted drug and gene de- livery [3, 4], and as artificial cells for hemoglobin encap- sulation and oxygen transport [5]. A widely accepted model for a vesicle proposes the fol- lowing assumptions: zero Reynolds number limit, mem- brane impermeability and local inextensibility, and the continuity of velocity field and mechanical stress across the membrane. The force exerted on the membrane by the liquids is balanced by the membrane rigidity force that is calculated from the energy of membrane bending taken as[6] One can also regard a vesicle, from mechanical point of view, as a simplified model of a living cell. The most prominent biological counterpart for a vesicle is the red blood cell (RBC). The vesicle system allows to iden- tify the elementary processes of visco-elastic properties of cells moving passively in a fluid. The study of a single vesicle under a shear flow is one of the simplest nonequi- librium example. Nevertheless, this problem gives rise to a very complicated dynamics with different types of vesi- cle motion because of the nontrivial competition between the interactions of straining and rotational parts of shear flow with the vesicle. Three main types of motion for a vesicle in a shear flow have been identified so far: (i) tank- treading (TT), when shape and orientation of the vesicle is a steady-state, (ii) tumbling (TB), when the vesicle makes full rotations, and (iii) vacillating-breathing (VB, aka trembling or swinging), when the longest axis of the vesicle oscillates around certain direction with the shape strongly changing during these oscillation cycles. Since many properties of a vesicle (e.g. the effective viscosity) where κ is the bending rigidity coefficient of the mem- brane, H is the mean curvature and Z is a Lagrange multiplier, ensuring the local membrane inextensibility. Under these assumptions, and after a certain choice of units for all variables, the vesicle dynamics can be com- pletely described by three dimensionless numbers (see for example [7]): the viscosity contrast E = κ 2 ∫ (2 H ) $^{2}$dA + ∫ ZdA, (1) λ = η$_{int}$ η$_{ext}$ , (2) Δ = Ar − 2 0 − 4 π, (3) the excess area relative to a sphere arXiv:1001.0159v2 [cond-mat.soft] 5 May 2010 I. INTRODUCTION