We analytically derive an equation describing vesicle evolution in a fluid where some stationary flow is excited regarding that the vesicle shape is close to a sphere. A character of the evolution is governed by two dimensionless parameters, S and Λ, depending on the vesicle excess area, viscosity contrast, membrane viscosity, strength of the flow, bending module, and ratio of the elongation and rotation components of the flow. We establish the "phase diagram" of the system on the S − Λ plane: we find curves corresponding to the tanktreading to tumbling transition (described by the saddle-node bifurcation) and to the tank-treading to trembling transition (described by the Hopf bifurcation). PACS numbers: 87.16.Dg, 87.17.Jj, 05.45.a Vesicles are closed membranes which separate two regions occupied by possibly different fluids. The vesicles are attracting significant attention not only due to their resemblance with biological objects but also because of their importance in different industries such as pharmaceutics where they are used for drug transportation. A natural problem which arises in these applications is understanding of how a single vesicle behaves in an external flow. This non-equilibrium problem has revealed a variety of new physical effects and became a subject of intense experimental and theoretical studies. Laboratory experiments [1,2,3,4,5] have shown that the vesicles immersed in a shear flow exhibit at least two qualitatively different types of behavior, either tank-treading or tumbling motion. In the tank-treading regime a vesicle shape is stationary, it is ellipsoid oriented at an angle with respect to the shear flow. In the tumbling regime the vesicle experiences periodic flipping in the shear plane. A novel type of behavior: trembling, discovered in the work [6] is an intermediate regime between tank-treading and tumbling in which a vesicle trembles around the flow direction.Constructing a phase diagram for all these regimes depending on the external parameters is a challenging and an extremely difficult task because the problem in consideration is both strongly non-linear and non-equilibrium. As long as no analytic solution of this problem exists theoretical studies were based either on numerical simulations or on some approximations allowing analytical treatment. Numerical investigations of this problem involved several different computational schemes, including boundary element method [7], mesoscopic particle-based approximation [8,9,10,11,12], and an advected field approach [13]. These approaches have shown qualitative agreement with experiments however did not solve the problem of constructing the vesicle dynamics phase diagram completely. Analytical studies of the problem can be divided in two major classes. In the first one [9,14,15], phenomenological models of a vesicle dynamics based on the classical work of Keller and Skallak [16] were proposed and proved themselves to be rather efficient in explaining the experiments. In the second series of works [17,18,19,20] the studies foc...

Tank-treading, tumbling and trembling are different types of the vesicle behavior in an external flow. We derive a dynamical equation for nearly spherical vesicles enabling to establish a phase diagram of the system predicting the regimes. The diagram is drawn in terms of two dimensionless parameters depending on the vesicle excess area, fluid viscosities, membrane viscosity and bending modulus, strength of the flow, and ratio of the elongational and rotational components of the flow. The tank-treading to tumbling transition occurs via a saddle-node bifurcation whereas the tanktreading to trembling transition occurs via a Hopf bifurcation. We establish a critical slowing near the merging point of the transition lines.

Traditional wave kinetics describes the slow evolution of systems with many degrees of freedom to equilibrium via numerous weak non-linear interactions and fails for very important class of dissipative (active) optical systems with cyclic gain and losses, such as lasers with non-linear intracavity dynamics. Here we introduce a conceptually new class of cyclic wave systems, characterized by non-uniform double-scale dynamics with strong periodic changes of the energy spectrum and slow evolution from cycle to cycle to a statistically steady state. Taking a practically important example-random fibre laser-we show that a model describing such a system is close to integrable non-linear Schrödinger equation and needs a new formalism of wave kinetics, developed here. We derive a non-linear kinetic theory of the laser spectrum, generalizing the seminal linear model of Schawlow and Townes. Experimental results agree with our theory. The work has implications for describing kinetics of cyclical systems beyond photonics.

Recent experiments by Kantsler et al. [Phys. Rev. Lett. 99, 178102 (2007)10.1103/PhysRevLett.99.178102] have shown that the relaxational dynamics of a vesicle in external elongation flow is accompanied by the formation of wrinkles on a membrane. Motivated by these experiments we present a theory describing the dynamics of a wrinkled membrane. The formation of wrinkles is related to the dynamical instability induced by negative surface tension of the membrane. For quasispherical vesicles we perform analytical study of the wrinkle structure dynamics. We derive the expression for the instability threshold and identify three stages of the dynamics. The scaling laws for the temporal evolution of wrinkling wavelength and surface tension are established, confirmed numerically, and compared to experimental results.