Transition to Tumbling andTwo Regimes of Tumbling Motion of a Vesicle in Shear Flow Vasiliy Kantsler and Victor Steinberg Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot, 76100 Israel (Received 4 October 2005; published 24 January 2006) Experimental results on the tank-treading-tumbling transition in the dynamics of a vesicle subjected to a shear flow as a function of a vesicle excess area, viscosity contrast, and the normalized shear rate are presented. Good agreement on the transition curve and scaling behavior with theory and numerical simulations was found. A new type of unsteady motion at a large degree of vesicle deformability was discovered and described as follows: a vesicle trembles around the flow direction, while the vesicle shape strongly oscillates. DOI: 10.1103/PhysRevLett.96.036001 PACS numbers: 83.50.v, 87.16.Dg, 87.17.Jj The dynamics of individual vesicles under nonequilib- rium conditions has received increasing attention in recent years in theory [1– 3], numerical simulations [4 – 6], as well as in experiments [7–10]. This interest is motivated, first, by the strong resemblance in dynamical behavior of real biological cells, and second, by a possibility to understand rheological properties of vesicle solution flows. A vesicle is a liquid droplet bounded by a closed phos- pholipid bilayer membrane (usually unilamellar) sus- pended in a fluid that can be either the same solvent as an inner one or different. Both the volume and the surface area of the vesicle are conserved. The former means that the vesicle membrane is considered to be impermeable, at least on the time scale of the experiment, and the latter means that the membrane dilatation is neglected [1–3]. It is known that vesicles in a shear flow reveal two types of motion, tank-treading and tumbling, and a range of existence of each of them depends on two main control parameters: the excess area (dimensionless), S=R 2 4, and the viscosity contrast, in = out . Here R is the effective vesicle radius related to its volume via V 4 3 R 3 , S is the vesicle surface area that exceeds that of a sphere with the same volume by , in and out are the dynamic viscosities of inner and outer fluids, respectively. At sufficiently low < c , a steady mean vesicle ori- entation angle with respect to the shear flow direction and a tank-treading membrane motion are found [1,2,8,10], whereas at > c according to theoretical predictions [1,3], a transition to a tumbling motion should occur, when a vesicle axis rotates with respect to the flow direction. This transition is of fundamental importance, since it should alter rheological properties of a vesicle solution by reducing dissipation [3,6,11]. In this Letter we report first experimental studies of the tank-treading-tumbling transition in a shear flow as a func- tion of the excess area and the viscosity contrast. We also present results on two different types of tumbling motion characterized by different dimensionless shear rate, _ out R 3 =, that determines the degree of vesicle deform- ability under the shear rate, _ . Here is the bending rigidity of the membrane. There were numerous attempts to theoretically describe both types of the vesicle motion and the transition between them, particularly in regards to the dynamics of red blood cells in a shear flow. Keller and Skalak [1], by making a bold assumption about a fixed ellipsoidal shape of a vesicle with a moving membrane and by neglecting thermal fluc- tuations, have suggested theory that provides amazingly accurate description of the vesicle dynamics in both re- gimes of motion and particularly of the transition to tum- bling. They derived a generic evolution equation for the vesicle inclination angle, , in a form [1]: _ 1 d dt A B; cos2; (1) where A 1=2 and B; is a function defined by a model. However, one gets two regimes and scaling in the transition region without going into details of a specific model. The steady solution for exists and is defined by cos2A=B only if jA=Bj < 1. Otherwise, no steady state solution exists, and a vesicle tumbles in a shear flow [1,3]. The value jA=Bj 1 corresponds to 0, the transition value to tumbling. For a rigid body, as found long time ago in Ref. [12], one has jA=Bj > 1; i.e., tum- bling occurs always without a threshold. In a general case, when a tank-treading motion is taken into account, the ratio jA=Bj is a function of and , and it can be larger or smaller than unity. According to Ref. [1], the tumbling occurs at c , where c is determined by jA=Bj 1. At < c and close to the transition the steady solution shows scaling / c p [1,5], where 0 is the threshold value. At > c the transition via a saddle- node bifurcation to tumbling occurs [1,3,5]. It is remark- able that both the more intricate theory [5] that considers a vesicle as a deformable object, which shape is not given a priori but is obtained as a result of an interplay between flow, bending rigidity, and the physical constraints, and the numerical simulations [6] based on the deformable vesicle model show rather close agreement with Ref. [1]. Very recently further refinement of the Keller-Skalak theory that considers also morphological changes, was suggested [11]. This phenomenological model consists of two coupled PRL 96, 036001 (2006) PHYSICAL REVIEW LETTERS week ending 27 JANUARY 2006 0031-9007= 06=96(3)=036001(4)$23.00 036001-1 2006 The American Physical Society