Dynamic modes of quasispherical vesicles: Exact analytical solutions

Abstract

In this paper we introduce a simple mathematical analysis to reexamine vesicle dynamics in the quasispherical limit (small deformation) under a shear flow. In this context, a recent paper [Misbah, Phys. Rev. Lett. 96, 028104 (2006)] revealed a dynamic referred to as the vacillating-breathing (VB) mode where the vesicle main axis oscillates about the flow direction and the shape undergoes a breathinglike motion, as well as the tank-treading and tumbling (TB) regimes. Our goal here is to identify these three modes by obtaining explicit analytical expressions of the vesicle inclination angle and the shape deformation. In particular, the VB regime is put in evidence and the transition dynamics is discussed. Not surprisingly, our finding confirms the Keller-Skalak solutions (for rigid particles) and shows that the VB and TB modes coexist, and whether one prevails over the other depends on the initial conditions. An interesting additional element in the discussion is the prediction of the TB and VB modes as functions of a control parameter Γ, which can be identified as a TB-VB parameter.

Membrane compression in tumbling and vacillating-breathing regimes for quasispherical vesicles

“…[4] and recall the main result reported in Ref. [15] which will serve for the present study. The vesicle membrane in a quasispherical geometry is parametrized in the usual way by spherical harmonics Y m l [2,5,13,29] …”

“…Very recently, the exact KS solution (15) has been successfully extended to the general case where the assumption of a shape-preserving motion is relaxed [15]. For the purpose of the present paper, we summarize the results reported in Ref.…”

“…Interestingly, the solvability of system (13) only requires simple analytical algebra [15]. The main idea is to introduce new coordinates in terms of which system (13) has a particularly simple form.…”

“…[1][2][3][4][5][6][7][8][9][10][11][12][13][14][15], and the references therein). The emerging view is that vesicles exhibit a variety of different regimes of motion depending on three control parameters: (i) the excess area = (A − 4πr 2 0 )/r 2 0 , or reduced volume ν defined as ν = 6 √ πV /A 3/2 [ = 4π (ν −2/3 − 1)]; (ii) the viscosity contrast λ = η int /η ext ; and (iii) the bending number (or dimensionless shear rate) C κ = η extγ r 3 0 /κ, where A is the vesicle area, r 0 = (3V /4π ) 1/3 , V being the volume of the vesicle, η int and η ext are the viscosities of the internal and the external fluids, respectively, and κ is the bending rigidity modulus.…”

“…Section II deals with a brief description of the small deformation theory [4], followed by a short description of the dynamics of the vesicle that result from the exact analytical solutions reported in Ref. [15]. Sections III and IV contain the main contribution of the present paper.…”