Dynamics of a viscous vesicle in linear flows

Abstract

An analytical theory is developed to describe the dynamics of a closed lipid bilayer membrane (vesicle) freely suspended in a general linear flow. Considering a nearly spherical shape, the solution to the creeping-flow equations is obtained as a regular perturbation expansion in the excess area. The analysis takes into account the membrane fluidity, incompressibility and resistance to bending. The constraint for a fixed total area leads to a non-linear shape evolution equation at leading order. As a result two regimes of vesicle behavior, tank-treading and tumbling, are predicted depending on the viscosity contrast between interior and exterior fluid. Below a critical viscosity contrast, which depends on the excess area, the vesicle deforms into a tank-treading ellipsoid, whose orientation angle with respect to the flow direction is independent of the membrane bending rigidity. In the tumbling regime, the vesicle exhibits periodic shape deformations with a frequency that increases with the viscosity contrast. Non-Newtonian rheology such as normal stresses is predicted for a dilute suspension of vesicles. The theory is in good agreement with published experimental data for vesicle behavior in simple shear flow.

Hydrodynamic mechanisms of cell and particle trapping in microfluidics

“…Here, C L denotes the dimensionless lift coefficient which has been formulated in various formats, for instance in terms of reduced volume , 116 as a function of viscosity ratio k and ellipsoid geometrical measures, 125 or in terms of excess area D and k. 119 Assuming flow with negligible inertia, the corresponding lift force can be calculated using Stokes law as F L ¼ 6plRv L . In these models, it is presumed that the lift force always acts in the opposite direction of the wall to push the particle towards the centerline.…”

“…and (ii) excess area 118,119 which denotes the area difference of the particle and a sphere with equivalent volume,…”

Dynamics of Nearly Spherical Vesicles in an External Flow

“…An account of the third order term changes the phase diagram completely since the second order term becomes vanishingly small (due to the surface tension adjustment) in the vicinity of the tumbling to tanktreading transition region predicted in Refs. [19,20]. An importance of the third-order term is discussed also by Noguchi and Gompper [15].…”

“…al. [20] are the inclusion of membrane viscosity and third order expansion term of Helfrich energy. An account of the third order term changes the phase diagram completely since the second order term becomes vanishingly small (due to the surface tension adjustment) in the vicinity of the tumbling to tanktreading transition region predicted in Refs.…”

“…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 focused on quasi-spherical vesicles whose shape can be parameterized by spherical harmonics expansion. A perturbation scheme around the Lamb solution for spherical particle in external flow allows one to derive the dynamic equations for the shape and the orientation of a vesicle and to investigate them analytically.…”