Lateral migration of a two-dimensional vesicle in unbounded Poiseuille flow

Abstract

The migration of a suspended vesicle in an unbounded Poiseuille flow is investigated numerically in the low Reynolds number limit. We consider the situation without viscosity contrast between the interior of the vesicle and the exterior. Using the boundary integral method we solve the corresponding hydrodynamic flow equations and track explicitly the vesicle dynamics in two dimensions. We find that the interplay between the nonlinear character of the Poiseuille flow and the vesicle deformation causes a cross-streamline migration of vesicles towards the center of the Poiseuille flow. This is in a marked contrast with a result [L.G. Leal, Ann. Rev. Fluid Mech. 12, 435 (1980)] according to which the droplet moves away from the center (provided there is no viscosity contrast between the internal and the external fluids). The migration velocity is found to increase with the local capillary number (defined by the time scale of the vesicle relaxation towards its equilibrium shape times the local shear rate), but reaches a plateau above a certain value of the capillary number. This plateau value increases with the curvature of the parabolic flow profile. We present scaling laws for the migration velocity.

Noninertial lateral migration of vesicles in bounded Poiseuille flow

“…We showed that the effects of the walls and of the curvature of the velocity field are coupled in a nonlinear manner: Curvature not only induces migration 27 but also affects the shape and orientation, which affects the lift force. The law v m ϳ ␥ ͑y͒ / y markedly differs from what the naive extrapolation of the results for a vesicle near a wall and in a linear shear flow would give v m ϳ ␥ ͑y͒ / y 2 .…”

“…[22][23][24][25] In agreement with the numerical and theoretical studies for simple shear flows, 22,24 we showed recently that the migration velocity decreases like 1 / y 2 , where y is the distance to the wall of the center of mass of the vesicle. 26 ͑ii͒ The nonconstant shear rate in a parabolic velocity profile ͑even unbounded͒ leads to a subtle interplay between the gradient of shear and the shape, 27 resulting in migration toward the center with a constant drift velocity except near the centerline. In a realistic channel, both effects coexist and we shall see that this leads to a new and nontrivial noninertial migration law.…”

Two-dimensional vesicle dynamics under shear flow: Effect of confinement

“…This has been achieved by using ad hoc membrane forces that penalize any deviation from the equilibrium configuration. In the present paper we use the precise analytical expression of the local membrane force as it has been derived [15] from the known Helfrich bending energy [16]. The perimeter conservation in our case is achieved by using a field of local Lagrangian multiplicators (equivalent to an effective tension).…”

“…Unlike RBCs, for vesicles we can vary their intrinsic characteristic parameters (e.g., size, degree of deflation, and nature of internal fluid). Despite the simplicity of their structure, vesicles have exhibited many features observed for red blood cells: equilibrium shapes [21], tank-treading motion [3,22], lateral migration [15,23,24], or slipperlike shapes [25,26]. Capsules (a model system incorporating shear elasticity) have also revealed some common features with vesicles [27,28].…”

Numerical simulations of the behaviour of a drop in a square pipe flow using the two-phase lattice Boltzmann method

“…The kinematic viscosity n and the surface tension s are given by 10) with x being the coordinate normal to the interface.…”