Lateral vesicle migration in a bounded shear flow: Viscosity contrast leads to off-centered solutions

The lateral migration of a suspended vesicle (a model of red blood cells) in a bounded shear flow is investigated numerically at vanishing Reynolds number (the Stokes limit) using a boundary integral method. We explore, among other parameters, the effect of the viscosity contrast lambda =eta(in)/eta(out), where eta(in), eta(out) denote the inner and the outer fluids' viscosities. It is found that a vesicle can either migrate to the center line or towards the wall depending on lambda. More precisely, below a critical viscosity contrast lambda(c), the terminal position is at the center line, whereas above lambda(c), the vesicle can be either centered or off-centered depending on initial conditions. It is found that the equilibrium lateral position of the vesicle exhibits a saddle-node bifurcation as a function of the bifurcation parameter lambda. When the shear stress increases the saddle-node bifurcation evolves towards a pitchfork bifurcation. A systematic analysis is first performed in two dimensions (due to numerical efficiency), and the overall picture is confirmed in three dimensions. This study can be exploited in the problem of cell sorting and can help understand the intricate nature of the dynamics and rheology of confined suspensions.