(81b) The Steady Motion of a Train of Vesicles in a Cylindrical Channel of Arbitrary Cross Section

The three-dimensional, inertialess motion of a train of lipid-bilayer vesicles flowing through a cylindrical channel of variable cross section is simulated using the boundary integral equation method. Steady-state results are reported for vesicles positioned concentrically inside cylindrical channels of circular, square, and rectangular cross sections. The vesicle translational velocity U and excess channel pressure drop ∆p⁺ are found to depend strongly on the ratio of the vesicle radius to the hydraulic radius λ and the vesicle reduced volume Ï…. “Deflated vesicles” of lower reduced volume Ï… < 1 have more streamlined bodies and, therefore, translate with greater velocity U relative to the mean velocity V of the flow. Increasing the vesicle size (λ) increases the wall friction force and extra pressure drop ∆p⁺, which in turn reduces the vesicle velocity U. Hydrodynamic interactions between vesicles in a periodic train are largely screened by the channel walls, in accordance with previous results for spheres and drops. Changing the channel cross section has little effect on the steady vesicle shape but changes the mass flux through a cross section, viz. a vesicle passing through a square channel has a higher translational velocity than in a circular channel of equal hydraulic radius. A simple correction factor proportional to the ratio of mean velocities in channels of different cross sections is proposed to unify the results. Nonlinear effects are observed when β – the ratio of membrane bending elasticity to viscous traction – is changed. Increasing β tends to streamline the vesicle shape – preventing the formation of “parachute-like shapes” with regions of high membrane curvature – which, in turn, tends to increase U and decrease ∆p⁺. This effect is dampened – and, under certain conditions, reversed – as confinement is increased. The impact of the viscosity contrast κ between the interior and exterior fluid is also explored. The simulation results show excellent agreement with experimental measurements of vesicles in microfluidic channels and circular capillaries. In addition, the simulation results are commensurate with a previously reported “small-gap theory” valid for large values of λ.