Motion of a two-dimensional elastic capsule in a branching channel flow

Abstract

The transit of a two-dimensional elastic fluid-filled capsule through a channel with a side branch is investigated numerically. The mathematical formulation allows for a capsule carried in a pressure-driven flow of fluid of generally different viscosity to that inside the capsule. Far upstream and downstream in the main channel, and downstream in the side branch, the fluid velocity profiles are assumed to adopt those of unidirectional Poiseuille flow with prescribed flow rates. The capsule boundary is treated as a two-dimensional elastic membrane developing elastic tensions and bending moments according to simple constitutive laws. A boundary-integral formulation allows for the explicit computation of the fluid pressures upstream and downstream of the branching. The novelty of the approach is the inclusion of a notional boundary at the entrance to the side branch, which avoids the need to collocate the channel ends. The deformation experienced by the capsule in the region of the junction is found to depend strongly on the branch angle. The deformation is ameliorated by increasing the membrane stiffness or lowering the viscosity of the suspending fluid relative to the encapsulated fluid. When a capsule exits the branch region, a distance of many decades of capsule diameters is required before the capsule relaxes to an equilibrium shape. Capsule residence times in the vicinity of the branch region can be considerable, depending on the line of approach into the junction and the capsule deformability. The path selection of a cell at a branch junction can depend crucially on capsule deformability: capsules with different elastic properties may follow different routes out of the junction in otherwise identical flow conditions.