Wave propagation, diffusive relaxation, and fragmentation instabilities in flow-driven drop chains under quasi-2D confinement*

Drop chains can spontaneously emerge in confined emulsion flows and are frequently present in microfluidic systems. Here we discuss the dependence of the drop-chain dynamics on flow symmetries. For an antisymmetric incident flow (Poiseuille flow) the leading-order interparticle hydrodynamic interactions occur via Hele--Shaw flow dipoles. The vector-like symmetry of these interactions results in density wave propagation in the direction of the dipole orientation. For a symmetric incident flow (Couette flow), the leading-order interparticle interactions are quadrupolar. Due to the fore--aft symmetry of Hele--Shaw quadrupoles, the macroscopic chain dynamics is diffusive, because there in no first-order spatial derivative in the evolution equation. The sign of the diffusion constant depends on the balance between the quadrupolar interactions and near-field hydrodynamic repulsion (the repulsion stems from the swapping-trajectory effect and drop deformation). Density perturbations decay for a positive diffusion constant; for a negative value they grow, eventually leading to the decomposition of a chain into stable chain fragments with a uniform drop spacing.