Solid particles moving parallel to a deformable liquid–liquid interface in a micro-channel: migration forces

Abstract

This work focuses on the dynamics of a train of solid particles, separated by a distance $L$ , flowing near a deformable interface formed by two co-flowing immiscible fluids in a microchannel of height $h$ . Our study includes a systematic analysis of the influence of the governing parameters (fluids viscosity ratio, interface and particle positions, Reynolds $Re$ and capillary $Ca$ numbers and the inter-particle distance $L$ ) on the hydrodynamic force $f$ exerted on the particle. In the pure inertial regime with non-deformable interfaces $Ca=0$ , the particle is driven towards the wall (interface) when the particle is close to the interface (wall). Up to three neutral equilibrium positions $f=0$ , two of them stable, are found in this limit. The contrary is obtained in the pure capillary regime $Re=0$ . In this limit, we also carried out an asymptotic analysis in the distinguished limits of very large and very small surface tension. In the latter case, the amplitude of the interface deformation induced by the particle is large, comparable to its diameter, but its influence is limited to a small region upstream and downstream of the particle. In the limit of very large surface tension, the amplitude of the interface deformation is small but the presence of the particle modifies the shape of the interface in a region of length $2\lambda$ , much larger than the particle diameter $d$ . The parameter $\lambda$ , introduces an additional characteristic length that determines the asymptotic behaviour of the flow properties in the limit of large surface tension.