Self-similarity in particle accumulation on the advancing meniscus

When a mixture of viscous oil and non-colloidal particles displaces air between two parallel plates, the shear-induced migration of particles leads to the gradual accumulation of particles on the advancing oil-air interface. This particle accumulation results in the fingering of an otherwise stable fluid-fluid interface. While previous works have focused on the resultant instability, one unexplored yet striking feature of the experiments is the self-similarity in the concentration profile of the accumulating particles. In this paper, we rationalise this self-similar behaviour by deriving a depth-averaged particle transport equation based on the suspension balance model, following the theoretical framework of Ramachandran (J. Fluid Mech., vol. 734, 2013, pp. 219-252). The solutions to the particle transport equation are shown to be self-similar with slight deviations, and in excellent agreement with experimental observations. Our results demonstrate that the combination of the shear-induced migration, the advancing fluid-fluid interface and Taylor dispersion yield the self-similar and gradual accumulation of particles.