Cross-stream migration and coalescence of droplets in a microchannel co-flow using magnetophoresis

Manipulation of aqueous droplets in microchannels has great significance in various emerging applications such as biological and chemical assays. Magnetic-field based droplet manipulation that offers unique advantages is consequently gaining attention. However, the physics of magnetic field-driven cross-stream migration and the coalescence of aqueous droplets with an aqueous stream are not well understood. Here, we unravel the mechanism of cross-stream migration and the coalescence of aqueous droplets flowing in an oil based ferrofluid with a coflowing aqueous stream in the presence of a magnetic field. Our study reveals that the migration phenomenon is governed by the advection (τ_a) and magnetophoretic (τ_m) time scales. Experimental data show that the dimensionless equilibrium cross-stream migration distance δ^* and the length $Lδ*$ required to attain equilibrium cross-stream migration depend on the Strouhal number, St = (τ_a/τ_m), as δ^* = 1.1 St^0.33 and $Lδ*=5.3St−0.50$⁠, respectively. We find that the droplet-stream coalescence phenomenon is underpinned by the ratio of the sum of magnetophoretic (τ_m) and film-drainage time scales (τ_fd) and the advection time scale (τ_a), expressed in terms of the Strouhal number (St) and the film-drainage Reynolds number (Re_fd) as ξ = (τ_m + τ_fd)/τ_a = (St⁻¹ + Re_fd). Irrespective of the flow rates of the coflowing streams, droplet size, and magnetic field, our study shows that droplet-stream coalescence is achieved for ξ ≤ 50 and ferrofluid stream width ratio w^* < 0.7. We utilize the phenomenon and demonstrated the extraction of microparticles and HeLa cells from aqueous droplets to an aqueous stream.