Lateral migration of a ferrofluid droplet in a plane Poiseuille flow under uniform magnetic fields

The lateral migration of a two-dimensional (2D) viscous ferrofluid droplet in a plane Poiseuille flow under a uniform magnetic field is studied numerically by using the level set method. Focusing on low droplet Reynolds number flows (Re_{d}≤0.05), several numerical simulations are carried out to analyze the effects of magnetic field direction and strength, droplet size, and viscosity ratio on the lateral migration behavior of the droplet. The results indicate that the magnetic field direction plays a pivotal role in the trajectory of lateral migration of the droplet and the final equilibrium position in the channel. When the magnetic field is parallel to the channel, i.e., α=0^{∘} (the direction of magnetic field), the droplet is found to settle closer to the wall with an increase in magnetic Bond number Bo_{m}, while at α=45^{∘}, the droplet settles closer to the channel center. Varying the initial droplet sizes at a fixed magnetic Bond number Bo_{m} and viscosity ratio λ results in different final equilibrium positions within the channel. Additionally, the effect of different viscosity ratios on the migration behavior of the droplet is examined at variable magnetic Bond numbers Bo_{m}. At α=45^{∘}, a critical steady state of deformation is found for λ=0.5 and 1 where the droplet changes its migration direction and shifts toward the center of the channel, while at λ=0.05, the droplet crosses the center. At α=90^{∘}, the droplet is found to settle exactly at the center of the flow domain irrespective of different magnetic Bond numbers, droplet sizes, and viscosity ratios.