Rectilinear magnetophoresic oscillation of oil droplet dispersed in immiscible paramagnetic carrier phase

Unbounded single oil droplet in paramagnetic rare-earth salt solution is studied superposing a magnetic field. The resulting Bond and Morton number is located in a parametric region where the droplet moves rectilinearly when the motion is driven by gravity along. When released from an initially quiescent carrier phase, the dominating buoyancy, counter acting the hydrodynamic forces, drives the droplet moving upward until it reaches the near field of a permanent magnetic assembly. At a critical position, the Kelvin force supersedes buoyancy and droplet starts to oscillate periodically. A hydrodynamically quiescent end state is expected after the momentum is fully damped and the droplet settles at an equilibrium position where the Kelvin force and buoyancy is equivalent. The damping kinematics allows us to quantify the history term. We find, surprisingly, the hydrodynamic force where the memory kernel decays faster, instead of t ^-1/2, following an e^-t with a pre-factor scaled by external viscous dissipation time scale and hydrodynamic length scale. Furthermore, a refined Kelvin force formulation is proposed to improve the current volumeless Lagrangian force model. The latter is routinely used for magnetophoresis modelling where an accurate prediction of the motion of the dispersed phase in magnetic field is of primary importance.