Overshoot of ferrofluid droplet deformation in shear flow under magnetic field

A droplet with a viscosity ratio lambda>1 shows an overshoot phenomenon in start-up shear flow, where its maximum deformation exceeds the steady-state deformation. We numerically investigate the overshoot phenomenon of a ferrofluid droplet in shear flow at moderate and high viscosity ratios under a uniform magnetic field perpendicular to the shear plane. The overshoot deformation can lead to droplet breakup for a moderate viscosity ratio ( lambda is an element of[1,4]). However, imposing the magnetic field suppresses this overshoot, allowing the droplet to reach a steady state. Moreover, the droplet deforms into another steady state after turning off the magnetic field. The temporary magnetic field suppresses the droplet breakup due to overshoot at lambda>2.5 for the magnetic Bond number Bo=4.0. The overshoot deformation exhibits an oscillatory decay for a high viscosity ratio ( lambda similar to O(10)). The effects of a magnetic field perpendicular to the shear plane on the damping response of droplet deformation are examined. When the amplitude of droplet deformation is normalized using the first peak and steady-state value, the decay curves of the normalized peaks can be organized as a single curve by employing the capillary number with the cross-sectional radius in the (x,z)-plane, which is reduced by the magnetic field. The damping response of the high viscosity ferrofluid droplet under a magnetic field is primarily governed by the droplet viscosity and reduced cross-sectional radius.