Effect of viscosity ratio on the dynamic response of droplet deformation under a steady electric field

The effect of the fluid viscosity ratio on the transient deformation of a droplet is investigated. A numerical model is developed by employing the phase field method to capture the interface. The model is validated in both steady and transient cases with literature data with good agreement. In the creeping flow regime, the droplet always undergoes monotonic deformation. When the viscosity of the suspending fluid dominates, the transient process of the droplet deformation is nearly independent of the viscosity ratio. When the viscosities of the droplet and suspending fluid are comparable, the damping effect of the droplet viscosity on the deformation is magnified and the time to reach the steady-state deformation increases with viscosity. When the effect of suspending fluid inertia prevails, the droplet will deform to the steady state either in a monotonic way or in an oscillating way depending on the viscosity ratio. A quasi-steady mode, which can be considered as an intermediate mode between the oscillating and the steady mode, is identified for the first time. When the droplet is in the quasi-steady mode, the increase in the electric capillary number can turn it into the steady mode. The flow field evolution is analyzed and it shows that the vortices inside the droplet play an important role in the transient deformation. The deformation process can be determined by the competition between the inner and outer vortices. It is found that the minimum deformation time can be obtained for the quasi-steady mode when the viscosity of the suspending fluid is low.