We investigate the dynamics of viscoplastic droplets under the combined action of electric field and shear flow by performing direct numerical simulations. The electro-hydrodynamic equations are solved in a two-dimensional finite volume framework, and the interface is captured using a volume-of-fluid approach. The rheology of the viscoplastic droplet is modeled as a Bingham plastic fluid. Both the drop and the surrounding medium are considered to be perfect dielectric fluids. The simulations reveal that in the sole presence of the shear flow, the plasticity of the fluid plays a pivotal role in deciding the magnitude of droplet deformation and orientation. The local viscosity inside the drop is significantly augmented for higher plasticity of the fluid. Under the action of the electric field, the droplet deformation and orientation can be suitably tuned by varying the magnitude of the permittivity contrast between the fluids. The droplets experience enhanced deformation and preferred orientation against the flow direction when the permittivity ratio is greater than unity. Increasing the droplet plasticity leads to reduction in the droplet deformation. Conversely, by increasing the electric field strength, the deformation of the droplets can be notably enhanced, with a stronger response observed for a permittivity ratio beyond unity. Finally, it is observed that by suitably manipulating the strength of the shear flow and the electric field, droplet breakup can be engendered. The mode of droplet disintegration differs due to variation of the parameters, which can be attributed to the competing influence of shear and electric forces on the droplet.
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