We investigate theoretically the buoyancy-driven motion of a viscous drop in a yield-stress material, incorporating elastic effects represented by the Saramito–Herschel–Bulkley constitutive equation. We solve the governing equations using an open-source finite volume solver and utilizing the volume of fluid technique to accurately capture the interface between the two fluids. To validate our numerical approach, we compare our results with data from previous experimental and numerical studies. We find quantitative agreement in terms of terminal velocities and drop shapes, affirming the accuracy of our model and its numerical solution. Notably, we observe that incorporating elastic effects into the modeling of the continuous phase is essential for predicting phenomena reported in experiments, such as the inversion of the flow field behind the sedimenting drop (i.e., the negative wake) or the formation of a teardrop shape. Due to the elastoviscoplastic nature of the continuous phase, we observe that small drops remain entrapped because the buoyancy force is insufficient to fluidize the surrounding material. We investigate entrapment conditions using two different protocols, which yield different outcomes due to the interplay between capillarity and elastoplasticity. Finally, we conduct an extensive parametric analysis to evaluate the impact of rheological parameters (yield stress, elastic modulus, and interfacial tension) on the dynamics of sedimentation.
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