Shapes and terminal velocities of a drop rising in stagnant liquids

• A single drop rising in stagnant liquids is modeled using a front tracking method.
• The effects of four parameters on the dynamics of the drop are studied systematically.
• Qualitative relations between the four parameters and drop dynamics are identified.
• Quantitative relations between terminal velocity and λ, Nf, and Eo are determined.

The effects of the density ratio η, the viscosity ratio λ, the inverse viscosity number Nf, and the Eötvös number Eo on the shapes and terminal velocities of a drop rising in stagnant liquids are studied numerically using a front tracking method. It is found that mild variation of η barely changes the drop shape and terminal velocity while keeping the other non-dimensional parameters fixed, whereas λ, Nf and Eo significantly affect the drop dynamics. With the increase of λ, Nf or Eo, the drop becomes either more dimpled or more crescent. The flow details inside and around the drop are also investigated. A recirculation zone is observed for small λ or large Eo, and it becomes larger with the decrease of λ and increase of Eo. In addition, the qualitative and quantitative relationships between Fr, the non-dimensional terminal velocity of the drop, and λ, Nf and Eo are proposed based on the simulation results.