Taylor drop in a closed vertical pipe

In this work, we study the ascent dynamics of a liquid Taylor drop formed from a lock-exchange configuration in a closed vertical pipe. We focus on the buoyancy-driven motion of an elongated drop surrounded by a denser fluid when viscous forces dominate over inertial and surface tension effects. While gaseous Taylor bubbles have been studied extensively, a liquid Taylor drop moving in a closed pipe is less well understood. We formulate an analytical model for estimating the ascent speed and drop thickness from first principles. First, we use a lubrication approximation to solve for the velocity profiles in the two fluids. Then, we analyse the mechanical energy balance of the whole system, including the effect of viscous dissipation, to understand how the ascent speed and drop thickness scale with the viscosity ratio. We show that a drop with density ratio reaches a stationary state with a uniform dimensionless thickness of in the absence of dissipation and in the dissipative regime. Through a comparison with existing experimental data, we demonstrate that our model correctly predicts the ascent speed of a Taylor drop if the material properties of the fluids and the geometry of the conduit are known. Our theoretical framework can be generalized to an isolated Taylor drop rising in a vertical pipe.