# Film thickness distribution in gravity-driven pancake-shaped droplets rising in a Hele-Shaw cell

## TL;DR

This study investigates the shape, velocity, and film thickness distribution of a buoyant droplet rising in a Hele-Shaw cell through experiments, simulations, and lubrication theory, revealing key insights into the film behavior and droplet shape.

## Contribution

It provides a comprehensive analysis of droplet dynamics and film thickness in a Hele-Shaw cell, combining experimental, numerical, and theoretical approaches, and introduces a model for the film distribution.

## Key findings

- Droplet velocity is unaffected by film thickness in the studied regime.
- Experimental and numerical film thickness distributions agree well for iso-viscous cases.
- The film thickness follows the Aussillous & Quéré model with fitted parameters.

## Abstract

We study here experimentally, numerically and using a lubrication approach; the shape, velocity and lubrication film thickness distribution of a droplet rising in a vertical Hele-Shaw cell. The droplet is surrounded by a stationary immiscible fluid and moves purely due to buoyancy. A low density difference between the two mediums helps to operate in a regime with capillary number $Ca$ lying between $0.03-0.35$, where $Ca=\mu_o U_d /\gamma$ is built with the surrounding oil viscosity $\mu_o$, the droplet velocity $U_d$ and surface tension $\gamma$. The experimental data shows that in this regime the droplet velocity is not influenced by the thickness of the thin lubricating film and the dynamic meniscus. For iso-viscous cases, experimental and three-dimensional numerical results of the film thickness distribution agree well with each other. The mean film thickness is well captured by the Aussillous & Qu\'er\'e (2000) model with fitting parameters. The droplet also exhibits the ''catamaran'' shape that has been identified experimentally for a pressure-driven counterpart (Huerre $\textit{et al}$. 2015). This pattern has been rationalized using a two-dimensional lubrication equation. In particular, we show that this peculiar film thickness distribution is intrinsically related to the anisotropy of the fluxes induced by the droplet's motion.