Theoretical analysis for flattening of a rising bubble in a Hele-Shaw cell

TL;DR

This paper provides a theoretical analysis of the shape and velocity of a rising bubble in a Hele-Shaw cell, incorporating boundary conditions that account for three-dimensional effects, and confirms the flattening behavior observed experimentally.

Contribution

It introduces a variational formulation to predict bubble shape and velocity, considering 3D effects and elliptic shape assumptions, advancing understanding of bubble dynamics in Hele-Shaw cells.

Findings

Bubble flattens as it rises in the Hele-Shaw cell.

Theoretical predictions agree with experimental observations for large cells.

Shape and velocity depend on bubble size, gap distance, and inclination angle.

Abstract

We calculate the shape and the velocity of a bubble rising in an infinitely large and closed Hele-Shaw cell using Park and Homsy's boundary condition which accounts for the change of the three dimensional structure in the perimeter zone. We first formulate the problem in the form of a variational problem, and discuss the shape change assuming that the bubble takes elliptic shape. We calculate the shape and the velocity of the bubble as a function of the bubble size, gap distance and the inclination angle of the cell. We show that the bubble is flattened as it rises. This result is in agreement with experiments for large Hele-Shaw cells.