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

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.