Differences in Path Instabilities Between a Bubble Rising in Water and in Aqueous Polymer Solution in a Hele-Shaw Cell in the Transient and Steady States

Abstract

The buoyancy-driven path instabilities of an air bubble rising in a Hele-Shaw cell filled with pure water are examined as a function of the Eötvös number (Eo) in terms of changes in the shape, trajectory, and peripheral length (L) of the bubble in the transient and steady states. The path instabilities in an aqueous solution of hydroxypropyl methyl cellulose (HPMC) are also studied as a function of the adsorption time of HPMC at a fixed Eo. The differences between the path instabilities in the transient state and those in the steady state are discussed. The path instabilities of a bubble in water can be summarized as follows: (1) below Eo = 5, an oblate ellipse bubble is accompanied by a vibrational trajectory with no changes in L in the steady state, while in the transient state, the bubble shape changes from a more elongated oblate ellipse to a less elongated one, and the bubble's rise velocity in the latter state is greater than in the former one; (2) above Eo = 5, a comma-shaped bubble is accompanied by a vibrational motion of both the path and L in the steady state, while in the transient state, an increase in the Eo produces a straight trajectory with a decrease in the amplitude of the vibration of L, and the rise velocity of a bubble in the transient state is less than it is in the steady state. The adsorption of HPMC on the surface of a bubble causes a more rounded bubble, a suppression of changes in the bubble's shape, and a decrease in its rise velocity.

Keywords:

• Bubble
• Eötvös number
• Hele-Shaw cell
• hydroxypropyl methyl cellulose (HPMC) solution
• path instabilities
• transient state
• steady state

Acknowledgments

This work was supported in part by both the Takahashi Industrial and Economic Research Foundation and a Grant-in-Aid for Scientific Research on Priority Area “Soft Matter Physics” (2006-2010) from the Ministry of Education, Culture, Sports, and Technology of Japan.