A two-dimensional cusp at the trailing edge of an air bubble rising in a viscoelastic liquid

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Abstract
When an air bubble rises in a viscoelastic fluid there is a critical capillary number for cusping and jump in velocity: when the capillary number is below critical, which is about 1 in our data, there is no cusp at the tail of a (smooth) air bubble. For larger volumes, a two-dimensional cusp, sharp in one view and broad in the orthogonal view, is in evidence. Measurements suggest that the cusp tip is in the generic form y = ax2/3 satisfied by analytic cusps. The intervals of volumes for which dramatic changes in air bubble shape take place is very small and the two to ten fold increase in the rise velocity which accompanies the small change of volume could be modelled as a discontinuity. A second drag transition and an orientational transition occurred when U/c > 1 where U is the rise velocity of an air bubble and c is the shear wave speed. For U/c < 1, U is proportional to d2, where d is the equivalent diameter for a sphere of diameter d having the same volume, and when U/c > 1 then U is proportional to d and the Deborah number does not change with U. Moreover the bubble shapes when U/c < 1 are overall prolate (with or without a cusped tail) with the long side parallel to gravity, in contrast to the oblate shapes which are always observed in Newtonian fluids and in viscoelastic fluids with U/c > 1 when inertia is dominant. The formation of cusps occurs in all kinds of columns of different sizes and shapes. Cusping is generic but the orientation of the broad edge with respect to the sidewalls is an issue. There is no preferred orientation in columns with round cross-sections, or in the case of walls far away from the rising bubble. In columns with rectangular cross-sections, three relatively stable configurations can be observed: the cusp can be observed in the wide window and the broad edge in the narrow window; the cusp can be observed in the narrow window and the broad edge in the wide window or, less frequently, the broad edge lies along a diagonal. These orientational and drag alternatives are directly analogous to those which are observed in the settling of long or broad solid bodies (Liu & Joseph 1993).

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