THE SHAPE CONTROL OF A GROWING AIR BUBBLE IN A HELE-SHAW CELL

The external Hele-Shaw problem when an air bubble is surrounded by oil is considered. It is assumed that the bubble expands with negligible surface tension. Howison [P. Roy. Soc. Edinburgh Sect. A, 102 (1986), pp. 141-148] proved that the elliptical initial shape of the bubble is the only shape that grows for all times and whose boundary crosses all of the points initially outside the bubble in the case of a single point sink at infinity in the oil domain. In this paper we study a possibility of a similar dynamics for bubbles with other than elliptical boundaries by considering some line sink distribution in the oil domain, an external mother body. We show examples when it is possible and when it is impossible to grow bubbles that fill the whole plane in an infinite time.