Volume 63, pp. 150-170, 2025.

A computational method for multiple steady Hele-Shaw bubbles in planar domains

Mohamed M. S. Nasser, Christopher C. Green, and El Mostafa Kalmoun

Abstract

We present a unified numerical method to determine the shapes of multiple Hele-Shaw bubbles in steady motion, and in the absence of surface tension, in three planar domains: free space, the upper half-plane, and an infinite channel. Our approach is based on solving the free boundary problem for the bubble boundaries using a fast and accurate boundary integral method. The main advantage of our method is that it allows for the treatment of a very high number of bubbles. The presented method is validated by recovering some existing results for steady bubbles in channels and free space. Several numerical examples are presented, many of which feature configurations of bubbles that have not appeared in the literature before.

Full Text (PDF) [9 MB], BibTeX , DOI: 10.1553/etna_vol63s150

Key words

Hele-Shaw flow, bubbles, free boundary problem, multiply connected domain, conformal mapping

AMS subject classifications

76D27, 65E10

Links to the cited ETNA articles

[16]	Vol. 44 (2015), pp. 189-229 Mohamed M.S. Nasser: Fast solution of boundary integral equations with the generalized Neumann kernel