Bubble interactions for the Mullins–Sekerka problem: Some case examples

The Mullins–Sekerka free boundary problem originates from the study of solidification and liquidation of materials where material is transported by diffusion. In the present paper we explore dynamics of bubbles for the Mullins–Sekerka problem. Using a set of ordinary differential equations for the radii and the centers, we numerically simulate the relevant interactions in both “two-dimensional” and “three-dimensional” settings. Our results illustrate how larger bubbles grow at the expense of smaller ones and highlight the role of additional factors such as the initial inter-bubble distance or weak asymmetries in the bubble position in the ensuing dynamics. One novel feature in comparison with earlier works is the possibility to continue for the three-dimensional case the simulation past the points where one of the bubbles disappears.