Optimal Control by Multipoles in the Hele-Shaw Problem

• Published in: Journal of mathematical fluid mechanics Vol. 17; no. 2; pp. 261 - 277
• Main Authors: Lokutsievskiy, Lev, Runge, Vincent
• Format: Journal Article
• Language: English
• Published: Basel Springer Basel 01.06.2015
• Subjects: Classical and Continuum Physics; Fluid- and Aerodynamics; Mathematical Methods in Physics; Physics; Physics and Astronomy; optimal control; Secondary 76D27; multipoles; controllability; Primary 49K15; Hele-Shaw flow
• ISSN: 1422-6928; 1422-6952
• DOI: 10.1007/s00021-015-0206-9

The two-dimensional Hele-Shaw problem for a fluid spot with free boundary can be solved using the Polubarinova–Galin equation. The main condition of its applicability is the smoothness of the spot boundary. In the sink-case, this problem is not well-posed and the boundary loses smoothness within finite time—the only exception being the disk centred on the sink. An extensive literature deals with the study of the Hele-Shaw problem with non-smooth boundary or with surface tension, but the problem remains open. In our work, we propose to study this flow from a control point of view, by introducing an analogue of multipoles (term taken from the theory of electromagnetic fields). This allows us to control the shape of the spot and to avoid non-smoothness phenomenon on its border. For any polynomial contours, we demonstrate how all the fluid can be extracted, while the border remains smooth until the very end. We find, in particular, sufficient conditions for controllability and a link between Richardson’s moments and Polubarinova–Galin equation.