Polubarinova-Galin equation for Hele-Shaw flows with two free boundaries (Q6544178)

This paper is an important improvement of two well-known results concerning the Hele-Shaw flow: [\textit{P. Ya Polubarinova-Kochina}, ``On the motion of the oil contour'' (Russian), Dokl. Akad. Nauk USSR 47, 254--257 (1945); \textit{L. A. Galin}, C. R. (Dokl.) Acad. Sci. URSS, n. Ser. 47, 246--249 (1945; Zbl 0061.46202)]. These paper studied the evolution of the free boundary appearing when the oil is displaced by water in a Hele-Shaw cell, by neglecting the surface tension effects. The present paper extends these results to the case of two free boundaries -- the water (or air) displaces an ``oil plug''. From the physical plane, the author switch to the plane of complex velocity potential, in two variants explained in Figure 2. A specific parametrization technique is used -- see also [\textit{M. I. Gurevich}, Theory of jets in ideal fluids. Translated from the Russian edition by Robert L. Street and Konstantin Zagustin. New York and London: Academic Press (1965; Zbl 0131.23702)]. Two particular conformal mapping are used, given by Formulas (4) and (5). The details for the analogue of the Polubarinova-Galin equation with two free boundaries are given in Section 3. A numerical study of the evolution of a circular ring of \textit{incompressible} fluid in a Hele-Shaw cell is given in Section 4. However, I think here is a problem. The ``radius'' of the ring increases during the process. Thus the ``distance'' between the two free boundaries becomes very small. A careful analysis of the two ``very close'' boundary conditions could be useful.