Entov, Vladimir M.; Etingof, Pavel I.; Kleinbock, Dmitry Ya. Hele-Shaw flows with a free boundary produced by multipoles. (English) Zbl 0780.76024 Eur. J. Appl. Math. 4, No. 2, 97-120 (1993). Summary: We study Hele-Shaw flows with a moving boundary and multipole singularities. We find that such flows can be defined only on a finite time interval. Using a complex variable approach, we construct a family of explicit solutions for a single multipole. These solutions turn out to have the maximal possible lifetime in a certain class of solutions. We also discuss the generalized Hele-Shaw model in which surface tension at the moving boundary is considered, and develop a method of finding steady shapes. This method yields new one-parameter families of stationary solutions. In the appendix we discuss a connection between these solutions and a variational problem of potential theory. Cited in 1 ReviewCited in 6 Documents MSC: 76D99 Incompressible viscous fluids 76D45 Capillarity (surface tension) for incompressible viscous fluids 30C20 Conformal mappings of special domains Keywords:finite time interval; complex variable approach; single multipole; surface tension; one-parameter families of stationary solutions; variational problem; potential theory PDFBibTeX XMLCite \textit{V. M. Entov} et al., Eur. J. Appl. Math. 4, No. 2, 97--120 (1993; Zbl 0780.76024) Full Text: DOI References: [1] Vasconcelos, Phys. Rev. A44 pp 6490– (1991) [2] Davis, Schwarz function and its applications (1974) [3] DOI: 10.1093/qjmam/12.3.265 · Zbl 0091.18502 [4] DOI: 10.1016/0022-247X(80)90045-1 · Zbl 0447.35016 [5] Lacey, J. Austr. Math. Soc. B24 pp 171– (1982) [6] Howison, Euro. J. Appl. Math. 3 pp 209– (1992) [7] Etingof, Soviet Phys. Dokl. 35 pp 625– (1990) [8] DOI: 10.1017/S0022112081002632 · Zbl 0451.76015 [9] DOI: 10.1017/S0022112072002551 · Zbl 0256.76024 [10] Kufarev, Tomskogo Universiteta 17 pp 129– (1952) [11] Kufarev, Dolk. Akad. Nauk. SSSR 75 pp 507– (1950) [12] Kufarev, Dolk. Akad. Nauk. SSSR 60 pp 1333– (1948) [13] DOI: 10.1098/rspa.1958.0085 · Zbl 0086.41603 [14] DOI: 10.1103/RevModPhys.58.977 [15] DOI: 10.1007/BF02392634 · Zbl 0041.20301 [16] DOI: 10.1093/qjmam/38.3.343 · Zbl 0591.35087 [17] DOI: 10.1137/0146003 · Zbl 0592.76042 [18] Howison, Proc. Royal Soc. 102 pp 141– (1986) · Zbl 0608.76085 [19] DOI: 10.1093/qjmam/44.4.507 · Zbl 0743.76020 [20] Varchenko, Why the Boundary of a Round Drop Becomes a Curve of Order Four (1992) · Zbl 0768.76073 [21] Millar, Proc. Workshop on Inverse Problems and Imaging (1989) [22] Goldstein, Qualitative Methods in Continuum Mechanics (1989) [23] Millar, Compl. Var. Th. Appl. 15 pp 1– (1990) · Zbl 0813.30022 [24] DOI: 10.1103/PhysRevLett.65.2986 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.