×

Plateau’s rotating drops and rotational figures of equilibrium. (English) Zbl 1370.37140

Summary: We give a detailed classification of all rotationally symmetric figures of equilibrium corresponding to rotating liquid masses subject to surface tension. When the rotation rate is zero, these shapes were studied by Delaunay who found six different qualitative types of complete connected interfaces (spheres, cylinders, unduloids, nodoids, catenoids, and planes). We find twenty-six qualitatively different interfaces providing a complete picture of symmetric equilibrium shapes, some of which have been studied by other authors. In particular, combining our work with that of A. Beer [“Note”, Ann. Phys., Berlin 96, 210 (1855)], S. Chandrasekhar [Proc. R. Soc. Lond., Ser. A 286, 1–26 (1965; Zbl 0137.23004)], R. Gulliver [in: Variational methods for equilibrium problems of fluids. Meet. Trento/Italy 1983. Astérisque 118, 167–179 (1984; Zbl 0581.76108)], D. R. Smith and J. E. Ross [Methods Appl. Anal. 1, No. 2, 210–228 (1994; Zbl 0835.34018)], we conclude that every compact equilibrium is in either a smooth connected one parameter family of spheroids or a smooth connected one parameter family of tori (possibly immersed in either case).

MSC:

37N10 Dynamical systems in fluid mechanics, oceanography and meteorology
76U05 General theory of rotating fluids
65D18 Numerical aspects of computer graphics, image analysis, and computational geometry
76B45 Capillarity (surface tension) for incompressible inviscid fluids
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Beer, A., Note, Pogg. Ann., 96, 210, (1855)
[2] Brown, R. A.; Scriven, L. E., The shape and stability of rotating liquid drops, Proc. R. Soc. Lond. Ser. A, 371, 1746, 331-357, (1980) · Zbl 0435.76073
[3] Chandrasekhar, S., The stability of a rotating liquid drop, Proc. R. Soc. Ser. A, 286, 1-26, (1965) · Zbl 0137.23004
[4] Gulliver, R., Tori of prescribed mean curvature and the rotating drop, Soc. Math. de France, Astérisque, 118, 167-179, (1984) · Zbl 0581.76108
[5] Heine, J., Computations of form and stability of rotating drops with finite elements, IMA J. Numer. Anal., 26, 723-751, (2006) · Zbl 1102.76034
[6] Hynd, R.; McCuan, J., On toroidal rotating drops, Pacific J. Math., 224, 2, 279-289, (2006) · Zbl 1125.49001
[7] Kapouleas, Nicolaos, Slowly rotating drops, Comm. Math. Phys., 129, 1990, 139-159, (1990) · Zbl 0694.76042
[8] López, R., Stationary rotating surfaces in Euclidean space, Calc. Var. Partial Differential Equations, 39, 3-4, 333-359, (2010) · Zbl 1203.53008
[9] Plateau, J. A.F., Mémoire sure LES phénomènes que présente une masse liquide libre et soustraite à l’action de la pesanteur, (Mémoires de l’Acad. Bruxelles, vol. 16, (1843)), 1-35
[10] Rayleigh, Lord, The equilibrium of revolving liquid under capillary force, Philos. Mag. Ser. 6, 28, 164, 161-170, (1914) · JFM 45.1053.01
[11] Smith, D.; Ross, J., Universal shapes and bifurcation for rotating incompressible fluid drops, Methods Appl. Anal., 1, 2, 210-228, (1994) · Zbl 0835.34018
[12] Wang, T. G.; Trinh, E. H.; Croonquist, A. P.; Elleman, D. D., Shapes of rotating free drops: spacelab experimental results, Phys. Rev. Lett., 56, 452-455, (1986)
[13] Wente, H., The symmetry of rotating fluid bodies, Manuscripta Math., 39, 287-296, (1982) · Zbl 0496.76103
[14] Wilkin-Smith, N., A class of stable energy minimisers for the rotating drop problem, J. Math. Anal. Appl., 332, 577-606, (2007) · Zbl 1166.76058
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.