Abstract
We will consider surfaces whose mean curvature at a point is a linear function of the square of the distance from that point to the vertical axis. We restrict ourselves here to surfaces which are cylinders over a curve in a horizontal plane. We describe the moduli space of the set of solutions which includes numerous properly immersed and embedded examples. We also analyze the stability of these cylindrical surfaces.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
R.A. Brown and L.E. Scriven. The shapes and stability of captive rotating drops. Philos. Trans. Roy. Soc. London Ser. A, 297(1429) (1980), 51–79.
S. Chandrasekhar. The stability of a rotating liquid drop. Proc. Roy. Soc. Ser. A, 286 (1965), 1–26.
E.A. Coddington and N. Levinson. Theory of Ordinary Differential Equations, Krieger Publishing Co., Malabar (1984).
M.G. Crandall and P.H. Rabinowitz. Bifurcation from simple eigenvalues. J. Func. Analysis, 54 (1971), 321–340.
R. Hynd and J. McCuan. On toroidal rotating drops. Pacific J. Math., 224(2) (2006), 279–289.
M. Koiso and B. Palmer. Stability of anisotropic capillary surfaces between parallel planes. Calculus of Variations and Partial Differential Equations, 25(3) (2006), 275–298.
R. López. Stationary rotating surfaces in Euclidean space. Calculus of Variations and Partial Differential Equations, 39(3-4) (2010), 333–359.
R. López. A criterion on instability of cylindrical rotating surfaces. Archiv der Mathematik, 94(1) (2010), 91–99.
P.A. Djondjorov, V.M. Vassilev, M. Ts. Hadzhilazova and I.M. Mladenov. Analytic description of the viscous fingering in rotating Hele-Shaw cells. Thirteenth International Conference on Geometry, Integrability and Quantization, June 3–8, 2011, Varna, Bulgaria, Avangard Prima, Sofia 2012, pp. 107–113.
P.A. Djondjorov, V.M. Vassilev and I.M. Mladenov. Analytic description and explicit parameterisation of the equilibrium shapes of elastic rings and tubes under uniform hydrostatic pressure. International Journal of Mechanical Sciences, 53(5) (2011), 355–364.
V.M. Vassilev, P.A. Djondjorov and I.M. Mladenov. Cylindrical equilibriumshapes of fluid membranes. Journal of Physics A: Mathematical and Theoretical, 41(43) (2008).
E.S. Leandro, R.M. Oliveira and J.A. Miranda. Geometric approach to stationary shapes in rotating Hele-Shaw flows. Physica D: Nonlinear Phenomena, 237(5) (2008), 652–664.
B. Palmer and O. Perdomo. Rotating surfaces with helicoidal symmetry, to appear in Pacific J. Math.
O. Perdomo. A dynamical interpretation of cmc Twizzlers surfaces. Pacific J. Math., 258(2) (March 2012), 459–585.
O. Perdomo. Embedded constant mean curvature hypersurfaces of spheres. Asian J. Math., 14(1) (March 2010), 73–108.
O. Perdomo. Hellicoidal minimal surfaces in R3. Illinois J. Math., 57(1) (2013), 87–104.
V.A. Solonnikov. On linear stability and instability of equilibrium figures of uniformly rotating liquid. Recent advances in elliptic and parabolic problems, 231–257, World Sci. Publ., Hackensack, NJ (2005).
Author information
Authors and Affiliations
Corresponding author
About this article
Cite this article
Palmer, B., Perdomo, O.M. Equilibrium shapes of cylindrical rotating liquid drops. Bull Braz Math Soc, New Series 46, 515–561 (2015). https://doi.org/10.1007/s00574-015-0103-0
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00574-015-0103-0