Bifurcation Analysis of a Capillarity Problem with Circular Symmetry

Both stable and unstable equilibrium shapes of small drops under gravity are well understood in the nonlinear formulation. These shapes are solutions to the capillarity equation, which we can find in series form using iterative methods. If the droplet is sufficiently large, or a potential acts inside it, then the convergence of the approximate solutions breaks down. In this case the solutions contradict physical experiments. The solvability of the capillary equation was proved by Uraltseva.
The surface adjusts under the action of a potential. The description of special states of the surface using the capillarity equation is complicated by the structure of this equation and its linearization. On the other hand, the capillarity problem is variational. The main term of the energy functional is the area functional studied by Fomenko, Borisovich, and Stenyukhin in connection with minimal surfaces. Sapronov, Darinskii, Tsarev, Sviridyuk, and other authors explored the extremals of similar nonlinear functionals on Banach and Hilbert spaces. As a result, in this paper we obtain sufficient conditions for the existence of special solutions to the capillary problem under the influence of external potential in terms of variational problems and normal bundle of perturbations. In an example we construct a new reduction of the capillarity equation near the center of symmetry of the drop. We find the critical value of the parameter, which depends on the Bond number, and determine the analytic form of the solution.

Keywords

capillarity problem; Bond number; bifurcation; special solution.

References

1. Wente H.C. The Symmetry of Sessile and Pendent Drops. Pasific Journal of Mathematics, 1980, vol. 88, no. 2, pp. 387-397. DOI: 10.2140/pjm.1980.88.387
2. Finn R. Equilibrium capillary surface. Mathematical theory. Edited by A.T Fomenko. Moscow, Mir, 1989. 310 p.
3. Sapronov Yu.I. About Modelling Diffuser Currents of Liquid by Means of Reduced Equations. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming & Computer Software, 2012, no. 27 (286), pp. 82-93. (in Russian)
4. Darinskii B.M., Sapronov Yu.I., Tsarev S.L. Bifurcations of Extremals of Fredholm Functionals. Journal of Mathematical Sciences, 2007, vol. 145, no. 6, pp. 5311-5453. DOI: 10.1007/s10958-007-0356-2
5. Sviridyuk G.A. Quasistationary Trajectories of Semilinear Dynamical Equations of Sobolev Type. Russian Academy of Sciences. Izvestiya Mathematics, 1994, vol. 42, no. 3, pp. 601-614. DOI: 10.1070/IM1994v042n03ABEH001547
6. Nirenberg L. Topics in Nonlinear Functional Analysis. American Mathematical Soc., 1974. 145 p.
7. Stenyukhin L.V. On Special Solutions of the Capillary with Circular Symmetry. Vestnik VGU. Seriya: Fizika. Matematika [Proceedings of the Voronezh State University. Series: Physics. Mathematics], 2012, no. 2, pp. 242-245. (in Russian)
8. Stenyukhin L.V. Minimal Surfaces with Constraints of Inequality Type. Russian Mathematics, 2012, no. 11, pp. 45-51. DOI: 10.3103/S1066369X12110047
9. Borisovich A.Yu. Plateau Operator and Bifurcations of Two-Dimensional Minimal Surfaces. Global'nyy analiz i matematicheskaya fizika [Global Analysis and Mathematical Physics], Voronezh, 1987, pp. 142-155. (in Russian)