A variational-difference method for numerical simulation of equilibrium capillary surfaces
https://doi.org/10.37661/1816-0301-2023-20-4-56-68
Abstract
Objectives. A variational-difference method for numerical simulation of equilibrium capillary surfaces based on the minimization of the energy functional is proposed. As a test task a well-known axisymmetric hydrostatic problem on equilibrium shapes of a drop adjacent to a horizontal rotating plane under gravity is considered. The mathematical model of the problem is built on the basis of the variational principle: the shape of the drop satisfies the minimum total energy for a given volume. The problem of the functional minimization is reduced to a system of nonlinear equations using the finite element method. To solve the system a Newton's iterative method is applied.
Methods. The variational-difference approach (the finite element method) is used. The finite linear functions are chosen as basic functions.
Results. Equilibrium shapes of a drop on a rotating plane are constructed by the finite element method in a wide range of defining parameters: Bond number, rotational Weber number and wetting angle. The influence of these parameters on the shape of a drop is investigated. The numerical results are matched with the results obtained using the iterative-difference approach over the entire range of physical stability with respect to axisymmetric perturbations.
Conclusion. The finite element method responds to the loss of stability of a drop with respect to axisymmetric perturbations. Therefore it can be used to study the stability of the equilibrium of axisymmetric capillary surfaces.
About the Authors
Yu. N. Gorbacheva
Belarusian State University
Belarus
Belarus
Yuliya N. Gorbacheva, Senior Lecturer of the Department of Computational Mathematics of the Faculty of Applied Mathematics and Informatics
av. Nezavisimosti, 4, Minsk, 220030
V. K. Polevikov
Belarusian State University
Belarus
Belarus
Viktor K. Polevikov, Ph. D. (Phys.-Math.), Assoc. Prof., Assoc. Prof. of the Department of Computational Mathematics of the Faculty of Applied Mathematics and Informatics
av. Nezavisimosti, 4, Minsk, 220030
References
1. Sokurov A. A. An analytical and numerical study of capillary menisci. Vestnik KRAUNC. Fiziko-matematičeskie nauki [Bulletin KRASEC. Physical and Mathematical Sciences], 2021, vol. 36, no. 3, pp. 80–93 (In Russ.). https://doi.org/10.26117/2079-6641-2021-36-3-80-93
2. Dingle N. M., Tjiptowidjojo K., Basaran O. A., Harris M. T. A finite element based algorithm for determining interfacial tension (γ) from pendant drop profiles. Journal of Colloid and Interface Science, 2005, vol. 286, no. 2, pp. 647–660. https://doi.org/10.1016/j.jcis.2005.01.052
3. Dingle N. M., Harris M. T. A robust algorithm for the simultaneous parameter estimation of interfacial tension and contact angle from sessile drop profiles. Journal of Colloid and Interface Science, 2005, vol. 286, no. 2, pp. 670–680. https://doi.org/10.1016/j.jcis.2005.01.087
4. Gille M., Gorbacheva Yu., Hahn A., Polevikov V., Tobiska L. Simulation of a pending drop at a capillary tip. Communications in Nonlinear Science and Numerical Simulation, 2015, vol. 26, pp. 137–151. https://doi.org/10.1016/j.cnsns.2015.02.007
5. Basaran O. A., Scriven L. E. Axisymmetric shapes and stability of pendant and sessile drops in an electric field. Journal of Colloid and Interface Science, 1990, vol. 140, no. 1, pp. 10–30. https://doi.org/10.1016/0021-9797(90)90316-G
6. Saad S. M. I., Neumann A. W. Total Gaussian curvature, drop shapes and the range of applicability of drop shape techniques. Advances in Colloid and Interface Science, 2014, vol. 204, pp. 1–14. https://doi.org/10.1016/j.cis.2013.12.001
7. Danov K. D., Dimova S. N., Ivanov T. B., Novev J. K. Shape analysis of a rotating axisymmetric drop in gravitational field: Comparison of numerical schemes for real-time data processing. Colloids and Surfaces A: Physicochemical and Engineering Aspects, 2016, vol. 489, pp. 75–85. https://doi.org/10.1016/j.colsurfa.2015.10.028
8. Audzeichyk Ya. V., Konon P. N. Numerical study of the relative equilibrium of a droplet with a simply connected free surface on a rotating plane. Zhurnal Belorusskogo gosudarstvennogo universiteta. Matematika. Informatika [Journal of the Belarusian State University. Mathematics and Informatics], 2022, no. 3, pp. 79–90 (In Russ.). https://doi.org/10.33581/2520-6508-2022-3-79-90
9. Polevikov V. K. Methods for numerical modeling of two-dimensional capillary surfaces. Computational Methods in Applied Mathematics, 2004, vol. 4, no. 1, pp. 66–93. https://doi.org/10.2478/cmam-2004-0005
10. Polevikov V. K., Denisenko V. M. Numerical study of equilibrium shapes of a drop rotating in gravitational field. Vestnik Belorusskogo gosudarstvennogo universiteta imeni V. I. Lenina [Bulletin of the Belarusian State University named after V. I. Lenin], 1985, no. 2, pp. 37–41 (In Russ.).
11. Chernous’ko F. L. The problem of the equilibrium of a fluid subjected to the action of gravity and surface tension. Vvedenie v dinamiku tela s zhidkost'ju v uslovijah nevesomosti [Introduction to the Dynamics of a Body with Liquid in Weightlessness]. Moscow, Vychislitel'nyj centr Akademii nauk Sojuza Sovetskih Socialisticheskih Respublik, 1968, p. 69–97 (In Russ.).
12. Myshkis A. D., Babskij V. G., Zhukov M. Ju., Kopachevskij N. D., Slobozhanin L. A., Tjupcov A. D. Metody reshenija zadach gidromehaniki dlja uslovij nevesomosti. Methods for Solving Problems in Hydromechanics in Zero Gravity Conditions. In A. D. Myshkis (ed.). Kiev, Naukova dumka, 1992, 592 p. (In Russ.).
13. Lebedev-Stepanov P. V., Karabut T. A., Chernyshov N. A., Rybak S. A. Investigation of the shape and stability of a liquid drop on a rotating substrate. Acoustical Physics, 2011, vol. 57, no. 3, pp. 320–325. https://doi.org/10.1134/S1063771011030122
14. Finn R. Equilibrium Capillary Surfaces. New York, Springer, 1986, 245 p.
Review
For citations:
Gorbacheva Yu.N., Polevikov V.K. A variational-difference method for numerical simulation of equilibrium capillary surfaces. Informatics. 2023;20(4):56-68. (In Russ.) https://doi.org/10.37661/1816-0301-2023-20-4-56-68
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