Exact solution for capillary waves on the surface of a liquid of finite depth
https://doi.org/10.26907/0021-3446-2023-9-58-75
Abstract
Using the Schwartz function method, we have obtained a new exact solution for the problem of stationary capillary waves of finite amplitude on the surface of a liquid that has a finite depth. The reliability of the solution was confirmed by the results of numerical verification of the main boundary equation. The obtained solution of the problem is general in the sense that for any Weber number one can find the corresponding wave configuration. Parametric analysis showed a nonmonotonic dependence of the wave-length and its amplitude on the Weber number. The fact that the problem has one more branch of the solution (the trivial solution) indicates the possibility of the existence of other branches. The Schwartz function method cannot guarantee finding all solutions of the problem even from the specified class of functions. Therefore, the question of reproducing the known exact solution of W. Kimmersley for this problem and its reliability remains open. Note that for the parameter ß included in the main boundary equation, W. Kimmersley preliminarily laid down assumption ß = 1. The found exact solution has the property that ß > 1 and cannot coincide with W. Kimmersley’s solution.
About the Author
M. M. Alimov
Kazan Federal University
Russian Federation
Russian Federation
Mars Myasumovich Alimov
18 Kremlyovskaya str., Kazan, 420008
References
1. Kinnersley W. Exact large amplitude capillary waves on sheets of fluid, J. Fluid Mech. 77 (2), 229–241 (1976).
2. Зубарев Н.М., Зубарева О.В. Равновесные конфигурации заряженной поверхности проводящей жидкости при конечном межэлектродном расстоянии, Письма в ЖТФ 30 (21), 39–43 (2004).
3. Crowdy D.G. Exact solutions for steady capillary waves on a fluid annulus, J. Nonlinear Sci. 9, 615–640 (1999).
4. Куфарев П.П. Решение задачи о контуре нефтеносности для круга, ДАН СССР 60 (8), 1333–1334 (1948). [5] Howison S.D. Complex variable methods in Hele–Shaw moving boundary problems, Europ. J. Appl. Math. 3 (3), 209–224 (1992).
5. Алимов М.М. Точное решение задачи Маскета–Лейбензона для растущего эллиптического пузыря, Изв. РАН. Механ. жидкости и газа (5), 86–98 (2016).
6. Алимов М.М. Обобщение решения Жуковского для пузыря в канале, Изв. РАН. Механ. жидкости и газа (3), 16–29 (2021).
7. Ламб Г. Гидродинамика (Гостехиздат, М.-Л., 1947). [9] Сретенский Л.Н. Теория волновых движений жидкости (Наука, М., 1977).
8. Crapper G.D. An exact solution for progressive capillary waves of arbitrary amplitude, J. Fluid Mech. 2, 532–540 (1957).
9. Гухман А.А., Зайцев А.А. Обобщенный анализ (Факториал, М., 1998).
10. Лаврентьев М.А., Шабат Б.В. Методы теории функции комплексного переменного (Наука, М., 1973). [13] Уиттекер Е.Т., Ватсон Г.Н. Курс современного анализа (ГТТИ, М.-Л., 1934).
11. Абрамовиц М., Стиган И. Справочник по специальным функциям с формулами, графиками и математическими таблицами (Наука, М., 1979).
Review
For citations:
Alimov M.M. Exact solution for capillary waves on the surface of a liquid of finite depth. Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika. 2023;(9):58-75. (In Russ.) https://doi.org/10.26907/0021-3446-2023-9-58-75
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