Exact formulas for estimating the area of flow regions in free boundary problems
https://doi.org/10.26907/0021-3446-2024-5-79-84
Abstract
An effective technique is proposed for obtaining exact formulas for estimating the area of flow regions in two-dimensional fluid flow problems with free boundaries, that allow an exact solution in terms of elliptic functions. The effectiveness of the technique is demonstrated using a specific example of the problem of capillary waves on the surface of a liquid of finite depth. This example is characterized by mirror symmetry of the flow region, but the technique can be generalized to the case of other symmetry of the flow region.
About the Author
M. M. Alimov
Kazan Federal University
Russian Federation
Russian Federation
Mars Myasumovich Alimov
18 Kremlyovskaya str., Kazan, 420008
References
1. Tanveer S. The effect of surface tension on the shape of the Hele-Shaw cell bubble, Fhys. Fluids 29 (11), 3537–3548 (1986).
2. Tanveer S. New solutions for steady bubbles in a Hele–Shaw cell, Fhys. Fluids 30 (3), 651–658 (1987).
3. Алимов М.М., Скворцов Э.В. Об оценках расходных характеристик в теории фильтрации и теплопроводности, ПММ 53 (3), 462–468 (1989).
4. Лаврентьев М.А., Шабат Б.В. Методы теории функции комплексного переменного (Наука, М., 1987).
5. Алимов М.М. Точное решение для капиллярных волн на поверхности жидкости конечной глубины, Изв. вузов. Матем. (9), 58–75 (2023).
6. Kinnersley W. Exact large amplitude capillary waves on sheets of fluid, J. Fluid Mech. 77 (2), 229–241 (1976).
7. Уиттекер Е.Т., Ватсон Г.Н. Курс современного анализа, Т. 2 (ГТТ, М.-Л., 1934).
8. Абрамовиц М., Стиган И. Справочник по специальным функциям с формулами, графиками и математическими таблицами (Наука, М., 1979).
9. Davis P.J. The Schwarz Function and its Applications (Amer. Math. Soc., Washington, 1974).
10. Howison S.D. Complex variable methods in Hele-Shaw moving boundary problems, Europ. J. Appl. Math. 3 (3), 209–224 (1992).
Review
For citations:
Alimov M.M. Exact formulas for estimating the area of flow regions in free boundary problems. Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika. 2024;(5):79–84. (In Russ.) https://doi.org/10.26907/0021-3446-2024-5-79-84
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