Geodesic
Exact formulas for estimating the area of flow regions in free boundary problems
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 5 (2024), pp. 79-84 Cet article a éte moissonné depuis la source Math-Net.Ru

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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.
Keywords: potential flow, elliptic function
Mots-clés : complex variable, residue. 
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M. M. Alimov. Exact formulas for estimating the area of flow regions in free boundary problems. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 5 (2024), pp. 79-84. http://geodesic.mathdoc.fr/item/IVM_2024_5_a6/

[1] Tanveer S., “The effect of surface tension on the shape of the Hele-Shaw cell bubble”, Fhys. Fluids, 29:11 (1986), 3537–3548 | MR | Zbl

[2] Tanveer S., “New solutions for steady bubbles in a Hele–Shaw cell”, Fhys. Fluids, 30:3 (1987), 651–658 | MR

[3] Alimov M.M., Skvortsov E.V., “Ob otsenkakh raskhodnykh kharakteristik v teorii filtratsii i teploprovodnosti”, PMM, 53:3 (1989), 462–468 | MR | Zbl

[4] Lavrentev M.A., Shabat B.V., Metody teorii funktsii kompleksnogo peremennogo, Nauka, M., 1987 | MR

[5] Alimov M.M., “Tochnoe reshenie dlya kapillyarnykh voln na poverkhnosti zhidkosti konechnoi glubiny”, Izv. vuzov. Matem., 2023, no. 9, 58–75

[6] Kinnersley W., “Exact large amplitude capillary waves on sheets of fluid”, J. Fluid Mech., 77:2 (1976), 229–241 | DOI | MR | Zbl

[7] Uitteker E.T., Vatson G.N., Kurs sovremennogo analiza, v. 2, GTT, M.-L., 1934

[8] Abramovits M., Stigan I., Spravochnik po spetsialnym funktsiyam s formulami, grafikami i matematicheskimi tablitsami, Nauka, M., 1979

[9] Davis P.J., The Schwarz Function and its Applications, Amer. Math. Soc., Washington, 1974 | MR | Zbl

[10] Howison S.D., “Complex variable methods in Hele-Shaw moving boundary problems”, Europ. J. Appl. Math., 3:3 (1992), 209–224 | DOI | MR | Zbl