Properties of solutions for the problem of a~joint slow motion of a~liquid and a~binary mixture in a~two-dimensional channel

V. K. Andreev ; M. V. Efimova

Sibirskij žurnal industrialʹnoj matematiki, Tome 21 (2018) no. 3, pp. 3-17.

Résumé

Under study is a conjugate boundary value problemdescribing a joint motion of a binary mixture and a viscous heat-conducting liquid in a two-dimensional channel, where the horizontal component of the velocity vector depends linearly on one of the coordinates. The problemis nonlinear and inverse because the systems of equations contain the unknown time functions – the pressure gradients in the layers. In the case of small Marangoni numbers (the so-called creeping flow) the problem becomes linear. For its solutions the two different integral identities are valid which allow us to obtain a priori estimates of the solution in the uniform metric. It is proved that if the temperature on the channel walls stabilizes with time then, as time increases, the solution of the nonstationary problem tends to a stationary solution by an exponential law.

Détail

Keywords: conjugate problem, inverse problem, a priori estimates, thermocapillarity, asymptotic behavior.
Mots-clés : surface tension

Bibliographie

[1] Nepomnyashii A., Simanovskii I., Legros J.-C., Interfacial Convection in Multilayer System, Springer-Verl., N.Y., 2006 | MR

[2] Narayanan R., Schwabe D., Interfacial Fluid Dynamics and Transport Processes, Springer-Verl., Berlin, 2003 | MR | Zbl

[3] Zeytovnian R. Kh., Convection in Fluids, Springer-Verl., Dordrecht, 2009 | MR

[4] Andreev V. K., Zakhvataev V. E., Ryabitskii E. A., Termokapillyarnaya neustoichivost, Nauka, Novosibirsk, 2000

[5] Daniel D. Joseph, Stability of fluid motions, Springer-Verl., Berlin–Heidelberg–New York, 1976 | MR | Zbl

[6] Andreev V. K., Gaponenko Yu. A., Goncharova O. N., Pukhnachov V. V., Mathematical Models of Convection, Walter de Gruyter, Berlin, 2012 | MR

[7] Hiemenz K., “Die Grenzschicht an einem in den gleichförmigen Flüssigkeitsstrom eingetauchten geraden Kreiszylinder”, Dinglers Poliytech. J., 326 (1911), 321–324

[8] Brady J. F., Acrivos A., “Steady flow in a channel or tube with an accelerating surface velocity”, J. Fluid Mech., 112 (1981), 127–150 | DOI | MR | Zbl

[9] Riabouchinsky D., “Quelques considerations sur les mouvements plans rotationnels d'un liquide”, C. R. Acad. Sci., 179 (1924), 1133–1136 | Zbl

[10] Petrov A. G., “Tochnoe reshenie uravnenii Nave–Stoksa v sloe zhidkosti mezhdu dvizhuschimisya parallelno plastinami”, Prikl. mekhanika i tekhn. fizika, 53:5 (2012), 13–18 | MR | Zbl

[11] Petrov A. G., “Postroenie reshenii uravnenii Nave–Stoksa dlya sloya zhidkosti mezhdu dvizhuschimisya parallelno plastinami pri malykhi umerennykh chislakh Reinoldsa”, Prikl. mekhanika i tekhn. fizika, 54:1 (2013), 51–56 | MR | Zbl

[12] Petrova A. G., Pukhnachëv V. V., Frolovskaya O. A., “Nestatsionarnoe dvizhenie vblizi kriticheskoi tochki”, Dokl. Mezhdunar. konf., Megaprint, Irkutsk, 2014, 385–388

[13] Pukhnachëv V. V., “Gruppovye svoistva uravnenii Nave–Stoksa v ploskom sluchae”, Prikl. mekhanika i tekhn. fizika, 1:1 (1960), 83–90

[14] Bobkov N. N., Gupalo Yu. P., “Struktura techeniya v zhidkom sloe i spektr kraevoi zadachi pri nelineinoi zavisimosti poverkhnostnogo natyazheniya ot temperatury”, Prikl. matematika i mekhanika, 60:6 (1996), 1021–1028 | MR | Zbl

[15] Gupalo Yu. P., Ryazantsev Yu. S., “O termokapillyarnom dvizhenii zhidkosti so svobodnoi poverkhnostyu pri nelineinoi zavisimosti poverkhnostnogo natyazheniya ot temperatury”, Izv. AN. SSSR. Mekhanika zhidkosti i gaza, 1988, no. 5, 132–137 | Zbl

[16] Gupalo Yu. P., Ryazantsev Yu. S., Skvortsova A. V., “Vliyanie termokapillyarnykh sil na techenie zhidkosti so svobodnoi granitsei”, Izv. AN. SSSR. Mekhanika zhidkosti i gaza, 1989, no. 5, 3–7 | Zbl

[17] Admaev O. V., “Statsionarnoe termokapillyarnoe dvizhenie v tsilindricheskom sloe”, Modelirovanie v mekhanike, 6:23 (1992), 5–7

[18] Admaev O. V., Andreev V. K., “Axisymmetric thermocapillary flow in cylinder and cylindrical layer”, Hydromech and Heat/ Mass Transfer in Microgravity, Gordon and Breach Publ., Amsterdam, 1992, 157–162

[19] Admaev O. V., Andreev V. K., “Razvitie termokapillyarnogo dvizheniya s tsilindricheskoi granitsei”, Modelirovanie v mekhanike, 4:21 (1990), 73–77 | Zbl

[20] Andreev V. K., Reshenie Birikha uravnenii konvektsii i nekotorye ego obobscheniya, Preprint No 1-10, IVM SO RAN, Krasnoyarsk, 2010

[21] Andreev V. K., Sobachkina N. L., Dvizhenie binarnoi smesi v ploskikhi tsilindricheskikh oblastyakh, izd. SFU, Krasnoyarsk, 2012

[22] Andreev V. K., “O neravenstve tipa Fridrikhsa dlya sostavnykh oblastei”, J. Siberian Federal Univ. Math. Phys., 2:2 (2009), 146–157

[23] Polyanin A. D., Spravochnik po lineinym uravneniyam matematicheskoi fiziki, Fizmatlit, M., 2001

[24] Efimova M. V., “On one two-dimensional stationary flow of a binary mixture and viscous fluid in a plane layer”, J. Siberian Federal Univ. Math. Phys., 9:1 (2016), 30–36 | DOI