On the conditions for existence of unidirectional motions of binary mixtures in the Oberbeck--Boussinesq model

V. K. Andreev ; I. V. Stepanova

Sibirskij žurnal industrialʹnoj matematiki, Tome 22 (2019) no. 2, pp. 3-12.

Résumé

Compatibility conditions are obtained for the nonstationary Oberbeck–Boussinesq equations describing the unidirectional motions of a liquid binary mixture in a horizontal strip. We examine the case of polynomial dependence of temperature on the longitudinal coordinate is considered, and the influence of the dependence of the kind on the remaining unknown functions from the original system. It is shown that a nonstationary unidirectional motion between two solid walls can be described by the Oberbeck–Boussinesq model only for the quadratic or linear law of temperature distribution along the horizontal coordinate. Some initial-boundary value problems are posed.

Détail

Keywords: Oberbeck–Boussinesq equations, binary mixture, compatibility conditions.
Mots-clés : unidirectional motion

Bibliographie

[1] Gershuni G. Z., Zhukhovitskii E. M., Nepomnyaschii A. A., Ustoichivost konvektivnykh techenii, Nauka, M., 1989

[2] Andreev V. K., Reshenie Birikha uravnenii konvektsii i nekotorye ego obobscheniya, Preprint, No 1–10, IVM SO RAN, Krasnoyarsk, 2010, 66 pp.

[3] Andreev V. K., Gaponenko Yu. A., Goncharova O. N., Pukhnachev V. V., Sovremennye matematicheskie modeli konvektsii, Fizmatlit, M., 2008

[4] Ryzhkov I. I., Termodiffuziya v smesyakh: uravneniya, simmetrii, resheniya i ikh ustoichivost, Izd-vo SO RAN, Novosibirsk, 2013

[5] Gnevanov N. V., Smorodin B. L., “Konvektivnaya neustoichivost techeniya binarnoi smesi v usloviyakh vibratsii i termodiffuzii”, Prikl. mekhanika i tekhn. fizika, 47:2 (2006), 77–84

[6] Ryzhkov I. I., Stepanova I. V., “Gruppovye svoistva i tochnye resheniya modeli vibratsionnoi konvektsii binarnoi smesi”, Prikl. mekhanika i tekhn. fizika, 52:4 (2011), 4. S. 72–82 | Zbl

[7] Andreev V. K., Sobachkina N. L., Dvizhenie binarnoi smesi v ploskikh i tsilindricheskikh oblastyakh, Red.-izd. tsentr SFU, Krasnoyarsk, 2012

[8] Andreev V. K., Stepanova I. V., “Ostroumov–Birikh solution of convection equations with nonlinear buoyancy force”, Appl. Math. Comput., 228 (2014), 59–67 | MR | Zbl

[9] Andreev V. K., Stepanova I. V., “Odnonapravlennye techeniya binarnykh smesei v modeli Oberbeka–Bussineska”, Izv. RAN. Mekhanika zhidkosti i gaza, 2016, no. 2, 13–24 | DOI

[10] Pukhnachev V. V., “Nestatsionarnye analogi resheniya Birikha”, Izv. AltGU. Matematika i mekhanika, 2011, no. 1–2(69), 62–69

[11] Efimova M. V., Darabi N., “Termokontsentratsionnaya konvektsiya v sisteme vyazkoi zhidkosti i binarnoi smesi v ploskom kanale pri malykh chislakh Marangoni”, Prikl. mekhanika i tekhn. fizika, 59:5 (2018), 93–103 | Zbl

[12] Andreev V. K., Efimova M. V., “Svoistva reshenii zadachi sovmestnogo medlennogo dvizheniya zhidkosti i binarnoi smesi v ploskom kanale”, Sib. zhurn. industr. matematiki, 21:3(75) (2018), 3–17

[13] Ostroumov G. A., Svobodnaya konvektsiya v usloviyakh vnutrennei zadachi, Gostekhizdat, M., 1952

[14] Birikh R. V., “O termokapillyarnoi konvektsii v gorizontalnom sloe zhidkosti”, Prikl. mekhanika i tekhn. fizika, 1966, no. 3, 69–72

[15] Goncharova O. N., Rezanova E. V., Lyulin Yu. V., Kabov O. A., “Izuchenie konvektivnykh techenii zhidkosti i sputnogo potoka gaza s uchetom ispareniya”, Teplofizika vysokikh temperatur, 55:6 (2017), 720–732 | DOI

[16] Bekezhanova V. B., Goncharova O. N., “Stability of the exact solutions describing the two-layer flows with evaporation at interface”, Fluid Dyn. Res., 48:6 (2016), 061408 | DOI | MR

[17] Bekezhanova V. B., Goncharova O. N., Rezanova E. V., Shefer I. A., “Ustoichivost dvukhsloinykh techenii zhidkosti s ispareniem na granitse razdela”, Izv. RAN. Mekhanika zhidkosti i gaza, 2017, no. 3, 23–35 | DOI