zbMATH — the first resource for mathematics

Two-dimensional plane steady-state thermocapillary flow. (English. Russian original) Zbl 1416.76041
Fluid Dyn. 54, No. 1, 33-41 (2019); translation from Izv. Ross. Akad. Nauk, Mekh. Zhidk. Gaza 2019, No. 1, 36-43 (2019).
Summary: The problem of a two-dimensional steady flow of a fluid in a flat channel with a free boundary when the surface tension coefficient depends linearly on the temperature is considered. On the channel bottom, a fixed temperature distribution is maintained. The temperature in the fluid is distributed in accordance with the quadratic law, which is consistent with the velocity field of the Xiemenz type. The arising boundary-value problem is strongly nonlinear and inverse with respect to the pressure gradient along the channel. The application of the tau-method shows that this problem has three different solutions. In the case of a thermally insulated free boundary, only one solution exists. Typical flow patterns are studied for each solution.
76D45 Capillarity (surface tension) for incompressible viscous fluids
76D27 Other free boundary flows; Hele-Shaw flows
76M25 Other numerical methods (fluid mechanics) (MSC2010)
80A20 Heat and mass transfer, heat flow (MSC2010)
Full Text: DOI
[1] Gupalo, Yu P.; Ryazantsev, Yu S., Thermocapillary motion of a liquid with a free surface with nonlinear dependence of the surface tension on the temperature, Fluid Dynamics, 23, 752-757, (1988) · Zbl 0677.76034
[2] Legro, J. K.; Limburg-Foutaine, M. S.; Petre, G., Influence of surface tension minimum as a function of temperature on the marangoni convection, Acta Astrounaut., 11, 143-147, (1984)
[3] Bobkov, N. N.; Gupalo, Yu P., The flow pattern in a liquid layer and the spectrum of the boundary-value problems when the surface tension depends non-linearly on the temperature, J. Appl. Mathem. Mech., 60, 999-1005, (1996) · Zbl 1040.76508
[4] Gupalo, Yu P.; Ryazantsev, Yu S.; Skvortsova, A. V., Effect of thermocapillary forces on free-surface fluid motion, Fluid Dynamics, 24, 657-661, (1989) · Zbl 0705.76034
[5] Hiemenz, K., Die Grenzschicht in Einem in Den Gleichf\(ö\)rmigen Fl\(ü\)ssigkeitsstrom Eingetauchten Geraden Kreiszylinder, Dinglers Poliytech., 326, 321-440, (1911)
[6] V. K. Andreev, V. E. Zakhvataev, and E. A. Ryabitskii, Thermocapillary Instability [in Russian] (Nauka, Novosibirsk, 2000).
[7] Zeytounian, R. H., The Benard-Marangoni thermocapillary instability problem, Phys. Usp., 41, 241-267, (1998)
[8] Andreev, V. K., Influence of interfacial energy on internal thermocapillary steady flow, J. Siberian Federal Univ. Mathematics/Physics, 10, 537-547, (2017)
[9] C. A. J. Fletcher, Computational Galerkin Methods (Springer Verlag, New-York, Berlin, 1984). · Zbl 0533.65069
[10] G. Sege, Orthogonal Polynomials [in Russian] (Fizmatlit, Moscow, 1962).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.