Abstract
The exact invariantOstroumov–Birikh solution of the Oberbeck–Bussinesq equationswhich describes two-layer advective thermocapillary flows in the inclined plane is analyzed. The spectrum of the characteristic perturbations of all classes of the flows is investigated and analytical representations of the eigennumbers and eigenfunctions of the corresponding spectral problem are obtained in the zeroth approximation. Stability of the flows with respect to longwave perturbations and the possibility of existence of oscillatory regimes are proved.
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References
L.G. Napolitano, “Plane Marangoni–Poiseuille Flow of Two Immiscible Fluids,” Acta Astronautica, No. 7, 461–478 (1980).
A.A. Nepomnyashchy, M.G. Velarde, and P. Colinet, Interfacial Phenomena and Convection (Chapman & Hall, Boca Raton, 2002).
A.A. Nepomnyashchy, I.B. Simanovskii, and J.-C. Legros, Interfacial Convection in Multilayer Systems (Springer Science + Business Media, New York, 2006).
L.P. Kholpanov and B.Ya. Shkadov, Fluid Dynamics and Heat- and Mass-Transport with an Interface (Nauka, Moscow, 1990) [in Russian].
L.A. Dávalos–Orozco, “Evaporation and Flow Dynamics of Thin, Shear–Driven Liquid Films in Microgap Channels,” Interfacial Phenomena and Heat Transfer 1(2), 93–138 (2013).
E.A. Chinnov and O.A. Kabov, “Two-Phase Flows in Horizontal Flat Microchannels,” Dokl. Ros. Akad. Nauk 44, No. 2, 190–194 (2012).
V.K. Andreev, Yu.A. Gaponenko, O.N. Goncharova, and V.V. Pukhnachev, Mathematical Models of Convection (Walter de Gruyter GmbH & CO KG, Berlin/Boston, 2012).
O.N. Goncharova and O.A. Kabov, “Mathematical and Numerical Modeling of Convection in a Horizontal Layer under Co-Current Gas Flow,” Int. J. Heat Mass Transfer 53, 2795–2807 (2010).
O.N.Goncharova O.A.Kabov, and V.V. Pukhnachev, “Solutions of Special Type Describing the Three Dimensional Thermocapillary Flows with an Interface,” Int. J. Heat Mass Transfer 55, 715–725
O.N. Goncharova and E.V. Rezanova, “Example of the Exact Solution of a Steady-State Problem of Two-Layer Flows with Evaporation on the Interface,” Z. Prikl. Mat. Tekh. Fiz., No. 2, 68–79 (2014).
A.A. Nepomnyashchy and I.B. Simanovskii, “The Influence of Gravity on the Dynamics of Non–Isothermic Ultra-Thin Two-Layer Films,” Microgravity Sci. Technol. 21, Suppl. 1, S261–S269 (2009).
A.A. Nepomnyashchy and I.B. Simanovskii, “Instabilities and Ordered Patterns in Nonisothermal Ultrathin Bilayer Fluid Films,” Phys. Rev. Letters 102, 164501 (2009).
G.A. Ostroumov, Free Convection under the Conditions of the Inner Problem (Gostekhizdat, Moscow, 1952) [in Russian].
R.V. Birikh, “ThermocapillaryConvection in a Horizontal Fluid Layer,” Z. Prikl. Mat. Tekh. Fiz., No. 3, 69–72 (1966).
V.V. Pukhnachev, “Group-Theoretical Nature of the Birikh Solution and its Generalizations,” in: Proceedings of Int. Conf. Symmetry and Differential Equations (Institute of ComputationalMathematics of the Siberian Branch of the RAS, Krasnoyarsk, 2000) [in Russian], pp. 180–183.
V.K. Andreev and V.B. Bekezhanova, Stability of Nonisothermal Fluids (Sib. Fed. University, Krasnoyarsk, 2010) [in Russian].
T. Doi and J.N. Koster, “Thermocapillary Convection in Two Immiscible Liquid Layers with Free Surface,” Phys. Fluids A 5, No. 8, 1914–1927 (1993).
V.K. Andreev and V.B. Bekezhanova, “Small Perturbations of Thermocapillary Steady-State Two-Layer Flows in a Plane Layer with a Movable Boundary,” J. Sib. Fed. University, No. 4(4), 434–444 (2011).
V.B. Bekezhanova, “Three-Dimensional Disturbances of a Plane-Parallel Two-Layer Flow of a Viscous, Heat- Conducting Fluid,” Fluid Dynamics 47(6), 702–708 (2012).
D.D. Joseph, Stability of Fluid Motions (Springer, Berlin, 1976; Mir, Moscow, 1981).
V.V. Pukhnachev, Motion of Viscous Fluid with Free Boundaries (Novosibirsk State University, Novosibirsk, 1989) [in Russian].
N.I. Lobov, D.V. Lyubimov, and T.P. Lyubimova, “Convective Instability of a System of Horizontal Layers of Immiscible Liquids with a Deformable Interface,” Fluid Dynamics 31(2), 186–192 (1996).
I.V. Repin, “Stability of Steady-State Thermocapillary Flow with a Plane Interface,” in: Abstracts of the All- Russian Conf. Theory and Applications of Problems with Free Boundaries (AGU, Barnaul, 2002) [in Russian], pp. 83–84.
I.V. Repin, “Steady-State Flows of Two-Layer Heat-Conducting Fluid in a Plane Layer,” in: Proceedings of Int. Conf. Mathematical Models and Methods of their Investigation (KGU, Krasnoyarsk, 2001) [in Russian], pp. 161–165.
V.B. Bekezhanova, “Change of the Types of Instability of a Steady Two-Layer Flow in an Inclined Channel,” Fluid Dynamics 46(4), 525–535 (2011).
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Original Russian Text © V.B. Bekezhanova, A.V. Rodionova, 2015, published in Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, 2015, Vol. 50, No. 6, pp. 3–19.
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Bekezhanova, V.B., Rodionova, A.V. Longwave stability of two-layer fluid flow in the inclined plane. Fluid Dyn 50, 723–736 (2015). https://doi.org/10.1134/S0015462815060010
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DOI: https://doi.org/10.1134/S0015462815060010