# Longwave stability of two-layer fluid flow in the inclined plane

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## 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.

## Keywords

stability interface spectrum of characteristic perturbations## Preview

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## References

- 1.L.G. Napolitano, “Plane Marangoni–Poiseuille Flow of Two Immiscible Fluids,” Acta Astronautica, No. 7, 461–478 (1980).zbMATHCrossRefADSGoogle Scholar
- 2.A.A. Nepomnyashchy, M.G. Velarde, and P. Colinet,
*Interfacial Phenomena and Convection*(Chapman & Hall, Boca Raton, 2002).zbMATHGoogle Scholar - 3.A.A. Nepomnyashchy, I.B. Simanovskii, and J.-C. Legros,
*Interfacial Convection in Multilayer Systems*(Springer Science + Business Media, New York, 2006).zbMATHGoogle Scholar - 4.L.P. Kholpanov and B.Ya. Shkadov,
*Fluid Dynamics and Heat- and Mass-Transport with an Interface*(Nauka, Moscow, 1990) [in Russian].Google Scholar - 5.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).CrossRefGoogle Scholar - 6.E.A. Chinnov and O.A. Kabov, “Two-Phase Flows in Horizontal Flat Microchannels,” Dokl. Ros. Akad. Nauk
**44**, No. 2, 190–194 (2012).Google Scholar - 7.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).zbMATHCrossRefGoogle Scholar - 8.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).zbMATHCrossRefGoogle Scholar - 9.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**, 715–725Google Scholar - 10.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).MathSciNetzbMATHGoogle Scholar
- 11.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).CrossRefGoogle Scholar - 12.A.A. Nepomnyashchy and I.B. Simanovskii, “Instabilities and Ordered Patterns in Nonisothermal Ultrathin Bilayer Fluid Films,” Phys. Rev. Letters
**102**, 164501 (2009).CrossRefADSGoogle Scholar - 13.G.A. Ostroumov,
*Free Convection under the Conditions of the Inner Problem*(Gostekhizdat, Moscow, 1952) [in Russian].Google Scholar - 14.R.V. Birikh, “ThermocapillaryConvection in a Horizontal Fluid Layer,” Z. Prikl. Mat. Tekh. Fiz., No. 3, 69–72 (1966).Google Scholar
- 15.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.Google Scholar - 16.V.K. Andreev and V.B. Bekezhanova,
*Stability of Nonisothermal Fluids*(Sib. Fed. University, Krasnoyarsk, 2010) [in Russian].zbMATHGoogle Scholar - 17.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).zbMATHCrossRefADSGoogle Scholar - 18.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).Google Scholar
- 19.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).zbMATHMathSciNetCrossRefADSGoogle Scholar - 20.D.D. Joseph,
*Stability of Fluid Motions*(Springer, Berlin, 1976; Mir, Moscow, 1981).zbMATHGoogle Scholar - 21.V.V. Pukhnachev,
*Motion of Viscous Fluid with Free Boundaries*(Novosibirsk State University, Novosibirsk, 1989) [in Russian].zbMATHGoogle Scholar - 22.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).zbMATHCrossRefADSGoogle Scholar - 23.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.Google Scholar - 24.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.Google Scholar - 25.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).zbMATHMathSciNetCrossRefADSGoogle Scholar

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