Abstract
The present study deals with the effects of floating flexible membrane on the instability of a gravity-driven flow down an incline. Linear stability of the flow system is explored using normal-mode analysis. Free surface gravity-driven flow is unstable at much lower Reynolds numbers. Instability of such a flow can be controlled either by changing behavioral of the lower wall or by altering the surface waves at the free surface which is done here by including a floating flexible membrane at the top of the liquid layer. Influence of membrane tension is taken into account in terms of stress jump at the free surface. The Orr-Sommerfeld system of the flow is solved numerically using spectral collocation method. The results displays a destabilizing role of membrane tension for a wide range of parameters. The growth rate of the perturbation waves increases with an increase of membrane tension and the critical Reynolds number becomes smaller. Therefore, it is possible to enhance the instability of the flow system with help of membrane properties, which may be useful in Ocean engineering and coating industries.
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Amaouche, M., Abderrahmane, H.A., Bourdache, L.: Hydromagnetic thin film flow: linear stability. Phys. Rev. E 88, 023028 (2013)
Anjalaiah, Usha, R., Millet, S.: Thin film flow down a porous substrate in the presence of an insoluble surfactant: stability analysis. Phys. Fluids 25, 022101 (2013)
Blyth, M., Pozrikidis, C.: Effect of surfactant on the stability of film flow down an inclined plane. J. Fluid Mech. 521, 241–250 (2004)
Canuto, C., Hussaini, M.Y., Quarteroni, A., Thomas Jr., A.: Spectral Methods in Fluid Dynamics. Springer Science & Business Media (2012)
Cho, I., Kim, M.: Interactions of a horizontal flexible membrane with oblique incident waves. J. Fluid Mech. 367, 139–161 (1998)
Cho, I., Kim, M.: Wave deformation by a submerged flexible circular disk. Appl. Ocean Res. 21, 263–280 (1999)
Criminale, W.O., Jackson, T.L., Joslin, R.D.: Theory and Computation of Hydrodynamic Stability. Cambridge University Press, Cambridge (2003)
Karmakar, D., Sahoo, T.: Gravity wave interaction with floating membrane due to abrupt change in water depth. Ocean Eng. 35, 598–615 (2008)
Karmakar, D., Soares, C.G.: Oblique scattering of gravity waves by moored floating membrane with changes in bottom topography. Ocean Eng. 54, 87–100 (2012)
Liu, R., Liu, Q.: Instabilities of a liquid film flowing down an inclined porous plane. Phys. Rev. E 80, 036316 (2009)
Mondal, R., Mandal, S., Sahoo, T.: Surface gravity wave interaction with circular flexible structures. Ocean Eng. 88, 446–462 (2014)
Sahoo, T., Yip, T.L., Chwang, A.T.: Scattering of surface waves by a semi-infinite floating elastic plate. Phys. Fluids 13, 3215–3222 (2001)
Skotheim, J.M., Thiele, U., Scheid, B.: On the instability of a falling film due to localized heating. J. Fluid Mech. 475, 1–19 (2003)
Tshehla, M.: The flow of a variable viscosity fluid down an inclined plane with a free surface. Math. Probl. Eng. 2013 (2013)
Ueno, H., Iguchi, T.: A mathematical justification of a thin film approximation for the flow down an inclined plane. J. Math. Anal. Appl. 444, 804–824 (2016)
Wang, C.: Thin film flowing down a curved surface. Z. Angew. Math. Phys. ZAMP 35, 532–544 (1984)
Yih, C.S.: Stability of liquid flow down an inclined plane. Phys. Fluids 6, 321–334 (1963)
Yih, C.S.: Stability of liquid flow down an inclined plane. In: Selected Papers By Chia-Shun Yih (In 2 Volumes). World Scientific, pp. 357–370 (1991)
Yip, T., Sahoo, T., Chwang, A.T.: Wave scattering by multiple floating membranes. In: The Eleventh International Offshore and Polar Engineering Conference. International Society of Offshore and Polar Engineers (2001)
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HB and SG are thankful to SERB, DST, Government of India (Award No. CRG/2018/004521) for the financial support.
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Sani, M., Behera, H., Ghosh, S. (2020). Stability Analysis of a Film Flow Down an Incline in the Presence of a Floating Flexible Membrane. In: Manna, S., Datta, B., Ahmad, S. (eds) Mathematical Modelling and Scientific Computing with Applications. ICMMSC 2018. Springer Proceedings in Mathematics & Statistics, vol 308. Springer, Singapore. https://doi.org/10.1007/978-981-15-1338-1_19
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DOI: https://doi.org/10.1007/978-981-15-1338-1_19
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