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Linear stability analysis of a thin liquid film on a horizontal wall under quasi-periodic oscillation

Published online by Cambridge University Press:  08 January 2025

Abdelouahab El Jaouahiry Open the ORCID record for Abdelouahab El Jaouahiry [Opens in a new window]  and
Saïd Aniss Open the ORCID record for Saïd Aniss [Opens in a new window]
Abdelouahab El Jaouahiry*
Affiliation:
Faculty of Sciences Aïn-Chock, Laboratory of Mechanics, University Hassan II, Casablanca 20100, Morocco
Saïd Aniss
Affiliation:
Faculty of Sciences Aïn-Chock, Laboratory of Mechanics, University Hassan II, Casablanca 20100, Morocco
*
†Email address for correspondence: [email protected]
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Abstract

We carry out a linear stability analysis of the flow of a thin layer of Newtonian fluid with a deformable free surface bounded at the bottom by a horizontal wall subjected to quasi-periodic oscillation in its own plane. Or's model (J. Fluid Mech., vol. 335, 1997, pp. 213–232), using a periodic oscillation, is extended to the configuration where oscillation has two incommensurate frequencies, $\omega _1$ and $\omega _2$, with an irrational ratio $\omega ={\omega _2}/{\omega _1}$. Using the long-wave expansion, we derive the asymptotic function involved in the long-wave instability criterion while taking into account the frequency ratio. It turns out that the maximum of this asymptotic function, as well as the frequency parameter at which long-wave instabilities occur, depend strongly on the frequency ratio. For arbitrary wavenumbers, the equations governing the problem under consideration are solved in space using Chebyshev's spectral collocation method, while the temporal resolution is performed using Floquet theory, knowing that an irrational number can be approximated by a rational number. For a large frequency ratio and for a velocity amplitude ratio equal to unity, we obtain, as in Or's work (J. Fluid Mech., vol. 335, 1997, pp. 213–232) considering the same frequency parameter interval, an alternation between the U shape and oblique shape referring respectively to instabilities of long wavelength and finite wavelength appearing in the diagram representing Reynolds number as a function of frequency parameter. By decreasing the frequency ratio towards $1/\sqrt {37}$, the three initial U-shaped and three oblique instabilities merge into a single U-shaped and a single oblique instability. This merging phenomenon also occurs when the ratio of the amplitudes of the superimposed velocities, linked to the introduction of the second frequency, increases from small values to unity. For a fixed frequency parameter, the effect of frequency ratio and velocity amplitude ratio on the marginal stability curves in terms of Reynolds number versus wavenumber is also investigated, focusing on the appearance of long wavelength instability and finite wavelength instability.

JFM classification

Flow Control: Instability control
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 1002 , 10 January 2025 , A39
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Copyright
© The Author(s), 2025. Published by Cambridge University Press

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