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Stability of laminar flows in an inclined open channel

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Abstract

We study the stability of laminar flows in a sheet of fluid (open channel) down an incline with constant slope angle \(\beta \). The basic motion is the velocity field \(U(z) \mathbf{i}\), where z is the coordinate of the axis orthogonal to the channel, and \(\mathbf{i}\) is the unit vector in the direction of the flow. U(z) is a parabolic function which vanishes at the bottom of the channel and whose derivative with respect to z vanishes at the top. We study the linear stability, and prove that the basic motion is linearly stable for any Reynolds number. We also study the nonlinear Lyapunov stability by solving the Orr equation for the associated maximum problem. As in Falsaperla et al. (Phys Rev E 100(1):013113, 2019. https://doi.org/10.1103/PhysRevE.100.013113) we finally study the nonlinear stability of tilted rolls. This work is a preliminary investigation to model debris flows down an incline (Introduction to the physics of landslides. Lecture notes on the dynamics of mass wasting. Springer, Dordrecht, 2011. https://doi.org/10.1007/978-94-007-1122-8).

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Acknowledgements

The results contained in the present paper have been partially presented in WASCOM 2019. The research that led to the present paper was partially supported by the following Grants: 2017YBKNCE of national project PRIN of Italian Ministry for University and Research, PTRDMI-53722122113 of the University of Catania. The authors acknowledge also support from the project PON SCN 00451 CLARA—CLoud plAtform and smart underground imaging for natural Risk Assessment, Smart Cities and Communities and Social Innovation. We also thank the group GNFM of INdAM for financial support.

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  1. Dipartimento di Matematica e Informatica, Università degli Studi di Catania, Viale A. Doria 6, 95125, Catania, Italy

    Paolo Falsaperla, Andrea Giacobbe & Giuseppe Mulone

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  1. Paolo Falsaperla
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  2. Andrea Giacobbe
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  3. Giuseppe Mulone
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Correspondence to Giuseppe Mulone.

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This paper is dedicated to Prof. G. Toscani and Prof. M. Sugiyama on the occasion of their 70th birthday.

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Falsaperla, P., Giacobbe, A. & Mulone, G. Stability of laminar flows in an inclined open channel. Ricerche mat 70, 67–79 (2021). https://doi.org/10.1007/s11587-020-00487-8

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