Abstract
We consider the evolution of linear waves of small perturbations of an unstable flow of a viscous fluid layer over a curved surface. The source of perturbations is assumed to be given by initial conditions defined in a small domain (in the limit, in the form of a \(\delta\)-function) or by an instantaneous localized external impact. The behavior of perturbations is described by hydrodynamic equations averaged over the thickness of the layer, with the gravity force and bottom friction taken into account (Saint-Venant equations). We study the asymptotic behavior of one-dimensional perturbations for large times. The inclination of the surface to the horizon is defined by a slowly varying function of the spatial variable. We focus on the perturbation amplitude as a function of time and the spatial variable. To study the asymptotics of perturbations, we use a simple generalization of the well-known method, based on the saddle-point technique, for finding the asymptotics of perturbations developing against a uniform background. We show that this method is equivalent to the one based on the application of the approximate WKB method for constructing solutions of differential equations. When constructing the asymptotics, it is convenient to assume that \(x\) is a real variable and to allow time \(t\) to take complex values.
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1134%2FS0081543823040120/MediaObjects/11501_2023_8354_Fig1_HTML.gif)
References
A. O. Akan, Open Channel Hydraulics (Elsevier, Oxford, 2006).
S. V. Alekseenko, V. E. Nakoriakov, and B. G. Pokusaev, Wave Flow of Liquid Films (Begell House, Redding, CT, 1994) [transl. from Russian (Nauka, Novosibirsk, 1992)].
H.-C. Chang and E. A. Demekhin, Complex Wave Dynamics on Thin Films (Elsevier, Amsterdam, 2002).
V. T. Chow, Open-Channel Hydraulics (McGraw-Hill, New York, 1959).
R. V. Craster and O. K. Matar, “Dynamics and stability of thin liquid films,” Rev. Mod. Phys. 81 (3), 1131–1198 (2009).
C. Di Cristo and A. Vacca, “On the convective nature of roll waves instability,” J. Appl. Math. 2005 (3), 259–271 (2005).
J. Heading, An Introduction to Phase-Integral Methods (J. Wiley and Sons, New York, 1962).
P. Huerre and P. A. Monkewitz, “Local and global instabilities in spatially developing flows,” Annu. Rev. Fluid Mech. 22, 473–537 (1990).
S. Kalliadasis, C. Ruyer-Quil, B. Scheid, and M. G. Velarde, Falling Liquid Films (Springer, London, 2012), Appl. Math. Sci. 176.
Yu. A. Kravtsov and Yu. I. Orlov, Geometrical Optics of Inhomogeneous Media (Springer, Berlin, 1990) [transl. from Russian (Nauka, Moscow, 1980)].
A. G. Kulikovskii, “Evolution of perturbations on a steady weakly inhomogeneous background. Complex Hamiltonian equations,” J. Appl. Math. Mech. 81 (1), 1–10 (2017) [transl. from Prikl. Mat. Mekh. 81 (1), 3–17 (2017)].
A. Kulikovskii and J. Zayko, “Asymptotic behavior of localized disturbance in a viscous fluid flow down an incline,” Phys. Fluids 34 (3), 034119 (2022).
E. M. Lifshitz and L. P. Pitaevskii, Physical Kinetics (Pergamon, Oxford, 1981), L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics 10 [transl. from Russian (Nauka, Moscow, 1979)].
V. P. Maslov, Perturbation Theory and Asymptotic Methods (Mosk. Gos. Univ., Moscow, 1965) [in Russian].
V. P. Maslov, Operator Methods (Mir, Moscow, 1982) [transl. from Russian (Nauka, Moscow, 1973)].
A. A. Rukhadze and V. P. Silin, “Method of geometrical optics in the electrodynamics of an inhomogeneous plasma,” Phys. Usp. 7 (2), 209–229 (1964) [transl. from Usp. Fiz. Nauk 82 (3), 499–535 (1964)].
O. Thual, L.-R. Plumerault, and D. Astruc, “Linear stability of the 1D Saint-Venant equations and drag parameterizations,” J. Hydraul. Res. 48 (3), 348–353 (2010).
J. H. Trowbridge, “Instability of concentrated free surface flows,” J. Geophys. Res. 92 (C9), 9523–9530 (1987).
V. V. Vedernikov, “Conditions at the front of a release wave that breaks the steady motion of a real fluid,” Dokl. Akad. Nauk SSSR 48 (4), 256–259 (1945).
B. Zanuttigh and A. Lamberti, “Instability and surge development in debris flows,” Rev. Geophys. 45 (3), RG3006 (2007).
Funding
The work of A. G. Kulikovskii (statement of the problem and description of general approaches to the solution of similar problems; Sections 1–4) was supported by the Russian Science Foundation under grant no. 20-11-20141, https://rscf.ru/project/20-11-20141/, and performed at the Steklov Mathematical Institute of Russian Academy of Sciences. The work of J. S. Zayko (specification of the problem as applied to the study of perturbations in a layer of flowing fluid, and calculation of the flow; Sections 4 and 5) was supported by a grant of the President of the Russian Federation for young scientists (project no. MK-4090.2022.4) and performed at the Institute of Mechanics of the Lomonosov Moscow State University.
Author information
Authors and Affiliations
Corresponding authors
Additional information
Translated from Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2023, Vol. 322, pp. 146–156 https://doi.org/10.4213/tm4346.
Translated by I. Nikitin
Rights and permissions
About this article
Cite this article
Kulikovskii, A.G., Zayko, J.S. On Waves on the Surface of an Unstable Layer of a Viscous Fluid Flowing Down a Curved Surface. Proc. Steklov Inst. Math. 322, 140–150 (2023). https://doi.org/10.1134/S0081543823040120
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0081543823040120