On estimates for short wave stability and long wave instability in three-layer Hele-Shaw flows |
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| Authors: | Prabir Daripa |
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| Institution: | 1. Transportation Institute, Shijiazhuang Tiedao University, Shijiazhuang 050043, China;2. Department of Mechanical Engineering, Shijiazhuang Tiedao University, Shijiazhuang 050043, China;3. Department of Mathematics and Physics, Shijiazhuang Tiedao University, Shijiazhuang 050043, China;1. Institute of Continuous Media Mechanics, Academician Korolev Street, 1, 614013, Perm, Russia;2. Perm State University, Bukireva Street, 15, 614990, Perm, Russia;1. Service de Physique de l’Etat Condensé, DSM, CEA-Saclay, CNRS UMR 3680, 91191 Gif-sur-Yvette, France;2. Departamento de Física, FCFM, Universidad de Chile, Casilla 487-3, Santiago, Chile;3. Laboratoire de Physique, ENS de Lyon, CNRS UMR 5672, 46 allée d’Italie, F69007 Lyon, France;1. Aix-Marseille Univ., Institut de Recherche sur les Phénomènes Hors Equilibre (IRPHE), UMR 7342 (CNRS, Aix-Marseille Univ., Ecole Centrale Marseille), 49 rue Frédéric Joliot-Curie, BP 146, 13384 Marseille Cedex 13, France;2. Université Paris-Est, Saint-Venant Hydraulics Laboratory (ENPC, EDF R&D, Cerema), 6 quai Watier, BP 49, 78401 Chatou, France;3. EDF R&D, Laboratoire National d’Hydraulique et Environnement, 6 quai Watier, BP 49, 78401 Chatou, France;4. Cerema, Water, Sea and Rivers, 134 rue de Beauvais, CS 60039, 60280 Margny-les-Compiègne, France;1. Prokhorov General Physics Institute of the Russian Academy of Sciences, 38 Vavilov str., 119991 Moscow, Russia;2. Joint International Laboratory LIA LICS, France-Russia;3. Univ. Lille, CNRS, Centrale Lille, ISEN, Univ. Valenciennes, UMR 8520 – IEMN, F-59000 Lille, France |
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| Abstract: | We consider the linear stability of three-layer Hele-Shaw flows with each layer having constant viscosity and viscosity increasing in the direction of a basic uniform flow. While the upper bound results on the growth rate of long waves are well known from our earlier works, lower bound results on the growth rate of short stable waves are not known to date. In this paper, we obtain such a lower bound. In particular, we show the following results: (i) the lower bound for stable short waves is also a lower bound for all stable waves, and the exact dispersion curve for the most stable eigenvalue intersects the dispersion curve based on the lower bound at a wavenumber where the most stable eigenvalue is zero; (ii) the upper bound for unstable long waves is also an upper bound for all unstable waves, and the exact dispersion curve for the most unstable eigenvalue intersects the dispersion curve based on the upper bound at a wavenumber where the most unstable eigenvalue is zero. Numerical results are provided which support these findings. |
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