Gravity Driven Flow Past the Bottom with Small Waviness

Conference paper
Part of the Trends in Mathematics book series (TM)

Abstract

We propose an introductory study of gravity driven Stokesian flow past the wavy bottom, based on Adler’s et al. papers. In examples the waviness is described by a sinus function and its amplitude is small, up to O(ε2). A correction to Hagen-Poiseuille’s type free-flow solution is found. A contribution of capillary surface tension is discussed.

Keywords

Stokes’ equation Asymptotic and Fourier’s expansions Roughness Obstacles 

Mathematics Subject Classification (2010)

Primary 35C20 35J05; Secondary 41A58 42B05 85A30 86A05 

Notes

Acknowledgements

The authors gratefully acknowledge many helpful suggestions from Professor Vladimir V. Mityushev.

This work was partially supported within statutory activities No 3841/E-41/S/2017 of the Ministry of Science and Higher Education of Poland.

References

  1. 1.
    G.A. Chechkin, A. Friedman, A.L. Piatnitski, The boundary-value problem in domains with very rapidly oscillating boundary. J. Math. Anal. Appl. 231(1), 213–234 (1999)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    R. Czapla, V.V. Mityushev, W. Nawalaniec, Macroscopic Conductivity of Curvilinear Channels (Department of Computer Science and Computational Methods, Pedagogical University of Cracow, Kraków, 2017, preprint)Google Scholar
  3. 3.
    F.M. Esquivelzeta-Rabell, B. Figueroa-Espinoza, D. Legendre, P. Salles, A note on the onset of recirculation in a 2D Couette flow over a wavy bottom. Phys. Fluids 27(1), 014108-1–014108-14 (2015)Google Scholar
  4. 4.
    K. Evangelos, The impact of vegetation on the characteristics of the flow in an inclined open channel using the piv method. Water Resour. Ocean Sci. 1(1), 1–6 (2012)CrossRefGoogle Scholar
  5. 5.
    W. Huai, W. Wang, Y. Zeng, Two-layer model for open channel flow with submerged flexible vegetation. J. Hydraul. Res. 51(6), 708–718 (2013)CrossRefGoogle Scholar
  6. 6.
    E. Kubrak, J. Kubrak, P.M. Rowiński, Application of one-dimensional model to calculate water velocity distributions over elastic elements simulating Canadian waterweed plants (Elodea canadensis). Acta Geophys. 61(1), 194–210 (2013)CrossRefGoogle Scholar
  7. 7.
    L.D. Landau, E.M. Lifshitz, Fluid Mechanics. Course of Theoretical Physics (transl. from the Russian by J.B. Sykes, W.H. Reid), vol. 6, 2nd edn. (Pergamon Press, Oxford, 1987)Google Scholar
  8. 8.
    A.E. Malevich, V.V. Mityushev, P.M. Adler, Stokes flow through a channel with wavy walls. Acta Mech. 182(3–4), 151–182 (2006)CrossRefMATHGoogle Scholar
  9. 9.
    A.E. Malevich, V.V. Mityushev, P.M. Adler, Couette flow in channels with wavy walls. Acta Mech. 197(3), 247–283 (2008)CrossRefMATHGoogle Scholar
  10. 10.
    A.E. Malevich, V.V. Mityushev, P.M. Adler, Electrokinetic phenomena in wavy channels. J. Colloid Interface Sci. 345, 72–87 (2010)CrossRefGoogle Scholar
  11. 11.
    G. Sobin. Luminous Debris: Reflecting on Vestige in Provence and Languedoc (University of California Press, Berkeley, 2000)Google Scholar
  12. 12.
    R. Wojnar, W. Bielski, Flow in the canal with plants on the bottom. In Complex Analysis and Potential Theory with Applications. Proceedings of the 9th ISAAC Congress, August, 2013, Kraków, ed. by T.A. Azerogly, A. Golberg, S.V. Rogosin (Cambridge Scientific Publishers, Cambridge, 2014), pp. 167–183Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of Fundamental Technological Research PASWarszawaPoland
  2. 2.Institute of GeophysicsPolish Academy of SciencesWarsawPoland

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