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Some Remarks on Avalanches Modelling: An Introduction to Shallow Flows Models

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Advances in Numerical Simulation in Physics and Engineering

Part of the book series: SEMA SIMAI Springer Series ((SEMA SIMAI,volume 3))

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Abstract

The main goal of these notes is to present several depth-averaged models with application in granular avalanches. We begin by recalling the classical Saint-Venant or Shallow Water equations and present some extensions like the Saint-Venant–Exner model for bedload sediment transport. The first part is devoted to the derivation of several avalanche models of Savage–Hutter type, using a depth-averaging procedure of the 3D momentum and mass equations. First, the Savage–Hutter model for aerial avalanches is presented. Two other models for partially fluidized avalanches are then described: one in which the velocities of both the fluid and the solid phases are assumed to be equal, and another one in which both velocities are unknowns of the system. Finally, a Savage–Hutter model for submarine avalanches is derived. The second part is devoted to non-newtonian models, namely viscoplastic fluids. Indeed, a one-phase viscoplastic model can also be used to simulate fluidized avalanches. A brief introduction to Rheology and plasticity is presented in order to explain the Herschel–Bulkley constitutive law. We finally present the derivation of a shallow Herschel–Bulkley model.

These notes are dedicated to D. Antonio Valle Sánchez (1930–2012). D. Antonio was the first Spanish PhD student of Jacques-Louis Lions. He can be considered as one of the founders of modern Applied Mathematics in Spain.

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Acknowledgements

The first author would like to thanks the organizers of the Jacques-Louis Lions Spanish-French school for the invitation. The second author would like to thank the Institute of Mathematics of the University of Seville (IMUS) for the financial support to work on the numerical analysis of models for visco-plastic avalanches.

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Authors and Affiliations

  1. Dpto. Matemática Aplicada I, Universidad de Sevilla, Sevilla, Spain

    Enrique D. Fernández-Nieto

  2. Unité de Mathématiques Pures et Appliquées, Ecole Normale Supérieure de Lyon, Lyon, France

    Paul Vigneaux

Authors
  1. Enrique D. Fernández-Nieto
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  2. Paul Vigneaux
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Corresponding author

Correspondence to Enrique D. Fernández-Nieto .

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Editors and Affiliations

  1. Facultad de Ciencias, Universidad de Málaga, Málaga, Málaga, Spain

    Carlos Parés

  2. Facultad de Informática, Universidade da Coruña, A Coruña, La Coruña, Spain

    Carlos Vázquez

  3. Centre de Mathématiques Appliquées, CNRS and Ecole Polytechnique, Paris, Paris, France

    Frédéric Coquel

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Fernández-Nieto, E.D., Vigneaux, P. (2014). Some Remarks on Avalanches Modelling: An Introduction to Shallow Flows Models. In: Parés, C., Vázquez, C., Coquel, F. (eds) Advances in Numerical Simulation in Physics and Engineering. SEMA SIMAI Springer Series, vol 3. Springer, Cham. https://doi.org/10.1007/978-3-319-02839-2_2

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