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Networks and Heterogeneous Media

2010, Volume 5, Issue 3: 385-404. doi: 10.3934/nhm.2010.5.385
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Small solids in an inviscid fluid

  • 1.
    Laboratoire de Mathématiques de Besançon, Université de Franche-Comté, 25030 Besançon Cedex
  • 2.
    Laboratoire de mathématiques, Université Paris-Sud, 91405 Orsay cedex
  • 3.
    UMR 7598 Laboratoire J.-L. Lions, UPMC Univ Paris 06, Paris, F-75005
  • 4.
    Institut Élie Cartan UMR 7502, INRIA, Nancy-Université, CNRS, 54506 Vandoeuvre-lès-Nancy Cedex
  • Received: 01 January 2010 Revised: 01 June 2010

  • Primary: 35F25, 35L80, 65M99.

  • We present in this paper several results concerning a simple model of interaction between an inviscid fluid, modeled by the Burgers equation, and a particle, assumed to be point-wise. It is composed by a first-order partial differential equation which involves a singular source term and by an ordinary differential equation. The coupling is ensured through a drag force that can be linear or quadratic. Though this model can be considered as a simple one, its mathematical analysis is involved. We put forward a notion of entropy solution to our model, define a Riemann solver and make first steps towards well-posedness results. The main goal is to construct easy-to-implement and yet reliable numerical approximation methods; we design several finite volume schemes, which are analyzed and tested.

    Citation: Boris Andreianov, Frédéric Lagoutière, Nicolas Seguin, Takéo Takahashi. Small solids in an inviscid fluid[J]. Networks and Heterogeneous Media, 2010, 5(3): 385-404. doi: 10.3934/nhm.2010.5.385

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  • We present in this paper several results concerning a simple model of interaction between an inviscid fluid, modeled by the Burgers equation, and a particle, assumed to be point-wise. It is composed by a first-order partial differential equation which involves a singular source term and by an ordinary differential equation. The coupling is ensured through a drag force that can be linear or quadratic. Though this model can be considered as a simple one, its mathematical analysis is involved. We put forward a notion of entropy solution to our model, define a Riemann solver and make first steps towards well-posedness results. The main goal is to construct easy-to-implement and yet reliable numerical approximation methods; we design several finite volume schemes, which are analyzed and tested.


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    1. G.M. Coclite, M. Garavello, Vanishing viscosity for mixed systems with moving boundaries, 2013, 264, 00221236, 1664, 10.1016/j.jfa.2013.01.010
    2. Nina Aguillon, Riemann problem for a particle–fluid coupling, 2015, 25, 0218-2025, 39, 10.1142/S0218202515500025
    3. John D. Towers, A fixed grid, shifted stencil scheme for inviscid fluid–particle interaction, 2016, 110, 01689274, 26, 10.1016/j.apnum.2016.08.002
    4. Sabrina Carpy, Hélène Mathis, Modeling binary alloy solidification by a random projection method, 2019, 35, 0749159X, 733, 10.1002/num.22322
    5. Boris Andreianov, Frédéric Lagoutière, Nicolas Seguin, Takéo Takahashi, Well-Posedness for a One-Dimensional Fluid-Particle Interaction Model, 2014, 46, 0036-1410, 1030, 10.1137/130907963
    6. Boris Andreianov, Nicolas Seguin, Analysis of a Burgers equation with singular resonant source term and convergence of well-balanced schemes, 2012, 32, 1553-5231, 1939, 10.3934/dcds.2012.32.1939
    7. Boris Andreianov, Abraham Sylla, Finite volume approximation and well-posedness of conservation laws with moving interfaces under abstract coupling conditions, 2023, 30, 1021-9722, 10.1007/s00030-023-00857-9
    8. Nicola De Nitti, Denis Serre, Enrique Zuazua, Pointwise constraints for scalar conservation laws with positive wave velocity, 2025, 76, 0044-2275, 10.1007/s00033-025-02459-0
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  • © 2010 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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