Small solids in an inviscid fluid

Boris Andreianov; Frédéric Lagoutière; Nicolas Seguin; Takéo Takahashi; Boris Andreianov; Frédéric Lagoutière; Nicolas Seguin; Takéo Takahashi

doi:10.3934/nhm.2010.5.385

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.
- Solid-fluid interaction,
- Burgers equation,
- singular source term,
- adapted entropy,
- well-balanced scheme,
- random-choice method
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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Abstract

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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