Small solids in an inviscid fluid

Pages: 385 - 404, Volume 5, Issue 3, September 2010

doi:10.3934/nhm.2010.5.385       Abstract        Full Text (688.0K)       Related Articles

Boris Andreianov - Laboratoire de Mathématiques de Besançon, Université de Franche-Comté, 25030 Besançon Cedex, France (email)
Frédéric Lagoutiére - Laboratoire de mathématiques, Université Paris-Sud, 91405 Orsay cedex, France (email)
Nicolas Seguin - UMR 7598 Laboratoire J.-L. Lions, UPMC Univ Paris 06, Paris, F-75005, France (email)
Takéo Takahashi - Institut Élie Cartan UMR 7502, INRIA, Nancy-Université, CNRS, 54506 Vandoeuvre-lès-Nancy Cedex, France (email)

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.

Keywords:  Solid-fluid interaction, Burgers equation, singular source term, adapted entropy, well-balanced scheme, random-choice method.
Mathematics Subject Classification:  Primary: 35F25, 35L80, 65M99.

Received: January 2010;      Revised: June 2010;      Published: July 2010.