Small solids in an inviscid fluid.

*(English)*Zbl 1262.35180Summary: We present 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.

##### MSC:

35Q35 | PDEs in connection with fluid mechanics |

35A35 | Theoretical approximation in context of PDEs |

35F25 | Initial value problems for nonlinear first-order PDEs |

35L80 | Degenerate hyperbolic equations |

65M08 | Finite volume methods for initial value and initial-boundary value problems involving PDEs |

74F10 | Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) |