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Small solids in an inviscid fluid. (English) Zbl 1262.35180
Summary: 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.)
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