Riemann problem for a particle–fluid coupling

Abstract

We present a model of coupling between a pointwise particle and a compressible inviscid fluid following the Euler equations. The interaction between the fluid and the particle is achieved through a drag force. It writes as the product of a discontinuous function and a Dirac measure. After defining the solution, we solve the Riemann problem with a fixed particle for arbitrary data. We exhibit a set of condition on the drag force under which there exists a unique self-similar solution.

Analysis of the Riemann Problem for a Shallow Water Model with Two Velocities

“…Let us first precise the similarities between the shallow water model with two velocity (1) and two other models mentioned in the Introduction, i.e. the model for shear shallow flow introduced in [22] and extended in 2D in [11] and the bilayer version of the layerwise model for hydrostatic flows introduced in [4].…”

“…Let us now consider the hyperbolicity of the shallow water model with two velocities (1). It can be written in quasi linear form…”

“…We refer to Section 3.3 in this case. We are able to propose a definition of the nonconservative products based on a regularization of the water depth inside the shock, see [1,15]. In the case of an isolated λ L -shock wave, the water depth is given by…”

“…Setting the standard deviation to zero, one retrieves the classical shallow water model for which the solution of the planar Riemann problem is well known [25]. The solution of the planar Riemann problem (1)- (2) is a major improvement in the understanding of Model (1). In particular it is closely related to the implementation of the finite volume method on two dimensional meshes since it can be used to construct Godunov type schemes [13,16,25].…”

“…One can for example use the theory developed in [8]. Here it is possible to use the simpler approach developed in [1,15] that gives a path-independent definition to the nonconservative products. Furthermore, it is well known that such a nonconservative system may lead to resonant solutions of the Riemann problem [7,12,14].…”