A coupled particle-continuum model: well-posedness and the limit of zero radius

Abstract

This paper treats a simplified model of combustion that takes into account only the exchange of momentum between the fuel particles and ambient gas. Even in the absence of particles there are no rigorous mathematical results available in three dimensions, so our attention is restricted to the two-dimensional case. Treating the particles as points leads to mathematical problems that are circumvented by an appropriate regularization. The latter sections of the paper study the behaviour of the model as this regularization parameter tends to zero under some appropriate scalings, showing that in the limit the standard Navier-Stokes equations are obtained, and that for numerical (Galerkin) approximations the paths of the particles will approach fluid streamlines.