Mathematical and Numerical Study of a Dusty Knudsen Gas Mixture: Extension to Non-spherical Dust Particles

Abstract

In this work, we consider the model introduced in [7] describing the movement of dust particles in a very rarefied atmosphere. The gas is treated as a Knudsen gas, whereas the interaction between dust particles and gas molecules is modeled by considering a moving domain free transport equation (including the boundary with the particles and the boundary of the domain). We here precise the proof of existence of solutions to the initial-boundary value problem annonced in [7]. Moreover, we introduce a new numerical strategy, based on a splitting between the transport of the gas molecules and the movement of the boundary. This strategy allow to perform 2d-numerical simulations with elliptical-shaped particles.

Mathematical and numerical study of a kinetic model describing the evolution of planetary rings

“…We are now ready to employ what we have previously presented to obtain an existence and uniqueness result for System (1)-( 5)- (6).…”

“…In this section we describe the numerical method used for the simulation of equations ( 1)-( 5)-( 6) in two dimensions in space and velocity. Our procedure, based on a particle method and a splitting strategy, is an evolution of the approach introduced in [6]. The substantial difference of our problem from the one studied in [6] is that the dust particles in planetary rings, unlike gas molecules, being affected by the gravitational acceleration due to the planet and moons, satisfy the Vlasov equation ( 1), which is more complicated to deal with than the free transport equation.…”

“…Our procedure, based on a particle method and a splitting strategy, is an evolution of the approach introduced in [6]. The substantial difference of our problem from the one studied in [6] is that the dust particles in planetary rings, unlike gas molecules, being affected by the gravitational acceleration due to the planet and moons, satisfy the Vlasov equation ( 1), which is more complicated to deal with than the free transport equation. On the other hand, the boundary condition are, in this problem, particularly easy to handle, because the surfaces of the satellites are modelled with an absorbing boundary: the particles that collide with it disappear from the problem's domain.…”

“…The next step consists in the reconstruction of the density f n+1 N 0 ,ε . As in [6], we use Bsplines of 3-order in two space dimensions as shape functions ϕ ε , with shape sizes…”

“…The precise assumptions that justify the mathematical structure of the model are described in detail in the next section. At this point, we recall that the mathematical study of kinetic equations in evolutionary domains is still in its early stages (see, for example, the pioneering paper [1] and [7,6,9,25]).…”