Coupling Euler and Vlasov Equations in the Context of Sprays: The Local-in-Time, Classical Solutions

Abstract

Sprays are complex flows made of liquid droplets surrounded by a gas. They can be modeled by introducing a system coupling a kinetic equation (for the droplets) of Vlasov type and a (Euler-like) fluid equation for the gas. In this paper, we prove that, for the so-called thin sprays, this coupled model is well-posed, in the sense that existence and uniqueness of classical solutions holds for small time, provided the initial data are sufficiently smooth and their support have suitable properties.

Asymptotic-preserving schemes for kinetic-fluid modeling of disperse two-phase flows

“…0 in (1)-(3) is due to [29] by means of relative entropy methods, see also [30]. It is also worth mentioning related works like the local existence of smooth solutions for the case without velocity-diffusion [31], several studies of coupling with the Euler system (i.e. viscosity is sensible only at the scale of the particles) [23,32,33] and systems with energy exchanges [34][35][36].…”

Simulation of fluid–particles flows: heavy particles, flowing regime, and asymptotic-preserving schemes

“…introduce different levels of difficulty. A first attempt was to prove the local existence of smooth solutions; see [3,38] for the analysis of the Euler-Vlasov systems. Another approach concerning classical solutions restricts to solutions close to the equilibrium by using perturbation techniques and energy estimates [28,11].…”

Simulation of fluid and particles flows: Asymptotic preserving schemes for bubbling and flowing regimes

“…The code can be adapted to treat density-dependent viscosity models, as in (2). Reproducing the Hilbert expansion reasoning, we get f ð0Þ ðt; x; nÞ ¼ qðt; xÞMðnÞ; f ð1Þ ðt; x; nÞ ¼ ÀnMðnÞ Therefore, the limit system as !…”

“…A rigorous existence result for a coupling involving the compressible Navier-Stokes equations (which means that the dissipative term lDu is added in the fluid equation in (1)) is established in [37] while the analysis of its asymptotic limit is performed in [38]. The local well-posedness of smooth solutions for the system (1) was investigated in [2] while asymptotic problems and stability properties are studied in [7].…”