Fluid-kinetic modelling for respiratory aerosols with variable size and temperature

Abstract

In this paper, we propose a coupled fluid-kinetic model taking into account the radius growth of aerosol particles due to humidity in the respiratory system. We aim to numerically investigate the impact of hygroscopic effects on the particle behaviour. The air flow is described by the incompressible Navier-Stokes equations, and the aerosol by a Vlasov-type equation involving the air humidity and temperature, both quantities satisfying a convection-diffusion equation with a source term. Conservations properties are checked and an explicit time-marching scheme is proposed. Twodimensional numerical simulations in a branched structure show the influence of the particle size variations on the aerosol dynamics.

Global existence of weak solutions to the incompressible Vlasov–Navier–Stokes system coupled to convection–diffusion equations

“…Let us first present the model we investigate in this paper. It extends the model proposed in [6] to a time-dependent domain. Let τ > 0 and Ω 0 ⊂ R 3 be an open bounded domain with Lipschitz boundary.…”

“…is bounded from below by a positive constant, and those assumptions are physically relevant in the temperature range for which the model is expected to hold. Note that in [6], the numerical simulations lead to a variation of D v of only 2%. We prescribe the Neumann boundary condition.…”

“…Similarly, we multiply (2) by m(r)|v| 2 /2 and integrate by parts over Π t . Using (1), (5), (6), (11), and (12), we obtain…”

“…The model studied in this article describes the transport and deposition of a therapeutic aerosol evolving in a Newtonian viscous incompressible fluid, the air, in the first generations of the human bronchial tree. The study of this phenomenon related to respiration began with [2] and [15] and the system we analyze in this paper was presented in [6] in a fixed domain. Numerous works have contributed to this study for several decades while taking into account various parameters and physical processes such as the compressibility or viscosity of the fluid, the transport of the particles, the interaction between the particles and the fluid or the walls, etc.…”

“…Another static point of view is assumed in [21] to tackle these hygroscopic effects: they are considered globally in space and are described by a differential equation on the mass fraction of the water vapor in the air. This model is made more precise in [6] by taking into account the fact that these effects are local and time-dependent, and leads to the present study.…”