Spray impingement on a wall in the context of the upper airways

ESAIM: PROCEEDINGS, June 2008, Vol. 23, p. 1-9 L. Boudin, C. Grandmont, Y. Maday, B. Maury, B. Sapoval & J.-F. Gerbeau, Editors SPRAY IMPINGEMENT ON A WALL IN THE CONTEXT OF THE UPPER AIRWAYS ∗ Laurent Boudin 1, 2 and Lisl Weynans 1, 3 Abstract. We here address the modelling of an aerosol hitting the walls of the airways or an endotracheal tube used for a mechanical ventilation, and the possible creation of secondary droplets that may follow. We present a kinetic modelling of the spray-wall interaction and propose a boundary term that takes into account the possible formation of secondary droplets. Next we answer the following question: when, modelling the delivery of solute therapeutic aerosols, is it necessary to take into account the apparition of secondary droplets? A study of empirical models of drop-wall interaction allows us to conclude that in usual respiratory conditions, no secondary droplets appear. Finally, we perform numerical simulations of an aerosol delivered in an endotracheal tube, in the mechanical ventilation case. The idea is to compare our numerical results to in silico experiments from aerosols specialists. We study the trajectories and the deposition locations of spray droplets. 1. Introduction In this work, we address the modelling of the aerosol impingement on walls, in the context of human breathing. We are especially interested in the human upper airways, where the air behaviour is classically modelled by the incompressible Navier-Stokes equations, see for example [12]. Aerosols are used as medical treatment for many respiratory diseases. When they are liquid, which is the case we aim to study here, they are sent into the airways to the lungs by jet nebulizers. The aerosol can be delivered directly in the upper airways (trachea, larynx, pharynx), or through an endotracheal tube in case of mechanically ventilated patients. Part of the aerosol is lost by deposition on the wall. The treatment efficiency is obviously related to the amount of the aerosol which reaches the lung. To estimate which amount of the aerosol deposits on the endotracheal tube or the upper airways, and which amount actually reaches the respiratory tract, in vitro studies, such as [30], can be complemented by numerical simulations. There are many ways to model an aerosol. If there are very few particles, one can treat each particle as a numerical individual and perform direct numerical simulations on it, as it is possible for particles in blood flows. Unfortunately, the medical aerosols are constituted of numerous particles, so that direct numerical simulations are too expensive. It then seems quite natural to use a kinetic model for the aerosols [3,11]. As a matter of fact, that kind of fluid, which is in the frame of rarefied gases, allows to consider particles from a statistical point of view. ∗ This work has been funded by the ACI lepoumonvousdisje (resp. B. Maury) and by the INRIA Paris-Rocquencourt project team REO (resp. J.-F. Gerbeau). 1 Univ. Pierre et Marie Curie, Lab. J.-L. Lions, 175 rue du Chevaleret, BC 187, F-75013 Paris, France; e-mail: 2 INRIA Paris-Rocquencourt, REO Project team, BP 105, F-78153 Le Chesnay Cedex, France 3 Univ. Bordeaux 1, IMB, 351 cours de la Libration, F-33405 Talence Cedex, France; e-mail: c EDP Sciences, SMAI 2008 Article published by EDP Sciences and available at http://www.edpsciences.org/proc or http://dx.doi.org/10.1051/proc:082301 2 ESAIM: PROCEEDINGS As already stated, this work is more specifically dedicated to the study of a particle impingement on the wall of the human airways. Droplets impinging on a solid surface (dry wall or thin fluid film) usually deposit or splash. In the latter case, secondary droplets are ejected from the wall. The boundary between deposition and splashing has been widely investigated in the last years, both empirically and theoretically. It is generally described using a parameter K, function of the Ohnesorge and Weber numbers of the impinging droplet [5, 6, 23, 25, 29]: K = We Oh−2/5 , with We = ̺DV02 , σ Oh = µ , (̺σD)1/2 where ̺, σ and µ respectively denote the density, surface tension and viscosity of the droplet, D its diameter and V0 its velocity. If K exceeds a threshold value Ks , then the splashing occurs. Sometimes, the Weber number of the droplet is used as the splashing parameter instead of K [16, 31]. For more details on the general droplet impingement phenomenon, we refer to the reviews of Rein [27] and Yarin [33]. Most models were built from observations made in the case of a one-droplet impingement, where the incoming velocity is normal to the wall [31,32]. However, some authors recently began to take into account the interactions between impinging droplets that exist in the case of a spray, as well as the effect of an inclined impingement. We prefered Kalantari and Tropea’s model [17], because it is, to our knowledge, the only one established from real spray impingement observations. In particular, it accounts for inclined impingements. Details about this model and two other ones [18, 24], quite comprehensive, are given in the appendix. Note that most existing models for secondary droplets are reviewed in [5]. Since the aerosol evolves in an ambient fluid (the air), there may be some interactions between them. A common assumption is that the aerosol has no effect on the air, in the context of the airways. Hence, the aerosol may be considered as a thin or a very thin spray. The difference between these two kinds of spray lies in the action of the aerosol on the air. We can assume, as in [11], that the aerosol is thin, i.e. it does globally act on the air. The numerical context of [11] mimics O’Rourke’s [26] and uses the Kiva-3V code [1]. Note that if we assumed that the aerosol was very thin, we would not eventually need a specific equation for the particles, for they would instantaneously get the air velocity. Hence we get a strongly coupled Vlasov-Navier-Stokes system, which is further detailed in Subsectn. 2.1. Such coupled fluid-kinetic models have been studied a lot recently from an analytical point of view. For instance, Hamdache [13] tackled the existence problem for the Vlasov-Stokes system. Baranger and Desvillettes [4] proved the existence of classical solutions in small times for the Vlasov-Euler system. And more recently, Mellet and Vasseur [20] studied the existence of global weak solutions to the Vlasov-Fokker-Planck equations coupled with the Navier-Stokes equations. There have been lots of numerics dedicated works too: see [2,14,15,19] for instance. Our model is further discussed in the next two sections, then we address the effect of the impingement of a droplet on a wall, and eventually we present a numerical test in an endotracheal tube. 2. Aerosol modelling 2.1. Fluid-kinetic modelling In the same way fluids are classically described by the Navier-Stokes equations, kinetic models are often used to model the behaviour of a set of numerous particles (...truncated)