Existence of solutions and continuous and semi-discrete stability estimates for 3D/0D coupled systems modelling airflows and blood flows - Archive ouverte HAL Accéder directement au contenu
Article Dans Une Revue ESAIM: Mathematical Modelling and Numerical Analysis Année : 2021

Existence of solutions and continuous and semi-discrete stability estimates for 3D/0D coupled systems modelling airflows and blood flows

Résumé

In this paper we analyse geometric multiscale models arising in the description of physiological flows such as blood flow in arteries or air flow in the bronchial tree. The geometrical complexity of the networks in which air/blood flows lead to a classical decomposition in two areas: a truncated 3D geometry corresponding to the largest contribution of the domain, and a 0D part connected to the 3D part, modelling air/blood flows in smaller airways/vessels. The fluid in the 3D part is described by the Stokes or the Navier–Stokes system which is coupled to 0D models or so-called Windkessel models. The resulting Navier–Stokes–Windkessel coupled system involves Neumann non-local boundary conditions that depends on the considered applications. We first show that the different types of Windkessel models share a similar formalism. Next we derive existence results and stability estimates for the continuous coupled Stokes–Windkessel or Navier–Stokes–Windkessel problem as well as stability estimates for the semi-discretized systems with either implicit or explicit treatment of the boundary conditions. In all the calculations, we pay a special attention to the dependency of the various constants and smallness conditions on the data with respect to the physical and numerical parameters. In particular we exhibit different kinds of behavior depending on the considered 0D model. Moreover even if no energy estimates can be derived in energy norms for the Navier–Stokes–Windkessel system, leading to possible and observed numerical instabilities for large applied pressures, we show that stability estimates for both the continuous and semi-discrete problems, can be obtained in appropriate norms for small enough data by introducing a new well chosen Stokes-like operator. These sufficient stability conditions on the data may give a hint on the order of magnitude of the data enabling stable computations without stabilization method for the problem. Numerical simulations illustrate some of the theoretical results.
Fichier principal
Vignette du fichier
m2an210004.pdf (1.82 Mo) Télécharger le fichier
Origine : Publication financée par une institution

Dates et versions

hal-03393268 , version 1 (21-10-2021)

Identifiants

Citer

Céline Grandmont, Sébastien Martin. Existence of solutions and continuous and semi-discrete stability estimates for 3D/0D coupled systems modelling airflows and blood flows. ESAIM: Mathematical Modelling and Numerical Analysis, 2021, 55 (5), pp.2365-2419. ⟨10.1051/m2an/2021055⟩. ⟨hal-03393268⟩
91 Consultations
55 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More