Continuous and semi-discrete stability estimates for 3d/0d coupled systems modelling airflows and blood flows

Céline Grandmont (1, 2) , Sébastien Martin (3)

1 LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions
2 REO - Numerical simulation of biological flows
3 MAP5 - UMR 8145 - Mathématiques Appliquées Paris 5

In this paper we analyse 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 leads 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 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 coupling. In all the calculations, we pay a special attention to the dependancy 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 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.

Mots clés: Airflows; Energy estimates; Navier-Stokes; 3D-0D coupling schemes; Bloodflows; Stability analysis; Windkessel models

Domaines: Equations aux dérivées partielles [math.AP]; Analyse numérique [math.NA]; Physiologie [q-bio.TO]

Céline Grandmont, Sébastien Martin. Continuous and semi-discrete stability estimates for 3d/0d coupled systems modelling airflows and blood flows. 2018. ⟨hal-01734706v2⟩