> C. Ruyer-Quil
Xiaofei Wang (DALEMBERT)
Direct computing of the fluid-structure interaction in a large network of blood vessels is still impossible. In the assumption of long wave length, a 1D model has been presented.
In the 1D governing equations, the primary unknowns are the flow rate Q, transmural pressure P and the cross section area A. There are two partial differential equations (PDE) which account the conservation of mass and momentum. These two PDE's are supplemented by the constitutive relation of the vessel wall, which relates P and A. In our study, a Kelvin-Voigt viscoelastic constitutive relation was adopted. After insertion of this relation into the PDE's, we obtained a nonlinear hyperbolic-parabolic system, which has similar properties with shallow water waves with a diffusive effect. For the numerical resolution, we considered four schemes, namely, MacCormack, second order Finite volume, Taylor-Galerkin and discontinuous Galerkin. We verified all of the schemes by asymptotic analytic results in the case with a small nonlinearity (small amplitude of wave). In the verification part, several behaviours of the wave were considered: propagation in a uniform tube, attenuation of the amplitude due to the skin friction, diffusion due to the viscosity of the wall, and reflection and transmission at a branching point. Moreover, the schemes were also tested in case with a larger nonlinearity. Thereafter, we applied all of the schemes on a relatively realistic arterial system with 55 arteries. The schemes were compared in four aspects: the accuracy, the ability to capture shock phenomena, the computation speed and the complexity of the implementation. The suitable conditions for the applications of the various schemes were discussed at the final.