We consider the coupling between three-dimensional
(3D) and one-dimensional (1D) fluid-structure interaction
(FSI) models describing blood flow inside compliant vessels.
The 1D model is a hyperbolic
system of partial differential equations.
The 3D model consists of the Navier-Stokes equations
for incompressible Newtonian fluids coupled with
a model for the vessel wall dynamics. A non standard formulation
for the Navier-Stokes equations is adopted to
have suitable boundary conditions for the coupling
of the models. With this we derive an energy estimate
for the fully 3D-1D FSI coupling. We consider several possible
models for the mechanics of the vessel wall in the 3D problem
and show how the 3D-1D coupling depends on them.
Several comparative numerical tests illustrating the coupling are presented.