  Blood flow in arterial systems can be described by the three-dimensional
Navier-Stokes equations within a time-dependent spatial domain that accounts
for the elasticity of the arterial walls. In this article blood is treated as
an incompressible Newtonian fluid that flows through compliant vessels of
general cross section. A three-dimensional semi-implicit finite difference and
finite volume model is derived so that numerical stability is obtained at a low
computational cost on a staggered grid. The key idea of the method consists in
a splitting of the pressure into a hydrostatic and a non-hydrostatic part,
where first a small quasi-one-dimensional nonlinear system is solved for the
hydrostatic pressure and only in a second step the fully three-dimensional
non-hydrostatic pressure is computed from a three-dimensional nonlinear system
as a correction to the hydrostatic one. The resulting algorithm is robust,
efficient, locally and globally mass conservative and applies to hydrostatic
and non hydrostatic flows in one, two and three space dimensions. These
features are illustrated on nontrivial test cases for flows in tubes with
circular or elliptical cross section where the exact analytical solution is
known. Test cases of steady and pulsatile flows in uniformly curved rigid and
elastic tubes are presented. Wherever possible, axial velocity development and
secondary flows are shown and compared with previously published results.
