The arterial system is a complex network of viscoelastic vessels, with complex geometries, nonlinear elastic properties. Additionally the blood rheology is complex and the blood has generally non-Newtonian properties. Pressure and flow waves are set up by the heart and interacts with the arterial system. Thus, the arterial system will continuously modify the pressure and flow waves from the heart.
In this section the 1D governing equations for pressure and flow waves will be derived on a basis where most of the complicating factors mentioned above will be discarded.
The first the section 7.1 Poiseuille flow in a compliant vessel, we consider Poiseuille flow in a compliant vessel, i.e., only a simplified version of the momentum equation is included, whereas convective terms, and thus wave propagation phenomena, are not accounted for. However, an important feature for flow in compliant vessels, namely chocked flow, is illustrated with this simple model.
The following sections (7.2 Infinitesimal derivation of the 1D governing equations for a compliant vessel) and (7.3 Integral derivation of the 1D governing equations for a compliant vessel) are devoted to derivation of 1D governing equations for wave propagation in compliant vessels. In the section 7.2 Infinitesimal derivation of the 1D governing equations for a compliant vessel a flat velocity profile is assumed, whereas a velocity profile is accounted for in section (7.3 Integral derivation of the 1D governing equations for a compliant vessel).