On the exact solution of the Riemann problem for blood flow in human veins, including collapse

We solve exactly the Riemann problem for the non-linear hyperbolic system governing blood flow in human veins and note that, as modeled here, veins do not admit complete collapse, that is zero cross-sectional area A. This means that the Cauchy problem will not admit zero cross-sectional areas as initial condition. In particular, rarefactions and shock waves (elastic jumps), classical waves in the conventional Riemann problem, cannot be connected to the zero state with A=0. Moreover, we show that the area A* between two rarefaction waves in the solution of the Riemann problem can never attain the value zero, unless the data velocity difference uRuL tends to infinity. This is in sharp contrast to analogous systems such as blood flow in arteries, gas dynamics and shallow water flows, all of which admitting a vacuum state. We discuss the implications of these findings in the modelling of the human circulation system that includes the venous system.