On the exact solution of the Riemann problem for blood flow in human veins, including collapse. (English) Zbl 1411.76182
Summary: 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 \(u_R - u_L\) 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.
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