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

Author

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  • Spiller, C.
  • Toro, E.F.
  • Vázquez-Cendón, M.E.
  • Contarino, C.

Abstract

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 uR−uL 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.

Suggested Citation

  • Spiller, C. & Toro, E.F. & Vázquez-Cendón, M.E. & Contarino, C., 2017. "On the exact solution of the Riemann problem for blood flow in human veins, including collapse," Applied Mathematics and Computation, Elsevier, vol. 303(C), pages 178-189.
    • Handle: RePEc:eee:apmaco:v:303:y:2017:i:c:p:178-189
    DOI: 10.1016/j.amc.2017.01.024
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    References listed on IDEAS

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    1. Toro, Eleuterio F., 2016. "Brain venous haemodynamics, neurological diseases and mathematical modelling. A review," Applied Mathematics and Computation, Elsevier, vol. 272(P2), pages 542-579.
    2. Strocchi, M. & Contarino, C. & Zhang, Q. & Bonmassari, R. & Toro, E.F., 2017. "A global mathematical model for the simulation of stenoses and bypass placement in the human arterial system," Applied Mathematics and Computation, Elsevier, vol. 300(C), pages 21-39.
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    Cited by:

    1. Anco, Stephen C. & Garrido, Tamara M. & Márquez, Almudena P. & Gandarias, María L., 2023. "Exact solutions and conservation laws of a one-dimensional PDE model for a blood vessel," Chaos, Solitons & Fractals, Elsevier, vol. 170(C).
    2. Krivovichev, Gerasim V., 2022. "Comparison of inviscid and viscid one-dimensional models of blood flow in arteries," Applied Mathematics and Computation, Elsevier, vol. 418(C).

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