The Riemann problem for a blood flow model in arteries. (English) Zbl 1476.35139
Summary: In this paper, the Riemann solutions of a reduced \(6 \times 6\) blood flow model in medium-sized to large vessels are constructed. The model is non-strictly hyperbolic and non-conservative in nature, which brings two difficulties of the Riemann problem. One is the appearance of resonance while the other one is loss of uniqueness. The elementary waves include shock wave, rarefaction wave, contact discontinuity and stationary wave. The stationary wave is obtained by solving a steady equation. We construct the Riemann solutions especially when the steady equation has no solution for supersonic initial data. We also verify that the global entropy condition proposed by C. Dafermos can be used here to select the physical relevant solution. The Riemann solutions may contribute to the design of numerical schemes, which can apply to the complex blood flows.
MSC:
| 35L65 | Hyperbolic conservation laws |
| 35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
| 35L60 | First-order nonlinear hyperbolic equations |
| 35L67 | Shocks and singularities for hyperbolic equations |
| 35Q92 | PDEs in connection with biology, chemistry and other natural sciences |
| 76N10 | Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics |
Keywords:
shock wave; rarefaction wave; contact discontinuity; stationary wave; Riemann problem; non-uniqueness; global entropy conditionReferences:
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