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Flow in Collapsible Tubes with Discontinuous Mechanical Properties: Mathematical Model and Exact Solutions

Published online by Cambridge University Press:  03 June 2015

Eleuterio F. Toro  and
Annunziato Siviglia
Eleuterio F. Toro*
Affiliation:
Laboratory of applied Mathematics, University of Trento, Via Mesiano 77, 1-38100 Trento, Italy
Annunziato Siviglia*
Affiliation:
Laboratory of applied Mathematics, University of Trento, Via Mesiano 77, 1-38100 Trento, Italy
*
Email:toroe@ing.unitn.it
Corresponding author.Email:nunzio.siviglia@ing.unitn.it
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Abstract

We formulate a one-dimensional time-dependent non-linear mathematical model for some types of physiological fluid flow in collapsible tubes with discontinuous material properties. The resulting 6 × 6 hyperbolic system is analysed and the associated Riemann problem is solved exactly. Although the solution algorithm deals with idealised cases, it is nonetheless uniquely well-suited for assessing the performance of numerical methods intended for simulating more general situations. Moreover, our model may be a useful starting point for numerical calculations of realistic flows involving rapid and discontinuous material property variations. One important example in mind is the simulation of blood flow in medium-to-large veins in humans. Finally, we also discuss some peculiarities of the model regarding the loss of strict hyperbolicity and uniqueness. In particular we show an example in which the solution of the Riemann problem is non unique.

Keywords

76Z05Collapsible tubesphysiological flowsblood flowsRiemann problem
Type
Research Article
Information
Communications in Computational Physics , Volume 13 , Issue 2 , February 2013 , pp. 361 - 385
Copyright
Copyright © Global Science Press Limited 2013

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