Commun. Comput. Phys., 13 (2013), pp. 361-385.


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

Eleuterio F. Toro 1, Annunziato Siviglia 1*

1 Laboratory of applied Mathematics, University of Trento, Via Mesiano 77, I-38100 Trento, Italy.

Received 21 June 2011; Accepted (in revised version) 24 February 2012
Available online 28 June 2012
doi:10.4208/cicp.210611.240212a

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 6x6 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.

AMS subject classifications: 76Z05

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Key words: Collapsible tubes, physiological flows, blood flows, Riemann problem.

*Corresponding author.
Email: toroe@ing.unitn.it (E. F. Toro), nunzio.siviglia@ing.unitn.it (A. Siviglia)
 

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