Acta mathematica scientia, Series B >
A HYPERBOLIC SYSTEM OF CONSERVATION LAWS FOR FLUID FLOWS THROUGH COMPLIANT XISYMMETRIC VESSELS
Received date: 2009-11-14
Online published: 2010-03-20
Supported by
Gui-Qiang Chen's research was supported in part by the National Science Foundation under Grants DMS-0935967, DMS-0807551, DMS-0720925, and DMS-0505473, the Natural Science Foundation of China under Grant NSFC-10728101, and the Royal Society-Wolfson Research Merit Award (UK).
We are concerned with the derivation and analysis of one-dimensional hyperbolic systems of conservation laws modelling fluid flows such as the blood flow through compliant axisymmetric vessels. Early models derived are nonconservative and/or nonhomogeneous with measure source terms, which are endowed with infinitely many Riemann solutions for some Riemann data. In this paper, we derive a one-dimensional hyperbolic system that is conservative and homogeneous. Moreover, there exists a unique global Riemann solution for the Riemann problem for two vessels with arbitrarily large Riemann data, under a natural stability entropy criterion. The
Riemann solutions may consist of four waves for some cases. The system can also be written as a 3×3 system for which strict hyperbolicity fails and the standing waves can be regarded as the contact discontinuities corresponding to the second family with zero eigenvalue.
Key words：
conservation laws; hyperbolic system; fluid flow; blood flow; ; vessel; ; hyperbolicity; Riemann problem; Riemann solution; wave
curve; shock wave; ; rarefaction wave; ; standing wave; ; stability
Gui-Qiang G. Chen , Weihua Ruan . A HYPERBOLIC SYSTEM OF CONSERVATION LAWS FOR FLUID FLOWS THROUGH COMPLIANT XISYMMETRIC VESSELS[J]. Acta mathematica scientia, Series B, 2010 , 30(2) : 391 -427 . DOI: 10.1016/S0252-9602(10)60056-2
[1] Anliker M, Rockwell R, Ogden E. Nonlinear analysis of flow pulses and shock waves in arteries. Z Angew Math Phys,1971, 22: 217--246
[2] Canic S. Blood flow through compliant vessels after endovascular repair: wall deformations induced by the discontinuous wall properties. Comput Visualization Sci, 2002, 4: 147--1186
[3] Canic S, Kim E H. Mathematical analysis of the quasilinear effects in a hyperbolic model of blood flow through compliant axi-symmetric vessels. Math Meth Appl Sci, 2003, 26(14): 1161--1186
[4] Chen G -Q. Euler Equations and Related Hyperbolic Conservation Laws//Dafermos C M, Feireisl E, eds.
Handbook of Differential Equations: Evolutionary Differential Equations, Vol 2. Amsterdam: Elsevier Science, 2005: 1--104
[5] Chen G -Q, Slemrod M, Wang D. Isometric immersions and compensated compactness. Commun Math Phys, 2010, 294(2): 411--437
[6] Chen G -Q, Dafermos C M, Slemrod M, Wang D. On two-dimensional sonic-subsonic flow. Commun Math Phys, 2007, 271: 635--647
[7] Clark M E, Kufahl R H. Simulation of the cerebral macrocirculation//Cardiovascular Systems Dynamics. Cambridge, MA: MIT Press, 1978: 380--390
[8] Courant R, Friedrichs K O. Supersonic Flow and Shock Waves. New York: Springer-Verlag, 1948
[9] Dafermos C M. Hyperbolic Conservation Laws in Continuum Physics. 2nd ed. Berlin: Springer-Verlag, 2005
[10] Glimm J. Solutions in the large for nonlinear hyperbolic systems of equations. Comm Pure Appl Math, 1965, 18: 697--715
[11] Glimm J, Lax P D. Decay of solutions of systems of nonlinear hyperbolic conservation laws. Memoirs Amer Math Soc, 1970, 101
[12] Kufahl R H, Clark M E. A circle of Willis simulation using distensible vessels and pulsatile flow. J Biomech Eng, 1985, 107: 112--122
[13] Potter M C, Foss J F. Fluid Mechanics. New York: The Ronald Press Co, 1975
[14] Quarteroni A, Veneziani A, Zunino P. Mathematical and numerical modeling of solute dynamics in blood flow
and arterial walls. SIAM J Numer Anal, 2001/02, 39: 1488--1511
[15] Ruan W, Clark M E, Zhao M, Curcio A. A hyperbolic system of equations of blood flow in an arterial network. SIAM J Appl Math, 2003, 64(2): 637--667
[16] Ruan W, Clark M E, Zhao M, Curcio A. Global solutions to a hyperbolic problem arising in the modeling of blood flow in circulatory systems. J Math Anal Appl, 2007, 331(2): 1068--1092
[17] Smith N P, Pullan A J, Hunter P J. An anatomically based model of transient coronary blood flow in the heart.
SIAM J Appl Math, 2002, 62(3): 990--1018
[18] Smoller J. Shock Waves and Reaction-Diffusion Equations. 2nd ed. New York: Springer-Verlag, 1994
[19] Yuan S W. Foundation of Fluid Mechanics. Englewood Cliffs, N J: Prentice-Hall, Inc, 1967
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