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Mathematical analysis of the quasilinear effects in a hyperbolic model blood flow through compliant axi‐symmetric vessels
Author(s) -
Čanić Sunčica,
Kim Eun Heui
Publication year - 2003
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.407
Subject(s) - tapering , shock (circulatory) , mathematics , flow (mathematics) , boundary (topology) , partial differential equation , boundary value problem , mathematical analysis , hyperbolic partial differential equation , compressibility , radius , mechanics , geometry , physics , computer science , medicine , computer graphics (images) , computer security
In this paper, we present a mathematical analysis of the quasilinear effects arising in a hyperbolic system of partial differential equations modelling blood flow through large compliant vessels. The equations are derived using asymptotic reduction of the incompressible Navier–Stokes equations in narrow, long channels. To guarantee strict hyperbolicity we first derive the estimates on the initial and boundary data which imply strict hyperbolicity in the region of smooth flow. We then prove a general theorem which provides conditions under which an initial–boundary value problem for a quasilinear hyperbolic system admits a smooth solution. Using this result we show that pulsatile flow boundary data always give rise to shock formation (high gradients in the velocity and inner vessel radius). We estimate the time and the location of the first shock formation and show that in a healthy individual, shocks form well outside the physiologically interesting region (2.8m downstream from the inlet boundary). In the end we present a study of the influence of vessel tapering on shock formation. We obtain a surprising result: vessel tapering postpones shock formation. We provide an explanation for why this is the case. Copyright © 2003 John Wiley & Sons, Ltd.