Biomedical Engineering Reference
In-Depth Information
20
0.01
10
0.005
0
0
−0.005
−10
−0.01
−20
−0.2
0
0.2
−0.2
0
0.2
x [m]
x [m]
8
x 10
−3
6
0
4
−100
2
−200
0
−0.2
0
0.2
−0.2
0
0.2
x [m]
x [m]
Fig. 2.6.
Exact solution of tube-collapse Riemann problem at time 0
008
s
with initial discontinuity
is positioned at
x
0
=
0
.
0 m. Initial conditions are:
A
L
=
1
.
2
×
10
−
4
m
2
,
u
L
=
−
20
.
0m/s,
A
R
=
1
.
2
×
10
−
4
m
2
,
u
R
=
20
.
0m/s
.
the tail of the rarefaction coalesces with the blood/no blood front. Fig. 2.6 shows a
typical solution constructed in Proposition 6.3 in which the middle portion of the
vessel becomes dry as time evolves.
2.7 Conclusions
A simple model for blood flow in arteries with discontinuous material properties has
been presented. The equations have been analysed and exact solutions have been
derived, including tube collapse. These solutions can be useful in assessing the per-
formance of numerical methods for solving more realistic problems. The formula-
tion presented also holds promise as the basis for numerical methods for treating the
more general initial-boundary value problem. Current work includes the formulation
of models with more physical parameters with discontinuous variations, a detailed
study of the resonance phenomenon and the implementation of numerical methods.
Acknowledgements.
This research has been partially funded by the Italian Ministry of University
and Research (MIUR) under the project PRIN 2007.
