Biomedical Engineering Reference
In-Depth Information
Fig. 2.2.
Structure of the solution of the Riemann problem (2.24) for the simplified 3
×
3 blood
flow model of this paper
2.4.1 Jump across shocks: Rankine-Hugoniot conditions
An important feature of the proposed model for variable vessel properties is that
the system cannot be expressed in conservation-law form. However, for the case
of the Riemann problem the vessel properties
h
0
and
E
, and thus
K
, are constant
across the non-linear waves. Hence, across the genuinely non-linear characteristic
fields (rarefactions and shocks) it is possible to express the equations in conservation-
law form. In fact it is sufficient to consider the reduced 2
2 conservative system,
excluding the equation for
K
in (2.12). The homogeneous part of the equations in
conservation-law form are
×
∂
t
Q
+
∂
x
F
(
Q
)=
0
,
(2.25)
in terms of the redefined vector of conserved variables
q
1
q
2
A
Au
Q
=
≡
(2.26)
and flux vector
f
1
f
2
BA
m
+
1
Au
F
(
Q
)=
≡
,
(2.27)
Au
2
+
with
mK
B
=
:
constant
.
(2.28)
A
0
ρ
(
m
+
1
)
Proposition 4.1 (left shock).
If the left
λ
1
-wave is a left-facing shock wave of speed
S
L
then
B
L
(
A
m
+
1
A
m
+
1
L
A
∗
L
−
A
L
)(
∗
L
−
)
u
∗
L
=
u
L
−
f
L
,
f
L
=
(2.29)
A
L
A
∗
L
