Biomedical Engineering Reference
In-Depth Information
Proposition 4.3 (left rarefaction wave).
Across a left rarefaction wave associated
with the characteristic field
λ
1
=
u
−
c
the following relations hold
2
m
(
u
∗
L
=
u
L
−
f
L
,
f
L
=
c
∗
L
−
c
L
)
.
(2.35)
Proof.
From the left generalized Riemann invariants (2.21) we can write
2
m
c
∗
L
+
2
m
c
L
+
u
∗
L
=
u
L
,
from which we have
2
m
c
∗
L
−
2
m
c
L
)
u
∗
L
=
u
L
−
(
and the result follows.
Proposition 4.4 (right rarefaction wave).
Across a right rarefaction wave associ-
ated with the characteristic field
λ
=
u
+
c
the following relations hold
3
2
m
(
u
∗
R
=
u
R
+
f
R
,
f
R
=
c
∗
R
−
c
R
)
.
(2.36)
Proof.
The proof uses the right generalized Riemann invariants (2.23) and is entirely
analogous to the previous case.
2.4.3 Jump conditions across the stationary contact
We wish to establish jump conditions across the stationary contact discontinuity as-
sociated with the eigenvalue
0. As stated earlier, for variable material properties
it is not possible to express the equations in conservation-law form and therefore it
is not possible to apply the classical Rankine-Hugoniot conditions. Thus to establish
jump conditions we follow two alternative approaches, leading to identical results.
λ
2
=
Proposition 4.5.
Across the contact discontinuity the following relations hold
1
2
ρ
u
2
Au
=
constant
,
+
ψ
=
constant
,
(2.37)
leading to
1
2
ρ
1
2
ρ
u
2
∗
u
2
∗
A
∗
L
u
∗
L
=
A
∗
R
u
∗
R
,
+
ψ
∗
L
=
+
ψ
∗
R
.
(2.38)
L
R
Proof.
We first apply generalized Riemann invariants across the linearly degenerate
field. This approach was advocated by Embid and Baer [6] to analyse the Baer-
Nunziato equation for two-phase compressible flow, a well-known non-conservative
system. We obtain
dA
d
(
Au
)
dK
1
.
−
c
2
=
=
(2.39)
ψ
K
u
2
A
ρ
0
