Biomedical Engineering Reference
In-Depth Information
and the shock speed is given as
B
L
A
L
A
∗
L
(
A
m
+
1
A
m
+
1
L
M
L
A
L
,
∗
L
−
)
S
L
=
u
L
−
M
L
=
.
(2.30)
A
∗
L
−
A
L
Proof.
In Fig. 2.2 we illustrate the function
f
L
that connects the velocity
u
∗
L
to the left
data state and the unknown
A
∗
L
. Let us assume that the left
λ
1
-wave is a left-facing
shock wave of speed
S
L
. We need to establish relations across the shock, for which
one uses standard techniques, see [21] and [22]. We first transform the equations to
a stationary frame via
=
−
,
=
−
.
u
L
u
L
S
L
u
∗
L
u
∗
L
S
L
(2.31)
Then the jump conditions become
A
∗
L
u
∗
L
=
A
L
u
L
,
(2.32)
A
∗
L
u
2
∗
L
B
L
A
m
+
1
∗
L
A
L
u
L
+
B
L
A
m
+
1
L
+
=
.
Now from the first equation in (2.32) define
M
L
=
−
A
L
u
L
. In fact this
is the mass flux through the wave, which is constant. Use of
M
L
into the second
equation in (2.32) followed by suitable manipulations leads to the sought relations
(2.29). Details of the calculation of the shock speed
S
L
are omitted.
A
∗
L
u
∗
L
=
−
Proposition 4.2 (right shock).
If the right
λ
3
-wave is a right-facing shock wave of
speed
S
R
then
B
R
(
A
m
+
1
∗
A
m
+
1
R
A
∗
R
−
A
R
)(
−
)
R
u
∗
R
=
u
R
+
f
R
,
f
R
=
(2.33)
A
R
A
∗
R
and the shock speed is given as
B
R
A
R
A
∗
R
(
A
m
+
1
A
m
+
1
R
M
R
A
R
,
∗
R
−
)
S
R
=
u
R
+
M
R
=
.
(2.34)
A
∗
R
−
A
R
Proof.
The proof follows the same methodology as for a left shock and details are
thus omitted.
2.4.2 Jump conditions across rarefactions
It is possible to establish jump relations across rarefactions waves by means of gen-
eralized Riemann invariants introduced in Sect. 3.3.
