Biomedical Engineering Reference
In-Depth Information
The second member of the above equation states immediately that across the contact
discontinuity
d
constant
, proving the first result in (2.37).
Relations (2.39) also state that across the contact wave, see Fig. 2,
A
and
K
do change.
Equating the first and third members gives
(
Au
)=
0 and thus
Au
=
ψ
A
ρ
K
=
dA
.
(2.40)
u
2
−
c
2
0 just proved and
c
2
A
ρ
ψ
A
from the definition of wave speed we
Using
d
(
Au
)=
=
may write
ψ
K
dK
=
−
ρ
udu
−
ψ
A
dA
,
(2.41)
or
ρ
udu
+
ψ
A
dA
+
ψ
K
dK
.
(2.42)
But
ψ
=
ψ
(
A
;
K
)
and thus
d
ψ
=
ψ
A
dA
+
ψ
K
dK
. Therefore
ρ
udu
+
d
ψ
=
0, which
after integration gives the second sought result in (2.37).
Now we adopt the thin-layer approach advocated by Schwendemann et al. [19],
also used to analyse the Baer-Nunziato equations; see also [2]. It is assumed that the
transition layer containing the contact discontinuity is vanishingly thin and that the
solution is smooth within the layer. Assuming the layer travels with constant speed
S
we define the independent variable
ξ
=
x
−
St
,
S
=
constant
,
(2.43)
which measures distance across the layer. We now study the governing equations
locally. For any function
G
(
x
,
t
)
we have
∂
G
x
=
∂
G
∂ξ
∂ξ
∂
∂
G
∂
t
=
∂
G
∂ξ
∂ξ
∂
x
,
t
.
(2.44)
∂
Then the continuity equation (2.1) gives
∂
A
∂ξ
∂ξ
∂
t
+
∂
Au
∂ξ
∂ξ
∂
x
=
0
,
(2.45)
or
d
((
u
−
S
)
A
)=
0
(2.46)
and thus with
S
0, which is the first sought result in (2.37).
Analogous manipulations for the momentum equation give
=
0 we obtain
d
(
Au
)=
1
2
u
2
+
ψ
=
ρ
constant
,
which is the second sought result in (2.37) and the result is thus proved.
It is worth noting that conditions (2.37) are identical to those proposed by [7] in
the context of treating the discontinuous coefficients case by a domain decomposi-
tion approach. They derived the same conditions adopting an energetic approach.
