Geology Reference
In-Depth Information
simplicity, we will take the position of an observer moving with the shock
speed, so that the model
u u/ x =
2
u/ x
2
(6.90)
applies locally in that coordinate system. We can certainly rewrite Equation
6.90 in the conservation form
{1/2 u
2
- u/ x}/ x = 0 (6.91)
If we integrate from one side of the shock to the other, where each side is
represented by uniform thermodynamic conditions with vanishing u/ x's, it is
clear that {1/2 u
2
-
u/ x}
upstream
= { 1 / 2 u
2
-
u/ x}
downstream
, or more
conveniently,
u
2
-
= u
2
+
(6.92)
This “jump condition” is analogous to the global mass conservation constraint
enforced through the Buckley-Leverett constraint in reservoir engineering, or the
enthalpy laws used in modeling gasdynamic discontinuities.
Now let us, for example, multiply Equation 6.90 by u(x) throughout, so
that u
2
u/ x = u
2
u/ x
2
. This can be recast in the form (1/3 u
3
)/ x =
u
2
u/ x
2
. If we now integrate by parts, we have
(1/3 u
3
)
+
- (1/3 u
3
)
-
=
[{u u/ x - ( u/ x)
2
dx }
+
- {u u/ x - ( u/ x)
2
dx }
-
] (6.93)
The u/ x terms on either side of the shock vanish identically, but the positive
definite integral does not. This leaves
(1/3 u
3
)
+
- (1/3 u
3
)
-
= - ( u/ x)
2
dx < 0 (6.94)
and hence the “entropy condition” (1/3 u
3
)
-
> ( 1 / 3 u
3
)
+
or
(u
3
)
-
> ( u
3
)
+
(6.95)
Thus, we have shown that entropy conditions need not be derived at
“independently” via thermodynamic considerations; they, and indeed all of the
physics, can be obtained naturally once the structure of the high-order derivative
is known with confidence. Additional entropy conditions can, obviously, be
generated by multiplying Equation 6.90 by other powers of u(x), or even other
functionals of u(x), and then integrating; this is left as an exercise to the reader.
We might note that the more complete model in Equation 6.88 furnishes the
complete physical structure of the narrow zone where the characteristics of
Equation 6.81 would have intersected. Its solution is expected to be unique,
whereas a unique solution to Equation 6.81 will involve additional constraints
(“jump conditions” or “shock-fitting” formulas) related to anticipated global
conserved quantities and entropy conditions.
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