Biomedical Engineering Reference
In-Depth Information
is sharpened, the flow remains smooth and differentiable for all the time. A smaller
will lead to sharper troughs. Until
−
=0, i.e.
F
=1, to solve the propagation
correctly, we must follow the entropy condition to avoid the “swallowtail.” The
weak solution following the entropy condition is shown in Figure 3b. Comparing
Figure 3a with Figure 3b, Sethian [4] pointed out that at any time
T
,
X
curvature
(
T
)=
X
En
lim
→
0
constant
(
T
)
.
(12)
Here
X
curvature
(
t
) is the solution obtained by evolving the curve with
F
κ
=1
−
κ
,
and
X
En
constant
(
t
) is the weak solution obtained with speed function
F
=1and the
entropy condition. Equation (12) makes the point that the limit of motion under
curvature, known as the“
viscosity solution
,” is the entropy solution for the case of
constant speed.
To examine the role of the curvature in the speed function, we now turn
to a discussion of the relationship between propagating fronts and the hyperbolic
conservation laws. For completeness, we present the basic idea below, and readers
are referred to [4][37] for details.
We call the equation for
u
t
(
x, t
) of the form
u
t
+[
G
(
u
)]
x
=0
(13)
the “hyperbolic conservation law.” There is an example in [4] to demonstrate the
relationship between the hyperbolic conservation law and the propagating front,
given Burger's equation as the example, which follows the hyperbolic conservation
law:
u
t
+
uu
x
=0
.
(14)
Its physical meaning describes themotion of a compressible fluid in one dimension.
Imagine there is a sudden expansion or compression of the fluid; then the solution
will appear as discontinuities or “shocks.” However if we change the right-hand
side 0 with a second derivative,
u
t
+
uu
x
=
εu
xx
,
(15)
and then fluid viscosity will appear to retain the smoothness of the fluid, and such
shocks are avoided.
Look at the propagating front again, which is given by
ψ
(Figure 4). Suppose
the front propagates from time
t
to time
t
+∆
t
. The right-hand plot in Figure 4
shows the relation among speed
F
, tangent (1
,ψ
x
), and change in height
V
, i.e.,
F
=
(1 +
ψ
x
)
1
/
2
V
.
(16)
1
Rewriting it, we can obtain the motion equation as
ψ
t
=
F
(1 +
ψ
x
)
1
/
2
.
(17)
