Graphics Reference
In-Depth Information
Figure 21.2.
Fluid flowing around an obstacle.
p
C
¢
A
v
t
C
v
p
0
Figure 21.3.
Irrotational fluids have well-defined circula-
tions along curves.
v
n
Fluid flowing around an infinitely high cylindrical obstacle is an example. See Figure
21.2. One calls the flow
stationary
or
steady
if the velocity of the fluid depends only
on the position (x,y) and not on time. Given a curve
C
in the plane, the integral
Ú
v
t
is called the
circulation
of the fluid along the curve
C
, where v
t
is the tangential com-
ponent of v along
C
. Assume that the fluid is
irrotational
or
circulation free
, that is, its
circulation is zero along any closed curve. In that case, one can show that for any
fixed point
p
0
, the line integral along a curve
C
from
p
0
to a point
p
depends only on
p
and not the curve
C
, so that there is a well-defined function
()
=
Ú
v
t
f
p
C
where
p
is the endpoint of
C
. It follows that v =—f. See Figure 21.3. Assume next that
the fluid is incompressible and nonviscous. (A fluid is
incompressible
if it has constant
density, which means mathematically that the integral
Ú
v
n
along any closed curve
C
is zero, where v
n
is the normal component of v along
C
, that
is, the amount of fluid leaving the region bounded by
C
equals the amount entering
the region. It is
nonviscous
if there is no internal friction, so that pressure forces on
a surface are perpendicular to the surface.) With these assumptions, we are mathe-
matically in a situation like in the steady temperature problem, except that our func-
tions have other names and interpretations. It follows that the function f satisfies the
Laplace equation.
