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(5.1) |
Let
be a fixed point in the
-
plane, and let
and
be two curves, also in the
-
plane,
that join
to an arbitrary point
. (See Figure 5.1.) Suppose that fluid is neither created nor destroyed in the region,
(say), bounded by
these curves. Because the fluid is incompressible, which essentially means that its density is uniform and constant,
fluid continuity requires that the rate at which the fluid flows into the region
, from right to left (in Figure 5.1) across the
curve
, is equal to the rate at which it flows out the of the region, from right to left across the
curve
. The rate of fluid flow across a surface is generally termed the flux. Thus, the flux (per unit length parallel to the
-axis) from right to left across
is equal to the flux from right to left across
. Because
is arbitrary, it follows that the flux from right to left across any curve
joining points
and
is equal to the flux from right to left across
. In fact, once the base point
has been chosen, this flux only depends on the position of point
, and the time
. In other words, if we denote the flux by
then
it is solely a function of the location of
and the time. Thus, if point
lies at the origin, and point
has Cartesian
coordinates (
,
), then we can write
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(5.2) |
Consider two points,
and
, in addition to the fixed point
. (See Figure 5.2.) Let
and
be the fluxes
from right to left across curves
and
. Using similar arguments to those employed previously, the
flux across
is equal to the flux across
plus the flux across
. Thus, the
flux across
, from right to left, is
. If
and
both lie on the same streamline
then the flux across
is zero, because the local fluid velocity is directed everywhere parallel to
. It follows that
. Hence, we conclude that the
stream function is constant along a streamline. The equation of a streamline is thus
, where
is an arbitrary constant.
Let
be an infinitesimal arc of a curve that is sufficiently short that it can be regarded as a straight-line.
The fluid velocity in the vicinity of this arc can be resolved into components parallel and perpendicular to the arc.
The component parallel to
contributes nothing to the flux across the arc from right to left. The component perpendicular to
contributes
to the flux. However, the flux is equal to
. Hence,
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(5.3) |
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(5.4) |
The vorticity in two-dimensional flow takes the form
![]() ![]() |
(5.8) |
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(5.9) |
When expressed in terms of cylindrical coordinates (see Section C.3), Equation (5.7) yields
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(5.12) |