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Stokes Stream Function
Consider a fixed point
lying on the symmetry axis, and an arbitrary point
. Let us join
to
via
two different curves,
and
, that both lie in the same plane. (See Figure 7.1.) We shall refer to this plane as the meridian
plane. The position of a given point in the meridian plane can be specified either in terms of the cylindrical coordinates (
,
),
or the spherical coordinates (
,
). If
the meridian curves
and
rotate about the symmetry axis then closed surfaces are formed.
Assuming that the flow pattern is incompressible, the flux of fluid from right to left (in Figure 7.1) across the surface
generated by
must match that in the same direction across the surface
generated by
. Let us denote the flux across either of these surfaces by
. Here,
is known as the Stokes stream function. If we keep
fixed, and replace
by any
other meridian curve joining
to
, then the value of
is clearly unaltered. Thus, the stream function
depends on the position of the arbitrary point
, and, possibly, on that of the fixed point
. In fact, if we
take another fixed point
on the symmetry axis, and draw the meridian curve
, then the flux across the
surface generated by
will be the same as that across the surface generated by
, because, by symmetry, there is no
flow across
. (See Figure 7.1.) It follows that the value of
does not depend on the particular
fixed point that is used in its definition, as long as this point lies on the symmetry axis. Hence, we conclude
that the value of the stream function at
depends solely on the position of
. Furthermore, if
lies
on the axis then
.
Figure 7.1:
Definition of the Stokes stream function.

Consider two neighboring points,
and
, lying in the meridian plane. (See Figure 7.1.) The flux from right to left across the surface
generated by revolving any line joining
to
about the symmetry axis is
.
If the distance
takes the infinitesimal value
then we can write

(7.1) 
where
is the normal flow velocity (from right to left) across the straightline
in the meridian plane. (See Figure 7.1.)
It follows that

(7.2) 
In particular, if we suppose that
is, in turn, equal to
,
,
, and
then we obtain the
following expressions for the inplane velocity components in cylindrical and spherical coordinates:
Next: Axisymmetric Velocity Fields
Up: Axisymmetric Incompressible Inviscid Flow
Previous: Axisymmetric Flow
Richard Fitzpatrick
20160122