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Uniform Flow
Consider a uniform steady stream of velocity
. Consider the flux (in the minus
direction) across a plane
circle of radius
that lies in the

plane, and whose center coincides with the
axis.
From the definition of the Stokes stream function (see Section 7.3), we
have
, or

(7.25) 
When expressed in terms of spherical coordinates, the previous expression yields

(7.26) 
Of course, uniform flow is irrotational [this is clear from a comparison of Equations (7.10) and (7.25)], so we can also represent the flow pattern in terms of a velocity potential: that is (see Section 5.4),

(7.27) 
or

(7.28) 
It follows, from the previous analysis, that the velocity field of a uniform stream, running parallel to the
axis,
can either be written
, with
specified by Equations (7.25)(7.26), or
, with
specified by Equations (7.27)(7.28).
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Richard Fitzpatrick
20160122