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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
![$\displaystyle \psi(\varpi,z) = - \frac{1}{2}\,V\,\varpi^{\,2}.$](img2600.png) |
(7.25) |
When expressed in terms of spherical coordinates, the previous expression yields
![$\displaystyle \psi(r,\theta)= -\frac{1}{2}\,V\,r^{\,2}\,\sin^2\theta.$](img2601.png) |
(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),
![$\displaystyle \phi(\varpi,z) = -V\,z,$](img2602.png) |
(7.27) |
or
![$\displaystyle \phi(r,\theta) = -V\,r\,\cos\theta.$](img2603.png) |
(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).
Next: Point Sources
Up: Axisymmetric Incompressible Inviscid Flow
Previous: Axisymmetric Irrotational Flow in
Richard Fitzpatrick
2016-01-22