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Uniform Flow

Consider a uniform steady stream of velocity $ {\bf v} = V\,{\bf e}_z$ . Consider the flux (in the minus $ z$ -direction) across a plane circle of radius $ \varpi$ that lies in the $ x$ -$ y$ plane, and whose center coincides with the $ z$ -axis. From the definition of the Stokes stream function (see Section 7.3), we have $ 2\pi\,\psi(\varpi,z) = -\pi\,\varpi^{\,2}\,V$ , or

$\displaystyle \psi(\varpi,z) = - \frac{1}{2}\,V\,\varpi^{\,2}.$ (7.25)

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

$\displaystyle \psi(r,\theta)= -\frac{1}{2}\,V\,r^{\,2}\,\sin^2\theta.$ (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),

$\displaystyle \phi(\varpi,z) = -V\,z,$ (7.27)

or

$\displaystyle \phi(r,\theta) = -V\,r\,\cos\theta.$ (7.28)

It follows, from the previous analysis, that the velocity field of a uniform stream, running parallel to the $ z$ -axis, can either be written $ {\bf v} = \nabla\varphi\times\nabla\psi$ , with $ \psi $ specified by Equations (7.25)-(7.26), or $ {\bf v}=-\nabla\phi$ , with $ \phi$ specified by Equations (7.27)-(7.28).


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Next: Point Sources Up: Axisymmetric Incompressible Inviscid Flow Previous: Axisymmetric Irrotational Flow in
Richard Fitzpatrick 2016-01-22