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

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

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

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

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

or

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