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Next: Dipole Point Sources Up: Axisymmetric Incompressible Inviscid Flow Previous: Uniform Flow

Point Sources

Consider a point source, coincident with the origin, that emits fluid isotropically at the steady rate of $ Q$ volumes per unit time. By symmetry, we expect the associated steady flow pattern to be isotropic, and everywhere directed radially away from the source. In other words,

$\displaystyle {\bf v} = v(r)\,{\bf e}_r,$ (7.29)

where $ r$ is a spherical coordinate. Consider a spherical surface $ S$ of radius $ r$ whose center coincides with the source. In a steady state, the rate at which fluid crosses this surface must be equal to the rate at which the source emits fluid. Hence,

$\displaystyle \int_S {\bf v} \cdot d{\bf S} = 4\pi\,r^{\,2}\,v_r(r) = Q,$ (7.30)

which implies that

$\displaystyle v_r(r) = \frac{Q}{4\pi\,r^{\,2}}.$ (7.31)

Of course, $ v_\theta=0$ .

According to Equations (7.4), the Stokes stream function associated with a point source at the origin is such that $ \psi=\psi(\theta)$ , and is obtained by integrating

$\displaystyle v_r = \frac{Q}{4\pi\,r^{\,2}} = -\frac{1}{r^{\,2}\,\sin\theta}\,\frac{\partial\psi}{\partial\theta}.$ (7.32)

It follows that

$\displaystyle \psi = \frac{Q}{4\pi}\,\cos\theta = \frac{Q}{4\pi}\,\frac{z}{(\varpi^{\,2}+z^{\,2})^{1/2}}.$ (7.33)

It is clear, from a comparison of Equations (7.10) and (7.33), that the previously specified flow pattern is irrotational. Hence, this pattern can also be derived from a velocity potential. In fact, by symmetry, we expect that $ \phi=\phi(r)$ . The potential itself is obtained by integrating

$\displaystyle v_r = \frac{Q}{4\pi\,r^{\,2}} =-\frac{\partial\phi}{\partial r}.$ (7.34)

It follows that

$\displaystyle \phi = \frac{Q}{4\pi\,r} = \frac{Q}{4\pi\,(\varpi^{\,2}+z^{\,2})^{1/2}}.$ (7.35)


next up previous
Next: Dipole Point Sources Up: Axisymmetric Incompressible Inviscid Flow Previous: Uniform Flow
Richard Fitzpatrick 2016-03-31