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,

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

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

which implies that

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

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

It follows that

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

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

It follows that

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