Source and Sink

Consider a radial flow going away from the origin at a velocity $v_r$ as shown in Fig.4.19. This constitutes a Source Flow. This is a purely radial flow with no component of velocity in the tangential direction, i.e., $v_\theta = 0$. If m is the volumetric flow rate we have

Figure 4.19: Source Flow and Sink Flow.

$$(2 \pi r) v_r = m$$ (4.74)

We can now write down velocity potential and stream function for this flow -

$$\phi = {m \over {2\pi}} \ln r$$

$$\psi = {m \over {2\pi}} \theta$$ (4.75)

It is easily verified that $v_\theta = 0$ for this flow. Further, the equation we started out with , namely, Eqn.4.74 is the continuity equation for the source flow. It states that the Volumetric flow rate (mass flow rate when multiplied by density) is constant in a radial direction and is equal to m, which is called the Strength of the source.

Another point to make is that the radial velocity $v_r$ becomes infinite at r = 0. So the origin is a singularity of the flow.

If m is negative we have a flow which flows inwards and is called a Sink flow, which again has a singularity at the origin.

(c) Aerospace, Mechanical & Mechatronic Engg. 2005
University of Sydney