Source and Sink Flows

A source is a point where fluid originates and spreads out radially in all directions. A sink is the opposite; fluid converges towards a point and is absorbed or removed from the flow.

$m$ is the volumetric flow rate per unit length perpendicular to the $x$-$y$ plane. Qualitatively, it is the strength of the source/sink.

$$m=2\pi r\cdot V_r$$

• For a source, $m>0$ (fluid is added)
• For a sink, $m<0$ (fluid is removed)

We can consider source and sink flows in terms of their radial and tangential velocity components, leveraging cylindrical coordinates.

Radial velocity component $V_r$:

$$V_r=\frac{m}{2\pi r}$$

• $V_r$ decreases inversely with the radius $r$; the further we move from the source, the slower the speed.

Tangential component $V_{\theta }$:

$$V_{\theta }=0$$

• There is no tangential velocity; the motion is purely radial.

The velocity components are gradients of the velocity potential and the stream function (see plane potential flows)

$$V_r=\frac{\partial \varphi }{\partial r}=\frac{1}{r}\frac{\partial \psi }{\partial \theta }$$

which in turn gives us the velocity potential as

$$\varphi =\frac{m}{2\pi }\ln r$$

• This tells us that equipotential lines ($\varphi =\text{const}$) are concentric circles.

The stream function $\psi$ can be solved to be

$$\psi =\frac{m}{2\pi }\theta$$

• This tells us that streamlines $\psi =\text{const}$ are radial lines emerging from the source or converging to the sink, as expected.