A two-dimensional source is an idealized point from which incompressible fluid flows radially outward in a plane. A sink is the same mathematical flow with the direction reversed.

Let $Q$ denote the volume flow rate per unit depth perpendicular to the plane. For a circular contour of radius $r$ centered on the source, radial symmetry makes the speed constant around the circle. Conservation of volume gives

$$Q=v_r(2\pi r),$$

so

$$\boxed{v_r=\frac{Q}{2\pi r}},$$

$$\boxed{v_\theta=0}.$$

A source has

$$Q>0,$$

while a sink has

$$Q<0.$$

Because this is a two-dimensional model,

$$[Q]=\mathrm{m^2/s}.$$

Flux interpretation of source strength

For any circle enclosing the origin,

$$\oint_C\mathbf v\cdot\mathbf n,ds =\left(\frac{Q}{2\pi r}\right)(2\pi r) =\boxed{Q}.$$

Thus the parameter $Q$ is exactly the outward volume flux per unit depth through any closed contour surrounding the singular source.

Away from the origin, there is no distributed volume creation: the velocity field is divergence-free. The nonzero net flux arises because the ideal source is singular at the excluded point $r=0$.

Velocity potential

For an axisymmetric planar potential flow,

$$v_r=\frac{\partial\phi}{\partial r}.$$

Therefore

$$\frac{\partial\phi}{\partial r} =\frac{Q}{2\pi r}.$$

Integrating,

$$\boxed{\phi(r)=\frac{Q}{2\pi}\ln\left(\frac r{r_0}\right)},$$

where $r_0$ is an arbitrary reference length that only fixes the additive constant of the potential.

For every point with $r>0$,

$$\nabla^2\phi=0.$$

The origin itself is excluded because the ideal source is singular there.

Stream function

With the convention

$$v_r=\frac1r\frac{\partial\psi}{\partial\theta},$$

$$v_\theta=-\frac{\partial\psi}{\partial r},$$

one convenient stream function is

$$\boxed{\psi=\frac{Q}{2\pi}\theta}.$$

Curves of constant $\psi$ have constant $\theta$, so the streamlines are radial rays.

The angular coordinate changes by $2\pi$ around the source, so $\psi$ changes by $Q$. That jump is consistent with the fact that $Q$ units of volume flow per unit depth pass between one side of a branch cut and the other.

Irrotationality away from the singularity

Since the velocity is the gradient of a potential,

$$\nabla\times\mathbf v=\mathbf0$$

for

$$r>0.$$

Thus a source or sink is both incompressible and irrotational everywhere in its regular flow domain, even though it contains a singular point representing injection or removal of fluid.

Worked example

A two-dimensional source has strength

$$Q=0.040,\mathrm{m^2/s}.$$

At radius

$$r=0.20,\mathrm m,$$

the radial speed is

$$v_r =\frac{0.040}{2\pi(0.20)} \approx\boxed{3.18\times10^{-2},\mathrm{m/s}}.$$

At twice the radius,

$$r=0.40,\mathrm m,$$

the speed is half as large:

$$v_r\approx1.59\times10^{-2},\mathrm{m/s}.$$

The $1/r$ decrease is exactly what is required for the same flux $Q$ to cross circles whose circumference grows in proportion to $r$.

Why sources and sinks are useful

An ideal source is not usually a literal point-sized pump. It is an elementary harmonic flow that can be superposed with other potential flows. Source-sink combinations generate useful models for stagnation flows, body shapes, doublets, jets, and many ideal-flow constructions.

Its singularity is part of the model and must be kept outside any region where a smooth physical velocity field is required.