A doublet is an elementary potential flow obtained by bringing a source and sink of equal strength increasingly close together while increasing their strength so that a finite dipole moment remains.

Place a source of strength $Q$ at $x=-a$ and a sink of strength $-Q$ at $x=+a$. The combined velocity potential is

$$\phi =\frac{Q}{2\pi} \ln r_+ -\frac{Q}{2\pi} \ln r_-,$$

where $r_+$ and $r_-$ are the distances from the source and sink.

For observation points with

$$r\gg a,$$

the leading nonvanishing contribution is proportional to

$$\frac{Qa\cos\theta}{r}.$$

Take the limit

$$a\to0,\qquad Q\to\infty$$

while holding

$$\boxed{\kappa=2Qa}$$

constant. The result is a doublet aligned with the $x$ axis.

Potential and stream function

With this convention, the doublet velocity potential is

$$\boxed{\phi_D=rac{\kappa}{2\pi r}\cos\theta}.$$

A compatible stream function is

$$\boxed{\psi_D=-\frac{\kappa}{2\pi r}\sin\theta}.$$

Both are harmonic for

$$r>0.$$

The origin is a singular point inherited from the limiting source-sink pair.

Velocity field

In polar coordinates,

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

$$v_\theta=\frac1r\frac{\partial\phi}{\partial\theta}.$$

Therefore

$$\boxed{v_r=-\frac{\kappa}{2\pi r^2}\cos\theta},$$

$$\boxed{v_\theta=-\frac{\kappa}{2\pi r^2}\sin\theta}.$$

The disturbance decays as

$$\boxed{|\mathbf v|\propto\frac1{r^2}},$$

faster than the $1/r$ velocity of an isolated two-dimensional source.

Zero net source strength

A doublet contains equal source and sink strengths, so it has zero net volume flux through a large closed contour:

$$\boxed{Q_{net}=0}.$$

Its importance therefore does not come from injecting net fluid into the domain. Instead, it represents the leading far-field disturbance produced by a closely spaced source-sink pair.

Worked evaluation

Take a doublet strength

$$\kappa=0.20,\mathrm{m^3/s}$$

in the two-dimensional convention above. At

$$r=0.50,\mathrm m,$$

on the positive $x$ axis,

$$\theta=0.$$

Then

$$v_\theta=0,$$

and

$$v_r =-\frac{0.20}{2\pi(0.50)^2} \approx\boxed{-0.127,\mathrm{m/s}}.$$

The negative sign means the local velocity points radially inward toward the origin on the positive $x$ axis.

At the same radius on the positive $y$ axis,

$$\theta=\frac\pi2,$$

so

$$v_r=0,$$

and

$$v_\theta =-\frac{0.20}{2\pi(0.50)^2} \approx-0.127,\mathrm{m/s}.$$

The same doublet therefore produces different velocity directions around the singularity while preserving its characteristic $1/r^2$ magnitude scale.

Why the doublet matters

The doublet becomes especially useful when superposed with a uniform flow. Because Laplace's equation is linear, the potentials can be added directly. With one particular doublet strength, the superposed field has a circular streamline on which normal velocity is zero.

That construction produces the exact incompressible irrotational flow around a circular cylinder.

The doublet is therefore a reusable building block: it converts the source-sink idea into a compact singularity whose superposition can represent the disturbance created by an impermeable body.