Editing Potential flow (section)

==== Power laws with {{math|''n'' {{=}} −1}}: doublet ==== <!-- [[Doublet (potential flow)]] redirects here]] -->
If {{math|''n'' {{=}} −1}}, the streamlines are given by

<math display="block">\psi = -\frac{A}{r}\sin\theta.</math>

This is more easily interpreted in terms of real and imaginary components:
<math display="block">\begin{align}
                 \psi = \frac{-A y}{r^2} &= \frac{-A y}{x^2 + y^2} \,, \\
            x^2 + y^2 + \frac{A y}{\psi} &= 0 \,, \\
  x^2 + \left(y+\frac{A}{2\psi}\right)^2 &= \left(\frac{A}{2\psi}\right)^2 \,.
\end{align}</math>

Thus the streamlines are [[circle]]s that are tangent to the x-axis at the origin. The circles in the upper half-plane thus flow clockwise, those in the lower half-plane flow anticlockwise.  Note that the velocity components are proportional to {{math|''r''<sup>−2</sup>}}; and their values at the origin is infinite. This flow pattern is usually referred to as a '''doublet''', or '''dipole''', and can be interpreted as the combination of a source-sink pair of infinite strength kept an infinitesimally small distance apart. The velocity field is given by

<math display="block">(u,v)=\left( \frac{\partial \psi}{\partial y}, - \frac{\partial \psi}{\partial x} \right) =  \left(A\frac{y^2-x^2}{\left(x^2+y^2\right)^2},-A\frac{2xy}{\left(x^2+y^2\right)^2}\right) \,.</math>

or in polar coordinates:

<math display="block">(u_r, u_\theta)=\left( \frac{1}{r} \frac{\partial \psi}{\partial \theta}, - \frac{\partial \psi}{\partial r} \right) = \left(-\frac{A}{r^2}\cos\theta, -\frac{A}{r^2}\sin\theta\right) \,.</math>