Editing Potential flow (section)

==== Power laws with {{math|''n'' {{=}} 2}} ====
If {{math|''n'' {{=}} 2}}, then {{math|''w'' {{=}} ''Az''<sup>2</sup>}} and the streamline corresponding to a particular value of {{mvar|ψ}} are those points satisfying

<math display="block">\psi=Ar^2\sin 2\theta \,,</math>

which is a system of [[hyperbola|rectangular hyperbolae]]. This may be seen by again rewriting in terms of real and imaginary components. Noting that {{math|[[List of trigonometric identities#Multiple-angle formulae|sin 2''θ'' {{=}} 2 sin ''θ'' cos ''θ'']]}} and rewriting {{math|sin ''θ'' {{=}} {{sfrac|''y''|''r''}}}} and {{math|cos ''θ'' {{=}} {{sfrac|''x''|''r''}}}} it is seen (on simplifying) that the streamlines are given by

<math display="block">\psi=2Axy \,.</math>

The velocity field is given by {{math|∇''φ''}}, or

<math display="block">\begin{pmatrix} u \\ v \end{pmatrix} =  \begin{pmatrix} \frac{\partial \varphi}{\partial x} \\[2px] \frac{\partial \varphi}{\partial y} 
\end{pmatrix} =  \begin{pmatrix} + {\partial \psi \over \partial y} \\[2px] - {\partial \psi \over \partial x} \end{pmatrix} = \begin{pmatrix} +2Ax \\[2px] -2Ay \end{pmatrix} \,.</math>

In fluid dynamics, the flowfield near the origin corresponds to a [[stagnation point]].  Note that the fluid at the origin is at rest (this follows on differentiation of {{math|''f''(z) {{=}} ''z''<sup>2</sup>}} at {{math|''z'' {{=}} 0}}). The {{math|''ψ'' {{=}} 0}} streamline is particularly interesting: it has two (or four) branches, following the coordinate axes, i.e. {{math|''x'' {{=}} 0}} and {{math|''y'' {{=}} 0}}. As no fluid flows across the {{mvar|x}}-axis, it (the {{mvar|x}}-axis) may be treated as a solid boundary. It is thus possible to ignore the flow in the lower half-plane where {{math|''y'' < 0}} and to focus on the flow in the upper halfplane. With this interpretation, the flow is that of a vertically directed jet impinging on a horizontal flat plate. The flow may also be interpreted as flow into a 90 degree corner if the regions specified by (say) {{math|''x'', ''y'' < 0}} are ignored.