Editing Potential flow (section)

==Examples of two-dimensional incompressible flows==
{{main|Potential flow around a circular cylinder|Rankine half body}}
Any differentiable function may be used for {{mvar|f}}. The examples that follow use a variety of [[elementary function]]s; [[special function]]s may also be used. Note that [[multi-valued function]]s such as the [[natural logarithm]] may be used, but attention must be confined to a single [[Riemann surface]].

===Power laws===
{{multiple image
 | header = Examples of conformal maps for the power law {{math|''w'' {{=}} ''Az<sup>n</sup>''}}
 | direction = vertical
 | total_width = 250
 | image1   = Conformal power half.svg
 | image2   = Conformal power two third.svg
 | image3   = Conformal power one.svg
 | image4   = Conformal power one and a half.svg
 | image5   = Conformal power two.svg
 | image6   = Conformal power three.svg
 | image7   = Conformal power minus one.svg
 | footer   = Examples of conformal maps for the power law {{math|''w'' {{=}} ''Az<sup>n</sup>''}}, for different values of the power {{mvar|n}}. Shown is the {{mvar|z}}-plane, showing lines of constant potential {{mvar|φ}} and streamfunction {{mvar|ψ}}, while {{math|''w'' {{=}} ''φ'' + ''iψ''}}.
}}

In case the following [[power (mathematics)|power]]-law conformal map is applied, from {{math|''z'' {{=}} ''x'' + ''iy''}} to {{math|''w'' {{=}} ''φ'' + ''iψ''}}:<ref name=B_409_413>Batchelor (1973) pp. 409–413.</ref>

<math display="block">w=Az^n \,,</math>

then, writing {{mvar|z}} in polar coordinates as {{math|''z'' {{=}} ''x'' + ''iy'' {{=}} ''re<sup>iθ</sup>''}}, we have<ref name=B_409_413/>

<math display="block">\varphi=Ar^n\cos n\theta \qquad \text{and} \qquad \psi=Ar^n\sin n\theta \,.</math>

In the figures to the right examples are given for several values of {{mvar|n}}. The black line is the boundary of the flow, while the darker blue lines are streamlines, and the lighter blue lines are equi-potential lines. Some interesting powers {{mvar|n}} are:<ref name=B_409_413/>
*{{math|''n'' {{=}} {{sfrac|1|2}}}}: this corresponds with flow around a semi-infinite plate,
*{{math|''n'' {{=}} {{sfrac|2|3}}}}: flow around a right corner,
*{{math|''n'' {{=}} 1}}: a trivial case of uniform flow,
*{{math|''n'' {{=}} 2}}: flow through a corner, or near a stagnation point, and
*{{math|''n'' {{=}} −1}}: flow due to a source doublet

The constant {{mvar|A}} is a scaling parameter: its [[absolute value]] {{math|{{abs|''A''}}}} determines the scale, while its [[arg (mathematics)|argument]] {{math|arg(''A'')}} introduces a rotation (if non-zero).

==== Power laws with {{math|''n'' {{=}} 1}}: uniform flow ==== <!-- [[Uniform flow]] redirects here -->
If {{math|''w'' {{=}} ''Az''<sup>1</sup>}}, that is, a power law with {{math|''n'' {{=}} 1}}, the streamlines (i.e. lines of constant {{mvar|ψ}}) are a system of straight lines parallel to the {{mvar|x}}-axis. This is easiest to see by writing in terms of real and imaginary components:

<math display="block">f(x+iy) = A\, (x+iy) = Ax + i Ay </math>

thus giving {{math|''φ'' {{=}} ''Ax''}} and {{math|''ψ'' {{=}} ''Ay''}}. This flow may be interpreted as '''uniform flow''' parallel to the {{mvar|x}}-axis.

==== 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.

==== Power laws with {{math|''n'' {{=}} 3}} ====
If {{math|''n'' {{=}} 3}}, the resulting flow is a sort of hexagonal version of the {{math|''n'' {{=}} 2}} case considered above.  Streamlines are given by, {{math|''ψ'' {{=}} 3''x''<sup>2</sup>''y'' − ''y''<sup>3</sup>}} and the flow in this case may be interpreted as flow into a 60° corner.

==== 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>

==== Power laws with {{math|''n'' {{=}} −2}}: quadrupole ====
If {{math|''n'' {{=}} −2}}, the streamlines are given by

<math display="block">\psi=-\frac{A}{r^2}\sin 2 \theta \,.</math>

This is the flow field associated with a [[quadrupole]].<ref>{{Cite book| publisher = Wiley-Interscience| isbn = 9780471511298| last = Kyrala| first = A.| title = Applied Functions of a Complex Variable| year = 1972| pages = 116–117}}</ref>

===Line source and sink===
A line source or sink of strength <math>Q</math> (<math>Q>0</math> for source and <math>Q<0</math> for sink) is given by the potential

<math display="block">w = \frac{Q}{2\pi} \ln z</math>

where <math>Q</math> in fact is the volume flux per unit length across a surface enclosing the source or sink. The velocity field in polar coordinates are

<math display="block">u_r = \frac{Q}{2\pi r},\quad u_\theta=0</math>

i.e., a purely radial flow.

===Line vortex===
A line vortex of strength <math>\Gamma</math> is given by

<math display="block">w=\frac{\Gamma}{2\pi i}\ln z</math>

where <math>\Gamma</math> is the [[circulation (fluid dynamics)|circulation]] around any simple closed contour enclosing the vortex. The velocity field in polar coordinates are

<math display="block">u_r = 0,\quad u_\theta=\frac{\Gamma}{2\pi r}</math>

i.e., a purely azimuthal flow.