Editing Potential flow (section)

==Analysis for two-dimensional incompressible flow==<!-- [[Potential flow in two dimensions]] redirects here -->
{{main|Conformal map}}
'''Potential flow in two dimensions''' is simple to analyze using [[conformal mapping]], by the use of [[transformation (geometry)|transformation]]s of the [[complex plane]]. However, use of complex numbers is not required, as for example in the classical analysis of fluid flow past a cylinder.  It is not possible to solve a potential flow using [[complex number]]s in three dimensions.<ref name=B_106_108>Batchelor (1973) pp. 106–108.</ref>

The basic idea is to use a [[Holomorphic function|holomorphic]] (also called [[analytic function|analytic]]) or [[meromorphic function]] {{mvar|f}}, which maps the physical domain {{math|(''x'', ''y'')}} to the transformed domain {{math|(''φ'', ''ψ'')}}. While {{mvar|x}}, {{mvar|y}}, {{mvar|φ}} and {{mvar|ψ}} are all [[real number|real valued]], it is convenient to define the complex quantities

<math display="block">\begin{align}
  z &= x + iy \,, \text{ and } &
  w &= \varphi + i\psi \,.
\end{align}</math>

Now, if we write the mapping {{mvar|f}} as<ref name=B_106_108/>

<math display="block">\begin{align}
  f(x + iy) &= \varphi + i\psi \,, \text{ or } &
       f(z) &= w \,.
\end{align}</math>

Then, because {{mvar|f}} is a holomorphic or meromorphic function, it has to satisfy the [[Cauchy–Riemann equations]]<ref name=B_106_108/>

<math display="block">\begin{align}
  \frac{\partial\varphi}{\partial x} &=  \frac{\partial\psi}{\partial y} \,, &
  \frac{\partial\varphi}{\partial y} &= -\frac{\partial\psi}{\partial x} \,.
\end{align}</math>

The velocity components {{math|(''u'', ''v'')}}, in the {{math|(''x'', ''y'')}} directions respectively, can be obtained directly from {{mvar|f}} by differentiating with respect to {{mvar|z}}. That is<ref name=B_106_108/>

<math display="block">\frac{df}{dz} = u - iv</math>

So the velocity field {{math|'''v''' {{=}} (''u'', ''v'')}} is specified by<ref name=B_106_108/>

<math display="block">\begin{align}
  u &= \frac{\partial\varphi}{\partial x} = \frac{\partial\psi}{\partial y}, &
  v &= \frac{\partial\varphi}{\partial y} = -\frac{\partial\psi}{\partial x} \,.
\end{align}</math>

Both {{mvar|φ}} and {{mvar|ψ}} then satisfy [[Laplace's equation]]:<ref name=B_106_108/>

<math display="block">\begin{align}
  \Delta\varphi &= \frac{\partial^2\varphi}{\partial x^2} + \frac{\partial^2\varphi}{\partial y^2} = 0 \,,\text{ and } &
     \Delta\psi &= \frac{\partial^2\psi}{\partial x^2} + \frac{\partial^2\psi}{\partial y^2} = 0 \,.
\end{align}</math>

So {{mvar|φ}} can be identified as the velocity potential and {{mvar|ψ}} is called the [[stream function]].<ref name=B_106_108/> Lines of constant {{mvar|ψ}} are known as [[Streamlines, streaklines, and pathlines#Streamlines|streamlines]] and lines of constant {{mvar|φ}} are known as equipotential lines (see [[equipotential surface]]).

Streamlines and equipotential lines are orthogonal to each other, since<ref name=B_106_108/>

<math display="block">
  \nabla \varphi \cdot \nabla \psi =
  \frac{\partial\varphi}{\partial x} \frac{\partial\psi}{\partial x} + \frac{\partial\varphi}{\partial y} \frac{\partial\psi}{\partial y} =
  \frac{\partial \psi}{\partial y} \frac{\partial \psi}{\partial x} - \frac{\partial \psi}{\partial x} \frac{\partial \psi}{\partial y} =
  0 \,.
</math>

Thus the flow occurs along the lines of constant {{mvar|ψ}} and at right angles to the lines of constant {{mvar|φ}}.<ref name=B_106_108/>

{{math|Δ''ψ'' {{=}} 0}} is also satisfied, this relation being equivalent to {{math|∇ × '''v''' {{=}} '''0'''}}. So the flow is irrotational. The automatic condition {{math|{{sfrac|∂<sup>2</sup>Ψ|∂''x'' ∂''y''}} {{=}} {{sfrac|∂<sup>2</sup>Ψ|∂''y'' ∂''x''}}}} then gives the incompressibility constraint {{math|∇ · '''v''' {{=}} 0}}.