Potential flow in two dimensions

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In fluid dynamics, potential flow in two dimensions is simple to analyse using complex numbers.

The basic idea is to define a holomorphic or meromorphic function $f$ . If we write

f (x + i y) = φ + i ψ

then the Cauchy-Riemann equations show that

$\frac{\partial\phi}{\partial x}=\frac{\partial\psi}{\partial y}, \qquad \frac{\partial\phi}{\partial y}=-\frac{\partial\psi}{\partial x}.$

(it is conventional to regard all symbols as real numbers; and to write $z = x + i y$ and $w = φ + i ψ$ ).

The velocity field $\underline{u}=(u,v)$ , specified by

$u=\frac{\partial\phi}{\partial x},\qquad v=\frac{\partial\phi}{\partial y}$

then satisfies the requirements for potential flow:

$\nabla\cdot\underline{u}= \nabla^2\phi= \frac{\partial^2\phi}{\partial x^2}+\frac{\partial^2\phi}{\partial y^2}= {\partial \over \partial x} {\partial \psi \over \partial y} + {\partial \over \partial y} \left( - {\partial \psi \over \partial x} \right) = 0$

and

$\left|\nabla\times\underline{u}\right|= \frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}= \frac{\partial^2\phi}{\partial x\partial y}- \frac{\partial^2\phi}{\partial y\partial x}=0.$

Lines of constant $ψ$ are known as streamlines and lines of constant $φ$ are known as equipotential lines (see equipotential surface). The two sets of curves intersect at right angles, for

$\nabla \phi \cdot \nabla \psi = \frac{\partial\phi}{\partial x}\frac{\partial\psi}{\partial x}+ \frac{\partial\phi}{\partial y}\frac{\partial\psi}{\partial y}=0$

showing that, at any point, a vector perpendicular to the $φ$ contour line has a dot product of zero with a vector perpendicular to the $ψ$ contour line (the two vectors thus intersecting at $90^\circ$ ). The identity may be proved by using the Cauchy-Riemann equations given above:

$\frac{\partial\phi}{\partial x}\frac{\partial\psi}{\partial x}+ \frac{\partial\phi}{\partial y}\frac{\partial\psi}{\partial y}= {\partial \phi \over \partial x} {\partial \psi \over \partial x}+ \left( - {\partial \psi \over \partial x} \right) \left( {\partial \phi \over \partial x} \right) = 0.$

Thus the flow occurs along the lines of constant ψ and at right angles to the lines of constant φ.

It is interesting to note that $\nabla^2\psi=0$ is also satisfied, this relation being equivalent to $\nabla\times\underline{u}=0$ (the automatic condition $\partial^2\psi/\partial x\partial y=\partial^2\psi/\partial y\partial x$ gives $\nabla\cdot\underline{u}=0$ ).