Potential flow in two dimensions

From Wikipedia, the free encyclopedia

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 $\nabla\cdot\underline{u}=0$ ).