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 <math>f<math>. If we write
- <math>
f(x+iy)=\phi+i\psi <math>
then the Cauchy-Riemann equations show that
- <math>
\frac{\partial\phi}{\partial x}=\frac{\partial\psi}{\partial y}, \qquad \frac{\partial\phi}{\partial y}=-\frac{\partial\psi}{\partial x}. <math> (it is conventional to regard all symbols as real numbers; and to write <math>z=x+iy<math> and <math>w=\phi+i\psi<math>).
The velocity field <math>\underline{u}=(u,v)<math>, specified by
- <math>
u=\frac{\partial\phi}{\partial x},\qquad v=\frac{\partial\phi}{\partial y} <math> then satisfies the requirements for potential flow:
- <math>
\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 <math> and
- <math>
\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. <math>
Lines of constant <math>\psi<math> are known as streamlines and lines of constant <math>\phi<math> are known as equipotential lines (see equipotential surface). The two sets of curves intersect at right angles, for
- <math> \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 <math> showing that, at any point, a vector perpendicular to the <math>\phi<math> contour line has a dot product of zero with a vector perpendicular to the <math>\psi<math> contour line (the two vectors thus intersecting at <math>90^\circ<math>). The identity may be proved by using the Cauchy-Riemann equations given above:
- <math>
\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. <math>
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 <math>\nabla^2\psi=0<math> is also satisfied, this relation being equivalent to <math>\nabla\times\underline{u}=0<math> (the automatic condition <math>\partial^2\psi/\partial x\partial y=\partial^2\psi/\partial y\partial x<math> gives <math>\nabla\cdot\underline{u}=0<math>).Template:Math-stub