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Potential flow in two dimensions



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 + iy) = φ + iψ

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 + iy and w = φ + iψ).

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).

 
This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Potential_flow_in_two_dimensions". A list of authors is available in Wikipedia.
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