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Potential flow in two dimensionsIn fluid dynamics, potential flow in two dimensions is simple to analyse using complex numbers. Additional recommended knowledgeThe basic idea is to define a holomorphic or meromorphic function f. If we write
then the Cauchy-Riemann equations show that (it is conventional to regard all symbols as real numbers; and to write z = x + iy and w = φ + iψ). The velocity field , specified by then satisfies the requirements for potential flow: and 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 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 ). The identity may be proved by using the Cauchy-Riemann equations given above: 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 is also satisfied, this relation being equivalent to (the automatic condition gives ). |
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. |