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Unformatted text preview: Appendix C: Streamfunctions If âˆ‡Â· u = 0, then it follows that there exists a vector field A ( x , t ) s.t. u = âˆ‡Ã— A . Conversely, it is clear from this representation that the flow is incompressible. The rep resentation is particularly useful if there are two independent coordinates, in which case A can be written in terms of a single scalar function, the streamfunction. By defini tion, the streamfunction is constant along streamlines. There are two important cases: twodimensional flows and threedimensional, axisymmetric flows. In the latter case the streamfunction is known as the Stokes streamfunction. C.1 Twodimensional flows (i) Cartesians : Consider Cartesian coordinates ( x, y, z ), and take the flow in the ( x, y ) plane. Then A = Ïˆ ( x, y, t )Ë† z , and u = âˆ‚Ïˆ âˆ‚y , v = âˆ’ âˆ‚Ïˆ âˆ‚x . Now confirm that Ïˆ is indeed constant along streamlines, which are defined by d x ds = Î» u , and thus dx = Î»uds and dy = Î»vds . It follows that d Ïˆ = âˆ‚Ïˆ âˆ‚x d x + âˆ‚Ïˆ âˆ‚y d y...
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 Fall '11
 Eggers
 Cartesian Coordinate System, Coordinate system, Polar coordinate system, Coordinate systems, cylindrical polars

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