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Unformatted text preview: Appendix C: Streamfunctions If u = 0, then it follows that there exists a vector field A ( x , t ) s.t. u = 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: two-dimensional flows and three-dimensional, axisymmetric flows. In the latter case the streamfunction is known as the Stokes streamfunction. C.1 Two-dimensional flows (i) Cartesians : Consider Cartesian coordinates ( x, y, z ), and take the flow in the ( x, y )- plane. Then A = ( x, y, t ) z , and u = y , v = x . Now confirm that is indeed constant along streamlines, which are defined by d x ds = u , and thus dx = uds and dy = vds . It follows that d = x d x + y d y...
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- Fall '11