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Obtain parametric equations for the streamlines of a

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Unformatted text preview: º, 12.7º, 15.3º, 18º (B) Plotting Flow Lines Let us look again at the flow around a cylinder (which we failed to draw precisely due to the complexity of the formulas). Recall that we had the following intuitive picture: 43 and that the above region maps to H via È 1˘ F(z) = AÍz + ˙ Î z˚ for some constant A. To invert this function, we set w = F(z) and solve for z: È 1˘ w = AÍz + ˙ Î z˚ w 1 =z+ A z Taking 1/A = B, we get the quadratic 2 z - Bwz + 1 = 0 Solving for z, 22 1/2 Bw ± (B w - 4) z= 2 so the inverse is 22 1/2 Bz ± (B z - 4) -1 F (z) = 2 Now you have to be careful, since there are two possible square roots to choose from (no such thing as a “positive” square root anymore). If z is in the first quadrant, then everything is in the upper half plane, and to get the inverse, we use the primitive square root (the one in the upper half plane) and also use the (+) sign. The issue is now: How do we express this in terms of Cartesian coordinates? 1/2 1/2 Answer First look at z = r [cos(ø/2) + i sin(ø/2)] where cos(ø/2) = (1 + cos ø)/2 and sin(ø/2) = (1 - cos ø)/2 so that, in Cartesian coordinates, 1/2 1/2 z = r [ (1 + cos...
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