Problem 2.12

# Problem 2.12 - [Difficulty 3 Problem 2.12 Given Flow field...

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[Difficulty: 3] Problem 2.12 Given: Flow field Find: Plot of velocity magnitude along axes, and y = x; Equation of streamlines Solution: On the x axis, y = 0, so u Ky 2 π x 2 y 2 + () = 0 = v Kx 2 π x 2 y 2 + = K 2 π x = 1 0.5 0 0.5 1 160 80 80 160 x (km) v( m/s) Plotting The velocity is perpendicular to the axis, is very high close to the origin, and falls off to zero. This can also be plotted in Excel. On the y axis, x = 0, so u 2 π x 2 y 2 + = K 2 π y = v 2 π x 2 y 2 + = 0 = 1 0.5 0 0.5 1 160 80 80 160 y (km) Plotting

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The velocity is perpendicular to the axis, is very high close to the origin, and falls off to zero. This can also be plotted in Excel. On the y = x axis u Kx 2 π x 2 x 2 + () = K 4 π x = v 2 π x 2 x 2 + = K 4 π x = The flow is perpendicular to line y = x: Slope of line y = x: 1 Slope of trajectory of motion: u v 1 = If we define the radial position: rx 2 y 2 + = then along y = x 2 x 2 + = 2 x = Then the magnitude of the velocity along y = x is Vu 2 v 2 + = K 4 π 1 x 2 1 x 2 + = K 2 π 2 x = K 2 π r = 1 0.5 0 0.5
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Problem 2.12 - [Difficulty 3 Problem 2.12 Given Flow field...

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