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Problem 2.31

# Problem 2.31 - Problem 2.31[Difficulty 4 Given 2D velocity...

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Problem 2.31 [Difficulty: 4] Given: 2D velocity field Find: Streamlines passing through (6,6); Coordinates of particle starting at (1,4); that pathlines, streamlines and streaklines coincide Solution: For streamlines v u dy dx = b ay 2 = or y 2 dx b d = Integrating 3 3 bx c + = For the streamline through point (6,6) c6 0 = and y 3 6x 180 + = For particle that passed through (1,4) at t = 0 u dx dt = 2 = x 1 x 0 = t 2 d = We need y(t) v dy dt = b = y 1 dt b d = yy 0 bt + = y 0 2t + = Then xx 0 0 t t 0 + () 2 d = 0 0 2 t by 0 t 2 + b 2 t 3 3 + + = Hence, with x 0 1 = y 0 4 = x 1 16 t + 8t 2 + 4 3 t 3 + = At t = 1 s x 26.3 m = y42 t + = y6 m = For particle that passed through (-3,0) at t = 1 y 1 b d = 0 bt t 0
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