Problem 2.29 - Problem 2.29 [Difficulty: 4] Given: Velocity...

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Problem 2.29 [Difficulty: 4] Given: Velocity field Find: Plot of streakline for t = 0 to 3 s at point (1,1); compare to streamlines through same point at the instants t = 0, 1 and 2 s Solution: Governing equations: For pathlines u p dx dt = v p dy dt = For streamlines v u dy dx = Following the discussion leading up to Eq. 2.10, we first find equations for the pathlines in form x p t () xtx 0 , y 0 , t 0 , = and y p t ytx 0 , y 0 , t 0 , = x st t 0 0 , y 0 , t 0 , = and y st t 0 0 , y 0 , t 0 , = which gives the streakline at t, where x 0 , y 0 is the point at which dye is released (t 0 is varied from 0 to t) Assumption: 2D flow For pathlines u p dx dt = ax 1b t + = a1 = 1 s b 1 5 = 1 s v p dy dt = cy = c1 = 1 s So, separating variables dx x a1 b t + dt = dy y cd t = Integrating ln x x 0 at t 0 b t 2 t 0 2 2 + = ln y y 0 ct t 0 = yy 0 e ctt 0 = xx 0 e att 0 b t 2 t 0 2 2 + =
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The pathlines are x p t () x 0 e att 0 b t 2 t 0 2 2 + = y p t y 0 e ctt 0 = where x 0 , y 0 is the position of the particle at t = t 0 . Re-interpreting the results as streaklines: The streaklines are then x st t 0 x 0 e 0
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Problem 2.29 - Problem 2.29 [Difficulty: 4] Given: Velocity...

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