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Problem 2.35 - Problem 2.35[Difficulty 4 Given Velocity...

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Problem 2.35 [Difficulty: 4] Given: Velocity field Find: Coordinates of particle at t = 2 s that was at (1,2) at t = 0; coordinates of particle at t = 3 s that was at (1,2) at t = 2 s; plot pathline and streakline through point (1,2) and compare with streamlines through same point at 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 ( ) x t x 0 , y 0 , t 0 , ( ) = and y p t ( ) y t x 0 , y 0 , t 0 , ( ) = x st t 0 ( ) x t x 0 , y 0 , t 0 , ( ) = and y st t 0 ( ) y t x 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 Given data a 0.2 = 1 s b 0.4 = m s 2 Hence for pathlines u p dx dt = a y = v p dy dt = b t = Hence dx a y dt = dy b t dt = y y 0 b 2 t 2 t 0 2 = For x dx a y 0 a b 2 t 2 t 0 2 + dt = Integrating x x 0 a y 0 t t 0 ( ) a b 2 t 3 3 t 0 3 3 t 0 2 t t 0 ( )
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