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Problem 2.30 - Problem 2.30[Difficulty 4 Given Velocity...

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Problem 2.30 [Difficulty: 4] Given: Velocity field Find: Plot of pathline for t = 0 to 3 s for particle that started at point (1,2) at t = 0; compare to streakline through same point at the instant t = 3 Solution: Governing equations: For pathlines u p dx dt = v p dy dt = 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 For pathlines u p dx dt = a x t = a 1 4 = 1 s 2 b 1 3 = m s v p dy dt = b = So, separating variables dx x a t dt = dy b dt = Integrating ln x x 0 a 2 t 2 t 0 2 = y y 0 b t t 0 ( ) = y y 0 b t t 0 ( ) + = x x 0 e a 2 t
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