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Problem 2.23 - Problem 2.23[Difficulty 3 Given Velocity...

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Problem 2.23 [Difficulty: 3] Given: Velocity field Find: Plot of pathline of particle for t = 0 to 1.5 s that was at point (1,1) at t = 0; compare to streamlines through same point at the instants t = 0, 1 and 1.5 s Solution: Governing equations: For pathlines u p dx dt = v p dy dt = For streamlines v u dy dx = Assumption: 2D flow Hence for pathlines u p dx dt = a x = a 1 5 = 1 s v p dy dt = b y t = b 1 25 = 1 s 2 So, separating variables dx x a dt = dy b y t dt = dy y b t dt = Integrating ln x x 0 a t = x 0 1 = m ln y y 0 b 1 2 t 2 = y 0 1 = m Hence x t ( ) x 0 e a t = y t ( ) y 0 e 1 2 b t 2 = Using given data x t ( ) e t 5 = y t ( ) e t 2 50 = For streamlines v u dy dx = b y t a x = So, separating variables dy y b t a x dx = which we can integrate for any given t (t is treated as a constant)
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