Problem 2.27

# Problem 2.27 - Problem 2.27 Given Velocity field...

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Problem 2.27 [Difficulty: 5] Given: Velocity field Find: Plot streakline for first second of flow Solution: 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 , ( ) = where x 0 , y 0 is the position of the particle at t = t 0 , and re-interprete the results as streaklines 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) For particle paths, first find x(t) dx dt u = u 0 = Separating variables and integrating dx u 0 dt = o r x x 0 u 0 t t 0 ( ) + = For y(t) we have dy dt v = v 0 sin ω t x u 0 = so dy dt v = v 0 sin ω t x 0 u 0 t t 0 ( ) + u 0 = and dy dt v = v 0 sin ω t 0 x 0 u 0 = Separating variables and integrating dy v 0 sin ω t 0 x 0 u 0
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