Problem 2.26 - Problem 2.26[Difficulty 4 Given Velocity...

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Problem 2.26 [Difficulty: 4] Given: Velocity field Find: Plot streamlines that are at origin at various times and pathlines that left origin at these times Solution: For streamlines v u dy dx = v 0 sin ω t x u 0 u 0 = So, separating variables (t=const) dy v 0 sin ω t x u 0 u 0 dx = Integrating y v 0 cos ω t x u 0 ω c + = Using condition y = 0 when x = 0 y v 0 cos ω t x u 0 cos ω t () ω = This gives streamlines y(x) at each time t For particle paths, first find x(t) dx dt u = u 0 = Separating variables and integrating dx u 0 dt = o r xu 0 t c 1 + = Using initial condition x = 0 at t = τ c 1 u 0 τ = 0 t τ = 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 u 0 t τ u 0 = and dy dt v = v 0 sin ωτ = Separating variables and integrating dy v 0 sin dt = yv 0 sin t c 2 + = Using initial condition y = 0 at t =
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Problem 2.26 - Problem 2.26[Difficulty 4 Given Velocity...

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