Problem 2.27[Difficulty: 5]Given:Velocity fieldFind:Plot streakline for first second of flowSolution:Following the discussion leading up to Eq. 2.10, we first find equations for the pathlines in form xpt()xtx0, y0, t0, =andyptytx0, y0, t0, =where x0, y0is the position of the particle at t = t0, and re-interprete the results as streaklinesxstt00, y0, t0, =andystt00, y0, t0, =which gives the streakline at t, where x0, y0is the point at which dye is released (t0is varied from 0 to t)For particle paths, first find x(t) dxdtu=u0=Separating variables and integratingdxu0dt⋅=orxx0u0tt0−⋅+=For y(t) we havedydtv=v0sinωtxu0−⎛⎜⎝⎞⎠⋅⎡⎢⎣⎤⎥⎦⋅=sodydtv=v0sinωtx0u00−⋅+u0−⎡⎢⎣⎤⎥⎦⋅⎡⎢⎣⎤⎥⎦⋅=anddydtv=v0sinωt0x0u0−⎛⎜⎝⎞⎠⋅⎡⎢⎣⎤⎥⎦⋅=Separating variables and integratingdyv0sinωt0x0u0−⎛⎜⎝⎞⎠⋅⎡⎢
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