 Solution

ID: past-exam-of-the-fluids-and-waves-course-of-the-university-of-oxford/2021/1/a/i/solution

X, Y

held fixed. Direct differentiation gives

\overset{x}{˙} \overset{y}{˙} = - e^{- t} (X cos Ω t + Y sin Ω t) + Ω e^{- t} (Y cos Ω t - X sin Ω t) = - x + Ω e^{- 2 t} y, = e^{t} (Y cos Ω t - X sin Ω t) - Ω e^{t} (X cos Ω t + Y sin Ω t) = y - Ω e^{2 t} x .

u (x, y, t) = (- x + Ω e^{- 2 t} y, y - Ω e^{2 t} x) .

The distinction between following a fluid element and differentiating at fixed Eulerian coordinates is what connects the given deformation map to this velocity field.

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