Past exam of the mathematics course of the University of Cambridge / 2021 / ib / Paper 1 / 16A / c

Solution

The dyed particle satisfies the Lagrangian trajectory equations

\overset{x}{˙} = ε y cos (x - t), \overset{y}{˙} = ε sin (x - t), x (0) = y (0) = 0.

For the proposed approximation,

x = O (ε^{2})

and

\overset{y}{˙} = - ε sin t = ε sin (x - t) + O (ε^{3}) .

Also

\overset{x}{˙} = ε^{2} (cos^{2} t - cos t) = ε y cos (x - t) + O (ε^{3}) .

The initial conditions hold, verifying

x = ε^{2} (\frac{1}{4} sin 2 t + \frac{t}{2} - sin t), y = ε (cos t - 1)

through order

ε^{2}

Over one period, the periodic terms return to their initial values while the secular term changes

x

ε^{2} π

. Hence the dyed particle has Stokes drift

\overline{v}_{particle} = (\frac{ε ^{2}}{2}, 0)

to this order, despite the zero Eulerian mean velocity.

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